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+++ b/.gitignore @@ -0,0 +1,3 @@ +/build +/output/imgs +/scripts/__pycache__ diff --git a/AGENTS.md b/AGENTS.md new file mode 100644 index 0000000..c885a30 --- /dev/null +++ b/AGENTS.md @@ -0,0 +1,82 @@ +# AGENTS.md + +## 项目目标 + +本仓库要构建一个**离线、以物理正确性为优先的数值相对论时空视频渲染器**。最终目标是把双黑洞（BBH）并合等动态数值相对论（NR）演化输出的真实四维时空渲染为 4K 视频。 + +渲染器必须直接消费时间依赖的 4D metric，对每条相机光线进行向过去的 null geodesic tracing；它不是仅渲染静态快照、波形或标量诊断的工具。第一阶段只处理两种 ray 终态：被黑洞捕获，或逃逸到无穷远天球。局域物质辐射、吸积盘、流体和等离子体均不在当前范围内。 + +设计依据是 [`nr_spacetime_movie_renderer_design.md`](nr_spacetime_movie_renderer_design.md)。实现前应先阅读与修改内容相关的章节；该文档是架构和物理取舍的权威来源。 + +## 不要偏离的方向 + +- 不以实时性能、游戏式视觉近似或 ShaderToy 式效果为目标。 +- frozen snapshot 只能作为数值演化代码内的廉价诊断，不能替代最终的动态 4D 渲染。 +- 不把恒星背景预烘焙成 RGB 天球纹理；恒星是带方向、温度和振幅的点源 catalog。 +- 相机不是固定三维坐标，而是 worldline 加 tetrad 的预生成轨迹。 +- 第一版只做 CPU；先验证物理与数据流，再考虑 GPU。 +- 不为复杂 OO/class hierarchy 增加抽象。首选 C 的 `struct`、明确 ownership、函数表和连续数组。 + +## 核心架构约束 + +### 时间组织 + +所有帧的 ray 必须按 coordinate time 汇总，沿时间从新到旧，通过可装入内存的 metric time slab 推进。不要按 frame 各自重复加载所需的 4D 数据。 + +每个 refinement pass 只追踪新请求的 samples：完整追踪后才能判断 image-plane triangle 是否需要细分，而早期 slab 此时已释放。不能在同一次 slab sweep 中临时创建新 ray 并回到相机时间重新追踪。 + +### Ray 与并行 + +- 以 coordinate time `t` 作为 geodesic ODE 自变量，使 ray 演化与 slab 边界自然对齐。 +- 批量 ray 状态采用 SoA `RayPool`，避免 per-ray allocation；支持 activation、终止和 stream compaction。 +- 用 OpenMP 对 active ray pool 做粗粒度循环并行。不要为单条 ray 或 triangle 创建 task，也不要引入细粒度 mutex。 +- metric evaluator 的可变缓存和插值 scratch space 必须是 thread-local，不能共享可变 cache。 +- 优先 bulk arrays、顺序 metric I/O、time-slab streaming 和 coarse-grained parallelism。 + +### 镜头映射与点源渲染 + +每帧在 image plane 上使用 adaptive triangle mesh。每个顶点保存 film 坐标、ray 状态、逃逸方向 `n_inf` 与频移信息。根据真实 mapping 与局部插值的误差、orientation consistency 和 Jacobian 奇异性进行局部细分。 + +最终从局部可逆 triangle 构造 inverse lens map：`sky direction -> image position`。同一恒星被多个局部 patch 覆盖时自然形成多像；不要依赖全局“第几阶像”分类。 + +catalog 内部数据保留 `(direction, temperature, amplitude)`，而非 RGB。对每个像计算局部 magnification 与频率比 `g`，用 `T_obs = g * T_emit` 处理黑体频移，并以保留亚像素位置的 PSF 直接 splat 到 HDR framebuffer。 + +## Backend 与模块边界 + +顶层 movie scheduler 不得依赖具体 metric 来源。时空 backend 至少应可替换为：Minkowski、解析 Schwarzschild，以及 nmesh 4D 数值时空；未来可加入 Kerr/BBH。 + +- `observer`：worldline/tetrad 读取、插值和正交化；不实现 geodesic 或渲染。 +- `movie` / `frame`：帧时间表、adaptive mesh、refinement 请求与 ray 结果回填。 +- `ray`：批量 ray 状态、初始化、生命周期和 compaction。 +- `geodesic`：3+1 null geodesic RHS、频移、ODE stepper 和 slab 内推进；不处理磁盘 I/O 或 frame mesh。 +- `spacetime`：backend 统一接口、slab、metric/导数插值以及 capture/infinity classification。 +- `nmesh_backend`：读取原生 DG node 输出、AMR lookup、Lagrange 空间插值及导数、时间插值和 puncture trajectories。 +- `catalog`：Gaia/2MASS 等 catalog 的读取、内部转换和天球索引。 +- `optics` / `psf`：黑体频移、linear RGB、flux、PSF 和 HDR 累积。 +- `output`：HDR、曝光、tone mapping、PNG/EXR 与视频编码接口。 + +对 nmesh 数据，直接在原生 DG nodes 上用 Lagrange basis 插值及求空间导数；不要先重采样到 Cartesian uniform grid，也不要求相邻时间 slice 的 AMR tree 拓扑对应。对固定全局点，应分别在各 slice 的空间 mesh 上求值，再进行时间插值。 + +优先从 3+1 identities、已知 gauge RHS 或 temporal interpolant 的解析导数获得时间导数；不要为已有插值量另行做低阶 finite difference，也不要在 geodesic RHS 中计算不会使用的量。 + +## 黑洞终止 + +对 moving-puncture 数据，生产渲染使用经 AH calibration 得到、保守地位于 apparent horizon 内部的 puncture-centered cutoff 判定捕获。不要假定每次生产演化都会运行昂贵的 AH finder。可在未来加入 common-horizon 终止优化，但不得改变物理分类。 + +## 开发与验证顺序 + +按以下层级实现和回归验证，后续阶段不得绕过前一阶段的基准： + +1. Minkowski + 星表：验证 observer/camera projection、天球方向、catalog、温度/振幅、PSF 和 mesh。 +2. 解析 Schwarzschild：验证 capture、Einstein ring、多像、refinement、局部 inverse map、放大率和频移。 +3. 数值单 Schwarzschild：验证动态 4D slab streaming，并与解析解比较逃逸方向、频移和捕获分类。 +4. FOZ4c BBH：在上层架构不重设计的前提下接入成熟的 BBH 演化。 + +新增物理、插值或优化时，优先添加能与前一阶段比较的收敛测试或 regression test。尚未由 prototype/convergence test 决定的参数（如时间输出 cadence、时间插值阶数、slab 大小、refinement 阈值、PSF、ODE stepper）不要伪装成既定事实。 + +## 修改原则 + +- 改动应保持 backend、observer、geodesic、movie mesh 与 optics 的职责分离。 +- 性能优化不得破坏 time-dependent tracing、局部可逆映射或点源的亚像素 PSF 渲染。 +- 新增缓存先明确 ownership、生命周期和线程归属；高开销可变缓存默认 thread-local。 +- 未在设计文档中明确的物理或数值取舍，应通过小型 prototype、基准或收敛实验确定，并同步更新设计文档。 diff --git a/Makefile b/Makefile new file mode 100644 index 0000000..1a06351 --- /dev/null +++ b/Makefile @@ -0,0 +1,64 @@ +CC ?= cc +CFLAGS ?= -std=c11 -O2 -Wall -Wextra -Wpedantic +OPENMP_FLAGS ?= -fopenmp +LDLIBS ?= -lm +ENABLE_PNG ?= 0 +SPACETIME ?= minkowski + +ifeq ($(ENABLE_PNG),1) +CPPFLAGS += -DENABLE_PNG +LDLIBS += -lpng +endif + +COMMON_SOURCES := $(filter-out src/main.c src/spacetime_minkowski.c src/spacetime_schwarzschild.c,$(wildcard src/*.c)) +PROVIDER_SOURCE := src/spacetime_$(SPACETIME).c +TARGET := build/$(SPACETIME)_sky +CORE_MINKOWSKI_SOURCES := $(COMMON_SOURCES) src/spacetime_minkowski.c +TEST_TARGET := build/test_geodesic +FRAME_TEST_TARGET := build/test_frame +SCHWARZSCHILD_TEST_TARGET := build/test_schwarzschild + +.PHONY: all clean run test minkowski schwarzschild + +ifeq ($(SPACETIME),minkowski) +BACKEND_CPPFLAGS := -DSPACETIME_MINKOWSKI +else ifeq ($(SPACETIME),schwarzschild) +BACKEND_CPPFLAGS := -DSPACETIME_SCHWARZSCHILD +else +$(error Unknown SPACETIME '$(SPACETIME)'; choose minkowski or schwarzschild) +endif + +all: $(TARGET) + +$(TARGET): $(COMMON_SOURCES) $(PROVIDER_SOURCE) src/main.c | build + $(CC) $(CPPFLAGS) $(BACKEND_CPPFLAGS) $(CFLAGS) $(OPENMP_FLAGS) -Isrc $^ $(LDLIBS) -o $@ + +build: + mkdir -p $@ + +run: $(TARGET) + mkdir -p output/imgs + ./$(TARGET) --catalog assets/sky_grid_5deg.csv --output output/imgs/$(SPACETIME)_sky.ppm + +minkowski: + $(MAKE) SPACETIME=minkowski all + +schwarzschild: + $(MAKE) SPACETIME=schwarzschild all + +$(TEST_TARGET): tests/test_geodesic.c $(CORE_MINKOWSKI_SOURCES) | build + $(CC) $(CPPFLAGS) $(CFLAGS) $(OPENMP_FLAGS) -Isrc $^ $(LDLIBS) -o $@ + +$(FRAME_TEST_TARGET): tests/test_frame.c $(CORE_MINKOWSKI_SOURCES) | build + $(CC) $(CPPFLAGS) $(CFLAGS) $(OPENMP_FLAGS) -Isrc $^ $(LDLIBS) -o $@ + +$(SCHWARZSCHILD_TEST_TARGET): tests/test_schwarzschild.c $(COMMON_SOURCES) src/spacetime_schwarzschild.c | build + $(CC) $(CPPFLAGS) $(CFLAGS) $(OPENMP_FLAGS) -Isrc $^ $(LDLIBS) -o $@ + +test: $(TEST_TARGET) $(FRAME_TEST_TARGET) $(SCHWARZSCHILD_TEST_TARGET) + ./$(TEST_TARGET) + ./$(FRAME_TEST_TARGET) + ./$(SCHWARZSCHILD_TEST_TARGET) + +clean: + rm -rf build diff --git a/README.md b/README.md new file mode 100644 index 0000000..77769ac --- /dev/null +++ b/README.md @@ -0,0 +1,125 @@ +# GR 4D ray tracing — Phase 0 prototype + +`minkowski_sky` is a deliberately small, CPU-only, single-frame Phase 0 +benchmark. It renders point sources from a sky catalog through an analytic +backend. It is not a sky texture: each source remains a direction, temperature, +and amplitude until its sub-pixel Gaussian PSF is splatted. + +Build and render the default 1280 x 720 image: + +```sh +make run +``` + +The program first creates `assets/sky_grid_5deg.csv` when it is missing. The +synthetic catalog places stars every 2 degrees on the union of longitude and +latitude lines spaced 10 degrees apart; the two poles are stored only once. +The eight octants (four 90-degree longitude sectors in each hemisphere) +alternate red `temperature_K = 3000` and blue `temperature_K = 12000`. +Red stars use `amplitude = 1`; blue stars use `amplitude = 0.00141095580387`, +which equalizes their CIE/linear-sRGB luminance under the renderer's blackbody +integration. Longitude boundaries belong to the sector to their east and the +equator to the northern hemisphere, so boundary stars have a deterministic +color. +The output is a binary PPM at `output/imgs/minkowski_sky.ppm`; it can be inspected by +most image viewers or converted to PNG with ImageMagick. + +PNG output is optional so the default build has no `libpng` dependency. Build +with `make ENABLE_PNG=1`, then select it with a `.png` output path: + +```sh +make clean && make ENABLE_PNG=1 +mkdir -p output/imgs +./build/minkowski_sky --output output/imgs/minkowski_sky.png +``` + +Run the flat-spacetime geodesic regression with: + +```sh +make test +``` + +This also checks that the Kerr--Schild metric remains finite at `r=2M` and +that the central ray from the default Schwarzschild camera is classified as +captured. + +Build an independent analytic Schwarzschild executable in Cartesian ingoing +Kerr--Schild coordinates (regular at the horizon), then render the test catalog +to PNG: + +```sh +make clean && make SPACETIME=schwarzschild ENABLE_PNG=1 +mkdir -p output/imgs +./build/schwarzschild_sky --catalog assets/sky_grid_5deg.csv \ + --width 640 --height 360 --coarse-cell-pixels 8 --fov-deg 60 \ + --output output/imgs/schwarzschild_test_catalog.png +``` + +`SPACETIME=minkowski` (the default) and `SPACETIME=schwarzschild` select source +files at compile time, so each executable contains exactly one metric provider. +The Schwarzschild demonstration uses mass `M=1`, a static camera at Cartesian +Kerr--Schild position `(30, 0, 0)`, directed at the hole, escapes at `r=256`, +and declares capture at `r=1.5`, safely inside the horizon at `r=2`. Those +rendering thresholds are Phase-1 demonstration values, not settled production +refinement or integration settings. + +For a local radial boost relative to that static camera, pass +`--observer-inward-speed V`, where `0 <= V < 1` is measured in the static +observer's orthonormal frame and positive values point toward the hole. The +default is `0`, preserving the static camera. + +For rays that asymptote to the future horizon in coordinate-time backward +integration, the Schwarzschild demo also terminates at +`log(alpha p^0) = 8`. This is the normalized-momentum horizon diagnostic +already evolved by the integrator; it is disabled by default and does not +replace the AH-calibrated spatial capture criterion planned for nmesh data. + +Useful options: + +```sh +./build/minkowski_sky --width 1920 --height 1080 --fov-deg 30 \ + --catalog assets/sky_grid_5deg.csv --output output/imgs/frame.ppm +./build/minkowski_sky --catalog assets/2mass/processed/2mass_psc_m31_0p5deg_stars.csv \ + --look-ra-deg 10.6847083 --look-dec-deg 41.26875 --fov-deg 1.8 \ + --exposure 1e15 --output output/imgs/2mass_m31.ppm +./build/minkowski_sky --catalog assets/2mass/processed/2mass_psc_m44_1p0deg_stars.csv \ + --look-ra-deg 129.99165 --look-dec-deg 19.54139 --fov-deg 2.0 \ + --exposure 1e15 --width 1920 --height 1920 --output output/imgs/2mass_m44.ppm +./build/minkowski_sky --write-catalog assets/sky_grid_5deg.csv +``` + +The camera is a fixed inertial observer at coordinate position `(0,0,0)`, +with a tetrad whose forward direction is coordinate `-Z` and whose vertical +direction is `+Y`. The frame first triangulates the image plane, then traces +only its vertices backwards. Escaped endpoints form a triangulation on the +source sky. For every locally invertible triangle, catalog stars inside its +spherical source triangle are interpolated back to the image triangle and +splatted as PSFs. Consequently multiple image triangles naturally create +multiple images of the same star. + +The ray state evolves `(x^i, Pi_i, log(alpha p^0))` in coordinate time with +RK4 using the 3+1 equations in Bohn et al. II.A, until the spacetime backend +classifies the ray. `spacetime.c` is the only module containing the Minkowski +metric or its infinity criterion; frame, observer, and integrator use only +`SpacetimeSource` and `MetricData`. The initial regular mesh size is exposed +as `--coarse-cell-pixels`; it is a Phase-0 sampling knob, not a settled +production refinement threshold. + +`--look-ra-deg` and `--look-dec-deg` rotate that fixed tetrad so its forward +axis is the corresponding catalog direction; their defaults reproduce the +original `-Z` view. `--exposure` converts a catalog's physical flux +normalization to the prototype HDR scale. Its default preserves the synthetic +catalog benchmark; a 2MASS blackbody normalization in steradians requires a +much larger display exposure such as the example above. The optics path +integrates each fitted Planck spectrum through CIE 1931 color-matching functions +and converts the resulting radiance to linear sRGB; it does not use an empirical +color-temperature RGB approximation. + +Point sources use a flux-normalized circular Moffat PSF by default +(`--psf-fwhm-pixels 2.7 --psf-moffat-beta 4.5`). The FWHM matches the former +1.15-pixel Gaussian core while the Moffat wings remain continuous; both values +are display/optics calibration parameters. + +Pass `--draw-mesh` to alpha-composite image-plane triangle edges as +one-pixel-wide 0.5 linear-gray diagnostic lines at 0.5 opacity. The line +rasterizer uses coverage-based antialiasing. diff --git a/assets/.gitkeep b/assets/.gitkeep new file mode 100644 index 0000000..8b13789 --- /dev/null +++ b/assets/.gitkeep @@ -0,0 +1 @@ + diff --git a/assets/2mass/README.md b/assets/2mass/README.md new file mode 100644 index 0000000..73c9333 --- /dev/null +++ b/assets/2mass/README.md @@ -0,0 +1,54 @@ +# 2MASS Point Source Catalog sample + +`raw/2mass_psc_m31_0p5deg.tbl` is an unmodified IPAC ASCII-table response from the +NASA/IPAC Infrared Science Archive (IRSA) Gator service, drawn from the **2MASS +All-Sky Point Source Catalog (PSC)**. + +It is a circular field of view centred on M31: + +- Centre: ICRS/J2000 RA 10.6847083 deg, Dec +41.2687500 deg +- Cone radius: 0.5 deg +- Retrieved rows: 9,047 +- Retrieval date: 2026-08-25 (recorded in the table header) + +The all-sky PSC has hundreds of millions of entries, so it is deliberately not +vendored. This real-survey slice keeps the repository lightweight while +providing a reproducible replacement for the synthetic latitude/longitude grid. +Run `bash scripts/download_2mass_psc_m31.sh` from any directory to refresh it. + +The table preserves the service response. Its relevant source fields are: + +| Field | Meaning | +| --- | --- | +| `ra`, `dec` | ICRS/J2000 right ascension and declination, in degrees | +| `j_m`, `h_m`, `k_m` | 2MASS J, H, and Ks default magnitudes, in mag | +| `dist` | Angular distance from the stated field centre, in arcsec | +| `rd_flg` | Per-band source/read flag needed to interpret the default magnitudes | + +`j_m`, `h_m`, and `k_m` may be null in the original PSC. No temperature is +included or inferred: the PSC provides photometry, not source temperatures. +Do not treat magnitudes as linear fluxes without applying the appropriate 2MASS +band zero points. + +`processed/2mass_psc_m31_0p5deg_stars.csv` is generated from the raw table by +`python3 scripts/process_2mass_psc.py`. It retains only `ra_deg`, `dec_deg`, +`temperature_K`, and `amplitude_sr`. See its directory README for the fitting +model and quality cut. + +`raw/2mass_psc_m44_1p0deg.tbl` is the corresponding unmodified 1.0-degree +cone around M44 (Beehive/Praesepe), centred on ICRS/J2000 RA 129.99165 deg and +Dec +19.54139 deg. It contains 8,790 retrieved PSC rows (2026-08-25) and is +downloaded by +`bash scripts/download_2mass_psc_m44.sh`; process it with: + +```sh +python3 scripts/process_2mass_psc.py \ + --input assets/2mass/raw/2mass_psc_m44_1p0deg.tbl \ + --output assets/2mass/processed/2mass_psc_m44_1p0deg_stars.csv +``` + +The resulting M44 CSV contains 7,802 sources after the documented three-band +photometry and `rd_flg` selection. + +Provenance: [IRSA Gator program interface](https://irsa.ipac.caltech.edu/docs/howto/gator_prog_interface.html), +[2MASS PSC column descriptions](https://irsa.ipac.caltech.edu/2MASS/download/allsky/format_psc.html). diff --git a/assets/2mass/processed/2mass_psc_m31_0p5deg_stars.csv b/assets/2mass/processed/2mass_psc_m31_0p5deg_stars.csv new file mode 100644 index 0000000..957a782 --- /dev/null +++ b/assets/2mass/processed/2mass_psc_m31_0p5deg_stars.csv @@ -0,0 +1,5466 @@ +ra_deg,dec_deg,temperature_K,amplitude_sr +10.684737,41.269035,3052.6009,5.798636774e-19 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+129.882369,20.535816,3186.9229,1.446040409e-21 +128.935991,19.443665,3809.1428,5.247481058e-22 diff --git a/assets/2mass/processed/README.md b/assets/2mass/processed/README.md new file mode 100644 index 0000000..137411d --- /dev/null +++ b/assets/2mass/processed/README.md @@ -0,0 +1,40 @@ +# Renderer-facing 2MASS star catalog + +`2mass_psc_m31_0p5deg_stars.csv` is made from the corresponding table in +`../raw/` by `python3 scripts/process_2mass_psc.py`. + +The processor accepts `--input` and `--output`, so the same documented quality +selection and fit are used for the M44 comparison field. + +The CSV is intentionally compatible with the current four-column catalog +loader: `ra_deg,dec_deg,temperature_K,amplitude_sr`. RA and Dec are ICRS/J2000 +degrees. The loader treats them as longitude and latitude when constructing +the source-sky direction. + +## Quality selection + +All three J/H/Ks magnitudes must be finite and every character of `rd_flg` must +be `1`, `2`, or `3`. These are the 2MASS default-magnitude origins that +generally indicate the best detections, photometry, and astrometry. Thus the +script rejects nondetections/upper limits (`0`), poor aperture photometry (`4`), +inconsistent band deblends (`6`), and missing brightness estimates (`9`). + +## Two-parameter fit + +For J, H, Ks effective wavelengths 1.235, 1.662, and 2.159 micrometres and +Vega zero points 1594, 1024, and 666.7 Jy, respectively, each magnitude is +converted to flux density using `F_nu = F_nu,0 * 10^(-0.4 m)`. + +The script then performs an unweighted least-squares fit in `log(F_nu)` to + +`F_nu = amplitude_sr * B_nu(T)`. + +`temperature_K` is constrained to 300--100000 K; `amplitude_sr` is the fitted +apparent solid angle in steradians. The fit treats each band as monochromatic +at its effective wavelength and does not deredden the source. This is the +same two-parameter thermal-source reduction described qualitatively in the +reference paper, but the paper does not publish its fit implementation; this +script therefore records the numerical choice explicitly. + +References: [2MASS `rd_flg` definitions](https://irsa.ipac.caltech.edu/data/2MASS/docs/releases/allsky/doc/sec1_6b.html), +[2MASS zero points](https://www.ipac.caltech.edu/2mass/releases/allsky/faq.html). diff --git a/assets/2mass/raw/2mass_psc_m31_0p5deg.tbl b/assets/2mass/raw/2mass_psc_m31_0p5deg.tbl new file mode 100644 index 0000000..002410c --- /dev/null +++ b/assets/2mass/raw/2mass_psc_m31_0p5deg.tbl @@ -0,0 +1,9090 @@ +\ +\fixlen = T +\RowsRetrieved = 9047 +\ORIGIN = 'IPAC Infrared Science Archive (IRSA), Caltech/JPL' +\SIMULATED_TABLE = 'n' +\DATETIME= '2026-08-25 11:49:28' +\DATABASE= '2MASS All-Sky Point Source Catalog (PSC) (fp_psc)' +\EQUINOX = 'J2000' +\SKYAREA = 'within 0.5 deg of ra=10.68471 dec=+41.26875 Eq J2000 ' +\StatusFile = '/workspace/TMP_is8j3L_264398/Gator/irsa/264398/log.264398.html' +\SQL = 'WHERE (no constraints) +\SQL = 'SELECT (9 column names follow in next row.)' +\ +\ designation +\ ___ source designation formed from sexigesimal coordinates +\ ra (deg) +\ ___ right ascension (J2000 decimal deg) +\ dec (deg) +\ ___ declination (J2000 decimal deg) +\ clon +\ ___ Right ascention in Sexagesimal format. +\ clat +\ ___ Declination in Sexagesimal format. +\ j_m (mag) +\ ___ J band selected "default" magnitude +\ h_m (mag) +\ ___ H band selected "default" magnitude +\ k_m (mag) +\ ___ K band selected "default" magnitude +\ rd_flg +\ ___ source of JHK "default" mags (AKA "read flag") +\ dist (arcsec) +\ ___ Distance between the target position and each source in arcsec. +\ angle (deg) +\ ___ Position Angle in degree. +\ +\ j_h -> Reference: Input column(s): j_m, h_m, k_m, rd_flg +\ h_k -> Reference: Input column(s): j_m, h_m, k_m, rd_flg +\ j_k -> Reference: Input column(s): j_m, h_m, k_m, rd_flg +| designation| ra| dec| clon| clat| j_m| h_m| k_m| rd_flg| dist| angle| j_h| h_k| j_k| +| char| double| double| char| char| double| double| double| char| double| double| double| double| double| +| | deg| deg| | | mag| mag| mag| | arcsec| deg| | | | +| null| null| null| null| null| null| null| null| null| null| null| -| -| -| + 00424433+4116085 10.684737 41.269035 00h42m44.34s 41d16m08.53s 9.453 8.668 8.475 222 1.028600 4.072939 0.7850 0.1930 0.9780 + 00424403+4116069 10.683469 41.268585 00h42m44.03s 41d16m06.91s 9.321 8.614 10.601 002 3.410092 259.968952 - - - + 00424455+4116103 10.685657 41.269550 00h42m44.56s 41d16m10.38s 10.773 8.532 8.254 200 3.854919 41.660119 - - - + 00424464+4116092 10.686026 41.269226 00h42m44.65s 41d16m09.21s 9.299 8.606 10.119 002 3.951746 64.301346 - - - + 00424403+4116108 10.683465 41.269676 00h42m44.03s 41d16m10.83s 11.507 8.744 8.489 200 4.739346 314.699888 - - - + 00424464+4116106 10.686015 41.269630 00h42m44.64s 41d16m10.67s 9.399 9.985 8.429 020 4.743936 48.102087 - - - + 00424446+4116016 10.685270 41.267124 00h42m44.46s 41d16m01.65s 12.070 9.301 9.057 206 6.046548 165.486523 - - - + 00424398+4116028 10.683263 41.267456 00h42m43.98s 41d16m02.84s 12.136 9.226 8.994 200 6.085314 220.047625 - - - + 00424420+4116009 10.684180 41.266941 00h42m44.20s 41d16m00.99s 10.065 9.374 11.504 002 6.668437 192.419261 - - - + 00424377+4116045 10.682383 41.267925 00h42m43.77s 41d16m04.53s 12.446 11.753 9.075 220 6.961851 244.748111 0.6930 - - + 00424386+4116123 10.682777 41.270111 00h42m43.87s 41d16m12.40s 9.977 11.683 11.839 022 7.166771 313.130573 - -0.1560 - + 00424460+4116141 10.685837 41.270599 00h42m44.60s 41d16m14.16s 9.880 12.051 8.934 020 7.321665 24.613203 - - - + 00424414+4116000 10.683944 41.266682 00h42m44.15s 41d16m00.06s 12.565 9.510 9.274 206 7.727948 195.558018 - - - + 00424416+4116152 10.684029 41.270901 00h42m44.17s 41d16m15.24s 10.063 9.359 11.409 002 7.959818 346.615172 - - - + 00424385+4116014 10.682713 41.267056 00h42m43.85s 41d16m01.40s 10.176 11.876 9.252 020 8.147998 221.544038 - - - + 00424497+4116034 10.687414 41.267632 00h42m44.98s 41d16m03.48s 12.371 9.627 9.379 200 8.350610 118.813694 - - - + 00424484+4116145 10.686846 41.270714 00h42m44.84s 41d16m14.57s 12.872 9.433 9.178 200 9.132050 39.263103 - - - + 00424487+4116005 10.686963 41.266827 00h42m44.87s 41d16m00.58s 10.450 12.094 11.728 622 9.224466 138.631440 - 0.3660 - + 00424502+4116130 10.687611 41.270302 00h42m45.03s 41d16m13.09s 13.055 9.504 9.246 200 9.634963 54.556372 - - - + 00424526+4116125 10.688605 41.270149 00h42m45.27s 41d16m12.54s 12.896 9.732 9.480 200 11.680719 64.456748 - - - + 00424380+4116181 10.682517 41.271721 00h42m43.80s 41d16m18.20s 12.933 9.900 11.862 202 12.231340 330.979754 - - 1.0710 + 00424345+4115598 10.681043 41.266632 00h42m43.45s 41d15m59.88s 10.511 9.815 11.861 002 12.513735 232.461259 - - - + 00424381+4115553 10.682581 41.265362 00h42m43.82s 41d15m55.30s 10.643 9.954 12.833 002 13.488879 205.283404 - - - + 00424312+4116032 10.679682 41.267559 00h42m43.12s 41d16m03.21s 10.684 9.985 12.268 002 14.264740 252.509618 - - - + 00424327+4115589 10.680326 41.266369 00h42m43.28s 41d15m58.93s 13.428 13.230 9.709 220 14.635393 234.150676 0.1980 - - + 00424487+4115529 10.686994 41.264702 00h42m44.88s 41d15m52.93s 10.907 10.225 12.453 002 15.829187 157.017378 - - - + 00424327+4115560 10.680320 41.265572 00h42m43.28s 41d15m56.06s 13.926 10.046 9.802 200 16.492467 226.077851 - - - + 00424469+4116234 10.686238 41.273193 00h42m44.70s 41d16m23.49s 13.343 10.069 9.820 266 16.520497 14.492297 - - - + 00424305+4115590 10.679384 41.266411 00h42m43.05s 41d15m59.08s 13.291 10.084 9.836 200 16.691232 239.704888 - - - + 00424435+4116247 10.684796 41.273548 00h42m44.35s 41d16m24.77s 13.194 10.192 9.951 200 17.274367 0.771797 - - - + 00424271+4116109 10.677969 41.269714 00h42m42.71s 41d16m10.97s 12.908 10.354 12.657 202 18.567186 280.774756 - - 0.2510 + 00424569+4115542 10.690409 41.265083 00h42m45.70s 41d15m54.30s 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14.775 14.605 626 1798.346977 271.081047 - - - + 00450212+4131189 11.258864 41.521935 00h45m02.13s 41d31m18.97s 15.318 14.438 14.181 222 1798.603306 59.362043 0.8800 0.2570 1.1370 + 00424743+4046091 10.697651 40.769203 00h42m47.44s 40d46m09.13s 16.444 15.535 15.594 222 1798.712672 178.876019 0.9090 -0.0590 0.8500 + 00402596+4101170 10.108169 41.021408 00h40m25.96s 41d01m17.07s 16.937 16.111 15.626 222 1798.822195 240.519738 0.8260 0.4850 1.3110 + 00443549+4137413 11.147912 41.628143 00h44m35.50s 41d37m41.31s 15.522 14.774 14.627 222 1798.938641 43.857784 0.7480 0.1470 0.8950 + 00421146+4046472 10.547763 40.779797 00h42m11.46s 40d46m47.27s 15.746 15.031 14.780 222 1799.097087 191.976238 0.7150 0.2510 0.9660 + 00444461+4135538 11.185885 41.598286 00h44m44.61s 41d35m53.83s 16.334 15.512 15.339 222 1799.190021 48.582818 0.8220 0.1730 0.9950 + 00401666+4127360 10.069423 41.460011 00h40m16.66s 41d27m36.04s 16.176 15.736 15.828 222 1799.373312 292.701279 0.4400 -0.0920 0.3480 + 00401523+4126557 10.063466 41.448807 00h40m15.23s 41d26m55.71s 13.800 13.167 12.973 222 1799.468261 291.318721 0.6330 0.1940 0.8270 + 00444364+4056159 11.181871 40.937775 00h44m43.65s 40d56m15.99s 14.190 13.550 13.400 222 1799.588544 131.296522 0.6400 0.1500 0.7900 + 00445898+4100057 11.245768 41.001591 00h44m58.98s 41d00m05.73s 13.573 12.941 12.895 626 1799.764253 122.117520 - - - + 00443968+4055269 11.165342 40.924149 00h44m39.68s 40d55m26.94s 15.811 14.954 14.940 222 1799.789721 133.414985 0.8570 0.0140 0.8710 + 00420069+4047167 10.502898 40.787979 00h42m00.70s 40d47m16.72s 17.029 16.049 15.440 222 1799.827939 195.982323 0.9800 0.6090 1.5890 + 00430751+4145484 10.781309 41.763466 00h43m07.51s 41d45m48.48s 16.284 15.702 15.456 222 1799.911778 8.286056 0.5820 0.2460 0.8280 diff --git a/assets/2mass/raw/2mass_psc_m44_1p0deg.tbl b/assets/2mass/raw/2mass_psc_m44_1p0deg.tbl new file mode 100644 index 0000000..9d5de0a --- /dev/null +++ b/assets/2mass/raw/2mass_psc_m44_1p0deg.tbl @@ -0,0 +1,8833 @@ +\ +\fixlen = T +\RowsRetrieved = 8790 +\ORIGIN = 'IPAC Infrared Science Archive (IRSA), Caltech/JPL' +\SIMULATED_TABLE = 'n' +\DATETIME= '2026-08-25 16:28:37' +\DATABASE= '2MASS All-Sky Point Source Catalog (PSC) (fp_psc)' +\EQUINOX = 'J2000' +\SKYAREA = 'within 1.0 deg of ra=129.99165 dec=+19.54139 Eq J2000 ' +\StatusFile = '/workspace/TMP_PoB6S8_1072533/Gator/irsa/1072533/log.1072533.html' +\SQL = 'WHERE (no constraints) +\SQL = 'SELECT (9 column names follow in next row.)' +\ +\ designation +\ ___ source designation formed from sexigesimal coordinates +\ ra (deg) +\ ___ right ascension (J2000 decimal deg) +\ dec (deg) +\ ___ declination (J2000 decimal deg) +\ clon +\ ___ Right ascention in Sexagesimal format. +\ clat +\ ___ Declination in Sexagesimal format. +\ j_m (mag) +\ ___ J band selected "default" magnitude +\ h_m (mag) +\ ___ H band selected "default" magnitude +\ k_m (mag) +\ ___ K band selected "default" magnitude +\ rd_flg +\ ___ source of JHK "default" mags (AKA "read flag") +\ dist (arcsec) +\ ___ Distance between the target position and each source in arcsec. +\ angle (deg) +\ ___ Position Angle in degree. +\ +\ j_h -> Reference: Input column(s): j_m, h_m, k_m, rd_flg +\ h_k -> Reference: Input column(s): j_m, h_m, k_m, rd_flg +\ j_k -> Reference: Input column(s): j_m, h_m, k_m, rd_flg +| designation| ra| dec| clon| clat| j_m| h_m| k_m| rd_flg| dist| angle| j_h| h_k| j_k| +| char| double| double| char| char| double| double| double| char| double| double| double| double| double| +| | deg| deg| | | mag| mag| mag| | arcsec| deg| | | | +| null| null| null| null| null| null| null| null| null| null| null| -| -| -| + 08395777+1932293 129.990746 19.541483 08h39m57.78s 19d32m29.34s 7.065 7.034 7.007 111 3.085166 276.230117 0.0310 0.0270 0.0580 + 08395827+1933109 129.992825 19.553053 08h39m58.28s 19d33m10.99s 16.351 15.182 15.207 200 42.175600 5.423184 - - - + 08400014+1931570 130.000606 19.532513 08h40m00.15s 19d31m57.05s 16.615 15.730 15.019 200 44.096830 136.442827 - - - + 08395649+1933107 129.985396 19.552986 08h39m56.50s 19d33m10.75s 6.907 6.859 6.789 111 46.827857 333.059518 0.0480 0.0700 0.1180 + 08395439+1932581 129.976647 19.549486 08h39m54.40s 19d32m58.15s 15.266 14.634 14.346 222 58.652571 299.798907 0.6320 0.2880 0.9200 + 08395983+1934003 129.999314 19.566765 08h39m59.84s 19d34m00.35s 10.106 9.630 9.484 222 94.977778 15.885617 0.4760 0.1460 0.6220 + 08395272+1931287 129.969703 19.524651 08h39m52.73s 19d31m28.74s 16.246 15.729 15.475 222 95.791053 231.021264 0.5170 0.2540 0.7710 + 08400501+1932446 130.020877 19.545723 08h40m05.01s 19d32m44.60s 16.133 15.500 15.497 222 100.374847 81.054799 0.6330 0.0030 0.6360 + 08395072+1932269 129.961350 19.540823 08h39m50.72s 19d32m26.96s 5.180 4.633 4.389 331 102.817448 268.867519 0.5470 0.2440 0.7910 + 08395084+1933020 129.961849 19.550581 08h39m50.84s 19d33m02.09s 10.421 10.085 9.996 222 106.377824 288.126812 0.3360 0.0890 0.4250 + 08395170+1933294 129.965419 19.558186 08h39m51.70s 19d33m29.47s 15.694 15.174 15.173 222 107.586723 304.199811 0.5200 0.0010 0.5210 + 08395142+1933466 129.964274 19.562956 08h39m51.43s 19d33m46.64s 12.718 12.415 12.388 222 121.047788 309.899361 0.3030 0.0270 0.3300 + 08395998+1934405 129.999942 19.577917 08h39m59.99s 19d34m40.50s 11.134 10.650 10.552 222 134.472046 12.072796 0.4840 0.0980 0.5820 + 08395864+1934488 129.994357 19.580223 08h39m58.65s 19d34m48.80s 16.060 15.402 15.374 222 140.100063 3.757658 0.6580 0.0280 0.6860 + 08400832+1933263 130.034675 19.557314 08h40m08.32s 19d33m26.33s 14.936 14.413 14.123 222 156.815094 68.550331 0.5230 0.2900 0.8130 + 08400914+1932424 130.038092 19.545137 08h40m09.14s 19d32m42.49s 16.271 15.959 14.603 200 158.135550 85.098870 - - - + 08400947+1931512 130.039474 19.530903 08h40m09.47s 19d31m51.25s 11.058 10.503 10.397 222 166.589188 103.090429 0.5550 0.1060 0.6610 + 08394633+1933500 129.943060 19.563913 08h39m46.33s 19d33m50.09s 15.168 14.530 14.350 222 183.699812 296.200571 0.6380 0.1800 0.8180 + 08395066+1929556 129.961105 19.498781 08h39m50.67s 19d29m55.61s 16.244 15.568 15.370 222 185.123918 214.050620 0.6760 0.1980 0.8740 + 08395175+1935119 129.965657 19.586655 08h39m51.76s 19d35m11.96s 16.668 16.039 15.700 220 185.279251 331.587042 0.6290 - - + 08400601+1935024 130.025055 19.584003 08h40m06.01s 19d35m02.41s 12.800 12.468 12.435 222 190.720223 36.446382 0.3320 0.0330 0.3650 + 08395704+1935408 129.987690 19.594687 08h39m57.05s 19d35m40.87s 16.155 15.888 15.682 222 192.338830 355.995957 0.2670 0.2060 0.4730 + 08394505+1933340 129.937735 19.559446 08h39m45.06s 19d33m34.01s 13.843 13.483 13.396 222 194.110976 289.573545 0.3600 0.0870 0.4470 + 08401180+1933086 130.049184 19.552404 08h40m11.80s 19d33m08.65s 16.325 15.712 15.555 220 199.172131 78.507442 0.6130 - - + 08400584+1935168 130.024362 19.588015 08h40m05.85s 19d35m16.85s 16.128 15.432 15.024 222 201.212908 33.462830 0.6960 0.4080 1.1040 + 08394321+1932460 129.930071 19.546133 08h39m43.22s 19d32m46.08s 14.075 13.452 13.236 222 209.608948 274.682806 0.6230 0.2160 0.8390 + 08401116+1930486 130.046511 19.513500 08h40m11.16s 19d30m48.60s 15.690 15.691 15.180 222 211.492213 118.333309 -0.0010 0.5110 0.5100 + 08395379+1929040 129.974130 19.484447 08h39m53.79s 19d29m04.01s 14.397 13.927 13.785 222 213.441127 196.175355 0.4700 0.1420 0.6120 + 08401053+1930154 130.043914 19.504286 08h40m10.54s 19d30m15.43s 16.153 15.796 17.123 200 222.011772 126.979755 - - - + 08394203+1932301 129.925160 19.541719 08h39m42.04s 19d32m30.19s 16.668 15.545 14.832 200 225.579537 270.311952 - - - + 08394238+1931272 129.926607 19.524248 08h39m42.39s 19d31m27.29s 16.767 16.185 15.540 220 229.145347 254.387646 0.5820 - - + 08394479+1934440 129.936637 19.578907 08h39m44.79s 19d34m44.07s 15.917 15.290 14.841 222 230.364200 305.903680 0.6270 0.4490 1.0760 + 08400108+1928310 130.004525 19.475290 08h40m01.09s 19d28m31.04s 14.652 14.088 14.019 222 241.937401 169.594275 0.5640 0.0690 0.6330 + 08394601+1935268 129.941723 19.590801 08h39m46.01s 19d35m26.88s 15.197 14.748 14.891 222 245.608271 316.414107 0.4490 -0.1430 0.3060 + 08400200+1928293B 130.008361 19.474825 08h40m02.01s 19d28m29.37s 15.676 15.132 15.075 002 246.251973 166.683846 - - - + 08400200+1928293A 130.008373 19.474819 08h40m02.01s 19d28m29.35s 16.230 15.546 15.046 222 246.282373 166.675782 0.6840 0.5000 1.1840 + 08394091+1933223 129.920478 19.556217 08h39m40.91s 19d33m22.38s 14.737 14.291 14.167 222 247.279565 282.477758 0.4460 0.1240 0.5700 + 08395629+1936375 129.984578 19.610418 08h39m56.30s 19d36m37.50s 13.439 12.770 12.614 222 249.655870 354.487543 0.6690 0.1560 0.8250 + 08400506+1928367 130.021116 19.476868 08h40m05.07s 19d28m36.72s 16.338 16.089 16.035 220 252.885599 156.705009 0.2490 - - + 08401589+1932275 130.066232 19.540974 08h40m15.90s 19d32m27.51s 16.637 16.331 15.266 200 253.034661 90.326637 - - - + 08401220+1935085 130.050859 19.585716 08h40m12.21s 19d35m08.58s 14.950 14.841 14.568 222 256.521634 51.522868 0.1090 0.2730 0.3820 + 08401434+1930368 130.059754 19.510229 08h40m14.34s 19d30m36.82s 13.019 12.382 12.299 222 256.865265 115.883754 0.6370 0.0830 0.7200 + 08393978+1932311 129.915774 19.541988 08h39m39.79s 19d32m31.16s 16.362 15.330 15.293 200 257.428504 270.491843 - - - + 08394670+1929057 129.944596 19.484921 08h39m46.70s 19d29m05.72s 14.149 13.679 13.667 222 258.493999 218.154387 0.4700 0.0120 0.4820 + 08394538+1929165 129.939084 19.487921 08h39m45.38s 19d29m16.52s 16.393 15.636 15.509 222 262.424437 222.828141 0.7570 0.1270 0.8840 + 08401701+1932186 130.070915 19.538527 08h40m17.02s 19d32m18.70s 14.067 13.564 13.476 222 269.117462 92.181624 0.5030 0.0880 0.5910 + 08393849+1933125 129.910407 19.553495 08h39m38.50s 19d33m12.58s 16.409 15.520 15.463 200 279.041740 278.998259 - - - + 08401784+1932184 130.074361 19.538452 08h40m17.85s 19d32m18.43s 15.325 14.882 14.425 222 280.810489 92.144739 0.4430 0.4570 0.9000 + 08401810+1931552 130.075436 19.532022 08h40m18.10s 19d31m55.28s 7.208 7.203 7.163 111 286.257559 96.751878 0.0050 0.0400 0.0450 + 08395560+1937176 129.981667 19.621561 08h39m55.60s 19d37m17.62s 15.041 14.482 14.214 222 290.595053 353.310334 0.5590 0.2680 0.8270 + 08400964+1936294 130.040193 19.608189 08h40m09.65s 19d36m29.48s 16.317 15.859 15.394 222 291.444868 34.391419 0.4580 0.4650 0.9230 + 08401427+1929289 130.059498 19.491365 08h40m14.28s 19d29m28.91s 16.589 15.698 15.140 222 292.289967 128.023079 0.8910 0.5580 1.4490 + 08394046+1935080 129.918623 19.585579 08h39m40.47s 19d35m08.08s 15.178 14.858 14.642 222 294.401045 302.719873 0.3200 0.2160 0.5360 + 08393737+1931469 129.905740 19.529716 08h39m37.38s 19d31m46.98s 15.613 15.139 15.157 222 294.486505 261.809627 0.4740 -0.0180 0.4560 + 08395441+1927372 129.976723 19.460339 08h39m54.41s 19d27m37.22s 14.242 13.608 13.358 222 296.147871 189.851082 0.6340 0.2500 0.8840 + 08393731+1931395 129.905475 19.527660 08h39m37.31s 19d31m39.58s 16.116 15.683 15.314 220 296.521875 260.418835 0.4330 - - + 08400523+1937079 130.021821 19.618881 08h40m05.24s 19d37m07.97s 14.666 14.306 14.272 222 297.145289 20.139693 0.3600 0.0340 0.3940 + 08394058+1935158 129.919116 19.587727 08h39m40.59s 19d35m15.82s 16.378 15.789 15.240 222 297.263332 304.148408 0.5890 0.5490 1.1380 + 08400705+1927590 130.029413 19.466410 08h40m07.06s 19d27m59.08s 13.240 12.588 12.384 222 298.801808 154.598064 0.6520 0.2040 0.8560 + 08394200+1929115 129.925025 19.486551 08h39m42.01s 19d29m11.58s 12.668 12.121 12.024 222 300.139675 228.881739 0.5470 0.0970 0.6440 + 08394781+1928033 129.949214 19.467596 08h39m47.81s 19d28m03.35s 16.559 16.139 15.711 222 302.177489 208.467466 0.4200 0.4280 0.8480 + 08401418+1929050 130.059087 19.484737 08h40m14.18s 19d29m05.05s 16.310 15.909 15.569 222 306.527180 131.698649 0.4010 0.3400 0.7410 + 08401981+1931522 130.082550 19.531178 08h40m19.81s 19d31m52.24s 14.030 13.424 13.222 222 310.584226 96.782718 0.6060 0.2020 0.8080 + 08394166+1929004 129.923616 19.483471 08h39m41.67s 19d29m00.50s 15.092 14.546 14.174 222 311.079355 227.923154 0.5460 0.3720 0.9180 + 08400851+1937022 130.035486 19.617294 08h40m08.52s 19d37m02.26s 14.384 13.707 13.643 222 311.087011 28.544413 0.6770 0.0640 0.7410 + 08402010+1932126 130.083789 19.536860 08h40m20.11s 19d32m12.70s 13.108 12.873 12.864 222 313.023972 92.970953 0.2350 0.0090 0.2440 + 08394737+1927514 129.947416 19.464300 08h39m47.38s 19d27m51.48s 16.543 15.592 15.677 222 315.517564 208.415244 0.9510 -0.0850 0.8660 + 08400911+1927541 130.037972 19.465050 08h40m09.11s 19d27m54.18s 16.499 15.561 15.314 222 316.602653 150.224014 0.9380 0.2470 1.1850 + 08401913+1934421 130.079713 19.578367 08h40m19.13s 19d34m42.12s 16.462 16.112 15.753 222 327.048780 65.967145 0.3500 0.3590 0.7090 + 08402046+1933480 130.085268 19.563356 08h40m20.46s 19d33m48.08s 16.221 15.872 15.779 222 327.287407 76.002450 0.3490 0.0930 0.4420 + 08401684+1929186 130.070199 19.488514 08h40m16.85s 19d29m18.65s 14.284 13.949 13.794 222 327.526882 125.520782 0.3350 0.1550 0.4900 + 08402038+1933574 130.084945 19.565947 08h40m20.39s 19d33m57.41s 13.929 13.650 13.626 222 328.607444 74.377848 0.2790 0.0240 0.3030 + 08400968+1937170 130.040338 19.621416 08h40m09.68s 19d37m17.10s 10.556 10.240 10.133 222 332.067914 29.813874 0.3160 0.1070 0.4230 + 08393574+1930426 129.898929 19.511852 08h39m35.74s 19d30m42.67s 11.746 11.405 11.365 222 332.083230 251.339797 0.3410 0.0400 0.3810 + 08400627+1927148 130.026140 19.454117 08h40m06.27s 19d27m14.82s 9.185 8.964 8.868 212 335.276111 159.562171 0.2210 0.0960 0.3170 + 08401246+1936578 130.051953 19.616076 08h40m12.47s 19d36m57.87s 15.228 14.643 14.553 222 337.826975 37.251488 0.5850 0.0900 0.6750 + 08401334+1928088 130.055617 19.469133 08h40m13.35s 19d28m08.88s 15.646 14.956 14.343 222 338.795757 140.145477 0.6900 0.6130 1.3030 + 08401296+1928042 130.054026 19.467850 08h40m12.97s 19d28m04.26s 15.145 14.257 13.828 222 338.957970 141.346630 0.8880 0.4290 1.3170 + 08393447+1931198 129.893659 19.522177 08h39m34.48s 19d31m19.84s 16.060 15.478 15.395 222 339.586535 258.264177 0.5820 0.0830 0.6650 + 08402015+1934426 130.083988 19.578508 08h40m20.16s 19d34m42.63s 16.751 16.221 16.314 200 340.544953 66.881553 - - - + 08393408+1931414 129.892032 19.528170 08h39m34.09s 19d31m41.41s 15.385 14.656 14.589 222 341.316202 262.001406 0.7290 0.0670 0.7960 + 08395093+1937571 129.962228 19.632528 08h39m50.93s 19d37m57.10s 14.260 13.643 13.347 222 342.936674 343.087884 0.6170 0.2960 0.9130 + 08400931+1937327 130.038828 19.625753 08h40m09.32s 19d37m32.71s 16.359 15.823 15.740 222 343.282643 27.775811 0.5360 0.0830 0.6190 + 08393553+1930165 129.898066 19.504610 08h39m35.54s 19d30m16.60s 16.636 16.088 15.414 220 344.033551 247.380008 0.5480 - - + 08401789+1929096 130.074551 19.486004 08h40m17.89s 19d29m09.61s 16.239 15.587 15.290 222 344.799557 125.315583 0.6520 0.2970 0.9490 + 08393676+1929311 129.903196 19.491987 08h39m36.77s 19d29m31.15s 17.075 16.187 15.832 222 348.875363 239.365392 0.8880 0.3550 1.2430 + 08394736+1927101 129.947365 19.452816 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08374640+1935575 129.443374 19.599319 08h37m46.41s 19d35m57.55s 10.725 10.341 10.244 222 1871.424397 276.489887 0.3840 0.0970 0.4810 + 08412196+1956371 130.341501 19.943657 08h41m21.96s 19d56m37.17s 13.554 13.221 13.268 222 1871.473343 39.244047 0.3330 -0.0470 0.2860 + 08374584+1930379 129.441038 19.510532 08h37m45.85s 19d30m37.92s 17.197 16.415 15.839 222 1871.506059 266.689083 0.7820 0.5760 1.3580 + 08380178+1917347 129.507418 19.292980 08h38m01.78s 19d17m34.73s 13.682 13.134 13.008 222 1871.560336 241.537322 0.5480 0.1260 0.6740 + 08374587+1934324 129.441136 19.575678 08h37m45.87s 19d34m32.44s 17.164 16.213 16.241 222 1871.571730 273.873723 0.9510 -0.0280 0.9230 + 08390909+2001289 129.787909 20.024704 08h39m09.10s 20d01m28.93s 16.039 15.272 15.155 222 1871.817547 338.397634 0.7670 0.1170 0.8840 + 08380215+1947390 129.508965 19.794184 08h38m02.15s 19d47m39.06s 16.161 15.663 15.262 222 1872.336733 299.162637 0.4980 0.4010 0.8990 + 08402478+1901541 130.103279 19.031696 08h40m24.79s 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0.4140 + 08420650+1924405 130.527094 19.411251 08h42m06.50s 19d24m40.50s 7.545 7.464 7.431 111 1876.717159 104.366730 0.0810 0.0330 0.1140 + 08420314+1922047 130.513101 19.367979 08h42m03.14s 19d22m04.72s 15.469 14.813 14.723 222 1876.905634 109.340349 0.6560 0.0900 0.7460 + 08374855+1939306 129.452327 19.658506 08h37m48.56s 19d39m30.62s 12.143 11.489 11.352 222 1877.027761 283.070878 0.6540 0.1370 0.7910 + 08420364+1922238 130.515176 19.373280 08h42m03.64s 19d22m23.81s 15.300 15.231 14.992 222 1877.283432 108.719495 0.0690 0.2390 0.3080 + 08381302+1951409 129.554277 19.861382 08h38m13.03s 19d51m40.98s 15.761 15.133 14.856 222 1877.353916 307.924797 0.6280 0.2770 0.9050 + 08405986+2000111 130.249419 20.003094 08h40m59.86s 20d00m11.14s 14.081 13.500 13.217 222 1877.569594 27.673248 0.5810 0.2830 0.8640 + 08382089+1953549 129.587072 19.898607 08h38m20.90s 19d53m54.99s 15.807 15.356 15.361 222 1879.773977 313.234053 0.4510 -0.0050 0.4460 + 08420352+1942549 130.514691 19.715250 08h42m03.53s 19d42m54.90s 16.948 16.138 15.684 222 1880.733148 70.473873 0.8100 0.4540 1.2640 + 08413731+1953227 130.405478 19.889643 08h41m37.31s 19d53m22.71s 14.377 14.028 14.068 222 1881.128313 48.135429 0.3490 -0.0400 0.3090 + 08411575+1957574 130.315625 19.965969 08h41m15.75s 19d57m57.49s 15.196 14.629 14.453 222 1881.795114 35.629320 0.5670 0.1760 0.7430 + 08374579+1936158 129.440811 19.604408 08h37m45.79s 19d36m15.87s 17.293 16.082 15.982 222 1882.155197 277.015167 1.2110 0.1000 1.3110 + 08414136+1952179 130.422368 19.871658 08h41m41.37s 19d52m17.97s 16.579 16.086 15.272 220 1882.699545 50.765164 0.4930 - - + 08420578+1941248 130.524095 19.690233 08h42m05.78s 19d41m24.84s 16.843 15.684 16.299 220 1883.390832 73.381470 1.1590 - - + 08410257+1905023 130.260709 19.083977 08h41m02.57s 19d05m02.32s 16.961 15.682 15.383 222 1883.391014 150.919930 1.2790 0.2990 1.5780 + 08374492+1930581 129.437179 19.516142 08h37m44.92s 19d30m58.11s 14.915 14.724 14.644 222 1883.461374 267.326622 0.1910 0.0800 0.2710 + 08413756+1911374 130.406515 19.193731 08h41m37.56s 19d11m37.43s 15.300 14.829 14.628 222 1884.594495 131.544828 0.4710 0.2010 0.6720 + 08382265+1910331 129.594395 19.175877 08h38m22.65s 19d10m33.16s 14.947 14.502 14.458 222 1884.662391 225.784350 0.4450 0.0440 0.4890 + 08395506+2003541 129.979453 20.065042 08h39m55.07s 20d03m54.15s 9.199 8.993 8.961 222 1885.599817 358.746641 0.2060 0.0320 0.2380 + 08413125+1910017 130.380212 19.167164 08h41m31.25s 19d10m01.79s 16.201 15.589 15.081 222 1885.941057 135.524967 0.6120 0.5080 1.1200 + 08385028+1959349 129.709515 19.993048 08h38m50.28s 19d59m34.97s 15.009 14.322 14.268 222 1886.104567 329.598181 0.6870 0.0540 0.7410 + 08374523+1935446 129.438498 19.595728 08h37m45.24s 19d35m44.62s 15.964 15.362 15.212 222 1886.498302 276.044460 0.6020 0.1500 0.7520 + 08392263+1902097 129.844292 19.036051 08h39m22.63s 19d02m09.78s 11.134 10.769 10.686 222 1886.868165 195.413081 0.3650 0.0830 0.4480 + 08392845+2003104 129.868567 20.052896 08h39m28.46s 20d03m10.43s 16.245 16.328 15.474 220 1888.027109 347.263710 -0.0830 - - + 08385240+1905040 129.718364 19.084455 08h38m52.41s 19d05m04.04s 11.717 11.516 11.451 222 1888.903314 209.486931 0.2010 0.0650 0.2660 + 08374758+1925402 129.448251 19.427849 08h37m47.58s 19d25m40.26s 14.847 14.305 14.210 222 1888.957825 257.593763 0.5420 0.0950 0.6370 + 08421162+1932177 130.548437 19.538263 08h42m11.62s 19d32m17.75s 13.931 13.522 13.429 222 1889.029066 90.248326 0.4090 0.0930 0.5020 + 08382157+1910420 129.589899 19.178358 08h38m21.58s 19d10m42.09s 15.914 15.259 15.162 222 1889.428337 226.302098 0.6550 0.0970 0.7520 + 08391108+2001598 129.796184 20.033304 08h39m11.08s 20d01m59.89s 16.422 15.619 15.659 222 1890.625370 339.532426 0.8030 -0.0400 0.7630 + 08405073+2001271 130.211415 20.024204 08h40m50.74s 20d01m27.13s 17.078 15.972 15.561 222 1890.850695 23.148925 1.1060 0.4110 1.5170 + 08420733+1924297 130.530578 19.408268 08h42m07.34s 19d24m29.76s 16.309 15.955 15.666 200 1890.879222 104.591675 - - - + 08393376+1901294 129.890685 19.024837 08h39m33.76s 19d01m29.41s 16.778 16.039 15.636 220 1890.973942 190.469867 0.7390 - - + 08412998+1955240 130.374925 19.923334 08h41m29.98s 19d55m24.00s 15.543 14.842 14.614 222 1891.404617 43.302384 0.7010 0.2280 0.9290 + 08393126+2003226 129.880262 20.056293 08h39m31.26s 20d03m22.65s 16.275 16.075 15.686 220 1891.657977 348.513974 0.2000 - - + 08374964+1923354 129.456838 19.393194 08h37m49.64s 19d23m35.50s 16.075 15.665 15.518 222 1892.030081 253.711120 0.4100 0.1470 0.5570 + 08393054+2003212 129.877252 20.055889 08h39m30.54s 20d03m21.20s 16.247 15.621 15.681 222 1892.294302 348.203182 0.6260 -0.0600 0.5660 + 08380188+1948141 129.507852 19.803930 08h38m01.88s 19d48m14.15s 16.276 15.964 15.137 222 1892.863832 300.036223 0.3120 0.8270 1.1390 + 08385953+1904057 129.748076 19.068260 08h38m59.54s 19d04m05.74s 16.542 16.513 15.589 222 1893.667566 205.953981 0.0290 0.9240 0.9530 + 08421179+1930560 130.549158 19.515575 08h42m11.80s 19d30m56.07s 14.995 14.310 14.006 222 1893.856056 92.719504 0.6850 0.3040 0.9890 + 08374539+1927564 129.439147 19.465672 08h37m45.40s 19d27m56.42s 16.464 16.128 15.914 222 1894.593806 261.820158 0.3360 0.2140 0.5500 + 08374660+1926181 129.444176 19.438368 08h37m46.60s 19d26m18.12s 9.591 9.330 9.276 222 1894.628144 258.802705 0.2610 0.0540 0.3150 + 08381451+1952350 129.560469 19.876390 08h38m14.51s 19d52m35.00s 13.932 13.409 13.177 222 1894.696965 309.604792 0.5230 0.2320 0.7550 + 08374754+1925101 129.448088 19.419481 08h37m47.54s 19d25m10.13s 13.318 13.082 13.009 222 1896.289103 256.709026 0.2360 0.0730 0.3090 + 08402944+1901450 130.122677 19.029169 08h40m29.44s 19d01m45.01s 15.437 14.939 14.999 222 1896.983448 166.404182 0.4980 -0.0600 0.4380 + 08384367+1906101 129.681992 19.102825 08h38m43.68s 19d06m10.17s 14.902 14.270 14.043 222 1897.199516 213.726893 0.6320 0.2270 0.8590 + 08421156+1929232 130.548187 19.489792 08h42m11.56s 19d29m23.25s 15.353 14.784 14.549 222 1897.544213 95.524740 0.5690 0.2350 0.8040 + 08374387+1933525 129.432819 19.564610 08h37m43.88s 19d33m52.60s 16.671 16.204 15.704 222 1897.617414 272.618249 0.4670 0.5000 0.9670 + 08394393+1901008 129.933066 19.016899 08h39m43.94s 19d01m00.84s 14.354 13.768 13.745 222 1898.633051 186.028325 0.5860 0.0230 0.6090 + 08374902+1941242 129.454252 19.690069 08h37m49.02s 19d41m24.25s 14.593 14.171 14.228 222 1899.333130 286.458127 0.4220 -0.0570 0.3650 + 08410420+2000032 130.267518 20.000906 08h41m04.20s 20d00m03.26s 16.706 15.306 15.287 202 1900.000985 29.417954 - - 1.4190 + 08382353+1955025 129.598044 19.917362 08h38m23.53s 19d55m02.50s 16.945 16.171 17.421 220 1900.256962 315.486242 0.7740 - - + 08415822+1918211 130.492590 19.305868 08h41m58.22s 19d18m21.12s 14.957 14.729 14.724 222 1900.375400 116.414352 0.2280 0.0050 0.2330 + 08412113+1957243 130.338062 19.956772 08h41m21.13s 19d57m24.38s 15.299 14.755 14.481 222 1900.995504 38.070261 0.5440 0.2740 0.8180 + 08374967+1923014 129.456990 19.383736 08h37m49.68s 19d23m01.45s 16.501 16.299 16.011 222 1901.469599 252.722760 0.2020 0.2880 0.4900 + 08404078+2002320 130.169927 20.042233 08h40m40.78s 20d02m32.04s 15.104 14.463 14.192 222 1901.475527 18.486994 0.6410 0.2710 0.9120 + 08374532+1927126 129.438858 19.453503 08h37m45.33s 19d27m12.61s 16.454 16.038 15.646 222 1902.427572 260.518973 0.4160 0.3920 0.8080 + 08415687+1947238 130.486987 19.789965 08h41m56.88s 19d47m23.87s 14.020 13.411 13.343 222 1902.762524 61.863053 0.6090 0.0680 0.6770 + 08374621+1938596 129.442556 19.649897 08h37m46.21s 19d38m59.63s 16.900 16.343 15.689 222 1902.778097 281.938576 0.5570 0.6540 1.2110 + 08410503+2000002 130.270997 20.000072 08h41m05.04s 20d00m00.26s 14.701 14.058 13.908 222 1903.221705 29.770819 0.6430 0.1500 0.7930 + 08404115+1902264 130.171469 19.040688 08h40m41.15s 19d02m26.48s 15.294 14.749 14.666 222 1903.266625 161.245143 0.5450 0.0830 0.6280 + 08380426+1915320 129.517788 19.258898 08h38m04.27s 19d15m32.03s 16.234 15.897 15.843 222 1903.483075 237.784596 0.3370 0.0540 0.3910 + 08401146+2004032 130.047752 20.067583 08h40m11.46s 20d04m03.30s 12.729 12.082 11.844 222 1903.801639 5.718844 0.6470 0.2380 0.8850 + 08384185+1958408 129.674397 19.978001 08h38m41.86s 19d58m40.80s 16.610 15.580 15.005 222 1904.172824 325.687551 1.0300 0.5750 1.6050 + 08420336+1944083 130.514006 19.735664 08h42m03.36s 19d44m08.39s 15.495 15.090 15.048 222 1904.184809 68.363859 0.4050 0.0420 0.4470 + 08420741+1941202 130.530916 19.688957 08h42m07.42s 19d41m20.25s 13.725 13.420 13.322 222 1904.297185 73.710891 0.3050 0.0980 0.4030 + 08402957+1901377 130.123232 19.027155 08h40m29.58s 19d01m37.76s 12.708 12.503 12.484 222 1904.474591 166.400054 0.2050 0.0190 0.2240 + 08404688+2002044 130.195335 20.034582 08h40m46.88s 20d02m04.50s 13.846 13.331 13.252 222 1904.841366 21.202067 0.5150 0.0790 0.5940 + 08401484+1900588 130.061842 19.016359 08h40m14.84s 19d00m58.89s 16.694 15.786 15.313 220 1905.101984 172.796015 0.9080 - - + 08420507+1921574 130.521151 19.365959 08h42m05.08s 19d21m57.45s 14.253 13.563 13.433 222 1905.105778 109.271807 0.6900 0.1300 0.8200 + 08401528+2003588 130.063669 20.066339 08h40m15.28s 20d03m58.82s 14.014 13.666 13.721 222 1905.494581 7.342813 0.3480 -0.0550 0.2930 + 08405505+2001168 130.229408 20.021357 08h40m55.06s 20d01m16.89s 15.674 15.026 14.424 222 1906.376489 24.951607 0.6480 0.6020 1.2500 + 08412584+1956369 130.357695 19.943600 08h41m25.85s 19d56m36.96s 9.659 9.396 9.329 222 1906.547139 40.521352 0.2630 0.0670 0.3300 + 08405433+2001223 130.226375 20.022884 08h40m54.33s 20d01m22.38s 16.678 15.815 14.805 200 1907.052065 24.602449 - - - + 08381098+1951523 129.545771 19.864555 08h38m10.99s 19d51m52.40s 16.596 15.939 15.664 220 1907.134457 307.666023 0.6570 - - + 08421266+1930148 130.552779 19.504133 08h42m12.67s 19d30m14.88s 16.490 16.020 15.639 222 1908.646077 93.935841 0.4700 0.3810 0.8510 + 08391908+2002573 129.829521 20.049257 08h39m19.09s 20d02m57.33s 15.324 14.709 14.680 222 1909.017691 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0.8580 - - + 08412488+1957275 130.353678 19.957653 08h41m24.88s 19d57m27.55s 16.181 15.586 15.293 222 1936.563018 39.241113 0.5950 0.2930 0.8880 + 08391397+1901551 129.808210 19.031998 08h39m13.97s 19d01m55.19s 13.265 12.957 12.855 222 1936.850523 198.803525 0.3080 0.1020 0.4100 + 08390461+1902453 129.769247 19.045929 08h39m04.62s 19d02m45.34s 10.625 10.402 10.352 222 1937.135858 202.997781 0.2230 0.0500 0.2730 + 08381490+1953474 129.562101 19.896505 08h38m14.90s 19d53m47.42s 16.762 15.863 15.712 222 1937.364403 311.362388 0.8990 0.1510 1.0500 + 08402707+1900554 130.112815 19.015402 08h40m27.08s 19d00m55.45s 16.483 15.985 16.501 220 1937.803173 167.712524 0.4980 - - + 08412552+1907386 130.356341 19.127390 08h41m25.52s 19d07m38.60s 14.951 14.345 14.188 222 1938.044539 140.205581 0.6060 0.1570 0.7630 + 08374094+1933290 129.420617 19.558064 08h37m40.95s 19d33m29.03s 15.556 15.263 15.192 222 1938.138400 271.870327 0.2930 0.0710 0.3640 + 08415437+1915266 130.476582 19.257416 08h41m54.38s 19d15m26.70s 10.026 9.732 9.643 222 1938.177115 121.753081 0.2940 0.0890 0.3830 + 08383684+1906276 129.653526 19.107668 08h38m36.85s 19d06m27.60s 16.415 15.712 15.392 222 1938.399320 216.396689 0.7030 0.3200 1.0230 + 08404663+2002420 130.194331 20.045004 08h40m46.64s 20d02m42.01s 15.929 15.198 15.095 662 1938.645350 20.706265 - - - + 08393042+2004087 129.876779 20.069092 08h39m30.43s 20d04m08.73s 9.140 8.839 8.807 112 1939.159922 348.444977 0.3010 0.0320 0.3330 + 08415174+1950350 130.465620 19.843079 08h41m51.75s 19d50m35.08s 14.434 14.060 13.912 222 1939.177172 55.859544 0.3740 0.1480 0.5220 + 08412635+1907469 130.359811 19.129709 08h41m26.35s 19d07m46.95s 14.415 13.866 13.651 222 1939.206992 139.779699 0.5490 0.2150 0.7640 + 08380277+1950041 129.511580 19.834478 08h38m02.78s 19d50m04.12s 15.073 14.735 14.556 222 1939.358117 303.040723 0.3380 0.1790 0.5170 + 08421482+1930028 130.561758 19.500801 08h42m14.82s 19d30m02.88s 16.205 15.996 15.867 222 1939.924437 94.224475 0.2090 0.1290 0.3380 + 08384254+1959306 129.677263 19.991835 08h38m42.54s 19d59m30.61s 16.644 15.927 15.647 222 1940.112039 326.755194 0.7170 0.2800 0.9970 + 08375558+1947090 129.481613 19.785856 08h37m55.59s 19d47m09.08s 14.323 13.891 13.761 222 1940.146762 297.061398 0.4320 0.1300 0.5620 + 08374494+1940290 129.437288 19.674726 08h37m44.95s 19d40m29.01s 12.822 12.262 11.986 222 1940.284005 284.416101 0.5600 0.2760 0.8360 + 08384409+1905139 129.683741 19.087200 08h38m44.10s 19d05m13.92s 15.492 14.821 14.500 222 1941.078875 212.661148 0.6710 0.3210 0.9920 + 08384798+1904394 129.699956 19.077629 08h38m47.99s 19d04m39.46s 14.343 13.855 13.759 222 1941.516917 210.741403 0.4880 0.0960 0.5840 + 08402777+1900534 130.115715 19.014835 08h40m27.77s 19d00m53.41s 15.558 15.051 14.982 222 1941.913528 167.440700 0.5070 0.0690 0.5760 + 08375728+1947587 129.488681 19.799662 08h37m57.28s 19d47m58.78s 12.192 11.775 11.715 222 1942.057096 298.688886 0.4170 0.0600 0.4770 + 08374060+1933032 129.419201 19.550900 08h37m40.61s 19d33m03.24s 13.718 13.052 12.803 222 1942.357218 271.105701 0.6660 0.2490 0.9150 + 08420249+1918497 130.510401 19.313812 08h42m02.50s 19d18m49.72s 14.753 14.406 14.195 222 1942.406318 114.860964 0.3470 0.2110 0.5580 + 08384093+1905416 129.670582 19.094915 08h38m40.94s 19d05m41.69s 14.618 14.179 14.177 222 1942.474723 214.215177 0.4390 0.0020 0.4410 + 08420415+1945217 130.517303 19.756046 08h42m04.15s 19d45m21.77s 13.830 13.385 13.338 222 1942.487319 66.469799 0.4450 0.0470 0.4920 + 08374070+1931063 129.419605 19.518433 08h37m40.71s 19d31m06.36s 7.779 7.731 7.662 111 1942.638781 267.657383 0.0480 0.0690 0.1170 + 08392084+2003406 129.836869 20.061287 08h39m20.85s 20d03m40.63s 16.678 15.930 15.547 222 1943.668515 344.377975 0.7480 0.3830 1.1310 + 08390946+1902106 129.789445 19.036301 08h39m09.47s 19d02m10.68s 15.565 15.362 14.965 222 1943.799746 200.733186 0.2030 0.3970 0.6000 + 08380784+1951550 129.532684 19.865288 08h38m07.84s 19d51m55.04s 15.924 15.357 15.248 222 1944.047655 306.932446 0.5670 0.1090 0.6760 + 08380359+1914320 129.514990 19.242231 08h38m03.60s 19d14m32.03s 13.284 12.818 12.813 222 1944.175719 236.441073 0.4660 0.0050 0.4710 + 08401970+1900289 130.082107 19.008043 08h40m19.71s 19d00m28.95s 14.060 13.621 13.517 222 1944.499335 170.889410 0.4390 0.1040 0.5430 + 08415064+1913543 130.461041 19.231754 08h41m50.65s 19d13m54.31s 15.844 15.250 15.294 222 1945.082805 124.887198 0.5940 -0.0440 0.5500 + 08410558+2000452 130.273286 20.012568 08h41m05.59s 20d00m45.24s 16.715 16.054 15.457 220 1946.152049 29.308985 0.6610 - - + 08415935+1917106 130.497324 19.286291 08h41m59.36s 19d17m10.65s 13.229 12.935 12.905 222 1947.097447 118.057467 0.2940 0.0300 0.3240 + 08413438+1909180 130.393262 19.155005 08h41m34.38s 19d09m18.02s 16.038 15.565 15.408 222 1948.263651 135.491405 0.4730 0.1570 0.6300 + 08375191+1919242 129.466328 19.323404 08h37m51.92s 19d19m24.25s 17.004 17.841 16.541 200 1948.446111 246.336950 - - - + 08401730+2004397 130.072108 20.077713 08h40m17.31s 20d04m39.77s 11.773 11.675 11.626 222 1949.898763 8.020081 0.0980 0.0490 0.1470 + 08400171+1859595 130.007130 18.999865 08h40m01.71s 18d59m59.51s 9.053 8.767 8.698 112 1950.199625 178.451730 0.2860 0.0690 0.3550 + 08415683+1916006 130.486828 19.266848 08h41m56.84s 19d16m00.65s 16.607 15.711 15.436 222 1950.354577 120.365420 0.8960 0.2750 1.1710 + 08381546+1954159 129.564417 19.904436 08h38m15.46s 19d54m15.97s 13.246 12.909 12.831 222 1950.462584 312.145062 0.3370 0.0780 0.4150 + 08374004+1933129 129.416845 19.553604 08h37m40.04s 19d33m12.97s 14.940 14.444 14.381 222 1950.527619 271.387869 0.4960 0.0630 0.5590 + 08421415+1927141 130.558985 19.453934 08h42m14.16s 19d27m14.16s 13.824 13.478 13.388 222 1950.856505 99.192613 0.3460 0.0900 0.4360 + 08400670+2004561 130.027923 20.082270 08h40m06.70s 20d04m56.17s 12.111 11.742 11.614 222 1951.039748 3.604074 0.3690 0.1280 0.4970 + 08383314+1958103 129.638091 19.969543 08h38m33.14s 19d58m10.35s 11.680 10.978 10.812 222 1952.105435 322.206139 0.7020 0.1660 0.8680 + 08375509+1947220 129.479570 19.789446 08h37m55.10s 19d47m22.01s 15.496 15.203 15.080 222 1952.182245 297.307843 0.2930 0.1230 0.4160 + 08420267+1946317 130.511138 19.775488 08h42m02.67s 19d46m31.76s 15.677 15.202 14.666 222 1952.405856 64.340601 0.4750 0.5360 1.0110 + 08374192+1926561 129.424672 19.448931 08h37m41.92s 19d26m56.15s 16.064 15.528 15.520 222 1952.679554 260.280169 0.5360 0.0080 0.5440 + 08381931+1909437 129.580494 19.162148 08h38m19.32s 19d09m43.73s 16.232 15.513 15.511 222 1953.014518 225.716868 0.7190 0.0020 0.7210 + 08384133+1959359 129.672238 19.993324 08h38m41.34s 19d59m35.97s 15.411 14.885 14.989 222 1953.968221 326.425216 0.5260 -0.1040 0.4220 + 08374002+1934312 129.416760 19.575335 08h37m40.02s 19d34m31.21s 15.553 15.326 15.082 222 1954.013859 273.681754 0.2270 0.2440 0.4710 + 08393516+2004366 129.896502 20.076838 08h39m35.16s 20d04m36.62s 16.574 16.224 15.661 222 1954.365416 350.524983 0.3500 0.5630 0.9130 + 08384755+1904277 129.698150 19.074362 08h38m47.56s 19d04m27.70s 15.444 15.287 14.938 222 1954.767697 210.720639 0.1570 0.3490 0.5060 + 08420724+1944074 130.530188 19.735413 08h42m07.25s 19d44m07.49s 14.715 14.086 13.740 222 1954.998667 68.976400 0.6290 0.3460 0.9750 + 08374172+1938093 129.423852 19.635942 08h37m41.72s 19d38m09.39s 16.954 15.949 15.628 222 1955.619465 280.118819 1.0050 0.3210 1.3260 + 08415659+1915439 130.485809 19.262211 08h41m56.59s 19d15m43.96s 16.658 16.254 15.393 220 1955.918164 120.838174 0.4040 - - + 08374670+1922122 129.444610 19.370068 08h37m46.71s 19d22m12.24s 14.172 13.876 13.776 222 1956.638997 251.717548 0.2960 0.1000 0.3960 + 08421009+1922476 130.542066 19.379902 08h42m10.10s 19d22m47.65s 16.116 15.331 15.224 222 1956.654923 107.192734 0.7850 0.1070 0.8920 + 08374428+1940582 129.434540 19.682858 08h37m44.29s 19d40m58.29s 14.804 14.190 13.957 222 1956.683211 285.180025 0.6140 0.2330 0.8470 + 08410737+1904164 130.280726 19.071245 08h41m07.37s 19d04m16.48s 8.995 8.669 8.636 112 1956.845239 149.825996 0.3260 0.0330 0.3590 + 08385269+2001154 129.719561 20.020950 08h38m52.69s 20d01m15.42s 10.573 10.370 10.334 222 1957.057628 331.948494 0.2030 0.0360 0.2390 + 08415353+1950300 130.473065 19.841684 08h41m53.54s 19d50m30.06s 16.264 15.683 15.110 220 1957.361237 56.393908 0.5810 - - + 08415099+1913397 130.462494 19.227711 08h41m51.00s 19d13m39.76s 15.322 14.691 14.299 222 1957.504764 125.152983 0.6310 0.3920 1.0230 + 08373947+1932294 129.414468 19.541510 08h37m39.47s 19d32m29.44s 16.058 15.258 15.264 222 1958.169354 270.109171 0.8000 -0.0060 0.7940 + 08404521+2003104 130.188402 20.052893 08h40m45.22s 20d03m10.41s 16.223 15.424 15.128 222 1958.299768 19.862996 0.7990 0.2960 1.0950 + 08380231+1950291 129.509635 19.841419 08h38m02.31s 19d50m29.11s 14.145 13.874 13.784 222 1958.531771 303.550046 0.2710 0.0900 0.3610 + 08421259+1940177 130.552482 19.671600 08h42m12.60s 19d40m17.76s 16.437 15.946 15.672 220 1958.845255 76.060662 0.4910 - - + 08390221+2002227 129.759218 20.039639 08h39m02.21s 20d02m22.70s 16.317 15.788 15.349 222 1958.886355 336.340429 0.5290 0.4390 0.9680 + 08420824+1943420 130.534340 19.728346 08h42m08.24s 19d43m42.05s 16.023 15.668 15.856 220 1959.307252 69.818120 0.3550 - - + 08375126+1919180 129.463588 19.321690 08h37m51.26s 19d19m18.08s 14.329 13.839 13.692 222 1959.455092 246.281920 0.4900 0.1470 0.6370 + 08420869+1921376 130.536230 19.360451 08h42m08.70s 19d21m37.62s 15.366 14.715 14.608 222 1960.000357 109.319843 0.6510 0.1070 0.7580 + 08374989+1945018 129.457894 19.750517 08h37m49.89s 19d45m01.86s 16.096 15.528 15.360 222 1960.019888 292.677853 0.5680 0.1680 0.7360 + 08400105+2005087 130.004388 20.085770 08h40m01.05s 20d05m08.77s 16.577 16.054 16.083 200 1960.242800 1.258945 - - - + 08401562+1900044 130.065103 19.001245 08h40m15.62s 19d00m04.48s 15.365 15.258 15.290 222 1960.477545 172.672929 0.1070 -0.0320 0.0750 + 08395309+2005083 129.971217 20.085657 08h39m53.09s 20d05m08.37s 14.956 14.369 14.222 222 1960.582944 357.980623 0.5870 0.1470 0.7340 + 08421397+1938589 130.558237 19.649696 08h42m13.98s 19d38m58.91s 16.392 15.565 15.329 220 1960.736846 78.435100 0.8270 - - + 08374078+1927444 129.419945 19.462345 08h37m40.79s 19d27m44.44s 16.002 15.493 15.424 222 1960.821023 261.751048 0.5090 0.0690 0.5780 + 08395825+2005106 129.992711 20.086296 08h39m58.25s 20d05m10.67s 16.371 15.701 15.389 222 1961.664891 0.104778 0.6700 0.3120 0.9820 + 08374370+1940478 129.432086 19.679962 08h37m43.70s 19d40m47.86s 14.688 14.306 14.265 222 1962.059366 284.823095 0.3820 0.0410 0.4230 + 08421142+1923308 130.547621 19.391911 08h42m11.43s 19d23m30.88s 16.916 16.507 16.398 222 1962.306816 105.823386 0.4090 0.1090 0.5180 + 08374913+1944421 129.454711 19.745033 08h37m49.13s 19d44m42.12s 14.959 14.383 14.144 222 1962.557034 292.024845 0.5760 0.2390 0.8150 + 08385076+1903524 129.711508 19.064564 08h38m50.76s 19d03m52.43s 11.497 11.180 11.113 222 1962.797046 209.054189 0.3170 0.0670 0.3840 + 08401078+2005038 130.044941 20.084394 08h40m10.79s 20d05m03.82s 14.541 13.993 13.788 222 1963.129143 5.266241 0.5480 0.2050 0.7530 + 08402845+2004245 130.118551 20.073486 08h40m28.45s 20d04m24.55s 14.561 14.159 14.061 222 1963.174535 12.625262 0.4020 0.0980 0.5000 + 08384128+1959471 129.672037 19.996433 08h38m41.29s 19d59m47.16s 15.225 14.607 14.274 222 1963.666862 326.589824 0.6180 0.3330 0.9510 + 08390507+1902135 129.771156 19.037106 08h39m05.08s 19d02m13.58s 13.285 13.003 13.036 222 1963.945293 202.462266 0.2820 -0.0330 0.2490 + 08402931+2004226 130.122147 20.072962 08h40m29.32s 20d04m22.66s 16.588 16.242 15.921 222 1964.039146 12.983416 0.3460 0.3210 0.6670 + 08420997+1942459 130.541582 19.712772 08h42m09.98s 19d42m45.98s 14.022 13.732 13.585 222 1964.145155 71.600159 0.2900 0.1470 0.4370 + 08420327+1918221 130.513640 19.306139 08h42m03.27s 19d18m22.10s 13.599 13.243 13.169 222 1964.171362 115.455222 0.3560 0.0740 0.4300 + 08420222+1917513 130.509287 19.297586 08h42m02.23s 19d17m51.31s 14.969 14.663 14.528 222 1964.451316 116.451569 0.3060 0.1350 0.4410 + 08421048+1942247 130.543698 19.706879 08h42m10.49s 19d42m24.76s 16.595 16.066 15.375 222 1964.453437 72.253314 0.5290 0.6910 1.2200 + 08402944+1900355 130.122688 19.009884 08h40m29.45s 19d00m35.58s 11.244 10.788 10.709 222 1964.552148 166.877617 0.4560 0.0790 0.5350 + 08380733+1952197 129.530557 19.872156 08h38m07.33s 19d52m19.76s 15.972 15.196 14.931 222 1964.679733 307.384257 0.7760 0.2650 1.0410 + 08374253+1925122 129.427220 19.420067 08h37m42.53s 19d25m12.24s 16.736 16.010 15.272 222 1964.785366 257.250338 0.7260 0.7380 1.4640 + 08412871+1957190 130.369654 19.955301 08h41m28.72s 19d57m19.08s 15.154 14.510 14.448 222 1964.871976 40.616593 0.6440 0.0620 0.7060 + 08385930+1902480 129.747092 19.046675 08h38m59.30s 19d02m48.03s 15.959 15.532 15.149 222 1965.288410 205.053165 0.4270 0.3830 0.8100 + 08373941+1935165 129.414223 19.587933 08h37m39.41s 19d35m16.56s 15.925 15.177 15.045 222 1965.872094 274.985992 0.7480 0.1320 0.8800 + 08421405+1925422 130.558578 19.428406 08h42m14.06s 19d25m42.26s 14.676 14.196 14.137 222 1966.576200 101.841854 0.4800 0.0590 0.5390 + 08390321+2002376 129.763416 20.043783 08h39m03.22s 20d02m37.62s 11.867 11.209 11.051 222 1966.919207 336.893964 0.6580 0.1580 0.8160 + 08374529+1942245 129.438738 19.706825 08h37m45.30s 19d42m24.57s 14.976 14.669 14.652 222 1967.186829 287.715554 0.3070 0.0170 0.3240 + 08402006+1900065 130.083612 19.001827 08h40m20.07s 19d00m06.58s 14.331 14.015 13.994 222 1967.405551 170.844945 0.3160 0.0210 0.3370 + 08414470+1911257 130.436282 19.190475 08h41m44.71s 19d11m25.71s 16.215 15.479 14.989 222 1968.839719 129.840536 0.7360 0.4900 1.2260 + 08405812+1902531 130.242188 19.048094 08h40m58.13s 19d02m53.14s 16.795 16.033 15.967 200 1969.356110 154.347380 - - - + 08411969+1905544 130.332056 19.098459 08h41m19.69s 19d05m54.45s 14.931 14.341 14.004 222 1969.762410 143.991986 0.5900 0.3370 0.9270 + 08395847+2005188 129.993663 20.088579 08h39m58.48s 20d05m18.88s 16.241 15.391 15.517 222 1969.892198 0.197959 0.8500 -0.1260 0.7240 + 08395990+2005194 129.999592 20.088732 08h39m59.90s 20d05m19.44s 15.606 15.113 14.798 222 1970.614784 0.780753 0.4930 0.3150 0.8080 + 08383786+1959231 129.657787 19.989754 08h38m37.87s 19d59m23.11s 7.700 7.663 7.610 111 1970.969248 325.035277 0.0370 0.0530 0.0900 + 08414076+1954432 130.419845 19.912022 08h41m40.76s 19d54m43.28s 14.686 14.361 14.371 222 1971.239598 47.328369 0.3250 -0.0100 0.3150 + 08411543+1905104 130.314296 19.086243 08h41m15.43s 19d05m10.47s 13.109 12.490 12.254 222 1971.377110 146.164456 0.6190 0.2360 0.8550 + 08383957+1959403 129.664895 19.994528 08h38m39.57s 19d59m40.30s 16.583 15.978 14.907 222 1971.436984 325.894309 0.6050 1.0710 1.6760 + 08420903+1943471 130.537640 19.729753 08h42m09.03s 19d43m47.11s 15.198 14.813 14.664 222 1971.548100 69.790940 0.3850 0.1490 0.5340 + 08405987+2001568 130.249476 20.032461 08h40m59.87s 20d01m56.86s 16.575 15.567 15.139 222 1971.823056 26.247188 1.0080 0.4280 1.4360 + 08375508+1916587 129.479512 19.282988 08h37m55.08s 19d16m58.76s 10.330 9.922 9.893 222 1972.074053 241.939965 0.4080 0.0290 0.4370 + 08412771+1907194 130.365494 19.122080 08h41m27.72s 19d07m19.49s 16.244 15.736 15.490 222 1972.667948 139.864030 0.5080 0.2460 0.7540 + 08394279+2005103 129.928333 20.086218 08h39m42.80s 20d05m10.38s 7.251 7.177 7.159 111 1973.069088 353.771078 0.0740 0.0180 0.0920 + 08373914+1929109 129.413098 19.486382 08h37m39.14s 19d29m10.98s 15.539 14.944 14.565 222 1973.114630 264.336575 0.5950 0.3790 0.9740 + 08380728+1952331 129.530340 19.875868 08h38m07.28s 19d52m33.12s 14.245 13.547 13.397 222 1973.375754 307.680287 0.6980 0.1500 0.8480 + 08401421+1859488 130.059245 18.996893 08h40m14.22s 18d59m48.81s 14.355 14.065 13.984 222 1973.602721 173.304956 0.2900 0.0810 0.3710 + 08373922+1928549 129.413450 19.481920 08h37m39.23s 19d28m54.91s 14.584 13.963 13.714 222 1973.630992 263.869133 0.6210 0.2490 0.8700 + 08403287+1900378 130.136983 19.010517 08h40m32.88s 19d00m37.86s 16.020 15.279 14.827 222 1973.922506 165.486832 0.7410 0.4520 1.1930 + 08412140+1958535 130.339194 19.981552 08h41m21.41s 19d58m53.59s 16.289 15.827 15.482 222 1974.168884 36.556782 0.4620 0.3450 0.8070 + 08374346+1941225 129.431119 19.689592 08h37m43.47s 19d41m22.53s 14.035 13.479 13.235 222 1974.261149 285.772587 0.5560 0.2440 0.8000 + 08421342+1924269 130.555927 19.407490 08h42m13.42s 19d24m26.96s 16.884 16.401 15.983 222 1974.912029 104.033390 0.4830 0.4180 0.9010 + 08400185+2005233 130.007712 20.089815 08h40m01.85s 20d05m23.33s 16.015 15.402 14.906 220 1975.079293 1.575575 0.6130 - - + 08380879+1911578 129.536659 19.199400 08h38m08.80s 19d11m57.84s 15.958 15.374 14.988 222 1975.739959 231.529858 0.5840 0.3860 0.9700 + 08380687+1952299 129.528648 19.874987 08h38m06.88s 19d52m29.95s 16.511 15.945 15.855 222 1975.996192 307.506178 0.5660 0.0900 0.6560 + 08375322+1917403 129.471772 19.294540 08h37m53.23s 19d17m40.34s 16.955 16.852 16.355 220 1976.182720 243.363144 0.1030 - - + 08412192+1958510 130.341346 19.980837 08h41m21.92s 19d58m51.01s 11.050 10.868 10.802 222 1976.466840 36.770635 0.1820 0.0660 0.2480 + 08421757+1934318 130.573232 19.575512 08h42m17.58s 19d34m31.84s 16.849 15.747 15.538 222 1976.709475 86.339841 1.1020 0.2090 1.3110 + 08383739+1905343 129.655808 19.092878 08h38m37.39s 19d05m34.36s 17.039 15.793 15.640 222 1977.084938 215.302103 1.2460 0.1530 1.3990 + 08413320+1956401 130.388347 19.944485 08h41m33.20s 19d56m40.15s 16.377 15.593 15.731 222 1978.022335 42.741446 0.7840 -0.1380 0.6460 + 08421796+1931509 130.574843 19.530811 08h42m17.96s 19d31m50.92s 16.779 16.100 15.715 220 1978.994474 91.005165 0.6790 - - + 08410839+2001006 130.284979 20.016851 08h41m08.39s 20d01m00.66s 15.716 14.990 14.858 222 1979.187143 30.087345 0.7260 0.1320 0.8580 + 08373809+1931071 129.408739 19.518656 08h37m38.10s 19d31m07.16s 10.301 9.844 9.756 222 1979.438304 267.727814 0.4570 0.0880 0.5450 + 08374497+1942514 129.437406 19.714302 08h37m44.98s 19d42m51.49s 11.461 10.799 10.586 222 1979.748095 288.419065 0.6620 0.2130 0.8750 + 08373799+1933317 129.408324 19.558821 08h37m38.00s 19d33m31.76s 14.319 13.948 13.939 222 1979.902189 271.913837 0.3710 0.0090 0.3800 + 08401520+2005140 130.063359 20.087227 08h40m15.21s 20d05m14.02s 13.567 12.971 12.687 222 1979.965025 7.033700 0.5960 0.2840 0.8800 + 08382486+1957106 129.603589 19.952969 08h38m24.86s 19d57m10.69s 17.029 16.562 15.716 222 1980.971089 318.479152 0.4670 0.8460 1.3130 + 08421685+1937040 130.570210 19.617798 08h42m16.85s 19d37m04.07s 11.743 11.435 11.361 222 1981.564474 81.923908 0.3080 0.0740 0.3820 + 08374210+1924163 129.425452 19.404533 08h37m42.11s 19d24m16.32s 17.273 16.354 15.787 222 1983.869255 255.714867 0.9190 0.5670 1.4860 + 08390069+1902182 129.752910 19.038412 08h39m00.70s 19d02m18.28s 16.719 15.876 15.403 222 1984.129648 204.172179 0.8430 0.4730 1.3160 + 08412805+1907075 130.366910 19.118755 08h41m28.06s 19d07m07.52s 15.532 15.180 15.035 222 1984.936614 139.979700 0.3520 0.1450 0.4970 + 08373849+1936189 129.410392 19.605265 08h37m38.49s 19d36m18.95s 16.161 15.738 15.590 222 1984.972138 276.749697 0.4230 0.1480 0.5710 + 08415008+1952270 130.458673 19.874176 08h41m50.08s 19d52m27.03s 5.240 5.026 4.683 131 1985.075145 52.799197 0.2140 0.3430 0.5570 + 08374340+1941587 129.430847 19.699642 08h37m43.40s 19d41m58.71s 15.292 15.072 14.753 222 1985.170762 286.771338 0.2200 0.3190 0.5390 + 08393518+2005081 129.896590 20.085608 08h39m35.18s 20d05m08.19s 13.989 13.467 13.351 222 1985.462261 350.683942 0.5220 0.1160 0.6380 + 08402141+1859512 130.089225 18.997578 08h40m21.41s 18d59m51.28s 16.544 16.105 15.409 222 1985.606059 170.370585 0.4390 0.6960 1.1350 + 08414546+1953500 130.439430 19.897243 08h41m45.46s 19d53m50.07s 16.664 16.113 15.766 222 1985.915523 49.753178 0.5510 0.3470 0.8980 + 08393591+2005106 129.899658 20.086279 08h39m35.92s 20d05m10.60s 15.886 15.506 15.494 222 1986.189362 350.990525 0.3800 0.0120 0.3920 + 08420243+1947544 130.510139 19.798466 08h42m02.43s 19d47m54.48s 12.859 12.530 12.479 222 1986.402767 62.144192 0.3290 0.0510 0.3800 + 08383481+1905489 129.645078 19.096931 08h38m34.82s 19d05m48.95s 15.386 14.906 14.858 222 1986.564326 216.405100 0.4800 0.0480 0.5280 + 08375830+1949532 129.492945 19.831457 08h37m58.31s 19d49m53.25s 16.942 16.154 16.022 220 1986.928639 301.789409 0.7880 - - + 08394015+1859380 129.917315 18.993910 08h39m40.16s 18d59m38.08s 12.681 12.024 11.911 222 1987.051009 187.316154 0.6570 0.1130 0.7700 + 08403320+1900252 130.138341 19.007027 08h40m33.20s 19d00m25.30s 16.807 16.129 15.805 222 1987.243256 165.448273 0.6780 0.3240 1.0020 + 08415447+1913581 130.476973 19.232807 08h41m54.47s 19d13m58.11s 11.186 10.972 10.924 222 1987.536103 123.901219 0.2140 0.0480 0.2620 + 08413492+1956310 130.395502 19.941969 08h41m34.92s 19d56m31.09s 15.594 15.028 15.056 222 1988.003783 43.430491 0.5660 -0.0280 0.5380 + 08384595+1904017 129.691494 19.067142 08h38m45.96s 19d04m01.71s 14.559 14.278 14.094 222 1988.680377 210.900667 0.2810 0.1840 0.4650 + 08402855+1900076 130.118976 19.002129 08h40m28.55s 19d00m07.66s 16.734 16.110 16.121 220 1988.973731 167.414157 0.6240 - - + 08392454+2004417 129.852273 20.078274 08h39m24.55s 20d04m41.79s 12.175 11.908 11.822 222 1989.595548 346.298299 0.2670 0.0860 0.3530 + 08385680+2002214 129.736703 20.039305 08h38m56.81s 20d02m21.50s 13.137 12.831 12.753 222 1989.682247 334.318862 0.3060 0.0780 0.3840 + 08404969+2003201 130.207068 20.055590 08h40m49.70s 20d03m20.12s 12.831 12.303 12.195 222 1989.736281 21.476705 0.5280 0.1080 0.6360 + 08375042+1918276 129.460122 19.307688 08h37m50.43s 19d18m27.68s 11.160 10.467 10.295 222 1991.067414 245.092849 0.6930 0.1720 0.8650 + 08374016+1939283 129.417350 19.657862 08h37m40.16s 19d39m28.30s 16.442 15.687 15.599 222 1992.310395 282.245454 0.7550 0.0880 0.8430 + 08373879+1927178 129.411643 19.454969 08h37m38.79s 19d27m17.89s 15.577 15.241 15.195 222 1992.716645 261.114751 0.3360 0.0460 0.3820 + 08375273+1917171 129.469712 19.288101 08h37m52.73s 19d17m17.16s 16.671 16.331 15.951 222 1992.962165 242.859112 0.3400 0.3800 0.7200 + 08383240+1906066 129.635004 19.101839 08h38m32.40s 19d06m06.62s 14.560 14.118 14.024 222 1992.969230 217.499959 0.4420 0.0940 0.5360 + 08421874+1930365 130.578119 19.510149 08h42m18.75s 19d30m36.54s 15.859 15.467 15.300 222 1993.045669 93.136882 0.3920 0.1670 0.5590 + 08411690+1904582 130.320454 19.082838 08h41m16.91s 19d04m58.22s 14.208 13.856 13.678 222 1993.232821 145.859239 0.3520 0.1780 0.5300 + 08420322+1947486 130.513421 19.796837 08h42m03.22s 19d47m48.61s 15.888 15.282 15.176 222 1993.549933 62.441878 0.6060 0.1060 0.7120 + 08411910+1905186 130.329591 19.088524 08h41m19.10s 19d05m18.69s 13.699 13.128 12.867 222 1994.012474 144.789772 0.5710 0.2610 0.8320 + 08400405+1859165 130.016877 18.987921 08h40m04.05s 18d59m16.52s 13.580 13.412 13.395 222 1994.331943 177.532052 0.1680 0.0170 0.1850 + 08392608+2004520 129.858701 20.081125 08h39m26.09s 20d04m52.05s 15.952 15.495 15.115 222 1994.539030 346.974850 0.4570 0.3800 0.8370 + 08375433+1948326 129.476379 19.809071 08h37m54.33s 19d48m32.66s 15.736 15.369 15.217 222 1994.865441 298.972356 0.3670 0.1520 0.5190 + 08375955+1914265 129.498129 19.240698 08h37m59.55s 19d14m26.51s 14.886 14.178 14.094 222 1995.092348 237.222931 0.7080 0.0840 0.7920 + 08375883+1914423 129.495129 19.245098 08h37m58.83s 19d14m42.35s 16.489 15.888 15.672 222 1995.125051 237.763938 0.6010 0.2160 0.8170 + 08374022+1939469 129.417607 19.663050 08h37m40.23s 19d39m46.98s 15.476 14.808 14.549 222 1995.443579 282.775144 0.6680 0.2590 0.9270 + 08405384+2003022 130.224352 20.050632 08h40m53.84s 20d03m02.28s 14.731 14.076 13.763 222 1995.537552 23.226143 0.6550 0.3130 0.9680 + 08405800+1902231 130.241685 19.039772 08h40m58.00s 19d02m23.18s 14.895 14.283 14.154 222 1995.694789 154.762957 0.6120 0.1290 0.7410 + 08384892+1903289 129.703865 19.058054 08h38m48.93s 19d03m28.99s 16.042 15.447 15.082 222 1995.927370 209.381632 0.5950 0.3650 0.9600 + 08391580+2004141 129.815841 20.070585 08h39m15.80s 20d04m14.11s 11.738 11.052 10.867 222 1995.995449 342.672174 0.6860 0.1850 0.8710 + 08373999+1939385 129.416652 19.660707 08h37m40.00s 19d39m38.55s 14.625 14.096 14.044 222 1996.785907 282.518686 0.5290 0.0520 0.5810 + 08404965+1901306 130.206895 19.025181 08h40m49.65s 19d01m30.65s 16.763 16.211 16.243 200 1997.104861 158.480797 - - - + 08400316+1859131 130.013208 18.986973 08h40m03.17s 18d59m13.10s 14.824 14.142 14.023 222 1997.245366 177.894231 0.6820 0.1190 0.8010 + 08375814+1950092 129.492275 19.835915 08h37m58.15s 19d50m09.29s 17.203 16.067 15.903 020 1997.314092 302.147322 - - - + 08380014+1950545 129.500617 19.848492 08h38m00.15s 19d50m54.57s 16.594 15.905 15.567 222 1998.046082 303.677862 0.6890 0.3380 1.0270 + 08404413+2003580 130.183881 20.066135 08h40m44.13s 20d03m58.09s 16.162 15.645 15.575 220 1998.139844 18.984886 0.5170 - - + 08413620+1908335 130.400869 19.142651 08h41m36.21s 19d08m33.54s 8.516 8.425 8.351 112 1998.185125 135.853030 0.0910 0.0740 0.1650 + 08382114+1908150 129.588122 19.137512 08h38m21.15s 19d08m15.04s 16.547 16.012 15.557 222 1998.221657 223.379226 0.5350 0.4550 0.9900 + 08393196+2005135 129.883177 20.087093 08h39m31.96s 20d05m13.53s 16.053 15.395 15.604 222 1998.587154 349.425902 0.6580 -0.2090 0.4490 + 08421641+1939201 130.568377 19.655603 08h42m16.41s 19d39m20.17s 16.812 16.300 15.663 220 1998.682366 78.031794 0.5120 - - + 08382873+1906397 129.619733 19.111055 08h38m28.74s 19d06m39.80s 15.938 15.380 15.299 222 1999.085604 219.260641 0.5580 0.0810 0.6390 + 08420356+1917122 130.514870 19.286736 08h42m03.57s 19d17m12.25s 15.655 15.004 14.741 222 1999.088937 117.208734 0.6510 0.2630 0.9140 + 08401526+2005333 130.063599 20.092602 08h40m15.26s 20d05m33.37s 16.412 15.666 15.408 222 1999.268717 6.988638 0.7460 0.2580 1.0040 + 08375514+1916000 129.479753 19.266685 08h37m55.14s 19d16m00.07s 16.951 17.310 15.670 200 1999.792845 240.447145 - - - + 08392556+1900022 129.856504 19.000628 08h39m25.56s 19d00m02.26s 16.605 16.082 15.770 220 2000.182726 193.296542 0.5230 - - + 08375397+1948357 129.474900 19.809935 08h37m53.98s 19d48m35.77s 16.273 15.780 15.522 222 2000.753611 298.981306 0.4930 0.2580 0.7510 + 08373792+1927446 129.408028 19.462399 08h37m37.93s 19d27m44.64s 15.878 15.042 14.840 222 2000.813470 261.926606 0.8360 0.2020 1.0380 + 08385903+1902093 129.745959 19.035929 08h38m59.03s 19d02m09.34s 11.456 11.057 10.966 222 2002.026321 204.685775 0.3990 0.0910 0.4900 + 08412323+1905508 130.346825 19.097467 08h41m23.24s 19d05m50.88s 16.980 15.850 15.916 222 2002.483249 142.887232 1.1300 -0.0660 1.0640 + 08381883+1956209 129.578486 19.939159 08h38m18.84s 19d56m20.97s 14.793 14.555 14.356 222 2002.615719 315.716723 0.2380 0.1990 0.4370 + 08382934+1906276 129.622268 19.107674 08h38m29.34s 19d06m27.63s 10.486 10.292 10.227 222 2003.134966 218.849579 0.1940 0.0650 0.2590 + 08420516+1947158 130.521541 19.787745 08h42m05.17s 19d47m15.88s 15.599 14.810 14.769 222 2003.354609 63.635002 0.7890 0.0410 0.8300 + 08380532+1912159 129.522169 19.204439 08h38m05.32s 19d12m15.98s 15.286 14.661 14.447 222 2003.407005 232.814709 0.6250 0.2140 0.8390 + 08421645+1939460 130.568555 19.662804 08h42m16.45s 19d39m46.09s 15.969 15.509 15.375 222 2004.722275 77.309966 0.4600 0.1340 0.5940 + 08421232+1943158 130.551347 19.721069 08h42m12.32s 19d43m15.85s 12.734 12.181 12.074 222 2004.997031 71.084932 0.5530 0.1070 0.6600 + 08420279+1916381 130.511628 19.277277 08h42m02.79s 19d16m38.20s 15.747 15.216 14.835 222 2005.278990 118.217521 0.5310 0.3810 0.9120 + 08420383+1917056 130.515986 19.284895 08h42m03.84s 19d17m05.62s 15.929 15.365 15.248 222 2005.508715 117.327037 0.5640 0.1170 0.6810 + 08374259+1942320 129.427475 19.708895 08h37m42.59s 19d42m32.02s 16.638 16.292 15.995 222 2005.835393 287.590147 0.3460 0.2970 0.6430 + 08411789+1904519 130.324548 19.081102 08h41m17.89s 19d04m51.97s 16.808 16.654 15.801 202 2006.223258 145.629464 - - 1.0070 + 08414934+1911471 130.455623 19.196424 08h41m49.35s 19d11m47.13s 11.649 11.013 10.834 222 2006.313515 128.164839 0.6360 0.1790 0.8150 + 08392619+2005048 129.859155 20.084675 08h39m26.20s 20d05m04.83s 11.442 11.241 11.213 222 2006.645607 347.100068 0.2010 0.0280 0.2290 + 08375163+1917156 129.465153 19.287693 08h37m51.64s 19d17m15.69s 15.958 15.551 15.666 222 2007.408419 243.024763 0.4070 -0.1150 0.2920 + 08392489+1859567 129.853715 18.999104 08h39m24.89s 18d59m56.77s 16.585 16.152 15.875 220 2007.715040 193.524364 0.4330 - - + 08410600+2001525 130.275002 20.031260 08h41m06.00s 20d01m52.54s 15.928 15.315 15.251 222 2007.819145 28.510452 0.6130 0.0640 0.6770 + 08384130+1904190 129.672097 19.071951 08h38m41.30s 19d04m19.02s 14.162 13.497 13.271 222 2008.671068 212.770910 0.6650 0.2260 0.8910 + 08412436+1905543 130.351506 19.098417 08h41m24.36s 19d05m54.30s 16.000 15.283 15.412 222 2009.388188 142.465887 0.7170 -0.1290 0.5880 + 08392244+2004548 129.843538 20.081894 08h39m22.45s 20d04m54.82s 13.360 12.751 12.506 222 2009.437160 345.568599 0.6090 0.2450 0.8540 + 08385993+2003037 129.749729 20.051052 08h38m59.93s 20d03m03.79s 16.604 16.564 15.736 220 2009.456851 335.974466 0.0400 - - + 08415365+1951595 130.473582 19.866547 08h41m53.66s 19d51m59.57s 12.232 11.930 11.842 222 2009.503548 54.291408 0.3020 0.0880 0.3900 + 08373785+1926528 129.407734 19.448025 08h37m37.86s 19d26m52.89s 12.544 12.043 11.953 222 2009.891414 260.470722 0.5010 0.0900 0.5910 + 08380045+1913397 129.501881 19.227716 08h38m00.45s 19d13m39.78s 15.676 15.159 14.910 222 2010.332254 235.907299 0.5170 0.2490 0.7660 + 08393552+2005346 129.898007 20.092964 08h39m35.52s 20d05m34.67s 13.699 13.309 13.297 222 2010.834270 350.941205 0.3900 0.0120 0.4020 + 08414881+1953318 130.453403 19.892178 08h41m48.82s 19d53m31.84s 16.148 15.898 15.338 222 2010.847541 51.018754 0.2500 0.5600 0.8100 + 08413844+1908467 130.410187 19.146324 08h41m38.44s 19d08m46.77s 14.879 14.422 14.288 222 2010.946683 134.941840 0.4570 0.1340 0.5910 + 08373666+1928283 129.402767 19.474546 08h37m36.66s 19d28m28.37s 16.200 15.590 15.508 222 2012.717424 263.231741 0.6100 0.0820 0.6920 + 08412649+1958471 130.360405 19.979753 08h41m26.50s 19d58m47.11s 15.127 14.830 14.782 222 2012.777584 38.305553 0.2970 0.0480 0.3450 + 08421549+1941156 130.564564 19.687679 08h42m15.50s 19d41m15.64s 9.031 8.845 8.773 112 2012.920784 74.737174 0.1860 0.0720 0.2580 + 08383753+2000109 129.656385 20.003031 08h38m37.53s 20d00m10.91s 16.069 15.528 15.115 222 2012.949029 325.706676 0.5410 0.4130 0.9540 + 08385660+2002460 129.735837 20.046129 08h38m56.60s 20d02m46.06s 15.308 14.748 14.662 222 2013.095151 334.547754 0.5600 0.0860 0.6460 + 08384545+2001226 129.689391 20.022970 08h38m45.45s 20d01m22.69s 15.527 14.986 14.646 222 2013.472661 329.484887 0.5410 0.3400 0.8810 + 08401460+2005491 130.060843 20.096987 08h40m14.60s 20d05m49.15s 16.446 15.775 15.303 222 2013.830075 6.670684 0.6710 0.4720 1.1430 + 08421640+1940296 130.568374 19.674906 08h42m16.41s 19d40m29.66s 13.137 12.779 12.638 222 2014.002321 76.096012 0.3580 0.1410 0.4990 + 08414640+1910433 130.443356 19.178701 08h41m46.41s 19d10m43.32s 13.744 13.270 13.129 222 2014.578818 130.324555 0.4740 0.1410 0.6150 + 08414116+1955410 130.421508 19.928074 08h41m41.16s 19d55m41.07s 14.986 14.330 13.971 222 2014.821337 46.225424 0.6560 0.3590 1.0150 + 08420098+1915325 130.504113 19.259047 08h42m00.99s 19d15m32.57s 15.871 15.304 15.081 222 2015.226818 120.204682 0.5670 0.2230 0.7900 + 08403865+1900157 130.161047 19.004381 08h40m38.65s 19d00m15.77s 14.835 14.352 14.291 222 2017.116804 163.390183 0.4830 0.0610 0.5440 + 08384799+2001477 129.699974 20.029932 08h38m47.99s 20d01m47.76s 13.965 13.682 13.548 222 2017.283087 330.722483 0.2830 0.1340 0.4170 + 08413758+1908256 130.406597 19.140459 08h41m37.58s 19d08m25.65s 15.893 15.338 15.167 222 2017.413600 135.610823 0.5550 0.1710 0.7260 + 08373570+1935115 129.398780 19.586554 08h37m35.71s 19d35m11.59s 16.234 15.693 15.707 222 2017.673784 274.721304 0.5410 -0.0140 0.5270 + 08405953+2002502 130.248081 20.047304 08h40m59.54s 20d02m50.29s 15.839 15.524 14.645 222 2017.812903 25.454011 0.3150 0.8790 1.1940 + 08394512+2005588 129.938023 20.099693 08h39m45.13s 20d05m58.89s 12.913 12.690 12.533 222 2018.079933 354.845659 0.2230 0.1570 0.3800 + 08420403+1916429 130.516796 19.278601 08h42m04.03s 19d16m42.96s 15.015 14.664 14.624 222 2018.500912 117.861463 0.3510 0.0400 0.3910 + 08413996+1908571 130.416535 19.149218 08h41m39.97s 19d08m57.18s 16.211 15.692 16.017 220 2018.939898 134.299213 0.5190 - - + 08414141+1909177 130.422551 19.154928 08h41m41.41s 19d09m17.74s 14.084 13.506 13.256 222 2019.362455 133.476315 0.5780 0.2500 0.8280 + 08400078+1858497 130.003253 18.980490 08h40m00.78s 18d58m49.76s 14.742 14.070 13.937 222 2019.624990 178.879323 0.6720 0.1330 0.8050 + 08390649+2003537 129.777059 20.064919 08h39m06.49s 20d03m53.71s 15.689 14.952 14.919 222 2020.001170 338.946978 0.7370 0.0330 0.7700 + 08401085+1858570 130.045213 18.982517 08h40m10.85s 18d58m57.06s 12.872 12.211 12.007 222 2020.160703 174.821331 0.6610 0.2040 0.8650 + 08392069+2004593 129.836248 20.083166 08h39m20.70s 20d04m59.40s 16.932 16.361 15.580 222 2020.162914 344.924195 0.5710 0.7810 1.3520 + 08380115+1951545 129.504803 19.865160 08h38m01.15s 19d51m54.58s 15.186 14.456 14.240 222 2020.187569 305.318981 0.7300 0.2160 0.9460 + 08373519+1931106 129.396637 19.519630 08h37m35.19s 19d31m10.67s 12.392 11.901 11.795 222 2020.319393 267.877333 0.4910 0.1060 0.5970 + 08373508+1933087 129.396170 19.552439 08h37m35.08s 19d33m08.78s 15.765 15.047 14.844 222 2020.570876 271.227589 0.7180 0.2030 0.9210 + 08390132+1901339 129.755517 19.026089 08h39m01.32s 19d01m33.92s 15.352 14.752 14.576 222 2021.175557 203.429167 0.6000 0.1760 0.7760 + 08391197+1900355 129.799915 19.009880 08h39m11.98s 19d00m35.57s 15.927 15.163 15.514 222 2021.324530 198.836155 0.7640 -0.3510 0.4130 + 08412881+1958322 130.370057 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Repo 目标 + +本项目的目标是开发一个**离线、尽可能物理正确的数值相对论时空视频渲染器**，最终用于渲染双黑洞（BBH）并合等动态数值时空的 4K 视频。 + +核心目标不是实时渲染，也不是构造视觉上“像双黑洞”的近似度规，而是： + +- 直接消费数值相对论演化输出的真实 4D 时空； +- 对每一帧的相机光线做完整的 time-dependent backward null ray tracing； +- 支持真实巡天恒星数据作为无穷远天球背景； +- 正确处理恒星点源的多像、放大、频移和最终 PSF； +- 允许相机沿一般 4D worldline 运动，并使用预先生成的 tetrad/标架轨迹； +- 让平直时空、解析时空、数值时空在同一渲染框架中作为可替换 backend； +- 最终能够“看到”每次 NR 代码实际跑出来的时空，而不是只看 waveform 或标量诊断。 + +第一阶段不考虑物质辐射、吸积盘、流体、等离子体等局域发射源。每条 ray 的终点暂时只有两类： + +1. 被黑洞捕获； +2. 到达无穷远天球。 + +--- + +## 2. 明确的非目标 + +当前 repo 不以以下内容为主要目标： + +- 实时渲染； +- ShaderToy / 游戏式视觉近似； +- frozen-snapshot 作为最终物理方案； +- 直接渲染吸积盘、喷流或其他 matter emission； +- GPU 加速； +- 通过 OO/class hierarchy 构建复杂抽象层； +- 将恒星背景预烘焙为 RGB 天球纹理； +- 依赖单个固定坐标位置的“相机”。 + +可以保留 frozen-snapshot 模式作为 nmesh 内部的廉价在线 diagnostic，但它与本 repo 的最终 4D renderer 是不同层次的工具。 + +--- + +## 3. 语言与总体实现风格 + +首选 **C**。 + +C++ 并不是当前项目的必要条件。需要的抽象主要可以通过： + +- `struct` +- 函数指针表 +- 显式 ownership +- 大块连续数组 +- thread-local workspace + +完成。 + +性能敏感部分尽量采用： + +- bulk array processing； +- SoA（structure of arrays）； +- 避免 per-ray malloc； +- 避免大量虚调用/函数指针 dispatch； +- 避免细粒度锁； +- 避免每个 triangle/ray 建 task； +- 尽量顺序读取 metric 数据； +- OpenMP 做 CPU 并行。 + +核心实现原则： + +> **bulk arrays + time-slab streaming + coarse-grained parallelism** + +--- + +# 4. 总体物理与数据流 + +整个程序不是“逐帧、逐像素独立 trace”，而是： + +1. 预先生成相机 worldline 与 tetrad 轨迹； +2. 建立所有视频帧的初始 image-plane adaptive mesh； +3. 从所有帧收集当前 refinement level 需要的新 ray samples； +4. 将这些 rays 组成一个全局 `RayPool`； +5. 从视频结束时刻向过去，按 time slab 顺序加载数值时空； +6. 在每个 slab 内，把所有 active rays 一起推进到 slab 左边界； +7. ray 若到达无穷远或进入黑洞，则立即终止； +8. 一轮 ray tracing 完成后，把 endpoint 数据回填到各帧 image mesh； +9. 根据局部 lens mapping 误差判断哪些 image-plane triangles 需要进一步细分； +10. 生成下一批新增 rays； +11. 重复若干 refinement passes； +12. 最终用局部 inverse lens maps 查询恒星 catalog； +13. 计算每个恒星像的位置、放大率和频移； +14. 将对应黑体 PSF 加到 HDR framebuffer； +15. tone mapping / 编码输出视频。 + +总结构： + +```text +Observer Track +z^μ(t), e_(a)^μ(t) + │ + ▼ +Movie / Frame Set + │ + ▼ +Adaptive image-plane meshes + │ + ▼ +Refinement analyzer + │ + ▼ +new ray samples + │ + ▼ +RayPool + │ + ▼ +Spacetime time-slab stream +(newest → oldest) + │ + ▼ +ray endpoint: +n∞, frequency shift, captured/escaped + │ + ├──────────► next refinement pass + │ + ▼ +Final local lens maps + │ + ▼ +Star catalog +(n∞, T, amplitude) + │ + ▼ +inverse mapping + μ + g + PSF + │ + ▼ +HDR video frames +``` + +--- + +# 5. 为什么必须按 time slab 组织，而不是按帧组织 + +最终 BBH 4D 数据很可能超过单机内存。 + +因此不能对每帧独立： + +```text +for frame: + load all required 4D metric data + trace all rays +``` + +否则同一段时空会被反复从硬盘读取。 + +更合理的是： + +```text +for time_slab = latest -> earliest: + load metric data for this slab + + activate all rays whose camera time enters this slab + + evolve every active ray backward through this slab + + remove terminated rays + + discard slab +``` + +这样： + +- metric I/O 主要取决于 NR 数据量； +- 不随 4K 分辨率、帧数、supersampling 线性重复； +- 数据访问可以是大块顺序读； +- 适合本地 NVMe； +- 适合未来超过 128 GB 的数据集。 + +--- + +# 6. Ray 生命周期与跨帧整合 + +不应一开始生成整个视频所有 rays 并常驻内存。 + +随着倒序时间扫描，在经过某个 frame 的 `t_camera` 时才激活该帧当前 refinement pass 的新 rays。 + +因此 active-ray pool 大小由以下因素决定： + +```text +单位相机时间产生的 ray 数 +× +ray 在 NR domain 内的典型 coordinate-time 停留长度 +``` + +而不是整个视频总 ray 数。 + +对于长时间停留在 strong-field region 的 rays，可以一直跨多个 slabs 保留。 + +--- + +# 7. Adaptive image-plane mesh + +最终点源渲染不适合简单的“每 pixel backward lookup sky color”。 + +恒星在无穷远天球上是点源： + +\[ +I(\hat n) \sim \sum_s F_s \delta(\hat n-\hat n_s) +\] + +因此需要先通过 backward ray tracing 得到局部 lens mapping： + +\[ +F:\ (x,y)_\text{image} \to \hat n_\infty +\] + +然后在每个局部可逆 patch 上建立 inverse： + +\[ +F^{-1}:\ \hat n_\infty \to (x,y)_\text{image} +\] + +### 基本单元：triangle + +每个 image-plane triangle 的三个顶点都保存： + +- image-plane 坐标 `(x,y)`； +- ray 是否 escaped/captured； +- 若 escaped：无穷远方向 `n_inf`； +- frequency shift / redshift accumulator。 + +示意： + +```c +typedef struct { + double x, y; + + double n_inf[3]; + double log_g; + + uint8_t ray_status; +} LensVertex; + +typedef struct { + uint32_t v[3]; + + uint8_t level; + uint8_t flags; +} LensTriangle; +``` + +--- + +# 8. 局部 adaptive refinement + +每个 triangle 只关心自身局部映射。 + +不需要先把整个像平面映射到天球、再全局分类“第几阶像”。 + +对一个 triangle： + +1. 已有三个 corner rays； +2. 计算或已有 edge midpoint / center rays； +3. 比较真实 mapping 与低阶插值； +4. 检查 orientation consistency； +5. 检查 Jacobian 是否接近奇异； +6. 若误差过大则 subdivide。 + +这意味着： + +- Schwarzschild 下无需显式按 winding number 分类； +- BBH 下也不需要定义全局“第几阶像”； +- 每个局部 patch 只需保持局部单值可逆； +- 多像会自然表现为同一个 source direction 被多个 image patches 覆盖。 + +--- + +# 9. 为什么 refinement 需要多 pass + +adaptive refinement 与 out-of-core time streaming 存在天然冲突。 + +只有当一条 ray 已经完整追到无穷远后，才知道某个 triangle 是否需要细分。 + +但此时早期 metric slab 已经释放。 + +因此不能在单次 time sweep 中随时创建新 ray 并从相机时刻重新追。 + +采用： + +```text +Pass 0: + trace coarse samples through whole spacetime + analyze maps + +Pass 1: + trace only newly requested midpoint/center rays + analyze maps + +Pass 2: + trace only newly requested rays + ... + +until converged +``` + +每个 pass： + +- metric data 仍然只顺序扫描； +- 只追踪新增样本； +- refinement 通常若干轮即可完成。 + +--- + +# 10. 相机不是固定坐标点 + +相机不应由： + +```c +double x[3]; +``` + +定义。 + +否则结果强依赖某个 NR 坐标系中的“固定点”。 + +应由独立 trajectory generator 预先生成： + +\[ +z^\mu(\tau), \qquad e_{(a)}{}^\mu(\tau) +\] + +其中： + +- \(z^\mu\)：observer worldline； +- \(e_{(0)}{}^\mu=u^\mu\)：observer 四速度； +- \(e_{(i)}{}^\mu\)：相机空间 tetrad。 + +trajectory generator 可以采用不同物理规则： + +- timelike geodesic； +- Fermi-Walker transport； +- parallel transport； +- 人为给定加速度； +- 人为叠加 camera pointing/rotation。 + +renderer 本身只读取轨迹。 + +示意： + +```c +typedef struct { + double t; + double tau; + + double x[3]; + double tetrad[4][4]; +} ObserverSample; + +typedef struct { + ObserverSample *samples; + size_t n; +} ObserverTrack; +``` + +每帧根据 `t_camera` 插值得到完整 observer state。 + +--- + +# 11. Ray 初始化 + +给定相机 film coordinate `(x,y)`： + +1. camera projection 得到 observer local frame 中的 photon direction； +2. 与 observer tetrad 相乘得到 spacetime 中的 \(k^\mu\)； +3. 转换到 geodesic integrator 所使用的 3+1 ray variables。 + +例如： + +\[ +k^\mu = e_{(a)}{}^\mu k^{(a)} +\] + +film model 只负责： + +\[ +(x,y) \to \hat n^{(i)}_\text{camera} +\] + +它与 gauge、metric backend、observer worldline 都解耦。 + +--- + +# 12. Ray 状态 + +采用 coordinate time \(t\) 作为 ODE 自变量，以便天然和 time slabs 对齐。 + +单 ray 概念结构： + +```c +typedef struct { + double t; + + double x[3]; + double Pi[3]; + + double log_E; + double h; + + uint32_t sample_id; + uint32_t frame_id; + + uint8_t status; +} Ray; +``` + +真正批量计算时建议 SoA： + +```c +typedef struct { + double *restrict t; + + double *restrict x0; + double *restrict x1; + double *restrict x2; + + double *restrict p0; + double *restrict p1; + double *restrict p2; + + double *restrict log_E; + double *restrict h; + + uint32_t *sample_id; + uint32_t *frame_id; + + uint8_t *status; + + size_t n; + size_t capacity; +} RayPool; +``` + +目的： + +- cache locality； +- SIMD； +- OpenMP； +- stream compaction； +- 避免 per-ray allocation。 + +--- + +# 13. Spacetime backend + +所有上层渲染逻辑都不应该知道 metric 来自哪里。 + +需要支持至少： + +- Minkowski； +- analytic Schwarzschild； +- 未来可加 analytic Kerr； +- numerical nmesh 4D spacetime。 + +核心不是“单点 metric 对象”，而是**当前已加载 time slab 的 evaluator**。 + +概念接口： + +```c +typedef struct SpacetimeSource SpacetimeSource; +typedef struct MetricSlab MetricSlab; + +typedef struct { + int (*load_slab)( + SpacetimeSource *src, + double t_hi, + size_t memory_budget, + MetricSlab **out); + + void (*free_slab)(MetricSlab *slab); + + int (*eval)( + const MetricSlab *slab, + double t, + const double x[3], + void *workspace, + MetricData *out); + + int (*classify)( + const MetricSlab *slab, + double t, + const double x[3]); +} SpacetimeOps; +``` + +解析 backend 的 `load_slab()` 可以近似 no-op。 + +nmesh backend 则真正加载一段时间的 3D outputs。 + +--- + +# 14. 统一的 3+1 metric 数据 + +初步采用： + +\[ +\alpha,\quad +\beta^i,\quad +\gamma_{ij},\quad +K_{ij} +\] + +作为主要 3+1 变量。 + +空间导数： + +\[ +\partial_i \alpha,\quad +\partial_i\beta^j,\quad +\partial_i\gamma_{jk} +\] + +由 backend evaluator 提供。 + +示意： + +```c +typedef struct { + double alpha; + + double beta[3]; + double gamma[6]; + double K[6]; + + double d_alpha[3]; + double d_beta[3][3]; + double d_gamma[3][6]; +} MetricData; +``` + +--- + +# 15. 时间导数 + +时间导数不能简单假设全部从文件直接输出。 + +应按量的性质选择来源。 + +例如： + +\[ +\partial_t \gamma_{ij} += +-2\alpha K_{ij} ++ +\mathcal L_\beta \gamma_{ij} +\] + +可以直接由 3+1 内部量构造。 + +对于 lapse/shift： + +- 若 gauge evolution equations 已知，可从 gauge RHS 构造； +- 或从 temporal interpolant 解析求导。 + +原则： + +> 若已经建立 \(u(t,\mathbf x)\) 的时间插值，就从同一个 interpolation polynomial 解析求 \(\partial_tu\)，而不是另做 finite difference。 + +且不要每个 geodesic RHS 都计算用不到的时间导数。 + +--- + +# 16. nmesh 数值时空 backend + +nmesh 当前 3D 输出格式已经非常适合 renderer： + +对每个需要输出的 timestep： + +- 每个 AMR leaf node 一个文件； +- 文件名按 node 的树状 refinement 命名； +- 从文件名可恢复 refinement tree； +- 文件内保存每个 nodal point 的 global coordinates； +- 保存所选物理量。 + +nmesh： + +- DG； +- node 内用 local coordinates 上 Legendre nodes 对应的 Lagrange basis； +- production 中每边一般最多约 12 个 nodal points； +- AMR 为 h-refinement； +- 典型网格为中心六面体 + 外层 cubic-sphere shells。 + +因此读入一个 node 文件后，可以直接构造 node 内： + +\[ +u(\xi,\eta,\zeta) += +\sum_{ijk} +u_{ijk} +L_i(\xi)L_j(\eta)L_k(\zeta) +\] + +空间导数直接对 Lagrange polynomial 求导。 + +不需要： + +- resample 到 Cartesian uniform grid； +- 额外输出 derivative fields。 + +--- + +# 17. nmesh time interpolation + +h-AMR 可能使相邻 timestep 的树结构不同。 + +但不要求时间方向上 node 一一对应。 + +对任意固定 global point \(\mathbf x\)： + +\[ +u_n(\mathbf x) += +I_{\mathcal M_n}[u](\mathbf x) +\] + +\[ +u_{n+1}(\mathbf x) += +I_{\mathcal M_{n+1}}[u](\mathbf x) +\] + +先分别在各自空间 mesh 上求值，再做： + +\[ +u(t,\mathbf x) += +T[u_n(\mathbf x),u_{n+1}(\mathbf x),\ldots] +\] + +因此： + +- 每个 time slice 可独立恢复空间函数； +- AMR topology 不需要跨时间匹配； +- time slab 只负责保存足够多邻近 slices 供时间插值。 + +slab 边界需要少量 overlapping temporal ghost slices。 + +--- + +# 18. 黑洞捕获判据 + +目标使用 moving-puncture BBH，而不是 excision。 + +production renderer 不希望依赖每次 NR run 都开启昂贵的 AH finder。 + +计划： + +1. 用低分辨率 single-BH / BBH calibration run 开 AH finder； +2. 测量 horizon 相对于 puncture 的最小 coordinate radius； +3. 选择明显保守、始终位于 AH 内部的 puncture-centered cutoff； +4. 正式 renderer 只根据 puncture trajectory 做判断。 + +形式： + +\[ +|\mathbf x-\mathbf x_p(t)|n; ++i) { + if (!ray_is_active(rays, i)) + continue; + + evolve_ray_to_slab_left_boundary( + rays, + i, + slab, + &ws + ); + } + + metric_workspace_free(&ws); + } + + compact_ray_pool(rays); + + spacetime_free_slab(slab); + } +} +``` + +--- + +# 27. repo 的几个主要模块 + +这里讨论的是代码职责，不强调具体文件树。 + +## `observer` + +负责： + +- 读取 worldline； +- 插值 observer event； +- 插值/正交化 tetrad； +- 给每个 frame 提供 observer state。 + +不负责： + +- geodesic； +- metric； +- image rendering。 + +--- + +## `movie / frame` + +负责： + +- frame 时间表； +- 每帧 adaptive image mesh； +- vertex / triangle； +- refinement passes； +- ray sample request； +- 安装 ray endpoint results。 + +--- + +## `ray` + +负责： + +- 批量 ray state； +- ray 初始化； +- active/inactive 状态； +- stream compaction； +- endpoint 保存。 + +--- + +## `geodesic` + +负责： + +- 3+1 null geodesic RHS； +- frequency shift evolution； +- ODE stepper； +- 在给定 metric slab 中推进 ray。 + +不负责： + +- disk I/O； +- frame mesh； +- star rendering。 + +--- + +## `spacetime` + +负责统一 backend： + +- Minkowski； +- analytic Schwarzschild； +- nmesh 4D numerical spacetime。 + +负责： + +- time slab； +- metric interpolation； +- spatial derivatives； +- 必要的 temporal derivatives； +- capture/infinity classification。 + +--- + +## `nmesh_backend` + +负责： + +- 读取 nmesh node 文件； +- 从 filename 恢复 AMR tree； +- node spatial lookup； +- global/local coordinate conversion； +- Lagrange interpolation； +- Lagrange derivatives； +- temporal interpolation； +- puncture trajectory。 + +--- + +## `catalog` + +负责： + +- 读取 Gaia / 2MASS 等 catalog； +- 转换为内部 `(direction, temperature, amplitude)`； +- 天球 spatial index； +- triangle/source-region 查询。 + +--- + +## `optics / psf` + +负责： + +- blackbody frequency shift； +- temperature → linear RGB； +- flux normalization； +- PSF splatting； +- HDR accumulation。 + +--- + +## `output` + +负责： + +- HDR frame； +- exposure； +- tone mapping； +- PNG/EXR； +- 最终视频编码接口。 + +--- + +# 28. 开发顺序 + +## Phase 0 — 平直时空 + 星表 + +目的： + +- 验证 observer/camera projection； +- 验证天球方向； +- 验证 Gaia/2MASS catalog； +- 验证 temperature/amplitude； +- 验证 PSF； +- 验证 adaptive image mesh 数据结构。 + +平直时空： + +\[ +(x,y)\leftrightarrow \hat n_\infty +\] + +可以作为严格 benchmark。 + +--- + +## Phase 1 — analytic Schwarzschild + +建议使用 horizon-penetrating coordinates。 + +验证： + +- capture； +- Einstein ring； +- multiple images； +- adaptive refinement； +- local inverse maps； +- magnification； +- frequency shift。 + +--- + +## Phase 2 — numerical single Schwarzschild + +使用当前已能跑的 FOCCZ4 moving-puncture Schwarzschild。 + +特点： + +- 开始存在 gauge relaxation； +- 随后 metric 趋于稳定； +- 可真实测试 4D metric streaming； +- 可和 analytic Schwarzschild 做 regression comparison。 + +比较： + +\[ +\Delta \hat n_\infty, +\quad +g, +\quad +\text{captured/escaped classification} +\] + +--- + +## Phase 3 — FOZ4c BBH + +等 FOZ4c： + +- 双曲性； +- constraint damping； +- 数值实现； +- BBH 稳定演化； + +成熟后接入。 + +renderer 顶层架构原则上不应为 BBH 重新设计。 + +--- + +# 29. 尚未决定的问题 + +需要以后通过 prototype / convergence test 决定： + +- 时间输出 cadence； +- time interpolation order； +- time slab 最佳内存大小； +- BBH production node 数； +- 4D 输出总数据量； +- AH calibration 后 puncture cutoff； +- adaptive triangle refinement criterion； +- critical curve 附近最大 refinement level； +- Gaia 与 2MASS 的最终组合； +- PSF 模型； +- ODE integrator； +- observer trajectory 标准； +- tone mapping / exposure 规则； +- 是否需要 diffuse Milky Way background； +- 是否将 ray redshift 变量定义为 `log(alpha p^0)` 或其他更方便的量。 + +--- + +# 30. 当前最核心的设计原则 + +整个 repo 可以压缩成以下几条： + +1. **最终目标是动态 4D NR 视频渲染，不是单帧 ray tracer。** +2. **所有帧的 rays 按 coordinate time 整合，通过 time slabs 倒序推进。** +3. **metric backend 可替换，但顶层 scheduler 与具体时空无关。** +4. **相机是 worldline + tetrad track，不是固定三维坐标。** +5. **恒星是点源 catalog，不是 RGB 天球纹理。** +6. **恒星存温度与 amplitude，频移通过温度缩放自然处理。** +7. **image-plane 使用 adaptive local triangles 构造局部 inverse lens maps。** +8. **adaptive refinement 采用多 pass，避免和 out-of-core metric streaming 冲突。** +9. **ray evolution 用 SoA + OpenMP bulk loops。** +10. **recursive refinement 用 thread-local iterative queues，不用细粒度 OpenMP tasks。** +11. **nmesh 原生 DG node 输出直接作为空间插值数据，不做 Cartesian resampling。** +12. **时间导数优先由 3+1 identities、gauge RHS 或 temporal interpolant 解析获得。** +13. **所有高开销 mutable cache 都 thread-local。** +14. **第一版 CPU-only，先把物理与数据流做正确，再谈 GPU。** +15. **从 Minkowski → analytic Schwarzschild → numerical Schwarzschild → BBH 逐级验证。** diff --git a/ref/arXiv1410.7775v3/arXiv1410.7775v3.md b/ref/arXiv1410.7775v3/arXiv1410.7775v3.md new file mode 100644 index 0000000..e2da72f --- /dev/null +++ b/ref/arXiv1410.7775v3/arXiv1410.7775v3.md @@ -0,0 +1,299 @@ +# What does a binary black hole merger look like? + +Andy Bohn,∗ William Throwe, Fran¸cois H´ebert, and Katherine Henriksson† Center for Radiophysics and Space Research, Cornell University, Ithaca, New York 14853, USA + +Darius Bunandar + +Theoretical Astrophysics 350-17, California Institute of Technology, Pasadena, California 91125, USA and MIT Kavli Institute, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA + +Mark A. Scheel and Nicholas W. Taylor + +Theoretical Astrophysics 350-17, California Institute of Technology, Pasadena, California 91125, USA (Dated: April 23, 2015) + +We present a method of calculating the strong-field gravitational lensing caused by many analytic and numerical spacetimes. We use this procedure to calculate the distortion caused by isolated black holes and by numerically evolved black hole binaries. We produce both demonstrative images illustrating details of the spatial distortion and realistic images of collections of stars taking both lensing amplification and redshift into account. On large scales the lensing from inspiraling binaries resembles that of single black holes, but on small scales the resulting images show complex and in some cases self-similar structure across different angular scales. + +PACS numbers: 95.30.Sf, 04.25.dg, 42.15.Dp Keywords: gravitational lensing, black holes, numerical relativity, ray tracing + +## I. INTRODUCTION + +Black holes are the most compact gravitating objects in the universe, with such strong gravitational fields that not even light can escape them. In the vicinity of a black hole, light rays can be very strongly deflected from a straight-line path, sometimes orbiting around the black hole before continuing on their way. It is now well-known that the bending of light by massive objects like galaxy clusters can create brightness amplification [1], deformed images, or even multiple images [2] of background objects such as quasars. These signatures have so far only been directly observed in cases where the deflection of light is very slight, up to approximately 11 arc seconds [3, 4]. However, here we are interested in the lensing effects associated with much more extreme bending of light near single or binary black holes, where the deflection angle is unbounded. + +The lensing effects near general-relativistic bodies were first studied in the 1970s, with Cunningham and Bardeen [5] looking at a star on an orbit in a Kerr spacetime, and Luminet [6] studying an accretion disk around a Schwarzschild black hole. More recently, open-source codes such as GYOTO [7] and GeoViS [8] have produced images of lensing in the neighborhood of various compact objects. While the lensing caused by an isolated black hole has been understood analytically, the case of lensing by a binary black hole (BBH) is much more challenging because of the difficulty of solving for the geometry of the spacetime. With some arguably unrealistic assumptions (e.g., two maximally charged black holes in static equilibrium), analytic solutions can be found and subsequently used for lensing [8–13]. + +![](images/4daadacdb5cb9fc339611262d0733a5bbb6ea09397eaffeb7aeac7f50b9c401a.jpg) +FIG. 1. A pair of black holes that are about to merge, with the Milky Way visible in the background. Supplementary images and movies can be found at [14]. + +For astrophysically relevant binaries, however, we must instead rely on numerical solutions. Solving these binary spacetimes numerically to high accuracy has been possible for the last decade (see [15, 16] for a review), motivated by the need to provide gravitational-wave templates used by experiments such as LIGO, VIRGO, and KAGRA to make detections. By using the spacetimes computed in such simulations, we gain the ability to solve for the lensing effects in BBH systems. + +In this paper, we focus on the question of what an observer in the vicinity of a BBH would actually see as the black holes orbit, spiral inward, and merge, with an example shown in figure 1. This is in contrast to most BBH visualizations, in which the positions or horizons of the two black holes are simply shown as a function of time in some coordinate system. We instead compute the paths of light rays that enter the observer’s eye or camera to find what would actually be seen. Furthermore, this path must be computed in the fully time-dependent spacetime, as the orbital velocities for a black-hole binary are typically large enough that the system cannot be approximated as time-independent during the time taken by the photons to travel across it. + +Because the black holes themselves do not emit light (we ignore Hawking radiation, which is significant only for microscopic black holes), the observer would see nothing unless there is some additional light source. For illustrative purposes, we will take an artificial background “painted on” at infinity (figure 3) as the light source for most of our examples; this will allow us to study in detail where each light ray originates. + +We begin by describing the problem setup and the methods that we use to generate lensing images in section II. In section III we show images of lensing by single and binary black holes, and we then conclude in section IV. + +## II. METHODS + +We set up the problem with our black hole(s) near the center of our chosen coordinate system. While any physical system representable by a spacetime metric can be used, we specialize in this paper to single and binary black holes. The observer (henceforth taken to be a camera) can be located anywhere in the space and is typically chosen to look towards the origin. A sphere with our light source encloses the black hole(s) and camera, infinitely far away. + +To recreate the image taken by the camera in this configuration, we must find the properties of the light that arrives at each point on the camera’s image plane. A na¨ıve approach would be to trace all possible light rays (i.e., null geodesics) emanating from the light source to determine which rays reach the camera and from what directions they arrive, but this is computationally infeasible. A more efficient approach is to reverse the problem by tracing light rays away from the camera and backwards in time (the computer graphics community calls this a ray-casting algorithm). This method identifies the origin of any light ray that illuminates the camera, from which we infer the color and intensity of the corresponding photons as detected by the camera. When black holes are present, some of the null geodesics traced backwards in time from the camera may approach arbitrarily close to an event horizon as $t \to - \infty$ ; these geodesics correspond to dark image regions. + +In what follows we describe how the light rays are traced from the camera using the geodesic language from general relativity. We show how we initialize these geodesics based on camera parameters such as position and viewing angle. Finally, we show how the origin of each light ray is determined and describe how the simulated image is constructed. + +## A. Geodesic tracing + +Our code can trace geodesics independently through either numerical or analytic metric data. It is common for numerical simulations to use the 3+1 decomposition [17], so we express the metric in the form + +$$ +d s ^ { 2 } = - \alpha ^ { 2 } d t ^ { 2 } + \gamma _ { i j } ( d x ^ { i } + \beta ^ { i } d t ) ( d x ^ { j } + \beta ^ { j } d t ) ,\tag{1} +$$ + +where α is the lapse function, $\beta ^ { i }$ is the shift vector, and $\gamma _ { i j }$ is the spatial metric.1 We obtain numerical data from simulations performed using the Spectral Einstein Code (SpEC) [18–22]. The geodesics are traced by evolving a solution to the geodesic equation + +$$ +{ \frac { d ^ { 2 } x ^ { \lambda } } { d \tau ^ { 2 } } } + \Gamma ^ { \lambda } { } _ { \mu \nu } { \frac { d x ^ { \mu } } { d \tau } } { \frac { d x ^ { \nu } } { d \tau } } = 0 ,\tag{2} +$$ + +where $x ^ { \lambda }$ is the four-position of the geodesic, τ is an affine parameter, and $\Gamma ^ { \lambda } { } _ { \mu \nu }$ are the Christoffel symbols describing the effective force caused by spacetime curvature. + +To facilitate the numerical geodesic evolution, we split this second-order differential equation into two first-order differential equations using an intermediate, momentumlike variable such as $p ^ { \lambda } \overset { \cdot } { = } d x ^ { \lambda } / d \tau$ As we have some freedom in the definition of this momentum variable, we look for one that helps to minimize computational time and numerical errors when evolving through spacetimes with black holes. + +We initially explored using the variable $p _ { \lambda } = g _ { \lambda \kappa } p ^ { \kappa }$ from Hughes et al. [23], along with converting the evolution equations from affine parameter $\tau$ to the coordinate time t of SpEC evolutions through the use of $p ^ { 0 } = d t / d \tau$ Although the resulting evolution equations are concise and have no time derivatives of metric variables, the variables $p ^ { 0 }$ and $p _ { i }$ grow exponentially near black hole horizons in typical coordinate systems used by SpEC simulations. This forces our time-stepper to take prohibitively small steps in order to achieve the desired accuracy. + +We therefore choose a momentum variable slightly different than $p _ { \lambda }$ to mitigate this time-stepping problem. Null geodesics satisfy $p \cdot p = 0$ , which can be rewritten as $p ^ { 0 } = \alpha ^ { - 1 } ( \gamma ^ { i j } p _ { i } p _ { j } ) ^ { 1 / 2 }$ using the metric (1). This expression shows that $p ^ { 0 }$ and $p _ { i }$ scale similarly, so we can eliminate the exponential behavior of these variables by evolving the ratio. Our intermediate variable thus becomes + +$$ +\Pi _ { i } \equiv \frac { p _ { i } } { \alpha p ^ { 0 } } = \frac { p _ { i } } { \sqrt { \gamma ^ { j k } p _ { j } p _ { k } } } ,\tag{3} +$$ + +where we also divide by α to reduce the number of terms in the resulting evolution equations. Using $\Pi _ { i }$ and the $3 { + 1 }$ decomposition (1), we can express the geodesic equation (2) in the form + +$$ +\begin{array} { l } { { \displaystyle \frac { d \Pi _ { i } } { d t } = - \alpha _ { , i } + ( \alpha _ { , j } \Pi ^ { j } - \alpha K _ { j k } \Pi ^ { j } \Pi ^ { k } ) \Pi _ { i } } } \\ { { \displaystyle ~ + \beta _ { , i } ^ { k } \Pi _ { k } - \frac { 1 } { 2 } \alpha \gamma ^ { j k } _ { , i } \Pi _ { j } \Pi _ { k } , } } \\ { { \displaystyle \frac { d x ^ { i } } { d t } = \alpha \Pi ^ { i } - \beta ^ { i } , } } \end{array}\tag{4} +$$ + +where $K _ { j k }$ is the extrinsic curvature $( \mathrm { s e e } , \mathrm { e . g . } , [ 1 7 ] )$ and $\Pi ^ { i }$ is defined via the inverse spatial metric as $\Pi ^ { i } \equiv \dot { \gamma } ^ { i j } \Pi _ { j }$ Note that the geodesic equation consists of four secondorder equations, yet we only have three pairs of coupled first-order equations in (4). Because we are evolving a normalized momentum (3), we have lost information about $p ^ { 0 }$ during evolution. Compared to Hughes et al. [23], we have introduced a time derivative of the threemetric inside $K _ { j k }$ , but we have significantly sped up the evolution near black holes by removing the exponential growth of $p ^ { 0 }$ and $p _ { i }$ + +The equations in (4) are similar to those in (28) of Vincent et al. [24]. In fact our intermediate evolution variable $\Pi _ { i }$ is related to their variable $V ^ { i }$ by the threemetric, such that $\Pi ^ { i } = V ^ { i }$ . But our (4) has a reduced number of both temporal and spatial derivatives of metric quantities compared to Vincent’s (28). + +During the backwards-in-time geodesic evolution, many geodesics are traced until they are far from the strong-field region, but some are traced until they encounter a black hole. These latter geodesics slowly converge towards the black hole’s event horizon, but as they can in principle be evolved indefinitely, we need some way of identifying them in finite time. We do this by monitoring $p ^ { 0 }$ for each geodesic, which (as discussed above) grows large near black hole horizons. Since our evolution equations (4) do not evolve $p ^ { 0 }$ , we must evolve another equation to keep track of it. However, we would still like to avoid the exponential growth of $p ^ { 0 }$ near the horizon. This can be accomplished by evolving the logarithm of $p ^ { 0 }$ . As was done in (3), we multiply $\bar { p } ^ { 0 }$ by the lapse to reduce the number of terms in the resulting equation, which gives the evolution variable $\ln ( \alpha p ^ { 0 } )$ . This leads to the evolution equation + +$$ +\frac { d \ln ( \alpha p ^ { 0 } ) } { d t } = - \alpha _ { , i } \Pi ^ { i } + \alpha K _ { i j } \Pi ^ { i } \Pi ^ { j } .\tag{5} +$$ + +When $p ^ { 0 }$ becomes too large, signaling a large energy, we flag the geodesic as originating from the black hole and we stop evolving it. + +The remaining geodesics are those that originate from infinity, so we need to determine the $( \theta , \phi )$ location at infinity where they come from. In section II C, we will need the gravitational redshift z of each photon, which can be calculated from the ratio of the photon’s energy at the two ends of its trajectory via + +$$ +1 + z = { \frac { E _ { \infty } } { E _ { \mathrm { c a m e r a } } } } ,\tag{6} +$$ + +where $E _ { \infty }$ is the photon’s energy at infinity, and $E _ { c }$ camera is the photon’s energy as measured by the camera. Therefore we will need to compute the energy that each photon would have at infinity. In practice, these geodesics are traced backwards in time until they reach a large distance R from the black hole(s), chosen so that the metric at R is equal to the flat space metric within about a percent error. We use the approximation that the metric is exactly flat for $r > R$ . Under this approximation, the geodesic’s direction and $p ^ { 0 }$ at infinity are the same as at $R .$ . The direction is used to calculate a $( \theta , \phi )$ location on the sky, while $p ^ { 0 }$ is the photon’s energy at infinity, $E _ { \infty }$ + +## B. Initial data + +Here we outline how we initialize our geodesic evolution variables. Because the geodesics are traced away from the camera, backwards in time, we initialize each geodesic’s evolution variables to their values at the camera. We have seven variables to set: three each for the initial position and momentum in (4), and one for the initial redshift in (5). + +The initial position for every geodesic is simply the camera’s position. The initial momentum, however, is different for each geodesic and is dependent on the angle at which it enters the camera. We express the momenta in terms of an orthonormal tetrad defined as + +$e _ { 0 } \colon$ The camera’s four-velocity, a timelike vector. For stationary cameras $e _ { 0 } \propto ( 1 , 0 , 0 , 0 )$ ; + +$e _ { 1 } \colon$ The direction in which the camera is pointing; + +$e _ { 2 } \colon$ The “upward” direction for the camera; + +$_ { e 3 } \colon$ The “rightward” direction for the camera. + +The four-vectors $e _ { 1 } , \ e _ { 2 }$ , and $e _ { 3 }$ are all spacelike, and their orientations in the camera’s reference frame are illustrated in figure 2. + +In order to specify this tetrad, we give guesses for the vectors $e _ { 0 } , e _ { 1 }$ , and $e _ { 2 }$ , with the condition that the guessed time components of $e _ { 1 }$ and $e _ { 2 }$ must be zero. We then apply the Gram-Schmidt process to the sequence $e _ { 0 } , \ : e _ { 1 }$ and $e _ { 2 }$ to transform these vectors into an orthonormal set. The final vector, $e _ { 3 } .$ , is found by calculating the generalized cross product of the other three; explicitly, + +$$ +e _ { 3 \rho } = \epsilon _ { \lambda \mu \nu \rho } e _ { 0 } { } ^ { \lambda } e _ { 1 } { } ^ { \mu } e _ { 2 } { } ^ { \nu } ,\tag{7} +$$ + +![](images/6dfe6cd95ddf9ac7a0a87df16c2822dcd96657ab28c968b18ad294c6fc45bbd0.jpg) +FIG. 2. Illustration of a pinhole camera in its rest frame with the three vectors $e _ { 1 } , e _ { 2 } ,$ , and $e _ { 3 }$ that describe its orientation. The inverted letter $^ { 6 } \mathrm { A } ^ { \dag }$ demonstrates the optical properties of the camera, which we correct for in the images we generate. + +where $\epsilon _ { \lambda \mu \nu \rho }$ is the Levi-Civita tensor (see [25, p. 202] for more details). + +Given the four orthonormal unit vectors, we can construct a null vector $\xi$ tangent to the geodesic that enters the camera from a given direction. The vector ξ will be proportional to the four-momentum of a photon following the geodesic; that is, $p = q \xi$ for some positive constant q. We define ξ by + +$$ +\begin{array} { r } { \xi ^ { \lambda } { } _ { ( a , b ) } = C e _ { 0 } { } ^ { \lambda } - e _ { 1 } { } ^ { \lambda } - [ ( 2 b - 1 ) \tan ( \alpha _ { v } / 2 ) ] e _ { 2 } { } ^ { \lambda } } \\ { - [ ( 2 a - 1 ) \tan ( \alpha _ { h } / 2 ) ] e _ { 3 } { } ^ { \lambda } , } \end{array}\tag{8} +$$ + +where $a , b \in [ 0 , 1 ]$ give the ray’s arrival direction in terms of fractions of the image’s horizontal and vertical lengths, respectively, and $\alpha _ { v } , \alpha _ { h }$ are the angular sizes of the camera aperture (field of view angles) in the vertical and horizontal directions. For the sign convention chosen in (8), $( a , b ) = ( 0 , 0 )$ corresponds to a photon seen at the bottom left corner of the image. We find C by requiring that ξ is null, i.e., $\xi \cdot \xi = 0 \colon$ + +$$ +C = \sqrt { 1 + ( 2 b - 1 ) ^ { 2 } \tan ^ { 2 } ( \alpha _ { v } / 2 ) + ( 2 a - 1 ) ^ { 2 } \tan ^ { 2 } ( \alpha _ { h } / 2 ) } .\tag{9} +$$ + +We then use the metric to lower the index on $\xi ,$ and we compute the initial value of our evolution variable $\Pi _ { i }$ using $\bar { \Pi _ { i } } = p _ { i } / ( \alpha p ^ { 0 } ) = \xi _ { i } / ( \alpha \xi ^ { 0 } )$ . Note that $\Pi _ { i }$ is independent of the proportionality constant $q$ relating $\xi$ and the actual photon momentum $p ;$ physically, this is because the photon trajectory is independent of the photon energy. The only place where $q$ enters is in the initial value of $\alpha p ^ { 0 }$ in (5). We fix the value of $q$ by demanding that the energy of the photon in the frame of the camera be unity when the photon strikes the camera, so $E _ { \mathrm { c a m e r a } } = 1$ in (6). + +## C. Image generation + +We create our image of the physical system by dividing the image plane into rectangular regions corresponding to the pixels of the output image and assigning an appropriate color to each region. Because each region has an extended size, there is no single source point we can look at to obtain its color, so we must adopt some prescription for assigning a single color to each pixel. We use two different prescriptions, based on the nature of the light source illuminating the system. + +![](images/7b5d24f0dd0f863d44572548b357babc528c1ec8bd6e436841d4d1fbf622ab3e.jpg) +FIG. 3. An illustration of our artificial background grid “painted $\mathrm { o n } ^ { \mathrm { 7 } }$ a sphere at infinity. This background is used for all the images with a grid in this paper. In the figure, we cut a window out of the sphere to show the inside. In addition to four colors differentiating the regions of the sphere, we include a white reference spot in the direction in which the camera is pointing. + +For extended sources, such as the artificial grid in figure 3, we use a subpixel sampling method. On each pixel we construct an evenly spaced grid of points, and at each of these points we determine where incident light rays originate, either from one of the holes or a location at infinity. We assign a color to each grid point based on that of the corresponding source point; the color of the pixel is then the average of these. We find that a grid of $4 \times 4$ sample points gives sufficiently smooth images without too much computational cost. For these images, we neglect the effects of redshift and focus on the spatial distortions. + +To create more astronomically relevant images, we wish to use a collection of point sources (i.e., stars) as our illumination. In this case we cannot determine a pixel’s color using sampling, but must instead sum the contributions from all the point sources contributing light there. For our list of sources, we use about $3 . 4 \times 1 0 ^ { 8 }$ stars from the Two Micron All Sky Survey (2MASS) [26]. To simplify computations, we approximate each star as a thermal source with temperature and brightness determined by fitting to the photometric information in the catalog. When we calculate the contribution of each star to the light arriving at the camera, we must account not only for its properties as a light source, but also for the effects of the spacetime curvature encountered by the photon. These effects come in two forms. First, the observed energies of photons at the camera will be modified by redshift effects, changing sources’ apparent brightnesses and temperatures. Second, the spatial convergence or divergence of nearby geodesics produces an overall adjustment to each source’s apparent brightness without affecting its spectrum. Both of these effects are discussed in detail in Mollerach and Roulet [27]. After we have drawn the entire image in this manner, we convolve it with a blurring function to make the stars more visible. This has the effect of transforming each star into a fuzzy circle with size dependent on its brightness. + +The result of this scheme can be seen in figure 1, which shows the BBH image from figure 11 in front of a background of stars. Note that by generating our starfield images from a catalog of point sources, we obtain a substantially more realistic image than would be generated by applying the lensing deformation to a raster image of the unlensed Milky Way stars. In such a raster image, each star is usually represented (whether as a result of camera optics or software rendering) as a blurred circle whose area depends on the star’s brightness. These circles are typically hundreds of arc seconds wide, and therefore lensing distortions applied to the image tend to produce stars that appear as smeared ellipses. In contrast, the angular sizes of real stars are many orders of magnitude smaller, so we expect them to remain as unresolved points under all but the most extreme lensing magnifications. These unresolved points can then be rendered as previously described, giving stars that better portray what an observer would actually see (as in figure 1). The difference between these methods lies in the non-commutativity between the lensing deformations and the blurring of each star. A minor shortcoming of our method arises at Einstein rings (discussed in section III A), where the magnification diverges. There a star could in principle (though with very low probability) appear as an extended object, but in our treatment it would remain point-like. On the other hand, blurring first and then lensing is almost guaranteed to produce unphysical extended streaks at the Einstein ring. + +## III. RESULTS + +Before applying our lensing code to binary black hole systems, we generate images of simpler analytic spacetimes. These serve both to provide checks that our images are consistent with earlier work, and also to illustrate general features of lensing around black holes that will appear again in BBH images. We then proceed to show two different configurations of BBH mergers. + +To help visualize the lensing, we divide our light source at infinity into colored quadrants with a superimposed grid. An external view of this sphere is shown in figure 3. In addition to the colored sections, our light source has a bright reference spot in the direction towards which we point our camera. This spot will prove useful in illustrating an important feature of black hole lensing called an Einstein ring. + +## A. Analytic spacetimes + +In figure 4, we compare a flat space image with the images obtained by lensing our light source through Schwarzschild and Kerr black hole spacetimes. The top row from left to right shows flat Minkowski space and a + +![](images/2325f7c2266821b4688da4f96aa91554be8d314a6b2506a23b233dd5ce315e09.jpg) +FIG. 4. Lensing caused by various analytic spacetimes. For all panels, we use figure 3 as a background, oriented such that the camera is pointed at the white reference dot. The camera has a 60◦ field of view and is at a distance of 15 Schwarzschild radii from the origin measured using Kerr-Schild coordinates [25]. The top row shows Minkowski and Schwarzschild spacetimes. The bottom row shows two views of the Kerr spacetime, with dimensionless spin χ = 0.95, viewed with the camera pointing parallel to the spin axis of the black hole (bottom left) and perpendicular to the spin axis (bottom right). + +Schwarzschild black hole. These spacetimes are spherically symmetric, so viewing them from different angles produces the same lensing effects. The bottom row shows a Kerr black hole, where in the left frame the spin vector is pointing out of the page and in the right frame it is pointing up. Here the spin breaks the spherical symmetry of the spacetime, leading to different lensing effects from different viewing directions. + +In Minkowski space in the top left image we expect no deflection of light, which is what we observe. The camera sees an upright image of the portion of the grid near the white dot. The bowing of the grid lines is an expected geometric effect of viewing a latitude-longitude grid. + +In the top right image, we see the lensing effects of a non-spinning black hole. The black circle in the center of the image is called the shadow of the black hole, where the hole prevents any light from reaching the camera. Alternatively, a shadow is a region of the image where geodesics are traced backwards in time from the camera to a black hole. Another prominent feature is that the white dot on our grid at infinity has been lensed into a large ring, called an Einstein ring [28]. Light from the point situated directly on the opposite side of the black hole, the antipodal point, will by symmetry be lensed into a ring around the black hole as observed by our camera. Regions inside the Einstein ring correspond to photons that are deflected by larger angles than are the Einstein ring photons; this results in an inverted image of the reference grid inside the Einstein ring. A second Einstein ring can be seen near the shadow, corresponding to light from a source behind the camera wrapping around the hole on its way to the camera. In fact, photons can wind an arbitrarily large number of times around the black hole, resulting in an infinite number of Einstein rings. + +The bottom row of figure 4 shows a single black hole with a large dimensionless spin of $\chi = 0 . 9 5$ . As in the Schwarzschild case, there is an Einstein ring around the black hole shadow as well as image inversion inside the Einstein ring. However, for the case of a Kerr spacetime, the light coming from the Einstein ring does not originate from a single point directly behind the black hole, but from a small region (unless the camera is pointing directly along the spin axis). The spin of the black hole causes frame dragging, where space is dragged in the direction of the rotation [29, 30]. In the bottom left image, the spin axis of the black hole is pointing out of the page, so space is dragged in a counterclockwise motion. The effect of the frame dragging on the photon trajectories produces an image in which the grid itself appears to be dragged by the spin, as is evident when compared to the non-spinning black hole in the top right image. The strength of frame dragging increases closer to the black hole, which can also be inferred from the deformation of the background grid. + +Frame dragging manifests differently in the bottom right image, where the spin axis is pointing up. The direction of frame dragging is out of the page on the left of the shadow of the black hole and into the page on the right. A photon traveling in the direction of the frame dragging can orbit closer to the black hole without being captured than a photon traveling opposite the frame dragging direction, resulting in an asymmetrical shadow about the spin axis. This causes the shadow to appear offset relative to the shadow of a Schwarzschild hole. + +## B. Binary black hole spacetimes + +Astrophysical black hole binaries are expected to radiate energy via gravitational waves, leading to a long inspiral followed by a merger, and finally a ringdown to a steady-state single black hole. Lensing by a final, steady-state black hole will look like the single black holes already seen in figure 4. However, the situation becomes more interesting when viewing these systems before merger. The first images we will present show an equal-mass BBH with non-spinning black holes—one of the simplest binary inspiral spacetimes to analyze— shortly before merger. The simulation we use is case 1 of Taylor et al. [31]. + +Figure 5 shows the image of our reference grid in the presence of this BBH, where the camera is situated such that the orbital angular momentum is pointing out of the page. This image bears a striking resemblance to the bottom left frame of figure 4, excluding the details near the shadows. This shows that, away from the shadows, the spacetime looks similar to a single rotating black hole, where the lensing is dominated by the mass monopole with corrections caused by the angular momentum of the system. In the single-hole case, the spin is responsible for frame dragging, whereas here the orbital angular momentum is responsible. + +![](images/aeb7cae1faa97beabefc660f166ad4e14656a276f6b89d558de607d0573915c0.jpg) +FIG. 5. A BBH system of equal-mass black holes with no spin, viewed near merger with the orbital angular momentum out of the page. + +![](images/230cd47271b2c003d205b9e36e1a69a3afcf4e3661d0468681122a2ce14962d4.jpg) +FIG. 6. A cropped version of figure 5 in order to show more detail near the black hole shadows. A small portion of the image (outlined) is enlarged and inset, where a smaller eyebrow is clearly visible. + +Focusing on the inner portion of the image, we observe that the binary lensing is markedly different from the Schwarzschild or Kerr cases. Figure 6 shows a cropped version of figure 5, emphasizing the structure of the shadows. As might be expected, there are two prominent shadows visible, each associated with one of the two black holes. We also see a narrow secondary shadow (an “eyebrow” [13]) close to the outside of each primary shadow. + +![](images/ed1362d5289f931038ac136001a94e8cb28a62ec74dc32a71325ec0212f3a07f.jpg) +FIG. 7. The same system as figure 6, viewed such that the orbital angular momentum of the system is pointing up. Note that the grid lines in the inset are shown in gray here to distinguish them from the black hole shadows. + +These secondary shadows correspond to one black hole (BH) casting a shadow which is lensed by the other BH on the way to the camera. Equivalently, they are image regions where geodesics are traced backwards from the camera to a BH, but bend around at least one BH on the way there. The first pair of eyebrows is evident in figure $6 ;$ however, we can resolve a pair of smaller eyebrows, shown in the inset. + +We show another view of the same system in figure 7. Here the camera is looking at the system edge on, such that the orbital angular momentum is pointing up. We see again an overall similarity with the corresponding orientation of Kerr spacetime (the bottom-right frame of figure 4), indicating the dominant effects of the mass and angular momentum in these images. We can see a primary shadow for each black hole, but in this configuration one black hole is located roughly behind the other and as a result its shadow gets lensed into a dark ring. Extending along the right side of this ring we see a long thin eyebrow, which is shown in the inset, along with another, smaller, eyebrow. + +To illustrate how photon trajectories behave near shadows, we plot trajectories of a few geodesics on the horizontal line passing through the middle of figure 7 near the eyebrow. Figure 8 shows four snapshots of these trajectories in time, with their current locations in each frame denoted by large dots. It is easiest to consider these trajectories as we evolve them, out of the camera and backwards in time, to see where they came from. In frames A–C, we see the trajectories under consideration start close together then diverge significantly, demonstrating how nearby pixels on the image can correspond to vastly different physical locations. In frame D we see the entire trajectories. A few extend to infinity, but most terminate on the black holes; these are denoted by solid lines and dotted lines, respectively. Only the trajectories extending to infinity result in a photon reaching the camera; those that reach the hole on the right of frame D correspond to the primary ring-like shadow in figure 7, while those that reach the left hole correspond to the larger eyebrow visible on the right side of figure 7. Note that the black holes are orbiting rapidly, so they move significantly while the photons pass through the system. + +![](images/fcc22782b34054804d64af365b410d596d7f5624d8b30a3e83ae456d6f38b57d.jpg) + +FIG. 8. Geodesic trajectories plotted in relation to the black hole event horizons during the lensing evolution for figure 7. Each frame shows a snapshot in time, with the dots representing the current positions of the geodesics, and the lines indicating the trajectories from the camera. The solid and dashed lines indicate whether the geodesics originate from infinity or from a black hole, respectively. +![](images/82cd399b7eff4fd05c559f73495fd64e9e73b8bc8c59c6e7c84c63687f6307ec.jpg) +FIG. 9. Plots identifying the origins of photons along the horizontal line through the center of figure 7. Photons coming from infinity are labeled $\infty ,$ and the shadows are labeled either BH 1 or BH 2. The first plot corresponds to the main portion of figure 7. The second plot focuses on the zoomed square in the inset of figure 7, showing a small feature of the first plot. The third plot zooms to a similar feature of the second plot. This figure demonstrates a striking self-similarity of the lensing structure of a binary black hole system. + +We can also uniquely identify which black hole casts each shadow, which enables us to show in figure 9 the origin of the photons along the horizontal line across the center of figure 7. We arbitrarily label the large shadow in the middle of figure 7 as BH 2, and the ring-like shadow as BH 1. Regions where photons reach the camera from infinity are labeled $\infty .$ . The top plot in figure 9 shows the origin of the photons that reach the camera along the entire middle horizontal line in figure 7. We see that each transition from $\infty$ to either of the BHs includes transitions to the other BH. Even though we cannot resolve them numerically, each vertical line in principle contains infinitely many transitions. To illustrate this idea, the second plot in figure 9 investigates the group of shadows indicated by the zoomed inset of figure 7. Here we find a structure which resembles the first plot. The third plot in figure 9 zooms to a similar group of shadows on the right side of the second plot to again reveal the same structure. This figure clearly shows evidence of self-similarity in the structure of BBH lensing, where the smaller length scales explore more photon orbits through the system. Furthermore, the structure of shadows in BBH lensing is more complex than figures 6 and 7 appear to suggest. The shadows these images focus on are merely some of the largest visible shadows, associated with simpler geodesic orbits around the binary. + +If we consider this equal-mass BBH earlier in the inspiral when its separation is large, the black holes are only weakly interacting. Therefore most camera viewpoints of this binary will yield images with two primary shadows, one for each black hole. Each shadow will be similar to an isolated Schwarzschild or Kerr shadow but with the addition of small eyebrows. However, when the binary is viewed edge-on and the black holes are nearly aligned with the camera, we see an interesting image. + +Figure 10 shows the equal-mass binary in this configuration, hundreds of orbits before merger. Just as in figure 7, the more distant black hole is lensed into a ringlike shadow; however, the ring is thinner here, primarily because of the large separation of the binary. The angular momentum causes the lensed grid outside the shadows to strongly resemble lensing by a Kerr black hole rather than lensing by a Schwarzschild black hole. In addition to the usual primary Einstein ring, another ring is visible between these shadows. Both of these rings correspond to the same source of light, which is in front of the camera and behind the BBH. The second Einstein ring is caused by photons following an “S”-shaped trajectory through the system. + +The second binary system we consider is a fully generic black hole binary with a mass ratio of $m _ { 1 } / m _ { 2 } = 3$ and black hole spins of $\chi _ { 1 } = 0 . 7$ and $\chi _ { 2 } = 0 . 3$ in arbitrary directions. This is case 4 of Taylor et al. [31]. In figure 11 we see a top view of this system, in analogy with what is presented in figure 6. Away from the shadows, the lensing is similar to a single black hole with spin, as was seen with the equal-mass binary images. This appears to be a generic feature of lensing from orbiting BBHs. We can clearly see that the symmetry present in the equal-mass system is gone. The unequal masses evidently change the relative sizes of not only the primary shadows, but all additional shadows as well. The inset in figure 11 zooms to show two successively smaller eyebrows near the small black hole’s primary shadow. However, the effects of the black holes’ spins are not at all clear from this viewpoint. + +![](images/4ff22eb24b8cb770638be617e7e423ccd5a6e268236f813c13c0e2f1770778ac.jpg) +FIG. 10. A BBH system of equal-mass black holes with no spin, viewed hundreds of orbits before merger, with the orbital angular momentum pointing up. The distance from the camera to the closer black hole in this figure is the same as in figure 7. Note that the grid lines are shown in gray here to distinguish them from the black hole shadows. + +![](images/612160cb16d05ec572c5c3a3a209bfcfab495c48573e5a9105a22aa793705f13.jpg) +FIG. 11. A view of a binary inspiral of mass ratio $m _ { 1 } / m _ { 2 } = 3$ near merger, with the orbital angular momentum approximately pointing out of the page. The black hole spins are $\chi _ { 1 } = 0 . 7$ and $\chi _ { 2 } = 0 . 3$ in arbitrary directions. This figure is analogous to figure 6. As in previous figures, a small portion of the image is enlarged and inset, displaying additional eyebrows. + +In figure 12 we see the same binary as in figure 11, viewed with the orbital angular momentum pointing upward, in analogy with figure 7. We again see that, away from the shadows, the system looks like a Kerr black hole. The unequal mass ratio is apparent here, with the smaller black hole lensing the shadow of the larger black hole into a partial ring. If it were not for the black hole spins, the lensing by the binary would be symmetric, giving either a ring-like shadow similar to figure 7 or a shadow and a very thick eyebrow. In this particular BBH, the effect of the individual black hole spins on the image depends strongly on the camera position. + +![](images/c97094653bf86cc65304ed4e55ce839021001518f9bc15938b31f33c2cfb8403.jpg) +FIG. 12. Another view of the BBH in figure 11, but with orbital angular momentum pointing up. The camera parameters are otherwise identical. This figure is analogous to figure 7; however, because of the asymmetry from the black hole spins, the larger black hole’s shadow is not lensed completely around the small black hole. + +## IV. CONCLUSIONS + +In this paper, we present the first images of gravitational lensing by astrophysically relevant binary black holes, thereby providing a realistic representation of what an observer near such a system would actually see. To accomplish this, we have developed a new set of equations that evolve photons efficiently near black hole horizons. Our images show there is a primary shadow—a region where the black hole prevents light from reaching the camera—for each black hole, as well as multiple secondary shadows (or eyebrows). + +We have found that, early in the inspiral, images of a BBH look similar to two separate Kerr black hole shadows, unless viewed when the holes are nearly collinear with the camera. Shortly before the merger, all camera angles yield interesting images of not just one shadow for each black hole, but a handful of smaller visible shadows. We showed for an equal-mass binary viewed edgeon that the lensing structure exhibits self-similarity on smaller scales, corresponding to photons taking an increasing number of orbits through the system. Lensing by a fully generic BBH illustrated that the spin of black holes in a binary can have a clear effect on the lensed shadows. + +We chose not to classify eyebrows and shadows into a hierarchy in this paper. In the inset of figure 6, for instance, identifying the largest eyebrow as the primary eyebrow and the next largest as the secondary eyebrow feels very natural, but the exact definition of such a hierarchy is not immediately clear. For example, simply specifying a geodesic winding number around each black hole is likely not to be sufficient. In addition to the trajectories not lying in a plane, the order that a geodesic orbits the black holes does not commute. Furthermore, the black holes are moving at comparable speeds to the geodesics. For these reasons, we leave the task of classifying shadows as future work. + +We have also shown in this paper that, away from the shadows, an image of a binary black hole system looks like that of an isolated black hole. Thus it is necessary to resolve individual shadows in order to discern the unique visual characteristics present in such images, which places limits on our ability to observe them. + +For systems involving matter, however, the combination of the lensing effects of strong gravity with the disruption and distortion of radiation-emitting matter might yield a unique optical signature. Generating lensed images of black hole-neutron star and neutron starneutron star mergers is an avenue of future investigation. The techniques presented here would allow us to produce detailed visualizations of these mergers; integrating over such images, we could predict the optical signature of an unresolved system. + +## ACKNOWLEDGMENTS + +We would like to thank Curran Muhlberger for providing the temperature fits to the 2MASS photometric data. This publication makes use of data products from the Two Micron All Sky Survey, which is a joint project of the University of Massachusetts and the Infrared Processing and Analysis Center/California Institute of Technology, funded by the National Aeronautics and Space Administration and the National Science Foundation. We would like to thank Daniel Hemberger and Saul Teukolsky for comments on an earlier version of this paper. The authors from Cornell would also like to thank Saul Teukolsky and Lawrence Kidder for general advice while writing this paper. + +This work was supported in part by NSF Grants PHY-1306125 and AST-1333129 at Cornell University, by NSF Grants PHY-1440083, AST-1333520, PHY-1005655, and DMS-1065438 at the California Institute of Technology, and by a grant from the Sherman Fairchild Foundation. FH acknowledges support by the NSF Graduate Research Fellowship under Grant No. DGE-1144153. DB acknowledges support from the LIGO Laboratory, with funding from the National Science Foundation under cooperative agreement PHY-0757058 and NSF REU award PHY-1062293. The binary black hole simulations were performed using the Zwicky computer system operated by the Caltech Center for Advanced Computing Research and funded by NSF MRI No. PHY-0960291 and the Sherman Fairchild Foundation. + +[1] Sjur Refsdal. The gravitational lens effect. 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zw;}#pm+Y@6gZox~l~Pp~OWmv3w?K#MZuaa^CgoRYo4=fw|G|uT=tBexQcG^T{@|o< zM#curl7jP+znm8I->%>9gZ-Pyf|Oo*`AI#deKuNjQoY0s#lw2+?RjA{{xQxMT>qc2 z?1na33j(H-@nuvHPTFS`339{#*QxjWg{`=K0-QK%bjv7N*AEI9M6z*BAb()?gEKBI d9NFi^7}MOB=zlob*&wCQx9 tuple[list[str], list[tuple[int, int]], int]: + """Return IPAC fixed-width column metadata and first data-line index.""" + lines = path.read_text(encoding="ascii").splitlines() + for index, line in enumerate(lines): + if line.startswith("|") and "designation" in line: + boundaries = [offset for offset, char in enumerate(line) if char == "|"] + names = [line[left + 1:right].strip() + for left, right in zip(boundaries, boundaries[1:])] + slices = [(left + 1, right) + for left, right in zip(boundaries, boundaries[1:])] + return names, slices, index + 4 + raise ValueError(f"no IPAC table header in {path}") + + +def load_required_columns(path: Path) -> dict[str, np.ndarray]: + names, slices, first_data_line = column_slices(path) + required = ("ra", "dec", "j_m", "h_m", "k_m", "rd_flg") + indices = {name: names.index(name) for name in required} + values: dict[str, list[str]] = {name: [] for name in required} + + with path.open(encoding="ascii") as table: + for line_number, line in enumerate(table): + if line_number < first_data_line or not line.strip(): + continue + for name, index in indices.items(): + left, right = slices[index] + values[name].append(line[left:right].strip()) + + result: dict[str, np.ndarray] = {} + for name in ("ra", "dec", "j_m", "h_m", "k_m"): + result[name] = np.array( + [float(value) if value else math.nan for value in values[name]], dtype=float + ) + result["rd_flg"] = np.array(values["rd_flg"], dtype="U3") + return result + + +def log_planck_nu_jy_per_sr(log_temperature: np.ndarray) -> np.ndarray: + """Evaluate log B_nu in Jy/sr at the three 2MASS effective wavelengths.""" + temperature = np.exp(log_temperature)[:, None] + frequency = LIGHT_C / WAVELENGTH_M + exponent = PLANCK_H * frequency / (BOLTZMANN_K * temperature) + radiance_si = 2.0 * PLANCK_H * frequency**3 / LIGHT_C**2 / np.expm1(exponent) + return np.log(radiance_si / 1e-26) + + +def fit_blackbody(log_flux_jy: np.ndarray) -> tuple[np.ndarray, np.ndarray]: + """Unweighted least-squares fit in log F_nu for T and apparent solid angle.""" + count = len(log_flux_jy) + lower = np.full(count, math.log(MIN_TEMPERATURE_K)) + upper = np.full(count, math.log(MAX_TEMPERATURE_K)) + golden = (math.sqrt(5.0) - 1.0) / 2.0 + first = upper - golden * (upper - lower) + second = lower + golden * (upper - lower) + + def objective(log_temperature: np.ndarray) -> np.ndarray: + model = log_planck_nu_jy_per_sr(log_temperature) + residual = (log_flux_jy - log_flux_jy.mean(axis=1, keepdims=True) + - (model - model.mean(axis=1, keepdims=True))) + return np.sum(residual * residual, axis=1) + + first_value = objective(first) + second_value = objective(second) + for _ in range(64): + keep_left = first_value <= second_value + upper = np.where(keep_left, second, upper) + second = np.where(keep_left, first, second) + second_value = np.where(keep_left, first_value, second_value) + lower = np.where(keep_left, lower, first) + first = np.where(keep_left, upper - golden * (upper - lower), second) + first_value = np.where(keep_left, objective(first), second_value) + second = np.where(keep_left, second, lower + golden * (upper - lower)) + second_value = np.where(keep_left, second_value, objective(second)) + + log_temperature = 0.5 * (lower + upper) + log_radiance = log_planck_nu_jy_per_sr(log_temperature) + log_solid_angle = np.mean(log_flux_jy - log_radiance, axis=1) + return np.exp(log_temperature), np.exp(log_solid_angle) + + +def main() -> None: + parser = argparse.ArgumentParser( + description="Fit a renderer-facing blackbody catalog from a 2MASS PSC IPAC table." + ) + parser.add_argument("--input", type=Path, default=INPUT, + help="raw 2MASS PSC IPAC table") + parser.add_argument("--output", type=Path, default=OUTPUT, + help="renderer-facing CSV to write") + args = parser.parse_args() + columns = load_required_columns(args.input) + magnitudes = np.column_stack((columns["j_m"], columns["h_m"], columns["k_m"])) + valid_photometry = np.isfinite(magnitudes).all(axis=1) + valid_rd_flag = np.array( + [len(flag) == 3 and all(value in GOOD_RD_FLAGS for value in flag) + for flag in columns["rd_flg"]] + ) + keep = valid_photometry & valid_rd_flag + if not np.any(keep): + raise ValueError("no sources retain valid J/H/Ks photometry") + + flux_jy = ZERO_POINT_JY * np.power(10.0, -0.4 * magnitudes[keep]) + temperature_k, amplitude_sr = fit_blackbody(np.log(flux_jy)) + + args.output.parent.mkdir(parents=True, exist_ok=True) + with args.output.open("w", newline="", encoding="ascii") as output: + writer = csv.writer(output, lineterminator="\n") + writer.writerow(("ra_deg", "dec_deg", "temperature_K", "amplitude_sr")) + for ra, dec, temperature, amplitude in zip( + columns["ra"][keep], columns["dec"][keep], temperature_k, amplitude_sr + ): + writer.writerow((f"{ra:.6f}", f"{dec:.6f}", f"{temperature:.8g}", + f"{amplitude:.9e}")) + + print(f"input_rows={len(keep)}") + print(f"discarded_invalid_photometry={np.count_nonzero(~valid_photometry)}") + print(f"discarded_rd_flg_not_123={np.count_nonzero(~valid_rd_flag)}") + print(f"retained_rows={np.count_nonzero(keep)}") + print(f"wrote={args.output}") + + +if __name__ == "__main__": + main() diff --git a/src/catalog.c b/src/catalog.c new file mode 100644 index 0000000..30150b8 --- /dev/null +++ b/src/catalog.c @@ -0,0 +1,107 @@ +#include "catalog.h" + +#include +#include +#include +#include + +#define PI 3.14159265358979323846 + +enum { + TEST_GRID_LINE_DEG = 10, + TEST_GRID_SAMPLE_DEG = 2, + TEST_GRID_RED_TEMPERATURE_K = 3000, + TEST_GRID_BLUE_TEMPERATURE_K = 12000, +}; + +/* + * With the renderer's 380--780 nm CIE integration and linear-sRGB luminance, + * B(3000 K) / B(12000 K) = 0.00141095580387. Red stars retain unit scale. + */ +#define TEST_GRID_BLUE_AMPLITUDE 0.00141095580387 + +static int test_grid_temperature_K(int longitude_deg, int latitude_deg) +{ + /* Boundaries belong to the octant immediately east/north of them. */ + const int longitude_sector = longitude_deg / 90; + const int hemisphere = latitude_deg < 0 ? 0 : 1; + /* Add the hemisphere bit to flip the color across the equator. */ + const int octant = hemisphere + longitude_sector; + return octant % 2 == 0 ? TEST_GRID_RED_TEMPERATURE_K + : TEST_GRID_BLUE_TEMPERATURE_K; +} + +static double test_grid_amplitude(int temperature_K) +{ + return temperature_K == TEST_GRID_BLUE_TEMPERATURE_K + ? TEST_GRID_BLUE_AMPLITUDE + : 1.0; +} + +int catalog_write_octant_grid(const char *path) +{ + FILE *file = fopen(path, "w"); + if (file == NULL) return -1; + fputs("longitude_deg,latitude_deg,temperature_K,amplitude\n", file); + for (int latitude = -90; latitude <= 90; latitude += TEST_GRID_SAMPLE_DEG) { + for (int longitude = 0; longitude < 360; + longitude += TEST_GRID_SAMPLE_DEG) { + const int on_longitude_line = longitude % TEST_GRID_LINE_DEG == 0; + const int on_latitude_line = latitude % TEST_GRID_LINE_DEG == 0; + if ((!on_longitude_line && !on_latitude_line) || + ((latitude == -90 || latitude == 90) && longitude != 0)) + continue; + const int temperature_K = + test_grid_temperature_K(longitude, latitude); + fprintf(file, "%d,%d,%d,%.12g\n", longitude, latitude, + temperature_K, test_grid_amplitude(temperature_K)); + } + } + return fclose(file) == 0 ? 0 : -1; +} + +int catalog_load_csv(StarCatalog *catalog, const char *path) +{ + FILE *file = fopen(path, "r"); + char line[256]; + size_t capacity = 0; + catalog->stars = NULL; + catalog->count = 0; + if (file == NULL || fgets(line, sizeof line, file) == NULL) goto fail; + + while (fgets(line, sizeof line, file) != NULL) { + double longitude, latitude, temperature, amplitude; + if (sscanf(line, "%lf,%lf,%lf,%lf", &longitude, &latitude, + &temperature, &litude) != 4) + goto fail; + if (catalog->count == capacity) { + size_t next = capacity == 0 ? 256 : capacity * 2; + Star *stars = realloc(catalog->stars, next * sizeof *stars); + if (stars == NULL) goto fail; + catalog->stars = stars; + capacity = next; + } + const double lon = longitude * PI / 180.0; + const double lat = latitude * PI / 180.0; + const double cos_lat = cos(lat); + Star *star = &catalog->stars[catalog->count++]; + star->direction[0] = cos_lat * cos(lon); + star->direction[1] = sin(lat); + star->direction[2] = cos_lat * sin(lon); + star->temperature_K = temperature; + star->amplitude = amplitude; + } + fclose(file); + return 0; +fail: + if (file != NULL) fclose(file); + catalog_destroy(catalog); + return -1; +} + +void catalog_destroy(StarCatalog *catalog) +{ + free(catalog->stars); + catalog->stars = NULL; + catalog->count = 0; +} diff --git a/src/catalog.h b/src/catalog.h new file mode 100644 index 0000000..fb83edc --- /dev/null +++ b/src/catalog.h @@ -0,0 +1,27 @@ +#ifndef CATALOG_H +#define CATALOG_H + +#include + +typedef struct { + double direction[3]; + double temperature_K; + double amplitude; +} Star; + +typedef struct { + Star *stars; + size_t count; +} StarCatalog; + +/* + * Synthetic lensing fixture: stars lie on the union of 10-degree longitude + * and latitude lines, sampled every 2 degrees. The eight longitude/hemisphere + * octants alternate between red and blue blackbody temperatures. Blue stars' + * scale compensates for their greater visible blackbody radiance. + */ +int catalog_write_octant_grid(const char *path); +int catalog_load_csv(StarCatalog *catalog, const char *path); +void catalog_destroy(StarCatalog *catalog); + +#endif diff --git a/src/frame.c b/src/frame.c new file mode 100644 index 0000000..ed09f53 --- /dev/null +++ b/src/frame.c @@ -0,0 +1,378 @@ +#include "frame.h" + +#include "optics.h" + +#include +#include +#include +#include +#include + +/* Private HDR buffers make splatting independent across workers. This is an + * allocation cap, not a rendering parameter: callers transparently fall back + * to the serial implementation when the frame is too large for two buffers. */ +#define FRAME_SPLAT_MAX_PRIVATE_HDR_BYTES ((size_t)512 * 1024 * 1024) + +static const double pi = 3.14159265358979323846; + +static double dot(const double a[3], const double b[3]) { + return a[0] * b[0] + a[1] * b[1] + a[2] * b[2]; +} + +static void cross(const double a[3], const double b[3], double out[3]) { + out[0] = a[1] * b[2] - a[2] * b[1]; + out[1] = a[2] * b[0] - a[0] * b[2]; + out[2] = a[0] * b[1] - a[1] * b[0]; +} + +static double normalize(double vector[3]) { + const double length = sqrt(dot(vector, vector)); + if (length > 0.0) + for (int i = 0; i < 3; ++i) + vector[i] /= length; + return length; +} + +static size_t vertex_index(int column, int row, int columns) { + return (size_t)row * (columns + 1) + column; +} + +int frame_lens_mesh_build_coarse(FrameLensMesh *mesh, int width, int height, + int cell_pixels, double horizontal_fov_deg) { + if (mesh == NULL || width <= 0 || height <= 0 || cell_pixels <= 0 || + horizontal_fov_deg <= 0.0 || horizontal_fov_deg >= 179.0) + return -1; + const int columns = (width + cell_pixels - 1) / cell_pixels; + const int rows = (height + cell_pixels - 1) / cell_pixels; + const size_t vertex_count = (size_t)(columns + 1) * (rows + 1); + const size_t triangle_count = (size_t)columns * rows * 2; + LensVertex *vertices = calloc(vertex_count, sizeof *vertices); + LensTriangle *triangles = malloc(triangle_count * sizeof *triangles); + if (vertices == NULL || triangles == NULL) { + free(vertices); + free(triangles); + return -1; + } + const double tan_half_x = tan(horizontal_fov_deg * pi / 360.0); + const double tan_half_y = tan_half_x * (double)height / width; + for (int row = 0; row <= rows; ++row) { + const double image_y = (double)row * height / rows; + for (int column = 0; column <= columns; ++column) { + LensVertex *vertex = &vertices[vertex_index(column, row, columns)]; + vertex->image_x = (double)column * width / columns; + vertex->image_y = image_y; + vertex->camera_direction[0] = 1.0; + vertex->camera_direction[1] = (0.5 - image_y / height) * 2.0 * tan_half_y; + vertex->camera_direction[2] = + (vertex->image_x / width - 0.5) * 2.0 * tan_half_x; + normalize(vertex->camera_direction); + } + } + size_t next_triangle = 0; + for (int row = 0; row < rows; ++row) + for (int column = 0; column < columns; ++column) { + const size_t top_left = vertex_index(column, row, columns); + const size_t top_right = vertex_index(column + 1, row, columns); + const size_t bottom_left = vertex_index(column, row + 1, columns); + const size_t bottom_right = vertex_index(column + 1, row + 1, columns); + triangles[next_triangle++] = + (LensTriangle){{top_left, bottom_left, bottom_right}}; + triangles[next_triangle++] = + (LensTriangle){{top_left, bottom_right, top_right}}; + } + *mesh = (FrameLensMesh){.vertices = vertices, + .triangles = triangles, + .vertex_count = vertex_count, + .triangle_count = triangle_count}; + return 0; +} + +int frame_lens_mesh_trace(FrameLensMesh *mesh, const SpacetimeSource *spacetime, + const ObserverState *observer, + const GeodesicTraceConfig *trace) { + if (mesh == NULL || spacetime == NULL || observer == NULL || trace == NULL) + return -1; + /* Each iteration exclusively owns one vertex. SpacetimeSource is shared + * read-only here; backends with mutable evaluation state must keep it + * thread-local. */ +#pragma omp parallel for schedule(static) + for (size_t i = 0; i < mesh->vertex_count; ++i) { + LensVertex *vertex = &mesh->vertices[i]; + RayEndpoint endpoint = geodesic_trace_past(spacetime, observer, + vertex->camera_direction, trace); + vertex->status = endpoint.status; + if (endpoint.status == RAY_ENDPOINT_ESCAPED) { + for (int axis = 0; axis < 3; ++axis) + vertex->n_infinity[axis] = endpoint.n_infinity[axis]; + vertex->log_frequency_ratio = log(endpoint.frequency_ratio); + } + } + return 0; +} + +static double spherical_area(const double a[3], const double b[3], + const double c[3]) { + double b_cross_c[3]; + cross(b, c, b_cross_c); + return 2.0 * atan2(fabs(dot(a, b_cross_c)), + 1.0 + dot(a, b) + dot(b, c) + dot(c, a)); +} + +static int spherical_barycentric(const double point[3], const double a[3], + const double b[3], const double c[3], + double weights[3]) { + const double area = spherical_area(a, b, c); + double edge_cross[3]; + if (area < 1e-14) + return -1; + const double *corners[3] = {a, b, c}; + for (int edge = 0; edge < 3; ++edge) { + const double *left = corners[edge]; + const double *right = corners[(edge + 1) % 3]; + const double *opposite = corners[(edge + 2) % 3]; + cross(left, right, edge_cross); + /* This is a sign test, so its tolerance must scale with the source + * triangle. A fixed absolute threshold turns sufficiently fine triangles + * into near-all-sky queries. */ + if (dot(edge_cross, point) * dot(edge_cross, opposite) < + -1e-14 * dot(edge_cross, edge_cross)) + return -1; + } + weights[0] = spherical_area(point, b, c) / area; + weights[1] = spherical_area(point, c, a) / area; + weights[2] = spherical_area(point, a, b) / area; + return 0; +} + +static int usable_triangle(const FrameLensMesh *mesh, + const LensTriangle *triangle, + const LensVertex *vertices[3]) { + for (int i = 0; i < 3; ++i) { + vertices[i] = &mesh->vertices[triangle->vertex[i]]; + if (vertices[i]->status != RAY_ENDPOINT_ESCAPED) + return 0; + } + return spherical_area(vertices[0]->n_infinity, vertices[1]->n_infinity, + vertices[2]->n_infinity) >= 1e-14; +} + +/* Give a source lying exactly on a shared source edge to one triangle only. + * Interior overlaps remain valid separate lens images. */ +static int owns_source_boundary(const LensTriangle *triangle, + const double weights[3]) { + for (int opposite = 0; opposite < 3; ++opposite) { + if (weights[opposite] > 1e-11) + continue; + const size_t left = triangle->vertex[(opposite + 1) % 3]; + const size_t right = triangle->vertex[(opposite + 2) % 3]; + if (left > right) + return 0; + } + return 1; +} + +static size_t splat_catalog_triangles(const FrameLensMesh *mesh, + const StarCatalog *catalog, double *hdr, + int width, int height, double exposure, + const PointSpreadFunction *psf, + size_t first_triangle, + size_t last_triangle) { + size_t images = 0; + for (size_t t = first_triangle; t < last_triangle; ++t) { + const LensVertex *vertex[3]; + if (!usable_triangle(mesh, &mesh->triangles[t], vertex)) + continue; + const double source_area = spherical_area( + vertex[0]->n_infinity, vertex[1]->n_infinity, vertex[2]->n_infinity); + const double image_area = + spherical_area(vertex[0]->camera_direction, vertex[1]->camera_direction, + vertex[2]->camera_direction); + const double magnification = image_area / source_area; + for (size_t s = 0; s < catalog->count; ++s) { + const Star *star = &catalog->stars[s]; + double weights[3]; + if (spherical_barycentric(star->direction, vertex[0]->n_infinity, + vertex[1]->n_infinity, vertex[2]->n_infinity, + weights)) + continue; + if (!owns_source_boundary(&mesh->triangles[t], weights)) + continue; + const double image_x = weights[0] * vertex[0]->image_x + + weights[1] * vertex[1]->image_x + + weights[2] * vertex[2]->image_x; + const double image_y = weights[0] * vertex[0]->image_y + + weights[1] * vertex[1]->image_y + + weights[2] * vertex[2]->image_y; + const double log_g = weights[0] * vertex[0]->log_frequency_ratio + + weights[1] * vertex[1]->log_frequency_ratio + + weights[2] * vertex[2]->log_frequency_ratio; + const LinearRgb color = + blackbody_to_linear_rgb(star->temperature_K * exp(log_g)); + splat_moffat(hdr, width, height, image_x, image_y, color, + exposure * star->amplitude * magnification, psf); + ++images; + } + } + return images; +} + +size_t frame_splat_catalog(const FrameLensMesh *mesh, + const StarCatalog *catalog, double *hdr, int width, + int height, double exposure, + const PointSpreadFunction *psf) { + if (mesh == NULL || catalog == NULL || hdr == NULL || exposure <= 0.0 || + psf == NULL || width <= 0 || height <= 0) + return 0; + + const size_t pixel_count = (size_t)width * height * 3; + if (pixel_count > SIZE_MAX / sizeof(double) || + pixel_count * sizeof(double) > FRAME_SPLAT_MAX_PRIVATE_HDR_BYTES / 2) + return splat_catalog_triangles(mesh, catalog, hdr, width, height, exposure, + psf, 0, mesh->triangle_count); + const size_t buffer_bytes = pixel_count * sizeof(double); + size_t worker_count = FRAME_SPLAT_MAX_PRIVATE_HDR_BYTES / buffer_bytes; + const int max_threads = omp_get_max_threads(); + if (worker_count > (size_t)max_threads) + worker_count = (size_t)max_threads; + if (worker_count < 2 || worker_count > INT_MAX) + return splat_catalog_triangles(mesh, catalog, hdr, width, height, exposure, + psf, 0, mesh->triangle_count); + + double **private_hdr = calloc(worker_count, sizeof *private_hdr); + if (private_hdr == NULL) + return splat_catalog_triangles(mesh, catalog, hdr, width, height, exposure, + psf, 0, mesh->triangle_count); + size_t allocated = 0; + for (; allocated < worker_count; ++allocated) { + private_hdr[allocated] = calloc(pixel_count, sizeof **private_hdr); + if (private_hdr[allocated] == NULL) + break; + } + if (allocated != worker_count) { + while (allocated > 0) + free(private_hdr[--allocated]); + free(private_hdr); + return splat_catalog_triangles(mesh, catalog, hdr, width, height, exposure, + psf, 0, mesh->triangle_count); + } + + size_t images = 0; +#pragma omp parallel num_threads((int)worker_count) reduction(+ : images) + { + const size_t worker = (size_t)omp_get_thread_num(); + const size_t first = mesh->triangle_count * worker / worker_count; + const size_t last = mesh->triangle_count * (worker + 1) / worker_count; + images += splat_catalog_triangles(mesh, catalog, private_hdr[worker], width, + height, exposure, psf, first, last); + } +#pragma omp parallel for schedule(static) + for (size_t pixel = 0; pixel < pixel_count; ++pixel) + for (size_t worker = 0; worker < worker_count; ++worker) + hdr[pixel] += private_hdr[worker][pixel]; + for (size_t worker = 0; worker < worker_count; ++worker) + free(private_hdr[worker]); + free(private_hdr); + return images; +} + +static void blend_gray(double *hdr, int width, int height, int x, int y, + double gray, double alpha) { + if (x < 0 || x >= width || y < 0 || y >= height) + return; + double *pixel = &hdr[3 * (y * width + x)]; + for (int channel = 0; channel < 3; ++channel) + pixel[channel] = (1.0 - alpha) * pixel[channel] + alpha * gray; +} + +static double fractional_part(double value) { return value - floor(value); } + +static void plot_aa(double *hdr, int width, int height, int steep, int x, int y, + double coverage, double gray, double opacity) { + if (coverage > 0.0) + blend_gray(hdr, width, height, steep ? y : x, steep ? x : y, gray, + coverage * opacity); +} + +/* Xiaolin Wu line rasterization: a one-pixel line with coverage-based alpha. */ +static void draw_line(double *hdr, int width, int height, + const LensVertex *from, const LensVertex *to, double gray, + double opacity) { + double x0 = from->image_x, y0 = from->image_y; + double x1 = to->image_x, y1 = to->image_y; + const int steep = fabs(y1 - y0) > fabs(x1 - x0); + if (steep) { + double swap = x0; + x0 = y0; + y0 = swap; + swap = x1; + x1 = y1; + y1 = swap; + } + if (x0 > x1) { + double swap = x0; + x0 = x1; + x1 = swap; + swap = y0; + y0 = y1; + y1 = swap; + } + const double dx = x1 - x0; + if (dx == 0.0) { + plot_aa(hdr, width, height, steep, (int)lround(x0), (int)floor(y0), 1.0, + gray, opacity); + return; + } + const double gradient = (y1 - y0) / dx; + double x_end = round(x0); + double y_end = y0 + gradient * (x_end - x0); + double x_gap = 1.0 - fractional_part(x0 + 0.5); + int x_pixel_start = (int)x_end; + int y_pixel = (int)floor(y_end); + plot_aa(hdr, width, height, steep, x_pixel_start, y_pixel, + (1.0 - fractional_part(y_end)) * x_gap, gray, opacity); + plot_aa(hdr, width, height, steep, x_pixel_start, y_pixel + 1, + fractional_part(y_end) * x_gap, gray, opacity); + double inter_y = y_end + gradient; + x_end = round(x1); + y_end = y1 + gradient * (x_end - x1); + x_gap = fractional_part(x1 + 0.5); + const int x_pixel_end = (int)x_end; + y_pixel = (int)floor(y_end); + plot_aa(hdr, width, height, steep, x_pixel_end, y_pixel, + (1.0 - fractional_part(y_end)) * x_gap, gray, opacity); + plot_aa(hdr, width, height, steep, x_pixel_end, y_pixel + 1, + fractional_part(y_end) * x_gap, gray, opacity); + for (int x = x_pixel_start + 1; x < x_pixel_end; ++x) { + y_pixel = (int)floor(inter_y); + plot_aa(hdr, width, height, steep, x, y_pixel, + 1.0 - fractional_part(inter_y), gray, opacity); + plot_aa(hdr, width, height, steep, x, y_pixel + 1, fractional_part(inter_y), + gray, opacity); + inter_y += gradient; + } +} + +void frame_draw_mesh(const FrameLensMesh *mesh, double *hdr, int width, + int height, double gray, double opacity) { + if (mesh == NULL || hdr == NULL || width <= 0 || height <= 0 || gray < 0.0 || + opacity < 0.0 || opacity > 1.0) + return; + for (size_t i = 0; i < mesh->triangle_count; ++i) { + const LensTriangle *triangle = &mesh->triangles[i]; + for (int edge = 0; edge < 3; ++edge) { + const size_t from_id = triangle->vertex[edge]; + const size_t to_id = triangle->vertex[(edge + 1) % 3]; + if (from_id < to_id) + draw_line(hdr, width, height, &mesh->vertices[from_id], + &mesh->vertices[to_id], gray, opacity); + } + } +} + +void frame_lens_mesh_destroy(FrameLensMesh *mesh) { + if (mesh == NULL) + return; + free(mesh->vertices); + free(mesh->triangles); + *mesh = (FrameLensMesh){0}; +} diff --git a/src/frame.h b/src/frame.h new file mode 100644 index 0000000..dd21063 --- /dev/null +++ b/src/frame.h @@ -0,0 +1,46 @@ +#ifndef FRAME_H +#define FRAME_H + +#include "catalog.h" +#include "geodesic.h" +#include "optics.h" +#include "observer.h" +#include "spacetime.h" + +#include + +typedef struct { + double image_x, image_y; + double camera_direction[3]; + double n_infinity[3]; + double log_frequency_ratio; + RayEndpointStatus status; +} LensVertex; + +typedef struct { + size_t vertex[3]; +} LensTriangle; + +typedef struct { + LensVertex *vertices; + LensTriangle *triangles; + size_t vertex_count, triangle_count; +} FrameLensMesh; + +int frame_lens_mesh_build_coarse(FrameLensMesh *mesh, int width, int height, + int cell_pixels, double horizontal_fov_deg); +int frame_lens_mesh_trace(FrameLensMesh *mesh, const SpacetimeSource *spacetime, + const ObserverState *observer, + const GeodesicTraceConfig *trace); + +/* Each locally invertible escaped triangle contributes one image per contained + * star. */ +size_t frame_splat_catalog(const FrameLensMesh *mesh, + const StarCatalog *catalog, double *hdr, int width, + int height, double exposure, + const PointSpreadFunction *psf); +void frame_draw_mesh(const FrameLensMesh *mesh, double *hdr, int width, + int height, double gray, double opacity); +void frame_lens_mesh_destroy(FrameLensMesh *mesh); + +#endif diff --git a/src/geodesic.c b/src/geodesic.c new file mode 100644 index 0000000..81bd245 --- /dev/null +++ b/src/geodesic.c @@ -0,0 +1,189 @@ +#include "geodesic.h" +#include + +typedef struct { + double x[3], Pi[3], log_alpha_p0; +} State; +typedef struct { + double x[3], Pi[3], log_alpha_p0; +} Derivative; + +static double dot(const double a[3], const double b[3]) { + return a[0] * b[0] + a[1] * b[1] + a[2] * b[2]; +} + +static int invert(double a[3][3], double b[3][3]) { + double d = a[0][0] * (a[1][1] * a[2][2] - a[1][2] * a[2][1]) - + a[0][1] * (a[1][0] * a[2][2] - a[1][2] * a[2][0]) + + a[0][2] * (a[1][0] * a[2][1] - a[1][1] * a[2][0]); + if (!isfinite(d) || fabs(d) < 1e-14) + return -1; + b[0][0] = (a[1][1] * a[2][2] - a[1][2] * a[2][1]) / d; + b[0][1] = (a[0][2] * a[2][1] - a[0][1] * a[2][2]) / d; + b[0][2] = (a[0][1] * a[1][2] - a[0][2] * a[1][1]) / d; + b[1][0] = (a[1][2] * a[2][0] - a[1][0] * a[2][2]) / d; + b[1][1] = (a[0][0] * a[2][2] - a[0][2] * a[2][0]) / d; + b[1][2] = (a[0][2] * a[1][0] - a[0][0] * a[1][2]) / d; + b[2][0] = (a[1][0] * a[2][1] - a[1][1] * a[2][0]) / d; + b[2][1] = (a[0][1] * a[2][0] - a[0][0] * a[2][1]) / d; + b[2][2] = (a[0][0] * a[1][1] - a[0][1] * a[1][0]) / d; + return 0; +} + +/* Equation (4) and (5) of Bohn et al., arXiv:1410.7775. */ +static int rhs(const SpacetimeSource *source, double t, const State *s, + Derivative *out) { + MetricData m; + double inv[3][3], up[3] = {0}, da_pi = 0, k_pi_pi = 0; + if (spacetime_eval(source, t, s->x, &m) || m.alpha <= 0 || + invert(m.gamma, inv)) + return -1; + for (int i = 0; i < 3; i++) + for (int j = 0; j < 3; j++) + up[i] += inv[i][j] * s->Pi[j]; + for (int i = 0; i < 3; i++) { + da_pi += m.d_alpha[i] * up[i]; + for (int j = 0; j < 3; j++) + k_pi_pi += m.K[i][j] * up[i] * up[j]; + } + for (int i = 0; i < 3; i++) { + double db_pi = 0, dg_pi_pi = 0; + out->x[i] = m.alpha * up[i] - m.beta[i]; + for (int j = 0; j < 3; j++) { + db_pi += m.d_beta[i][j] * s->Pi[j]; + for (int k = 0; k < 3; k++) { + /* Equation (4) requires partial_i gamma^{jk}, while MetricData + * deliberately stores partial_i gamma_jk because that is what metric + * backends interpolate naturally. Differentiate gamma^{-1}: + * partial_i gamma^{jk} = -gamma^{ja}(partial_i gamma_ab)gamma^{bk}. + */ + double d_inverse_gamma = 0.0; + for (int a = 0; a < 3; ++a) + for (int b = 0; b < 3; ++b) + d_inverse_gamma -= inv[j][a] * m.d_gamma[i][a][b] * inv[b][k]; + dg_pi_pi += d_inverse_gamma * s->Pi[j] * s->Pi[k]; + } + } + out->Pi[i] = -m.d_alpha[i] + (da_pi - m.alpha * k_pi_pi) * s->Pi[i] + + db_pi - 0.5 * m.alpha * dg_pi_pi; + } + out->log_alpha_p0 = -da_pi + m.alpha * k_pi_pi; + return 0; +} + +static State add(const State *s, const Derivative *d, double h) { + State r = *s; + for (int i = 0; i < 3; i++) { + r.x[i] += h * d->x[i]; + r.Pi[i] += h * d->Pi[i]; + } + r.log_alpha_p0 += h * d->log_alpha_p0; + return r; +} + +static int rk4(const SpacetimeSource *source, double t, double h, State *s) { + Derivative a, b, c, d; + State q; + if (rhs(source, t, s, &a)) + return -1; + q = add(s, &a, h / 2); + if (rhs(source, t + h / 2, &q, &b)) + return -1; + q = add(s, &b, h / 2); + if (rhs(source, t + h / 2, &q, &c)) + return -1; + q = add(s, &c, h); + if (rhs(source, t + h, &q, &d)) + return -1; + for (int i = 0; i < 3; i++) { + s->x[i] += h * (a.x[i] + 2 * b.x[i] + 2 * c.x[i] + d.x[i]) / 6; + s->Pi[i] += h * (a.Pi[i] + 2 * b.Pi[i] + 2 * c.Pi[i] + d.Pi[i]) / 6; + } + s->log_alpha_p0 += h * + (a.log_alpha_p0 + 2 * b.log_alpha_p0 + 2 * c.log_alpha_p0 + + d.log_alpha_p0) / + 6; + return 0; +} + +static int initialize(const SpacetimeSource *source, const ObserverState *o, + const double n[3], State *s) { + MetricData m; + double k[4] = {o->tetrad[0][0], o->tetrad[0][1], o->tetrad[0][2], + o->tetrad[0][3]}; + if (spacetime_eval(source, o->coordinate_time, o->coordinate_position, &m) || + m.alpha <= 0) + return -1; + for (int a = 0; a < 3; a++) + for (int mu = 0; mu < 4; mu++) + k[mu] -= n[a] * o->tetrad[a + 1][mu]; + if (k[0] <= 0) + return -1; + for (int i = 0; i < 3; i++) { + s->x[i] = o->coordinate_position[i]; + s->Pi[i] = 0; + for (int j = 0; j < 3; j++) + s->Pi[i] += m.gamma[i][j] * (k[j + 1] + m.beta[j] * k[0]); + s->Pi[i] /= m.alpha * k[0]; + } + s->log_alpha_p0 = log(m.alpha * k[0]); + return isfinite(s->log_alpha_p0) ? 0 : -1; +} + +static int escaped_direction(const SpacetimeSource *source, double t, + const State *s, double n[3]) { + MetricData m; + double inv[3][3], norm = 0; + if (spacetime_eval(source, t, s->x, &m) || invert(m.gamma, inv)) + return -1; + for (int i = 0; i < 3; i++) { + n[i] = 0; + for (int j = 0; j < 3; j++) + n[i] -= inv[i][j] * s->Pi[j]; + norm += n[i] * n[i]; + } + if (norm <= 0) + return -1; + for (int i = 0; i < 3; i++) + n[i] /= sqrt(norm); + return 0; +} + +RayEndpoint geodesic_trace_past(const SpacetimeSource *source, + const ObserverState *observer, + const double n[3], + const GeodesicTraceConfig *config) { + RayEndpoint out = {.frequency_ratio = 0, + .magnification = 1, + .status = RAY_ENDPOINT_INTEGRATION_FAILURE}; + State s; + double t; + if (!source || !observer || !config || config->coordinate_time_step <= 0 || + !config->max_steps || fabs(dot(n, n) - 1) > 1e-10 || + initialize(source, observer, n, &s)) + return out; + t = observer->coordinate_time; + for (unsigned int i = 0; i < config->max_steps; i++) { + if (config->capture_log_alpha_p0 > 0.0 && + s.log_alpha_p0 >= config->capture_log_alpha_p0) { + out.status = RAY_ENDPOINT_CAPTURED; + return out; + } + SpacetimeRayStatus status = spacetime_classify(source, t, s.x); + if (status != SPACETIME_RAY_ACTIVE) { + out.status = status == SPACETIME_RAY_ESCAPED ? RAY_ENDPOINT_ESCAPED + : RAY_ENDPOINT_CAPTURED; + if (out.status == RAY_ENDPOINT_ESCAPED && + escaped_direction(source, t, &s, out.n_infinity) == 0) + out.frequency_ratio = exp(-s.log_alpha_p0); + else if (out.status == RAY_ENDPOINT_ESCAPED) + out.status = RAY_ENDPOINT_INTEGRATION_FAILURE; + return out; + } + if (rk4(source, t, -config->coordinate_time_step, &s)) + return out; + t -= config->coordinate_time_step; + } + out.status = RAY_ENDPOINT_MAX_STEPS; + return out; +} diff --git a/src/geodesic.h b/src/geodesic.h new file mode 100644 index 0000000..8a31273 --- /dev/null +++ b/src/geodesic.h @@ -0,0 +1,37 @@ +#ifndef GEODESIC_H +#define GEODESIC_H + +#include "observer.h" +#include "spacetime.h" + +typedef enum { + RAY_ENDPOINT_ESCAPED, + RAY_ENDPOINT_CAPTURED, + RAY_ENDPOINT_MAX_STEPS, + RAY_ENDPOINT_INTEGRATION_FAILURE +} RayEndpointStatus; + +typedef struct { + double n_infinity[3]; + double frequency_ratio; /* E_camera / E_infinity */ + double magnification; /* Filled by the future local inverse lens map. */ + RayEndpointStatus status; +} RayEndpoint; + +typedef struct { + double coordinate_time_step; + unsigned int max_steps; + /* A positive value terminates a backwards ray whose horizon redshift has + * made log(alpha p^0) reach this value. Zero disables this analytic/demo + * criterion; numerical moving-puncture backends use their AH-calibrated + * spatial cutoff instead. */ + double capture_log_alpha_p0; +} GeodesicTraceConfig; + +/* camera_direction is a unit vector in the observer's (forward, up, right) + * tetrad. */ +RayEndpoint geodesic_trace_past(const SpacetimeSource *source, + const ObserverState *observer, + const double camera_direction[3], + const GeodesicTraceConfig *config); +#endif diff --git a/src/main.c b/src/main.c new file mode 100644 index 0000000..4d7a6d1 --- /dev/null +++ b/src/main.c @@ -0,0 +1,200 @@ +#include "catalog.h" +#include "frame.h" +#include "optics.h" +#include "spacetime.h" + +#include +#include +#include +#include + +typedef struct { + int width, height; + int coarse_cell_pixels; + int draw_mesh; + double horizontal_fov_deg, look_ra_deg, look_dec_deg, exposure; + double observer_inward_speed; + PointSpreadFunction psf; + const char *catalog_path; + const char *output_path; +} Settings; + +static int parse_int(const char *text, int *value) { + char *end; + errno = 0; + long parsed = strtol(text, &end, 10); + if (errno || *end || parsed <= 0 || parsed > 16384) + return -1; + *value = (int)parsed; + return 0; +} + +static int parse_double(const char *text, double *value) { + char *end; + errno = 0; + *value = strtod(text, &end); + return errno || *end || *value <= 0.0 || *value >= 179.0 ? -1 : 0; +} + +static int parse_ra_deg(const char *text, double *value) { + char *end; + errno = 0; + *value = strtod(text, &end); + return errno || *end || *value < 0.0 || *value >= 360.0 ? -1 : 0; +} + +static int parse_dec_deg(const char *text, double *value) { + char *end; + errno = 0; + *value = strtod(text, &end); + return errno || *end || *value < -90.0 || *value > 90.0 ? -1 : 0; +} + +static int parse_positive(const char *text, double *value) { + char *end; + errno = 0; + *value = strtod(text, &end); + return errno || *end || *value <= 0.0 ? -1 : 0; +} + +static int parse_speed(const char *text, double *value) { + char *end; + errno = 0; + *value = strtod(text, &end); + return errno || *end || *value < 0.0 || *value >= 1.0 ? -1 : 0; +} + +static int parse_moffat_beta(const char *text, double *value) { + char *end; + errno = 0; + *value = strtod(text, &end); + return errno || *end || *value <= 1.0 ? -1 : 0; +} + +static int parse_args(int argc, char **argv, Settings *s, + const char **write_path) { + *s = (Settings){1280, + 720, + 16, + 0, + 30.0, + 270.0, + 0.0, + 100.0, + 0.0, + {2.7, 4.5}, + "assets/sky_grid_5deg.csv", + "output/imgs/minkowski_sky.ppm"}; + *write_path = NULL; + for (int i = 1; i < argc; ++i) { + if (!strcmp(argv[i], "--catalog") && i + 1 < argc) + s->catalog_path = argv[++i]; + else if (!strcmp(argv[i], "--output") && i + 1 < argc) + s->output_path = argv[++i]; + else if (!strcmp(argv[i], "--width") && i + 1 < argc && + !parse_int(argv[++i], &s->width)) { + } else if (!strcmp(argv[i], "--height") && i + 1 < argc && + !parse_int(argv[++i], &s->height)) { + } else if (!strcmp(argv[i], "--coarse-cell-pixels") && i + 1 < argc && + !parse_int(argv[++i], &s->coarse_cell_pixels)) { + } else if (!strcmp(argv[i], "--draw-mesh")) { + s->draw_mesh = 1; + } else if (!strcmp(argv[i], "--fov-deg") && i + 1 < argc && + !parse_double(argv[++i], &s->horizontal_fov_deg)) { + } else if (!strcmp(argv[i], "--look-ra-deg") && i + 1 < argc && + !parse_ra_deg(argv[++i], &s->look_ra_deg)) { + } else if (!strcmp(argv[i], "--look-dec-deg") && i + 1 < argc && + !parse_dec_deg(argv[++i], &s->look_dec_deg)) { + } else if (!strcmp(argv[i], "--exposure") && i + 1 < argc && + !parse_positive(argv[++i], &s->exposure)) { + } else if (!strcmp(argv[i], "--observer-inward-speed") && i + 1 < argc && + !parse_speed(argv[++i], &s->observer_inward_speed)) { + } else if (!strcmp(argv[i], "--psf-fwhm-pixels") && i + 1 < argc && + !parse_positive(argv[++i], &s->psf.fwhm_pixels)) { + } else if (!strcmp(argv[i], "--psf-moffat-beta") && i + 1 < argc && + !parse_moffat_beta(argv[++i], &s->psf.moffat_beta)) { + } else if (!strcmp(argv[i], "--write-catalog") && i + 1 < argc) + *write_path = argv[++i]; + else + return -1; + } + return 0; +} + +static int render_frame(const Settings *s, const StarCatalog *catalog, + const SpacetimeSource *spacetime) { +#ifdef SPACETIME_SCHWARZSCHILD + ObserverState observer; + if (observer_inward_schwarzschild_ks(1.0, 30.0, + s->observer_inward_speed, &observer)) + return -1; + const GeodesicTraceConfig trace = {.coordinate_time_step = 0.1, + .max_steps = 4096, + .capture_log_alpha_p0 = 8.0}; +#else + const ObserverState observer = + observer_fixed_at_origin_look_at(s->look_ra_deg, s->look_dec_deg); + const GeodesicTraceConfig trace = {.coordinate_time_step = 1.0, + .max_steps = 2048}; +#endif + FrameLensMesh mesh = {0}; + double *hdr = calloc((size_t)s->width * s->height * 3, sizeof *hdr); + if (hdr == NULL || + frame_lens_mesh_build_coarse(&mesh, s->width, s->height, + s->coarse_cell_pixels, + s->horizontal_fov_deg) || + frame_lens_mesh_trace(&mesh, spacetime, &observer, &trace)) { + frame_lens_mesh_destroy(&mesh); + free(hdr); + return -1; + } + size_t images = frame_splat_catalog(&mesh, catalog, hdr, s->width, s->height, + s->exposure, &s->psf); + if (s->draw_mesh) + frame_draw_mesh(&mesh, hdr, s->width, s->height, 0.5, 0.5); + int result = write_tonemapped_image(s->output_path, hdr, s->width, s->height); + fprintf(stderr, "Rendered %zu images from %zu catalog stars to %s (%s)\n", + images, catalog->count, s->output_path, + result == 0 ? "ok" : "write failed"); + frame_lens_mesh_destroy(&mesh); + free(hdr); + return result; +} + +int main(int argc, char **argv) { + Settings settings; + const char *write_path; + if (parse_args(argc, argv, &settings, &write_path)) { + fprintf(stderr, + "Usage: %s [--catalog PATH] [--output PATH] [--width N] [--height " + "N] [--fov-deg D] [--look-ra-deg D] [--look-dec-deg D] " + "[--exposure E] [--observer-inward-speed V] " + "[--psf-fwhm-pixels N] [--psf-moffat-beta N] " + "[--coarse-cell-pixels N] [--draw-mesh] " + "[--write-catalog PATH]\n", + argv[0]); + return 2; + } + if (write_path != NULL) + return catalog_write_octant_grid(write_path) == 0 ? 0 + : (perror(write_path), 1); + StarCatalog catalog; + if (catalog_load_csv(&catalog, settings.catalog_path)) { + if (catalog_write_octant_grid(settings.catalog_path) || + catalog_load_csv(&catalog, settings.catalog_path)) { + perror(settings.catalog_path); + return 1; + } + fprintf(stderr, "Created test catalog: %s\n", settings.catalog_path); + } + SpacetimeSource spacetime = {0}; + if (spacetime_create_default(&spacetime)) { + fputs("Could not create spacetime source\n", stderr); + catalog_destroy(&catalog); + return 1; + } + int result = render_frame(&settings, &catalog, &spacetime); + spacetime_destroy(&spacetime); + catalog_destroy(&catalog); + return result == 0 ? 0 : 1; +} diff --git a/src/observer.c b/src/observer.c new file mode 100644 index 0000000..9552b84 --- /dev/null +++ b/src/observer.c @@ -0,0 +1,67 @@ +#include "observer.h" + +#include +#include + +static const double pi = 3.14159265358979323846; + +ObserverState observer_fixed_at_origin(void) { + return (ObserverState){.coordinate_time = 0.0, + .coordinate_position = {0.0, 0.0, 0.0}, + .tetrad = {{1.0, 0.0, 0.0, 0.0}, + {0.0, 0.0, 0.0, -1.0}, + {0.0, 0.0, 1.0, 0.0}, + {0.0, 1.0, 0.0, 0.0}}}; +} + +ObserverState observer_fixed_at_origin_look_at(double ra_deg, double dec_deg) { + const double ra = ra_deg * pi / 180.0; + const double dec = dec_deg * pi / 180.0; + const double cos_ra = cos(ra), sin_ra = sin(ra); + const double cos_dec = cos(dec), sin_dec = sin(dec); + const double forward[3] = {cos_dec * cos_ra, sin_dec, cos_dec * sin_ra}; + const double up[3] = {-sin_dec * cos_ra, cos_dec, -sin_dec * sin_ra}; + const double right[3] = {-sin_ra, 0.0, cos_ra}; + return (ObserverState){.coordinate_time = 0.0, + .coordinate_position = {0.0, 0.0, 0.0}, + .tetrad = {{1.0, 0.0, 0.0, 0.0}, + {0.0, forward[0], forward[1], forward[2]}, + {0.0, up[0], up[1], up[2]}, + {0.0, right[0], right[1], right[2]}}}; +} + +int observer_static_schwarzschild_ks(double mass, double radius, + ObserverState *out) { + if (out == NULL || mass <= 0.0 || radius <= 2.0 * mass) + return -1; + const double f = 2.0 * mass / radius; + const double normalization = sqrt(1.0 - f); + *out = (ObserverState){ + .coordinate_time = 0.0, + .coordinate_position = {radius, 0.0, 0.0}, + /* e_(0) is the static four-velocity. e_(1) points inward; its time + * component makes the tetrad orthonormal in the KS metric. */ + .tetrad = {{1.0 / normalization, 0.0, 0.0, 0.0}, + {-f / normalization, -normalization, 0.0, 0.0}, + {0.0, 0.0, 1.0, 0.0}, + {0.0, 0.0, 0.0, 1.0}}}; + return 0; +} + +int observer_inward_schwarzschild_ks(double mass, double radius, + double inward_speed, ObserverState *out) { + ObserverState static_observer; + if (inward_speed < 0.0 || inward_speed >= 1.0 || + observer_static_schwarzschild_ks(mass, radius, &static_observer)) + return -1; + + const double gamma = 1.0 / sqrt(1.0 - inward_speed * inward_speed); + *out = static_observer; + for (int mu = 0; mu < 4; ++mu) { + const double e0 = static_observer.tetrad[0][mu]; + const double forward = static_observer.tetrad[1][mu]; + out->tetrad[0][mu] = gamma * (e0 + inward_speed * forward); + out->tetrad[1][mu] = gamma * (inward_speed * e0 + forward); + } + return 0; +} diff --git a/src/observer.h b/src/observer.h new file mode 100644 index 0000000..17aed91 --- /dev/null +++ b/src/observer.h @@ -0,0 +1,24 @@ +#ifndef OBSERVER_H +#define OBSERVER_H + +typedef struct { + double coordinate_time; + double coordinate_position[3]; + /* e_(0), then spatial (forward, up, right), in coordinate components. */ + double tetrad[4][4]; +} ObserverState; + +ObserverState observer_fixed_at_origin(void); +/* Point the fixed inertial observer at an ICRS-style RA/Dec direction. + * The local spatial axes remain (forward, celestial north, increasing RA). */ +ObserverState observer_fixed_at_origin_look_at(double ra_deg, double dec_deg); +/* Static camera at (radius, 0, 0) in Cartesian Kerr--Schild coordinates, + * directed toward the Schwarzschild black hole at the origin. */ +int observer_static_schwarzschild_ks(double mass, double radius, + ObserverState *out); +/* Camera at (radius, 0, 0), moving inward at local speed inward_speed with + * respect to the static Schwarzschild observer. */ +int observer_inward_schwarzschild_ks(double mass, double radius, + double inward_speed, ObserverState *out); + +#endif diff --git a/src/optics.c b/src/optics.c new file mode 100644 index 0000000..8e7befb --- /dev/null +++ b/src/optics.c @@ -0,0 +1,183 @@ +#include "optics.h" + +#include +#include +#include +#include + +#ifdef ENABLE_PNG +#include +#endif + +static double clamp(double value, double low, double high) +{ + return value < low ? low : value > high ? high : value; +} + +/* Wyman, Sloan, and Shirley (2013), Eq. 4: analytic fits to the CIE 1931 + * 2-degree color-matching functions. Wavelength is in nanometres. */ +static void cie_1931_2deg(double wavelength_nm, double xyz[3]) +{ + const double x1 = (wavelength_nm - 442.0) * + (wavelength_nm < 442.0 ? 0.0624 : 0.0374); + const double x2 = (wavelength_nm - 599.8) * + (wavelength_nm < 599.8 ? 0.0264 : 0.0323); + const double x3 = (wavelength_nm - 501.1) * + (wavelength_nm < 501.1 ? 0.0490 : 0.0382); + const double y1 = (wavelength_nm - 568.8) * + (wavelength_nm < 568.8 ? 0.0213 : 0.0247); + const double y2 = (wavelength_nm - 530.9) * + (wavelength_nm < 530.9 ? 0.0613 : 0.0322); + const double z1 = (wavelength_nm - 437.0) * + (wavelength_nm < 437.0 ? 0.0845 : 0.0278); + const double z2 = (wavelength_nm - 459.0) * + (wavelength_nm < 459.0 ? 0.0385 : 0.0725); + xyz[0] = 0.362 * exp(-0.5 * x1 * x1) + + 1.056 * exp(-0.5 * x2 * x2) - 0.065 * exp(-0.5 * x3 * x3); + xyz[1] = 0.821 * exp(-0.5 * y1 * y1) + + 0.286 * exp(-0.5 * y2 * y2); + xyz[2] = 1.217 * exp(-0.5 * z1 * z1) + + 0.681 * exp(-0.5 * z2 * z2); +} + +static double planck_radiance_lambda(double wavelength_m, double temperature_K) +{ + const double h = 6.62607015e-34; + const double c = 299792458.0; + const double k = 1.380649e-23; + const double exponent = h * c / (wavelength_m * k * temperature_K); + return 2.0 * h * c * c / + (pow(wavelength_m, 5.0) * expm1(exponent)); +} + +LinearRgb blackbody_to_linear_rgb(double temperature_K) +{ + /* Integrate Planck spectral radiance from 380 to 780 nm into CIE XYZ, + * then transform XYZ to linear sRGB. Results are W m^-2 sr^-1 in each + * linear-primary channel, before the catalog amplitude and exposure. */ + double xyz[3] = {0.0, 0.0, 0.0}; + const double wavelength_step_m = 5e-9; + if (!isfinite(temperature_K) || temperature_K <= 0.0) + return (LinearRgb){0.0, 0.0, 0.0}; + for (int wavelength_nm = 380; wavelength_nm <= 780; wavelength_nm += 5) { + double matching[3]; + const double radiance = + planck_radiance_lambda(wavelength_nm * 1e-9, temperature_K); + cie_1931_2deg(wavelength_nm, matching); + for (int channel = 0; channel < 3; ++channel) + xyz[channel] += radiance * matching[channel] * wavelength_step_m; + } + return (LinearRgb){ + clamp(3.24096994 * xyz[0] - 1.53738318 * xyz[1] - 0.49861076 * xyz[2], + 0.0, INFINITY), + clamp(-0.96924364 * xyz[0] + 1.87596750 * xyz[1] + 0.04155506 * xyz[2], + 0.0, INFINITY), + clamp(0.05563008 * xyz[0] - 0.20397696 * xyz[1] + 1.05697151 * xyz[2], + 0.0, INFINITY)}; +} + +void splat_moffat(double *hdr, int width, int height, double x, double y, + LinearRgb color, double flux, + const PointSpreadFunction *psf) +{ + /* The tail omitted outside this radius contains 1e-8 of the Moffat's + * total flux. Unlike the old fixed 3-sigma box, this is both circular and + * far below the displayed HDR precision for the chosen beta. */ + const double tail_fraction = 1e-8; + if (hdr == NULL || psf == NULL || flux <= 0.0 || + psf->fwhm_pixels <= 0.0 || psf->moffat_beta <= 1.0) + return; + const double beta = psf->moffat_beta; + const double alpha = psf->fwhm_pixels / + (2.0 * sqrt(pow(2.0, 1.0 / beta) - 1.0)); + const double support_radius = alpha * sqrt( + pow(tail_fraction, 1.0 / (1.0 - beta)) - 1.0); + const double support_radius_squared = support_radius * support_radius; + const int min_x = (int)floor(x - support_radius); + const int max_x = (int)ceil(x + support_radius); + const int min_y = (int)floor(y - support_radius); + const int max_y = (int)ceil(y + support_radius); + const double normalization = flux * (beta - 1.0) / + (3.14159265358979323846 * alpha * alpha); + for (int py = min_y; py <= max_y; ++py) for (int px = min_x; px <= max_x; ++px) { + if (px < 0 || px >= width || py < 0 || py >= height) continue; + double dx = (px + 0.5) - x, dy = (py + 0.5) - y; + const double radius_squared = dx * dx + dy * dy; + if (radius_squared > support_radius_squared) continue; + const double w = normalization * + pow(1.0 + radius_squared / (alpha * alpha), -beta); + double *pixel = &hdr[3 * (py * width + px)]; + pixel[0] += color.r * w; pixel[1] += color.g * w; pixel[2] += color.b * w; + } +} + +static unsigned char tonemap_channel(double hdr_value) +{ + /* Reinhard tone mapping followed by the sRGB display transfer curve. */ + const double linear = hdr_value / (1.0 + hdr_value); + const double display = linear <= 0.0031308 ? 12.92 * linear + : 1.055 * pow(linear, 1.0 / 2.4) - 0.055; + return (unsigned char)lround(255.0 * clamp(display, 0.0, 1.0)); +} + +static int write_tonemapped_ppm(const char *path, const double *hdr, int width, + int height) +{ + FILE *file = fopen(path, "wb"); + if (file == NULL) return -1; + fprintf(file, "P6\n%d %d\n255\n", width, height); + for (int i = 0; i < width * height * 3; ++i) { + const unsigned char value = tonemap_channel(hdr[i]); + if (fwrite(&value, 1, 1, file) != 1) { fclose(file); return -1; } + } + return fclose(file) == 0 ? 0 : -1; +} + +#ifdef ENABLE_PNG +static int write_tonemapped_png(const char *path, const double *hdr, int width, + int height) +{ + FILE *file = fopen(path, "wb"); + png_structp png = NULL; + png_infop info = NULL; + unsigned char *pixels = NULL; + int result = -1; + if (file == NULL) return -1; + png = png_create_write_struct(PNG_LIBPNG_VER_STRING, NULL, NULL, NULL); + if (png == NULL) goto done; + info = png_create_info_struct(png); + if (info == NULL || setjmp(png_jmpbuf(png))) goto done; + pixels = malloc((size_t)width * height * 3); + if (pixels == NULL) goto done; + for (int i = 0; i < width * height * 3; ++i) + pixels[i] = tonemap_channel(hdr[i]); + png_init_io(png, file); + png_set_IHDR(png, info, (png_uint_32)width, (png_uint_32)height, 8, + PNG_COLOR_TYPE_RGB, PNG_INTERLACE_NONE, + PNG_COMPRESSION_TYPE_DEFAULT, PNG_FILTER_TYPE_DEFAULT); + png_write_info(png, info); + for (int row = 0; row < height; ++row) + png_write_row(png, &pixels[(size_t)row * width * 3]); + png_write_end(png, info); + result = 0; +done: + free(pixels); + png_destroy_write_struct(&png, &info); + if (fclose(file) != 0) result = -1; + return result; +} +#endif + +int write_tonemapped_image(const char *path, const double *hdr, int width, int height) +{ + const size_t path_length = strlen(path); + if (path_length >= 4 && strcmp(path + path_length - 4, ".png") == 0) { +#ifdef ENABLE_PNG + return write_tonemapped_png(path, hdr, width, height); +#else + fputs("PNG output is disabled; rebuild with make ENABLE_PNG=1.\n", stderr); + return -1; +#endif + } + return write_tonemapped_ppm(path, hdr, width, height); +} diff --git a/src/optics.h b/src/optics.h new file mode 100644 index 0000000..f618806 --- /dev/null +++ b/src/optics.h @@ -0,0 +1,20 @@ +#ifndef OPTICS_H +#define OPTICS_H + +typedef struct { double r, g, b; } LinearRgb; +typedef struct { + double fwhm_pixels; + double moffat_beta; +} PointSpreadFunction; + +/* Integrate a Planck spectrum into absolute linear-sRGB spectral radiance + * (W m^-2 sr^-1), before catalog amplitude and display exposure. */ +LinearRgb blackbody_to_linear_rgb(double temperature_K); +void splat_moffat(double *hdr, int width, int height, double x, double y, + LinearRgb color, double flux, + const PointSpreadFunction *psf); +/* Writes PPM by default; a .png path requires a build with ENABLE_PNG=1. */ +int write_tonemapped_image(const char *path, const double *hdr, int width, + int height); + +#endif diff --git a/src/spacetime.h b/src/spacetime.h new file mode 100644 index 0000000..01a4051 --- /dev/null +++ b/src/spacetime.h @@ -0,0 +1,48 @@ +#ifndef SPACETIME_H +#define SPACETIME_H + +typedef struct { + double alpha; + double beta[3]; + double gamma[3][3]; + double K[3][3]; + double d_alpha[3]; + double d_beta[3][3]; /* d_beta[spatial derivative][component] */ + double d_gamma[3][3][3]; /* d_gamma[spatial derivative][j][k] */ +} MetricData; + +typedef enum { + SPACETIME_RAY_ACTIVE, + SPACETIME_RAY_ESCAPED, + SPACETIME_RAY_CAPTURED +} SpacetimeRayStatus; + +typedef struct SpacetimeSource SpacetimeSource; + +typedef struct { + int (*eval)(const SpacetimeSource *source, double t, const double x[3], + MetricData *metric); + SpacetimeRayStatus (*classify)(const SpacetimeSource *source, double t, + const double x[3]); + void (*destroy)(SpacetimeSource *source); +} SpacetimeOps; + +struct SpacetimeSource { + const SpacetimeOps *ops; + void *context; +}; + +/* The selected build provides spacetime_create_default(). Named constructors + * remain available to backend-specific tests and tools. */ +int spacetime_create_default(SpacetimeSource *source); +int spacetime_create_minkowski(SpacetimeSource *source, double escape_radius); +int spacetime_create_schwarzschild_ks(SpacetimeSource *source, double mass, + double escape_radius, + double capture_radius); +void spacetime_destroy(SpacetimeSource *source); +int spacetime_eval(const SpacetimeSource *source, double t, const double x[3], + MetricData *metric); +SpacetimeRayStatus spacetime_classify(const SpacetimeSource *source, double t, + const double x[3]); + +#endif diff --git a/src/spacetime_common.c b/src/spacetime_common.c new file mode 100644 index 0000000..be121b5 --- /dev/null +++ b/src/spacetime_common.c @@ -0,0 +1,22 @@ +#include "spacetime.h" + +#include + +void spacetime_destroy(SpacetimeSource *source) { + if (source != NULL && source->ops != NULL) + source->ops->destroy(source); +} + +int spacetime_eval(const SpacetimeSource *source, double t, const double x[3], + MetricData *metric) { + return source == NULL || source->ops == NULL + ? -1 + : source->ops->eval(source, t, x, metric); +} + +SpacetimeRayStatus spacetime_classify(const SpacetimeSource *source, double t, + const double x[3]) { + return source == NULL || source->ops == NULL + ? SPACETIME_RAY_CAPTURED + : source->ops->classify(source, t, x); +} diff --git a/src/spacetime_minkowski.c b/src/spacetime_minkowski.c new file mode 100644 index 0000000..2058754 --- /dev/null +++ b/src/spacetime_minkowski.c @@ -0,0 +1,56 @@ +#include "spacetime.h" + +#include + +typedef struct { + double escape_radius; +} MinkowskiContext; + +static int minkowski_eval(const SpacetimeSource *source, double t, + const double x[3], MetricData *metric) { + (void)source; + (void)t; + (void)x; + *metric = (MetricData){ + .alpha = 1.0, + .gamma = {{1.0, 0.0, 0.0}, {0.0, 1.0, 0.0}, {0.0, 0.0, 1.0}}}; + return 0; +} + +static SpacetimeRayStatus minkowski_classify(const SpacetimeSource *source, + double t, const double x[3]) { + const MinkowskiContext *context = source->context; + const double radius_squared = x[0] * x[0] + x[1] * x[1] + x[2] * x[2]; + (void)t; + return radius_squared >= context->escape_radius * context->escape_radius + ? SPACETIME_RAY_ESCAPED + : SPACETIME_RAY_ACTIVE; +} + +static void minkowski_destroy(SpacetimeSource *source) { + free(source->context); + source->context = NULL; + source->ops = NULL; +} + +static const SpacetimeOps minkowski_ops = { + .eval = minkowski_eval, + .classify = minkowski_classify, + .destroy = minkowski_destroy, +}; + +int spacetime_create_minkowski(SpacetimeSource *source, double escape_radius) { + if (source == NULL || escape_radius <= 0.0) + return -1; + MinkowskiContext *context = malloc(sizeof *context); + if (context == NULL) + return -1; + context->escape_radius = escape_radius; + source->ops = &minkowski_ops; + source->context = context; + return 0; +} + +int spacetime_create_default(SpacetimeSource *source) { + return spacetime_create_minkowski(source, 1024.0); +} diff --git a/src/spacetime_schwarzschild.c b/src/spacetime_schwarzschild.c new file mode 100644 index 0000000..ffdc0ab --- /dev/null +++ b/src/spacetime_schwarzschild.c @@ -0,0 +1,131 @@ +#include "spacetime.h" + +#include +#include + +typedef struct { + double mass; + double escape_radius; + double capture_radius; +} SchwarzschildKsContext; + +/* Schwarzschild in ingoing Cartesian Kerr--Schild coordinates: + * g_mu_nu = eta_mu_nu + (2 M / r) l_mu l_nu, l_mu = (1, x_i / r). + * These slices are regular at r = 2 M; only the physical r = 0 singularity + * is excluded by the conservative capture cutoff. */ +static int schwarzschild_ks_eval(const SpacetimeSource *source, double t, + const double x[3], MetricData *metric) { + const SchwarzschildKsContext *context = source->context; + double r2 = 0.0; + (void)t; + for (int i = 0; i < 3; ++i) + r2 += x[i] * x[i]; + if (!isfinite(r2) || r2 <= 0.0) + return -1; + const double r = sqrt(r2); + const double m = context->mass; + const double f = 2.0 * m / r; + const double alpha = 1.0 / sqrt(1.0 + f); + const double beta_scale = 2.0 * m / (r * (r + 2.0 * m)); + const double dbeta_scale_dr = + -4.0 * m * (r + m) / (r2 * (r + 2.0 * m) * (r + 2.0 * m)); + *metric = (MetricData){.alpha = alpha}; + for (int i = 0; i < 3; ++i) { + metric->beta[i] = beta_scale * x[i]; + metric->d_alpha[i] = m * alpha * alpha * alpha * x[i] / (r2 * r); + for (int j = 0; j < 3; ++j) { + metric->gamma[i][j] = (i == j ? 1.0 : 0.0) + + 2.0 * m * x[i] * x[j] / (r2 * r); + metric->d_beta[i][j] = beta_scale * (i == j ? 1.0 : 0.0) + + dbeta_scale_dr * x[i] * x[j] / r; + for (int k = 0; k < 3; ++k) + metric->d_gamma[i][j][k] = + 2.0 * m * (((i == j ? 1.0 : 0.0) * x[k] + + (i == k ? 1.0 : 0.0) * x[j]) / + (r2 * r) - + 3.0 * x[i] * x[j] * x[k] / (r2 * r2 * r)); + } + } + /* K_ij = (D_i beta_j + D_j beta_i)/(2 alpha) for these stationary slices. + * Build the connection from the analytic spatial-metric derivatives above; + * this avoids finite differences in the geodesic RHS. */ + for (int i = 0; i < 3; ++i) + for (int j = 0; j < 3; ++j) { + double d_beta_cov_i_j = + 2.0 * m * ((i == j ? 1.0 : 0.0) / r2 - + 2.0 * x[i] * x[j] / (r2 * r2)); + double d_beta_cov_j_i = d_beta_cov_i_j; + double connection_term_ij = 0.0; + double connection_term_ji = 0.0; + for (int ell = 0; ell < 3; ++ell) { + double gamma_inverse_ell_k[3]; + for (int k = 0; k < 3; ++k) + gamma_inverse_ell_k[k] = (ell == k ? 1.0 : 0.0) - + f / (1.0 + f) * x[ell] * x[k] / r2; + double gamma_ell_ij = 0.0, gamma_ell_ji = 0.0; + for (int k = 0; k < 3; ++k) { + gamma_ell_ij += 0.5 * gamma_inverse_ell_k[k] * + (metric->d_gamma[i][k][j] + + metric->d_gamma[j][k][i] - + metric->d_gamma[k][i][j]); + gamma_ell_ji += 0.5 * gamma_inverse_ell_k[k] * + (metric->d_gamma[j][k][i] + + metric->d_gamma[i][k][j] - + metric->d_gamma[k][j][i]); + } + const double beta_cov_ell = 2.0 * m * x[ell] / r2; + connection_term_ij += gamma_ell_ij * beta_cov_ell; + connection_term_ji += gamma_ell_ji * beta_cov_ell; + } + metric->K[i][j] = (d_beta_cov_i_j - connection_term_ij + + d_beta_cov_j_i - connection_term_ji) / + (2.0 * alpha); + } + return 0; +} + +static SpacetimeRayStatus schwarzschild_ks_classify( + const SpacetimeSource *source, double t, const double x[3]) { + const SchwarzschildKsContext *context = source->context; + double r2 = 0.0; + (void)t; + for (int i = 0; i < 3; ++i) + r2 += x[i] * x[i]; + if (!isfinite(r2) || r2 <= context->capture_radius * context->capture_radius) + return SPACETIME_RAY_CAPTURED; + return r2 >= context->escape_radius * context->escape_radius + ? SPACETIME_RAY_ESCAPED + : SPACETIME_RAY_ACTIVE; +} + +static void schwarzschild_ks_destroy(SpacetimeSource *source) { + free(source->context); + source->context = NULL; + source->ops = NULL; +} + +static const SpacetimeOps schwarzschild_ks_ops = { + .eval = schwarzschild_ks_eval, + .classify = schwarzschild_ks_classify, + .destroy = schwarzschild_ks_destroy, +}; + +int spacetime_create_schwarzschild_ks(SpacetimeSource *source, double mass, + double escape_radius, + double capture_radius) { + if (source == NULL || mass <= 0.0 || escape_radius <= 2.0 * mass || + capture_radius <= 0.0 || capture_radius >= 2.0 * mass || + capture_radius >= escape_radius) + return -1; + SchwarzschildKsContext *context = malloc(sizeof *context); + if (context == NULL) + return -1; + *context = (SchwarzschildKsContext){mass, escape_radius, capture_radius}; + source->ops = &schwarzschild_ks_ops; + source->context = context; + return 0; +} + +int spacetime_create_default(SpacetimeSource *source) { + return spacetime_create_schwarzschild_ks(source, 1.0, 256.0, 1.5); +} diff --git a/tests/test_frame.c b/tests/test_frame.c new file mode 100644 index 0000000..1c93223 --- /dev/null +++ b/tests/test_frame.c @@ -0,0 +1,115 @@ +#include "frame.h" +#include "optics.h" + +#include +#include +#include +#include +#include + +int main(void) { + const int width = 100, height = 100; + const PointSpreadFunction psf = {.fwhm_pixels = 2.7, .moffat_beta = 4.5}; + const GeodesicTraceConfig trace = {.coordinate_time_step = 0.25, + .max_steps = 100}; + const ObserverState observer = observer_fixed_at_origin(); + Star star = { + .direction = {0.0, 0.0, -1.0}, .temperature_K = 7000.0, .amplitude = 1.0}; + StarCatalog catalog = {.stars = &star, .count = 1}; + SpacetimeSource spacetime = {0}; + FrameLensMesh mesh = {0}; + double *hdr = calloc((size_t)width * height * 3, sizeof *hdr); + int result = 1; + if (hdr == NULL || spacetime_create_minkowski(&spacetime, 10.0) || + frame_lens_mesh_build_coarse(&mesh, width, height, 20, 30.0) || + frame_lens_mesh_trace(&mesh, &spacetime, &observer, &trace)) + goto done; + const size_t images = + frame_splat_catalog(&mesh, &catalog, hdr, width, height, 100.0, &psf); + if (images != 1 || hdr[3 * (50 * width + 50)] <= 0.0) { + fputs("flat-space inverse lens-map regression failed\n", stderr); + goto done; + } + /* Private HDR accumulation must preserve the serial splat result. */ + double *serial_hdr = calloc((size_t)width * height * 3, sizeof *serial_hdr); + double *parallel_hdr = calloc((size_t)width * height * 3, sizeof *parallel_hdr); + if (serial_hdr == NULL || parallel_hdr == NULL) { + free(serial_hdr); + free(parallel_hdr); + goto done; + } + const int original_threads = omp_get_max_threads(); + omp_set_dynamic(0); + omp_set_num_threads(1); + const size_t serial_images = frame_splat_catalog( + &mesh, &catalog, serial_hdr, width, height, 100.0, &psf); + omp_set_num_threads(4); + const size_t parallel_images = frame_splat_catalog( + &mesh, &catalog, parallel_hdr, width, height, 100.0, &psf); + omp_set_num_threads(original_threads); + for (int value = 0; value < width * height * 3; ++value) + if (fabs(serial_hdr[value] - parallel_hdr[value]) > + 1e-12 * fmax(1.0, fabs(serial_hdr[value]))) { + fputs("parallel catalog splat regression failed\n", stderr); + free(serial_hdr); + free(parallel_hdr); + goto done; + } + free(serial_hdr); + free(parallel_hdr); + if (serial_images != 1 || parallel_images != serial_images) { + fputs("parallel catalog image-count regression failed\n", stderr); + goto done; + } + const LinearRgb cool = blackbody_to_linear_rgb(3000.0); + const LinearRgb hot = blackbody_to_linear_rgb(10000.0); + if (!(cool.r > cool.b && hot.b > hot.r && + hot.r + hot.g + hot.b > cool.r + cool.g + cool.b)) { + fputs("blackbody spectral-color regression failed\n", stderr); + goto done; + } + /* The Moffat is flux-normalized and retains a measurable, continuous wing + * beyond the former Gaussian's 3-sigma raster box. */ + memset(hdr, 0, (size_t)width * height * 3 * sizeof *hdr); + splat_moffat(hdr, width, height, 50.5, 50.5, + (LinearRgb){1.0, 1.0, 1.0}, 1.0, &psf); + double moffat_flux = 0.0; + for (int pixel = 0; pixel < width * height; ++pixel) + moffat_flux += hdr[3 * pixel]; + if (fabs(moffat_flux - 1.0) > 0.01 || + hdr[3 * (50 * width + 62)] <= 0.0) { + fputs("Moffat normalization or wing regression failed\n", stderr); + goto done; + } + frame_draw_mesh(&mesh, hdr, width, height, 0.5, 0.5); + if (hdr[3 * (10 * width + 20)] != 0.25) { + fputs("mesh diagnostic overlay regression failed\n", stderr); + goto done; + } + /* A fixed absolute edge tolerance used to make tiny source triangles claim + * sources far outside their field. */ + FrameLensMesh fine_mesh = {0}; + Star fine_stars[2] = {{.direction = {0.0, 0.0, -1.0}, + .temperature_K = 7000.0, + .amplitude = 1.0}, + {.direction = {0.01, 0.0, -0.9999499987499375}, + .temperature_K = 7000.0, + .amplitude = 1.0}}; + StarCatalog fine_catalog = {.stars = fine_stars, .count = 2}; + memset(hdr, 0, (size_t)width * height * 3 * sizeof *hdr); + if (frame_lens_mesh_build_coarse(&fine_mesh, width, height, 1, 0.1) || + frame_lens_mesh_trace(&fine_mesh, &spacetime, &observer, &trace) || + frame_splat_catalog(&fine_mesh, &fine_catalog, hdr, width, height, + 100.0, &psf) != 1) { + fputs("fine source-triangle containment regression failed\n", stderr); + frame_lens_mesh_destroy(&fine_mesh); + goto done; + } + frame_lens_mesh_destroy(&fine_mesh); + result = 0; +done: + frame_lens_mesh_destroy(&mesh); + spacetime_destroy(&spacetime); + free(hdr); + return result; +} diff --git a/tests/test_geodesic.c b/tests/test_geodesic.c new file mode 100644 index 0000000..ba2556f --- /dev/null +++ b/tests/test_geodesic.c @@ -0,0 +1,43 @@ +#include "geodesic.h" + +#include +#include + +static int nearly_equal(double a, double b) { return fabs(a - b) < 1e-12; } + +static int check_ray(const SpacetimeSource *source, + const ObserverState *observer, + const double local_direction[3], + const double expected[3]) { + const GeodesicTraceConfig config = {.coordinate_time_step = 0.25, + .max_steps = 100}; + RayEndpoint ray = + geodesic_trace_past(source, observer, local_direction, &config); + if (ray.status != RAY_ENDPOINT_ESCAPED || + !nearly_equal(ray.frequency_ratio, 1.0) || + !nearly_equal(ray.n_infinity[0], expected[0]) || + !nearly_equal(ray.n_infinity[1], expected[1]) || + !nearly_equal(ray.n_infinity[2], expected[2])) { + fprintf(stderr, "flat-space geodesic regression failed\n"); + return 1; + } + return 0; +} + +int main(void) { + SpacetimeSource source = {0}; + const ObserverState observer = observer_fixed_at_origin(); + if (spacetime_create_minkowski(&source, 10.0)) + return 1; + int result = check_ray(&source, &observer, (double[]){1.0, 0.0, 0.0}, + (double[]){0.0, 0.0, -1.0}) || + check_ray(&source, &observer, (double[]){0.0, 0.0, 1.0}, + (double[]){1.0, 0.0, 0.0}); + const ObserverState look_at_ra_zero = + observer_fixed_at_origin_look_at(0.0, 0.0); + result = result || check_ray(&source, &look_at_ra_zero, + (double[]){1.0, 0.0, 0.0}, + (double[]){1.0, 0.0, 0.0}); + spacetime_destroy(&source); + return result; +} diff --git a/tests/test_schwarzschild.c b/tests/test_schwarzschild.c new file mode 100644 index 0000000..911ab2d --- /dev/null +++ b/tests/test_schwarzschild.c @@ -0,0 +1,52 @@ +#include "geodesic.h" + +#include +#include + +int main(void) { + SpacetimeSource spacetime = {0}; + MetricData metric; + ObserverState observer; + ObserverState inward_observer; + const GeodesicTraceConfig trace = {.coordinate_time_step = 0.1, + .max_steps = 4096, + .capture_log_alpha_p0 = 8.0}; + int result = 1; + if (spacetime_create_schwarzschild_ks(&spacetime, 1.0, 256.0, 1.5) || + spacetime_eval(&spacetime, 0.0, (double[]){2.0, 0.0, 0.0}, &metric) || + !isfinite(metric.alpha) || !isfinite(metric.gamma[0][0]) || + !isfinite(metric.K[0][0]) || + observer_static_schwarzschild_ks(1.0, 30.0, &observer) || + observer_inward_schwarzschild_ks(1.0, 30.0, 0.5, + &inward_observer) || + fabs(inward_observer.tetrad[0][0] - + (2.0 / sqrt(3.0)) * (observer.tetrad[0][0] + + 0.5 * observer.tetrad[1][0])) > + 1e-12 || + fabs(inward_observer.tetrad[1][1] - + (2.0 / sqrt(3.0)) * (0.5 * observer.tetrad[0][1] + + observer.tetrad[1][1])) > + 1e-12 || + !observer_inward_schwarzschild_ks(1.0, 30.0, 1.0, + &inward_observer)) + goto done; + const RayEndpoint central = geodesic_trace_past( + &spacetime, &observer, (double[]){1.0, 0.0, 0.0}, &trace); + const RayEndpoint inside_shadow = geodesic_trace_past( + &spacetime, &observer, (double[]){cos(0.10), sin(0.10), 0.0}, &trace); + const RayEndpoint outside_shadow = geodesic_trace_past( + &spacetime, &observer, (double[]){cos(0.30), sin(0.30), 0.0}, &trace); + if (central.status != RAY_ENDPOINT_CAPTURED || + inside_shadow.status != RAY_ENDPOINT_CAPTURED || + outside_shadow.status != RAY_ENDPOINT_ESCAPED) { + fprintf(stderr, + "Schwarzschild KS shadow regression failed (center=%d, inside=%d, " + "outside=%d)\n", + central.status, inside_shadow.status, outside_shadow.status); + goto done; + } + result = 0; +done: + spacetime_destroy(&spacetime); + return result; +}

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