diff --git a/.devcontainer/Dockerfile.math-api b/.devcontainer/Dockerfile.math-api index ce48e7fe..0f130e8e 100644 --- a/.devcontainer/Dockerfile.math-api +++ b/.devcontainer/Dockerfile.math-api @@ -1,11 +1,11 @@ -# syntax=docker/dockerfile:1.5-labs -FROM node:18-alpine AS deps +# syntax=docker/dockerfile:1 +FROM node:20-alpine AS deps WORKDIR /usr/src/app ADD https://github.com/bryanwweber/math-api.git . RUN yarn install --frozen-lockfile -FROM node:18-alpine AS builder +FROM node:20-alpine AS builder WORKDIR /usr/src/app COPY --from=deps /usr/src/app/server.ts . COPY --from=deps /usr/src/app/index.ts . @@ -15,7 +15,7 @@ COPY --from=deps /usr/src/app/package.json . COPY --from=deps /usr/src/app/node_modules ./node_modules RUN npm run build -FROM node:18-alpine as runner +FROM node:20-alpine as runner WORKDIR /usr/src/app ENV NODE_ENV=production diff --git a/.devcontainer/Dockerfile.workspace b/.devcontainer/Dockerfile.workspace index 01fb584e..0306701e 100644 --- a/.devcontainer/Dockerfile.workspace +++ b/.devcontainer/Dockerfile.workspace @@ -1,71 +1,51 @@ -FROM python:3.10-alpine as pdm +FROM julia:1.10-bookworm as julia-builder -RUN mkdir -p /tmp/pdm -COPY pyproject.toml pdm.lock /tmp/pdm/ +COPY Project.toml Manifest.toml /root/.julia/environments/v1.10/ -WORKDIR /tmp/pdm +RUN julia -e "using Pkg; Pkg.instantiate(); Pkg.precompile();" + +FROM python:3.12-slim-bookworm as python-builder + +ENV PIP_ROOT_USER_ACTION=ignore RUN --mount=type=cache,target=/root/.cache \ - pip3 --disable-pip-version-check install pdm \ - && pdm export -f requirements > /tmp/pdm/requirements.txt - -FROM alpine:3 as julia -ARG JULIA_VERSION=1.9.0-rc1 -ARG JULIA_ARCH="x86_64" - -RUN mkdir -p /tmp/julia -WORKDIR /tmp - -RUN set -x && \ - if [ "$JULIA_ARCH" == "aarm64" ]; then \ - JULIA_ARCH="aarch64"; \ - S3_ARCH="aarch64"; \ - elif [ "$JULIA_ARCH" == "aarch64" ]; then \ - S3_ARCH="aarch64"; \ - elif [ "$JULIA_ARCH" == "x86_64" ]; then \ - S3_ARCH="x64"; \ - fi; \ - JULIA_INSTALLER="julia-${JULIA_VERSION}-linux-${JULIA_ARCH}.tar.gz" \ - && wget --quiet "https://julialang-s3.julialang.org/bin/checksums/julia-${JULIA_VERSION}.sha256" \ - && wget --quiet "https://julialang-s3.julialang.org/bin/linux/${S3_ARCH}/${JULIA_VERSION:0:3}/${JULIA_INSTALLER}" \ - && grep ${JULIA_INSTALLER} "julia-${JULIA_VERSION}.sha256" | sha256sum -cs \ - && tar -xf "${JULIA_INSTALLER}" -C /tmp/julia --strip-components=1 \ - && rm "${JULIA_INSTALLER}" - -FROM python:3.10-bullseye as workspace + python -m pip install pdm + +COPY pyproject.toml pdm.lock /project/ +WORKDIR /project +RUN --mount=type=cache,target=/root/.cache \ + mkdir __pypackages__ \ + && pdm sync + +FROM python:3.12-bookworm as workspace ARG USERNAME=vscode ARG USER_UID=1000 ARG USER_GID=$USER_UID -ARG JULIA_DIR=/opt/julia -ENV PATH="${JULIA_DIR}/bin:${PATH}" \ - HOME="/home/${USERNAME}" +ENV HOME="/home/${USERNAME}" \ + PYTHONPATH=/project/pkgs \ + JULIA_PATH=/usr/local/julia \ + JULIA_HOME=/home/${USERNAME}/.julia + +# Has to be a separate instruction to get the JULIA_PATH variable +ENV PATH="${JULIA_PATH}/bin:${PATH}" + COPY .devcontainer/fix-permissions /usr/local/bin/fix-permissions RUN chmod a+rx /usr/local/bin/fix-permissions -# Create USERNAME with name vscode user with UID=1000 and in the 'users' group -# and make sure these dirs are writable by the `users` group. +# Create USERNAME with name vscode with UID=1000 and in the 'users' group RUN groupadd --gid ${USER_GID} ${USERNAME} \ - && useradd --uid $USER_UID --gid ${USERNAME} --shell /bin/bash --create-home ${USERNAME} \ - && mkdir -p "${JULIA_DIR}" \ - && chown "${USER_UID}:${USER_GID}" "${JULIA_DIR}" \ - && chmod g+w /etc/passwd \ - && fix-permissions "${HOME}" \ - && fix-permissions "${JULIA_DIR}" - -COPY --from=pdm /tmp/pdm/requirements.txt /tmp/pip-tmp/ -RUN --mount=type=cache,target=/root/.cache \ - pip3 --disable-pip-version-check install -r /tmp/pip-tmp/requirements.txt \ - && rm -rf /tmp/pip-tmp + && useradd --uid ${USER_UID} --gid ${USER_GID} --shell /bin/bash --create-home ${USERNAME} -USER ${USERNAME} +COPY --from=python-builder /project/__pypackages__/3.12/lib /project/pkgs +COPY --from=python-builder /project/__pypackages__/3.12/bin/* /bin/ +COPY --from=julia-builder /usr/local/julia ${JULIA_PATH} +COPY --from=julia-builder /root/.julia ${HOME}/.julia -COPY --from=julia /tmp/julia/ "${JULIA_DIR}/" -COPY .devcontainer/packages.jl /tmp/packages.jl +RUN chown -R "${USER_UID}:${USER_GID}" "${JULIA_HOME}" \ + && fix-permissions "${JULIA_HOME}" -RUN fix-permissions "${JULIA_DIR}" -RUN julia "/tmp/packages.jl" \ - && fix-permissions "${HOME}/.julia" +USER ${USERNAME} WORKDIR /workspaces/orbital-mechanics-notes diff --git a/.devcontainer/devcontainer.json b/.devcontainer/devcontainer.json index 4e5a942f..affc648c 100644 --- a/.devcontainer/devcontainer.json +++ b/.devcontainer/devcontainer.json @@ -2,28 +2,28 @@ // README at: https://github.com/devcontainers/templates/tree/main/src/docker-existing-docker-compose { "name": "Orbital Mechanics Notes", - // Update the 'dockerComposeFile' list if you have more compose files or use different names. // The .devcontainer/docker-compose.yml file contains any overrides you need/want to make. - "dockerComposeFile": ["docker-compose.yml"], - + "dockerComposeFile": [ + "docker-compose.yml" + ], // The 'service' property is the name of the service for the container that VS Code should // use. Update this value and .devcontainer/docker-compose.yml to the real service name. "service": "workspace", - // The optional 'workspaceFolder' property is the path VS Code should open by default when // connected. This is typically a file mount in .devcontainer/docker-compose.yml "workspaceFolder": "/workspaces/${localWorkspaceFolderBasename}", - // Use 'forwardPorts' to make a list of ports inside the container available locally. - "forwardPorts": [8080], - + "forwardPorts": [ + 8080 + ], // Uncomment the next line if you want start specific services in your Docker Compose config. - "runServices": ["http-server", "math-api"], - + "runServices": [ + "http-server", + "math-api" + ], // Uncomment the next line if you want to keep your containers running after VS Code shuts down. "shutdownAction": "stopCompose", - // Configure tool-specific properties. "customizations": { "vscode": { @@ -31,19 +31,15 @@ "DavidAnson.vscode-markdownlint", "ms-python.python", "ms-toolsai.jupyter", - "redhat.vscode-yaml", "streetsidesoftware.code-spell-checker", "tamasfe.even-better-toml", "VisualStudioExptTeam.vscodeintellicode" ], - "settings": { - } + "settings": {} } }, - // Uncomment to connect as an existing user other than the container default. More info: https://aka.ms/dev-containers-non-root. "containerUser": "vscode", "remoteUser": "vscode", "updateRemoteUserUID": true - } diff --git a/.devcontainer/docker-compose.yml b/.devcontainer/docker-compose.yml index 5d0ed79d..62648133 100644 --- a/.devcontainer/docker-compose.yml +++ b/.devcontainer/docker-compose.yml @@ -41,7 +41,7 @@ services: - ..:/workspaces/orbital-mechanics-notes:cached command: /bin/sh -c "while sleep 1000; do :; done" http-server: - image: python:3.10-alpine + image: python:3.12-alpine ports: ["8080:8080"] command: python3 -m http.server 8080 -d /html volumes: diff --git a/.devcontainer/packages.jl b/.devcontainer/packages.jl deleted file mode 100644 index 5f5a946a..00000000 --- a/.devcontainer/packages.jl +++ /dev/null @@ -1,16 +0,0 @@ -using Pkg - -dependencies = [ - "Luxor", - "IJulia", - "Thebes", - "LightXML", - "NLsolve", - "DifferentialEquations", - "Plots", - "Colors", - "Rotations", -] - -Pkg.add(dependencies) -Pkg.precompile() diff --git a/.github/workflows/deploy-book.yml b/.github/workflows/deploy-book.yml index 7fd2df17..c672fe51 100644 --- a/.github/workflows/deploy-book.yml +++ b/.github/workflows/deploy-book.yml @@ -20,7 +20,7 @@ jobs: uses: pdm-project/setup-pdm@v4 with: cache: true - python-version: "3.11" + python-version: "3.12" - name: Install dependencies run: pdm sync --prod # Build the book diff --git a/.markdownlint-cli2.jsonc b/.markdownlint-cli2.jsonc index 9b305b25..94a61fe7 100644 --- a/.markdownlint-cli2.jsonc +++ b/.markdownlint-cli2.jsonc @@ -1,7 +1,18 @@ { - "globs": ["**/*.md", "!.venv"], + "globs": [ + "**/*.md", + "!.venv", + "!_build", + "!_static" + ], "config": { + "MD003": false, "MD013": false, - "MD037": false + "MD024": false, + "MD033": false, + "MD037": false, + "MD041": false, + "MD055": false, + "MD056": false } } diff --git a/.mise.toml b/.mise.toml index 350f455e..5b6fdf1f 100644 --- a/.mise.toml +++ b/.mise.toml @@ -1,8 +1,28 @@ [tools] -python = { version = "3.11", virtualenv = ".venv" } +python = { version = "3.12", virtualenv = ".venv" } markdownlint-cli2 = "latest" pdm = "latest" +julia = "latest" [plugins] pdm = "https://github.com/1oglop1/asdf-pdm" markdownlint-cli2 = "https://github.com/paulo-ferraz-oliveira/asdf-markdownlint-cli2" +julia = "https://github.com/rkyleg/asdf-julia.git" + +[tasks] +lint = { depends = [ + "ruff_check_lint", + "ruff_format_lint", + "markdownlint_lint", +], description = "Run all the linter tools", env = { MISE_JOBS = "1" } } +ruff_check_lint = { description = "Run the ruff linter", run = "ruff check ." } +ruff_format_lint = { run = "ruff format --check .", description = "Run the ruff formatter in check mode" } +markdownlint_lint = { run = "markdownlint-cli2", description = "Run markdownlint" } +ruff_check_format = { run = "ruff check --fix .", description = "Run the ruff linter in fix mode" } +ruff_format_format = { run = "ruff format .", description = "Run the ruff formatter" } +markdownlint_format = { run = "markdownlint-cli2 --fix", description = "Run markdownlint in fix mode" } +format = { depends = [ + "ruff_check_format", + "ruff_format_format", + "markdownlint_format", +], description = "Run all the formatting tools", env = { MISE_JOBS = "1" } } diff --git a/.mise/tasks/build b/.mise/tasks/build new file mode 100755 index 00000000..653182c9 --- /dev/null +++ b/.mise/tasks/build @@ -0,0 +1,6 @@ +#!/usr/bin/env bash +# mise description="Build the book" +# mise env={PYDEVD_DISABLE_FILE_VALIDATION = "1"} +# mise source=["**/*.md", "**/*.ipynb", "**/*.svg"] +# mise outputs=["_build/html/**/*.html"] +jupyter-book build -W . diff --git a/.vscode/settings.json b/.vscode/settings.json index e638ce01..7aa38fd6 100644 --- a/.vscode/settings.json +++ b/.vscode/settings.json @@ -25,6 +25,7 @@ // "---([\\n\\(\\)]|[^(\\$\\$)])*---" // ] } - ] + ], + "editor.formatOnSave": true } diff --git a/INSTALL.md b/INSTALL.md index dcdf9924..7fa4b9bc 100644 --- a/INSTALL.md +++ b/INSTALL.md @@ -1,5 +1,7 @@ # Install/Build the Book -The repo is set up to use [mise](https://mise.jdx.dev) to install required tools. Once `mise` is installed, changing into this directory should install the correct tools with the right versions. +The repo is set up to use [mise](https://mise.jdx.dev) to install required tools. Once `mise` is installed, changing into this directory should install the correct tools with the right versions. If not, run `mise install` to install everything. After that, running `pdm install` should install the dependencies. Then `doit build_jb` will build the book. + +If you want to work on the book, you may also need to install Julia dependencies. `mise` should install Julia. Running `julia --project=. -e 'using Pkg; Pkg.instantiate(); Pkg.precompile();'` should install dependencies in a local project. diff --git a/Manifest.toml b/Manifest.toml new file mode 100644 index 00000000..3e55ee5e --- /dev/null +++ b/Manifest.toml @@ -0,0 +1,2237 @@ +# This file is machine-generated - editing it directly is not advised + +julia_version = "1.10.2" +manifest_format = "2.0" +project_hash = "c58a762052c2959d818e0fa9f26be5bfad7d33fb" + +[[deps.ADTypes]] +git-tree-sha1 = "016833eb52ba2d6bea9fcb50ca295980e728ee24" +uuid = "47edcb42-4c32-4615-8424-f2b9edc5f35b" +version = "0.2.7" + +[[deps.Adapt]] +deps = ["LinearAlgebra", "Requires"] +git-tree-sha1 = "cea4ac3f5b4bc4b3000aa55afb6e5626518948fa" +uuid = "79e6a3ab-5dfb-504d-930d-738a2a938a0e" +version = "4.0.3" +weakdeps = ["StaticArrays"] + + [deps.Adapt.extensions] + AdaptStaticArraysExt = "StaticArrays" + +[[deps.ArgTools]] +uuid = "0dad84c5-d112-42e6-8d28-ef12dabb789f" +version = "1.1.1" + +[[deps.ArnoldiMethod]] +deps = ["LinearAlgebra", "Random", "StaticArrays"] +git-tree-sha1 = "62e51b39331de8911e4a7ff6f5aaf38a5f4cc0ae" +uuid = "ec485272-7323-5ecc-a04f-4719b315124d" +version = "0.2.0" + +[[deps.ArrayInterface]] +deps = ["Adapt", "LinearAlgebra", "SparseArrays", "SuiteSparse"] +git-tree-sha1 = "44691067188f6bd1b2289552a23e4b7572f4528d" +uuid = "4fba245c-0d91-5ea0-9b3e-6abc04ee57a9" +version = "7.9.0" + + [deps.ArrayInterface.extensions] + ArrayInterfaceBandedMatricesExt = "BandedMatrices" + ArrayInterfaceBlockBandedMatricesExt = "BlockBandedMatrices" + ArrayInterfaceCUDAExt = "CUDA" + ArrayInterfaceChainRulesExt = "ChainRules" + ArrayInterfaceGPUArraysCoreExt = "GPUArraysCore" + ArrayInterfaceReverseDiffExt = "ReverseDiff" + ArrayInterfaceStaticArraysCoreExt = "StaticArraysCore" + ArrayInterfaceTrackerExt = "Tracker" + + [deps.ArrayInterface.weakdeps] + BandedMatrices = "aae01518-5342-5314-be14-df237901396f" + BlockBandedMatrices = "ffab5731-97b5-5995-9138-79e8c1846df0" + CUDA = "052768ef-5323-5732-b1bb-66c8b64840ba" + ChainRules = "082447d4-558c-5d27-93f4-14fc19e9eca2" + GPUArraysCore = "46192b85-c4d5-4398-a991-12ede77f4527" + ReverseDiff = "37e2e3b7-166d-5795-8a7a-e32c996b4267" + StaticArraysCore = "1e83bf80-4336-4d27-bf5d-d5a4f845583c" + Tracker = "9f7883ad-71c0-57eb-9f7f-b5c9e6d3789c" + +[[deps.ArrayLayouts]] +deps = ["FillArrays", "LinearAlgebra"] +git-tree-sha1 = "e46675dbc095ddfdf2b5fba247d5a25f34e1f8a2" +uuid = "4c555306-a7a7-4459-81d9-ec55ddd5c99a" +version = "1.6.1" +weakdeps = ["SparseArrays"] + + [deps.ArrayLayouts.extensions] + ArrayLayoutsSparseArraysExt = "SparseArrays" + +[[deps.Artifacts]] +uuid = "56f22d72-fd6d-98f1-02f0-08ddc0907c33" + +[[deps.BandedMatrices]] +deps = ["ArrayLayouts", "FillArrays", "LinearAlgebra", "PrecompileTools"] +git-tree-sha1 = "fe7ed7b33416db994f706f66e31bbd1b378f71b7" +uuid = "aae01518-5342-5314-be14-df237901396f" +version = "1.6.0" +weakdeps = ["SparseArrays"] + + [deps.BandedMatrices.extensions] + BandedMatricesSparseArraysExt = "SparseArrays" + +[[deps.Base64]] +uuid = "2a0f44e3-6c83-55bd-87e4-b1978d98bd5f" + +[[deps.BenchmarkTools]] +deps = ["JSON", "Logging", "Printf", "Profile", "Statistics", "UUIDs"] +git-tree-sha1 = "f1dff6729bc61f4d49e140da1af55dcd1ac97b2f" +uuid = "6e4b80f9-dd63-53aa-95a3-0cdb28fa8baf" +version = "1.5.0" + +[[deps.BitFlags]] +git-tree-sha1 = "2dc09997850d68179b69dafb58ae806167a32b1b" +uuid = "d1d4a3ce-64b1-5f1a-9ba4-7e7e69966f35" +version = "0.1.8" + +[[deps.BitTwiddlingConvenienceFunctions]] +deps = ["Static"] +git-tree-sha1 = "0c5f81f47bbbcf4aea7b2959135713459170798b" +uuid = "62783981-4cbd-42fc-bca8-16325de8dc4b" +version = "0.1.5" + +[[deps.BoundaryValueDiffEq]] +deps = ["ADTypes", "Adapt", "ArrayInterface", "BandedMatrices", "ConcreteStructs", "DiffEqBase", "FastAlmostBandedMatrices", "ForwardDiff", "LinearAlgebra", "LinearSolve", "NonlinearSolve", "PreallocationTools", "PrecompileTools", "Preferences", "RecursiveArrayTools", "Reexport", "SciMLBase", "Setfield", "SparseArrays", "SparseDiffTools", "Tricks", "TruncatedStacktraces", "UnPack"] +git-tree-sha1 = "3ff968887be48760b0e9e8650c2d05c96cdea9d8" +uuid = "764a87c0-6b3e-53db-9096-fe964310641d" +version = "5.6.3" + + [deps.BoundaryValueDiffEq.extensions] + BoundaryValueDiffEqODEInterfaceExt = "ODEInterface" + BoundaryValueDiffEqOrdinaryDiffEqExt = "OrdinaryDiffEq" + + [deps.BoundaryValueDiffEq.weakdeps] + ODEInterface = "54ca160b-1b9f-5127-a996-1867f4bc2a2c" + OrdinaryDiffEq = "1dea7af3-3e70-54e6-95c3-0bf5283fa5ed" + +[[deps.Bzip2_jll]] +deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"] +git-tree-sha1 = "9e2a6b69137e6969bab0152632dcb3bc108c8bdd" +uuid = "6e34b625-4abd-537c-b88f-471c36dfa7a0" +version = "1.0.8+1" + +[[deps.CEnum]] +git-tree-sha1 = "389ad5c84de1ae7cf0e28e381131c98ea87d54fc" +uuid = "fa961155-64e5-5f13-b03f-caf6b980ea82" +version = "0.5.0" + +[[deps.CPUSummary]] +deps = ["CpuId", "IfElse", "PrecompileTools", "Static"] +git-tree-sha1 = "601f7e7b3d36f18790e2caf83a882d88e9b71ff1" +uuid = "2a0fbf3d-bb9c-48f3-b0a9-814d99fd7ab9" +version = "0.2.4" + +[[deps.Cairo]] +deps = ["Cairo_jll", "Colors", "Glib_jll", "Graphics", "Libdl", "Pango_jll"] +git-tree-sha1 = "d0b3f8b4ad16cb0a2988c6788646a5e6a17b6b1b" +uuid = "159f3aea-2a34-519c-b102-8c37f9878175" +version = "1.0.5" + +[[deps.Cairo_jll]] +deps = ["Artifacts", "Bzip2_jll", "CompilerSupportLibraries_jll", "Fontconfig_jll", "FreeType2_jll", "Glib_jll", "JLLWrappers", "LZO_jll", "Libdl", "Pixman_jll", "Xorg_libXext_jll", "Xorg_libXrender_jll", "Zlib_jll", "libpng_jll"] +git-tree-sha1 = "a4c43f59baa34011e303e76f5c8c91bf58415aaf" +uuid = "83423d85-b0ee-5818-9007-b63ccbeb887a" +version = "1.18.0+1" + +[[deps.Calculus]] +deps = ["LinearAlgebra"] +git-tree-sha1 = "f641eb0a4f00c343bbc32346e1217b86f3ce9dad" +uuid = "49dc2e85-a5d0-5ad3-a950-438e2897f1b9" +version = "0.5.1" + +[[deps.ChainRulesCore]] +deps = ["Compat", "LinearAlgebra"] +git-tree-sha1 = 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"0c46a032-eb83-5123-abaf-570d42b7fbaa" +IJulia = "7073ff75-c697-5162-941a-fcdaad2a7d2a" +LightXML = "9c8b4983-aa76-5018-a973-4c85ecc9e179" +Luxor = "ae8d54c2-7ccd-5906-9d76-62fc9837b5bc" +NLsolve = "2774e3e8-f4cf-5e23-947b-6d7e65073b56" +Plots = "91a5bcdd-55d7-5caf-9e0b-520d859cae80" +Rotations = "6038ab10-8711-5258-84ad-4b1120ba62dc" +Thebes = "8b424ff8-82f5-59a4-86a6-de3761897198" diff --git a/_config.yml b/_config.yml index 2b304ae2..a858f0d6 100644 --- a/_config.yml +++ b/_config.yml @@ -29,6 +29,7 @@ sphinx: html_show_copyright: false html_extra_path: ["assets"] html_theme_options: + navigation_with_keys: false logo: image_light: "_static/hohmann-light.svg" image_dark: "_static/hohmann-dark.svg" @@ -59,7 +60,6 @@ parse: - colon_fence - dollarmath - html: baseurl: "https://orbital-mechanics.space" favicon: "_static/favicon-16x16.png" diff --git a/chapter-4/7-effects-of-the-earths-oblateness.md b/chapter-4/7-effects-of-the-earths-oblateness.md index 7aec4eea..e70d3246 100644 --- a/chapter-4/7-effects-of-the-earths-oblateness.md +++ b/chapter-4/7-effects-of-the-earths-oblateness.md @@ -30,6 +30,7 @@ Therefore, it is sufficient to define any two of $a$, $b$, or $f$ to define the The parameters specified in the WGS84 standard are: + | Parameter | Value | |----------------|----------------------------------------------------------------------------------| | $a$ | 6,378,137.0 m | @@ -38,6 +39,7 @@ The parameters specified in the WGS84 standard are: | Zero Longitude | [IERS Reference Meridian](https://en.wikipedia.org/wiki/IERS_Reference_Meridian) | | $\mu$ | 3986004.418×108 m³/s². | | $\omega_E$ | 72.92115×10−6 rad/s | + For even more accurate orbit calculations, particularly over longer time scales, it is necessary to map the variations in the earth's gravitational field due to things such as the tides, mountain ranges, oceans, and more. These effects are represented in a model called the [**Earth Gravitational Model**](https://en.wikipedia.org/wiki/Earth_Gravitational_Model) (EGM). The most recent EGM is from 2008 and relies on high accuracy satellite measurements of Earth's gravitational field. @@ -45,7 +47,9 @@ For even more accurate orbit calculations, particularly over longer time scales, ## Effect of Oblateness on Orbits + The effect of planetary oblateness on orbits is captured in the dimensionless parameter called $J_2$, the **second zonal harmonic**. The zonal harmonics are values that depend on the particular planet. For Earth, $J_2 =$ 1.08263×10-3 {cite}`Vallado2013` (Page 1039). + The effect of oblateness is to cause the right ascension of the ascending node and the argument of the periapsis to change as a function of time. The average rate of change of these variables is: diff --git a/classical-orbital-elements/orbital-elements-and-the-state-vector.md b/classical-orbital-elements/orbital-elements-and-the-state-vector.md index 7ab1cc69..82ff4942 100644 --- a/classical-orbital-elements/orbital-elements-and-the-state-vector.md +++ b/classical-orbital-elements/orbital-elements-and-the-state-vector.md @@ -83,7 +83,9 @@ Next, we need to calculate the orbital angular momentum. By definition, $\vector ::: :::: + We find $\vector{h} =$ {glue:text}`orbital-elements-h_vec-I:.0f` $\uvec{I}$ + {glue:text}`orbital-elements-h_vec-J:.0f` $\uvec{J}$ - {glue:text}`orbital-elements-h_vec-K:.0f` $\uvec{K}$ km2/s and $h =$ {glue:text}`orbital-elements-h:.3f` km2/s. Notice that the $Z$ component of the angular momentum is negative. This means the momentum vector is pointing down and the orbit is retrograde. The magnitude of the orbital angular momentum is the first orbital element. + ### Step 3—Inclination @@ -328,8 +330,10 @@ The sequence of rotations to convert from the perifocal frame to the inertial fr ::::{grid} 1 --- + gutter: 2 --- + :::{grid-item-card} Step 2.1—Rotate Until $\uvec{p}$ is Aligned With the Node Line By definition, the $\uvec{p}$ vector points towards periapsis. Therefore, it is also aligned with the eccentricity vector. If we rotate the frame around the $\uvec{w}$ axis until $\uvec{p}$ is aligned with the node line, this will align $\vector{e}$ with $\vector{N}$. This rotation is the negative of the argument of periapsis, $\omega$. @@ -353,6 +357,7 @@ These three angles ($\omega$, $i$, and $\Omega$) are called [**Euler angles**](h ::::{tab-item} PYTHON --- + sync: PYTHON --- @@ -371,8 +376,10 @@ To actually perform the rotation, we need to multiply the position and velocity ::::{tab-item} MATLAB --- + sync: MATLAB --- + :::{literalinclude} scripts/orbital_elements_and_the_state_vector.m :start-after: "[section-9]" :end-before: "[section-10]" diff --git a/classical-orbital-elements/scripts/definition_of_argument_of_periapsis.py b/classical-orbital-elements/scripts/definition_of_argument_of_periapsis.py index d6eee381..bee57d68 100644 --- a/classical-orbital-elements/scripts/definition_of_argument_of_periapsis.py +++ b/classical-orbital-elements/scripts/definition_of_argument_of_periapsis.py @@ -1,18 +1,17 @@ #!/usr/bin/env python -# coding: utf-8 -# **The canonical version of this is in `classical-orbital-elements/scripts/definition-of-raan.py`.** +# **The canonical version of this is in +# `classical-orbital-elements/scripts/definition-of-raan.py`.** # # This Notebook is here for interactive experimentation. # In[1]: -import plotly.graph_objects as go import numpy as np +import plotly.graph_objects as go from scipy.spatial.transform import Rotation as R - # In[2]: @@ -70,9 +69,9 @@ def arrow(start, end, fig=None, **kwargs): a = 100 e = 0.4 -b = a * np.sqrt(1 - e ** 2) +b = a * np.sqrt(1 - e**2) r_p = a * (1 - e) -p = a * (1 - e ** 2) +p = a * (1 - e**2) inclination = 30 raan = 30 @@ -83,7 +82,7 @@ def arrow(start, end, fig=None, **kwargs): theta = np.arange(0, 2 * np.pi, step=0.01) phi = 0 # https://math.stackexchange.com/a/819533 -r = a * (1 - e ** 2) / (1 - e * np.cos(theta - phi)) +r = a * (1 - e**2) / (1 - e * np.cos(theta - phi)) x = r * np.cos(theta) y = r * np.sin(theta) z = np.zeros_like(x) diff --git a/classical-orbital-elements/scripts/definition_of_inclination.py b/classical-orbital-elements/scripts/definition_of_inclination.py index fce76b3a..1f2e5af1 100644 --- a/classical-orbital-elements/scripts/definition_of_inclination.py +++ b/classical-orbital-elements/scripts/definition_of_inclination.py @@ -1,14 +1,12 @@ #!/usr/bin/env python -# coding: utf-8 # In[1]: -import plotly.graph_objects as go import numpy as np +import plotly.graph_objects as go from scipy.spatial.transform import Rotation as R - # In[2]: @@ -66,7 +64,7 @@ def arrow(start, end, fig=None, **kwargs): a = 100 e = 0.4 -b = a * np.sqrt(1 - e ** 2) +b = a * np.sqrt(1 - e**2) r_p = a * (1 - e) inclination = 30 @@ -77,7 +75,7 @@ def arrow(start, end, fig=None, **kwargs): theta = np.arange(0, 2 * np.pi, step=0.01) phi = 0 -r = a * (1 - e ** 2) / (1 - e * np.cos(theta - phi)) +r = a * (1 - e**2) / (1 - e * np.cos(theta - phi)) x = r * np.cos(theta) y = r * np.sin(theta) z = np.zeros_like(x) diff --git a/classical-orbital-elements/scripts/definition_of_raan.py b/classical-orbital-elements/scripts/definition_of_raan.py index 35959bcc..c27608ca 100644 --- a/classical-orbital-elements/scripts/definition_of_raan.py +++ b/classical-orbital-elements/scripts/definition_of_raan.py @@ -1,18 +1,17 @@ #!/usr/bin/env python -# coding: utf-8 -# **The canonical version of this is in `classical-orbital-elements/scripts/definition-of-raan.py`.** +# **The canonical version of this is in +# `classical-orbital-elements/scripts/definition-of-raan.py`.** # # This Notebook is here for interactive experimentation. # In[1]: -import plotly.graph_objects as go import numpy as np +import plotly.graph_objects as go from scipy.spatial.transform import Rotation as R - # In[2]: @@ -70,9 +69,9 @@ def arrow(start, end, fig=None, **kwargs): a = 100 e = 0.4 -b = a * np.sqrt(1 - e ** 2) +b = a * np.sqrt(1 - e**2) r_p = a * (1 - e) -p = a * (1 - e ** 2) +p = a * (1 - e**2) inclination = 30 raan = 30 @@ -83,7 +82,7 @@ def arrow(start, end, fig=None, **kwargs): theta = np.arange(0, 2 * np.pi, step=0.01) phi = 0 # https://math.stackexchange.com/a/819533 -r = a * (1 - e ** 2) / (1 - e * np.cos(theta - phi)) +r = a * (1 - e**2) / (1 - e * np.cos(theta - phi)) x = r * np.cos(theta) y = r * np.sin(theta) z = np.zeros_like(x) diff --git a/classical-orbital-elements/scripts/orbital_elements_and_the_state_vector.py b/classical-orbital-elements/scripts/orbital_elements_and_the_state_vector.py index 2b65a477..58de8c44 100644 --- a/classical-orbital-elements/scripts/orbital_elements_and_the_state_vector.py +++ b/classical-orbital-elements/scripts/orbital_elements_and_the_state_vector.py @@ -8,7 +8,7 @@ r = np.linalg.norm(r_vec) v = np.linalg.norm(v_vec) v_r = np.dot(r_vec / r, v_vec) -v_p = np.sqrt(v ** 2 - v_r ** 2) +v_p = np.sqrt(v**2 - v_r**2) # [section-2] h_vec = np.cross(r_vec, v_vec) @@ -34,7 +34,7 @@ nu = np.arccos(np.dot(r_vec / r, e_vec / e)) # [section-8] -r_w = h ** 2 / mu / (1 + e * np.cos(nu)) * np.array((np.cos(nu), np.sin(nu), 0)) +r_w = h**2 / mu / (1 + e * np.cos(nu)) * np.array((np.cos(nu), np.sin(nu), 0)) v_w = mu / h * np.array((-np.sin(nu), e + np.cos(nu), 0)) # [section-9] diff --git a/constants-of-orbital-motion/angular-momentum-is-conserved.md b/constants-of-orbital-motion/angular-momentum-is-conserved.md index ec3a1676..cc813022 100644 --- a/constants-of-orbital-motion/angular-momentum-is-conserved.md +++ b/constants-of-orbital-motion/angular-momentum-is-conserved.md @@ -1,4 +1,5 @@ (sec:conservation-of-angular-momentum)= + # Angular Momentum Is Conserved In Orbital Motion The two masses $m_1$ and $m_2$ in the two-body problem form a _system_, so they must follow all of the conservation laws: diff --git a/constants-of-orbital-motion/energy-is-conserved-in-orbital-motion.md b/constants-of-orbital-motion/energy-is-conserved-in-orbital-motion.md index 6b0ea398..83c98701 100644 --- a/constants-of-orbital-motion/energy-is-conserved-in-orbital-motion.md +++ b/constants-of-orbital-motion/energy-is-conserved-in-orbital-motion.md @@ -1,4 +1,5 @@ (sec:conservation-of-energy)= + # Energy Is Conserved In Orbital Motion The two masses $m_1$ and $m_2$ in the two-body problem form a _system_, so they must follow all of the conservation laws: diff --git a/dodo.py b/dodo.py index 794aa11c..0d72b6e8 100644 --- a/dodo.py +++ b/dodo.py @@ -1,8 +1,9 @@ from pathlib import Path -from doit.action import CmdAction -import yaml from textwrap import dedent +import yaml +from doit.action import CmdAction + from raw_svg import render_math_svg HERE = Path(__file__).parent diff --git a/drawing-scripts/definition-of-argument-of-periapsis.ipynb b/drawing-scripts/definition-of-argument-of-periapsis.ipynb index 01c4ee10..73c61f84 100644 --- a/drawing-scripts/definition-of-argument-of-periapsis.ipynb +++ b/drawing-scripts/definition-of-argument-of-periapsis.ipynb @@ -17,8 +17,8 @@ "metadata": {}, "outputs": [], "source": [ - "import plotly.graph_objects as go\n", "import numpy as np\n", + "import plotly.graph_objects as go\n", "from scipy.spatial.transform import Rotation as R" ] }, @@ -62,7 +62,7 @@ " fig.add_trace(line)\n", " fig.add_trace(cone)\n", " else:\n", - " return line, cone\n" + " return line, cone" ] }, { @@ -99,9 +99,9 @@ "raan = 30\n", "omega = 60\n", "\n", - "rot = R.from_euler('ZYZ', [raan, inclination, omega], degrees=True)\n", + "rot = R.from_euler(\"ZYZ\", [raan, inclination, omega], degrees=True)\n", "\n", - "theta = np.arange(0, 2*np.pi, step=0.01)\n", + "theta = np.arange(0, 2 * np.pi, step=0.01)\n", "phi = 0\n", "# https://math.stackexchange.com/a/819533\n", "r = a * (1 - e**2) / (1 - e * np.cos(theta - phi))\n", @@ -129,15 +129,40 @@ " )\n", ")\n", "inclination_range = np.arange(0, np.radians(inclination), step=0.01)\n", - "arc = vector_scale / 2 * np.vstack((np.sin(inclination_range), np.zeros_like(inclination_range), np.cos(inclination_range))).T\n", + "arc = (\n", + " vector_scale\n", + " / 2\n", + " * np.vstack(\n", + " (\n", + " np.sin(inclination_range),\n", + " np.zeros_like(inclination_range),\n", + " np.cos(inclination_range),\n", + " )\n", + " ).T\n", + ")\n", "arc_rot = R.from_euler(\"Z\", [raan], degrees=True)\n", "arc = arc_rot.apply(arc)\n", - "arc_2 = -r_p * np.vstack((np.cos(inclination_range), np.zeros_like(inclination_range), -np.sin(inclination_range))).T\n", + "arc_2 = (\n", + " -r_p\n", + " * np.vstack(\n", + " (\n", + " np.cos(inclination_range),\n", + " np.zeros_like(inclination_range),\n", + " -np.sin(inclination_range),\n", + " )\n", + " ).T\n", + ")\n", "arc_2 = arc_rot.apply(arc_2)\n", "N_vec = np.cross([0, 0, 1], rot_vec[1, :])\n", "N = np.linalg.norm(N_vec)\n", "u_N = N_vec / N\n", - "node_line = np.vstack((np.linspace(-100, 100, 2), u_N[1] / u_N[0] * np.linspace(-100, 100, 2), np.zeros(2))).T\n", + "node_line = np.vstack(\n", + " (\n", + " np.linspace(-100, 100, 2),\n", + " u_N[1] / u_N[0] * np.linspace(-100, 100, 2),\n", + " np.zeros(2),\n", + " )\n", + ").T\n", "\n", "ascending_node = rotated[52]\n", "descending_node = rotated[367]\n", @@ -145,7 +170,11 @@ "# nodes = rot.apply(nodes)\n", "\n", "raan_range = np.arange(0, np.radians(90 + raan), step=0.01)\n", - "raan_arc = vector_scale / 2 * np.vstack((np.cos(raan_range), np.sin(raan_range), np.zeros_like(raan_range))).T\n", + "raan_arc = (\n", + " vector_scale\n", + " / 2\n", + " * np.vstack((np.cos(raan_range), np.sin(raan_range), np.zeros_like(raan_range))).T\n", + ")\n", "\n", "r_rot = np.linalg.norm(rotated, axis=1) + 5\n", "t_rot = np.arctan2(np.linalg.norm(rotated[:, :2], axis=1), rotated[:, 2])\n", @@ -178,7 +207,7 @@ "x": [ -8.121778264910676, -9.38036518844823, - -10.637221155606285, + -10.637221155606284, -11.891969765843314, -13.144235905644774, -14.39364605115673, @@ -194,7 +223,7 @@ -26.650219921378223, -27.844309499980604, -29.031290222357907, - -30.210835023504586, + -30.210835023504583, -31.382622198608946, -32.54633562203173, -33.70166495769521, @@ -228,7 +257,7 @@ -61.60100886395519, -62.409525088367715, -63.20368917896815, - -63.983421952962196, + -63.98342195296219, -64.74865169119127, -65.4993140492383, -66.23535196252205, @@ -414,7 +443,7 @@ -39.290340077951555, -38.761172628866674, -38.23131029326116, - -37.700775245747764, + -37.70077524574776, -37.16958921807414, -36.637773507908264, -36.10534898751826, @@ -433,11 +462,11 @@ -29.137202944859045, -28.5982282246596, -28.058912320569135, - -27.519270396510354, + -27.51927039651035, -26.97931732337251, -26.43906768596926, - -25.898535789912398, - -25.357735668403322, + -25.8985357899124, + -25.35773566840332, -24.816681088943117, -24.275385559963276, -23.733862337377747, @@ -455,18 +484,18 @@ -17.222331729324264, -16.678951350387006, -16.135505841780475, - -15.592005409176641, + -15.59200540917664, -15.048460101118126, -14.50487981446552, -13.961274299792862, - -13.417653166733261, + -13.41765316673326, -12.87402588927596, -12.330401811016111, -11.786790150359357, -11.243200005681912, - -10.699640360448479, + -10.69964036044848, -10.156120088288638, - -9.612647958034055, + -9.612647958034056, -9.069232638717097, -8.525882704533238, -7.982606639767834, @@ -482,7 +511,7 @@ -2.5557207717665733, -2.0137963571671875, -1.4720423622212286, - -0.9304665373919001, + -0.9304665373919, -0.3890766093329878, 0.15211971453716897, 0.693114741474099, @@ -490,7 +519,7 @@ 1.7744701796419675, 2.3148152378377285, 2.85492828396814, - 3.3948016305545154, + 3.394801630554515, 3.934427577595374, 4.47379840804971, 5.012906383320673, @@ -503,7 +532,7 @@ 8.778645608115545, 9.315368073643388, 9.851756320036644, - 10.387802143928491, + 10.387802143928493, 10.923497270372147, 11.45883334822546, 11.99380194551862, @@ -512,13 +541,13 @@ 13.596417224107665, 14.129829799688574, 14.662831358926866, - 15.195412886465471, - 15.727565253094243, + 15.195412886465473, + 15.727565253094244, 16.259279210930018, 16.790545388567473, 17.32135428619871, 17.851696270700916, - 18.381561570690078, + 18.38156157069008, 18.910940271540035, 19.43982231036488, 19.96819747096403, @@ -535,7 +564,7 @@ 25.74377946501179, 26.26521233536167, 26.78598918833549, - 27.306097106272006, + 27.30609710627201, 27.82552293694479, 28.34425328772479, 28.862274519683336, @@ -570,14 +599,14 @@ 43.504771741261955, 43.99369583100714, 44.48133051255338, - 44.967650735641556, + 44.96765073564155, 45.45263096858961, 45.93624518978281, 46.41846687907857, 46.89926900912519, 47.37862403659581, 47.85650389333687, - 48.332879977432604, + 48.33287997743261, 48.80772314418545, 49.28100369701332, 49.7526913782649, @@ -610,7 +639,7 @@ 61.75858658931413, 62.17080741325345, 62.58026997297789, - 62.986922574975836, + 62.98692257497584, 63.39071265091904, 63.79158674729722, 64.18949051510613, @@ -745,8 +774,8 @@ 59.451956798975026, 58.699695479586495, 57.93351631221121, - 57.153466421192476, - 56.359599611554124, + 57.15346642119248, + 56.35959961155413, 55.55197647521866, 54.73066449260617, 53.895738129321046, @@ -774,7 +803,7 @@ 32.32656932564533, 31.216073120943665, 30.09598306280602, - 28.966559728826454, + 28.966559728826457, 27.82807061761143, 26.680790023246225, 25.52499889989518, @@ -787,10 +816,10 @@ 17.221093455522876, 16.00807930918488, 14.789355819329622, - 13.565260997376953, + 13.565260997376951, 12.336137408538471, 11.102331929484569, - 9.864195498749385, + 9.864195498749384, 8.622082860229638, 7.376352300149914, 6.127365377889416, @@ -819,7 +848,7 @@ 135.12862241003344, 134.99611099527243, 134.84136638430348, - 134.66448104513987, + 134.6644810451399, 134.46555811627118, 134.24471127388938, 134.00206458714945, @@ -838,7 +867,7 @@ 128.93813895878873, 128.4087047929536, 127.86044833903767, - 127.29363201850819, + 127.2936320185082, 126.70852474879072, 126.10540165681562, 125.48454378948726, @@ -846,40 +875,40 @@ 124.19077576171884, 123.5184546565445, 122.82957629425826, - 122.12444690735525, + 122.12444690735524, 121.40337687544626, - 120.66668042850753, - 119.91467535104283, + 120.66668042850752, + 119.91467535104285, 119.14768268760892, - 118.36602645014315, - 117.57003332751593, + 118.36602645014317, + 117.57003332751592, 116.760032397716, 115.93635484306114, - 115.09933366880863, + 115.09933366880864, 114.2493034255251, - 113.38659993555517, - 112.51156002391271, - 111.62452125389929, + 113.38659993555515, + 112.51156002391272, + 111.62452125389927, 110.72582166773546, 109.8157995324737, 108.89479309144104, - 107.96314032144245, + 107.96314032144244, 107.02117869593648, 106.06924495437691, - 105.10767487789421, + 105.1076748778942, 104.13680307147489, 103.15696275277634, 102.16848554769923, 101.17170129282042, 100.16693784477432, 99.15452089665254, - 98.13477380147665, - 97.10801740278299, - 96.07456987234319, + 98.13477380147664, + 97.108017402783, + 96.0745698723432, 95.03474655503057, - 93.98885982082717, - 92.93721892395419, - 91.88012986909501, + 93.98885982082716, + 92.9372189239542, + 91.880129869095, 90.81789528466808, 89.75081430309574, 88.67918244800312, @@ -910,10 +939,10 @@ 61.19032522338667, 60.09050916495845, 58.992516596658625, - 57.896520746584436, - 56.802690431563796, + 57.89652074658443, + 56.80269043156379, 55.711190091365594, - 54.622179825427324, + 54.62217982542733, 53.535815431956195, 52.45224844926285, 51.37162619918973, @@ -938,7 +967,7 @@ 31.566079433607275, 30.569247445822644, 29.57757256847965, - 28.591123068138106, + 28.591123068138103, 27.60996430608498, 26.634158799658973, 25.66376628334793, @@ -957,9 +986,9 @@ 13.563731624957002, 12.673976944983908, 11.79023767462079, - 10.912536984114785, + 10.912536984114784, 10.040896226475752, - 9.175334988386433, + 9.175334988386432, 8.315871140343974, 7.462520886023131, 6.615298810852971, @@ -983,24 +1012,24 @@ -7.578639454857716, -8.308478831937368, -9.032170455718266, - -9.749723923794173, - -10.461149585418745, + -9.749723923794171, + -10.461149585418744, -11.166458510979034, -11.865662462182817, -12.55877386295032, -13.245805771000494, -13.926771850122032, - -14.601686343118903, - -15.270564045420203, + -14.601686343118905, + -15.270564045420205, -15.93342027934391, -16.590270869003973, -17.24113211585026, -17.886020774830477, -18.52495403116355, - -19.157949477713426, - -19.785025092952726, + -19.157949477713423, + -19.78502509295273, -20.406199219505176, - -21.021490543256242, + -21.02149054325624, -21.63091807302083, -22.234501120757564, -22.83225928231868, @@ -1019,7 +1048,7 @@ -30.08237382896159, -30.600758615811156, -31.113634947791873, - -31.621024741434038, + -31.62102474143404, -32.122949941776625, -32.61943251100657, -33.11049441744569, @@ -1063,7 +1092,7 @@ -47.96424866574614, -48.26015171480297, -48.55142593585872, - -48.838087334403916, + -48.83808733440392, -49.120151677711576, -49.39763449240222, -49.67055106210145, @@ -1090,7 +1119,7 @@ -54.37137589512737, -54.54718559896438, -54.71870927735793, - -54.885955870338876, + -54.88595587033888, -55.04893403847315, -55.207652162023514, -55.362118340142885, @@ -1100,7 +1129,7 @@ -55.93761569983894, -56.07093374647937, -56.20004249776619, - -56.324948064870796, + -56.3249480648708, -56.445656272382195, -56.562172657728205, -56.67450247061626, @@ -1125,10 +1154,10 @@ -58.01818105936971, -58.047434526532605, -58.07254438206654, - -58.093509435356516, + -58.09350943535651, -58.11032819912962, -58.12299888919474, - -58.131519424191204, + -58.1315194241912, -58.13588742534701, -58.13610021624629, -58.132154822606445, @@ -1157,7 +1186,7 @@ -56.88244395233546, -56.77703257124644, -56.66729142703484, - -56.553209763210596, + -56.55320976321059, -56.43477652466277, -56.311980357931226, -56.18480961151068, @@ -1186,7 +1215,7 @@ -52.02128356562391, -51.78486186787003, -51.54367919936883, - -51.297716415706596, + -51.29771641570659, -51.046954101955976, -50.791372574749545, -50.530951884458645, @@ -1218,7 +1247,7 @@ -41.99019104942883, -41.59068058264478, -41.18567730546668, - -40.775155130515564, + -40.77515513051557, -40.35908784193597, -39.9374491033619, -39.510212466175545, @@ -1241,7 +1270,7 @@ -31.364730565080933, -30.832139079528975, -30.293439263044053, - -29.748604685755478, + -29.74860468575548, -29.19760906270103, -28.640426270912375, -28.07703036702432, @@ -1264,10 +1293,10 @@ -17.524825298942527, -16.845482221081365, -16.15949489259133, - 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100.01460463859067, + 100.01460463859068, 98.94645362219126, 97.87184802145408, 96.79109996246312, - 95.70451836059367, - 94.61240879349423, - 93.51507338106063, - 92.41281067233581, - 91.30591553926173, + 95.70451836059368, + 94.61240879349424, + 93.51507338106065, + 92.4128106723358, + 91.30591553926172, 90.19467907719677, 89.07938851210471, 87.96032711431303, @@ -4829,7 +4858,7 @@ 66.32743878266089, 65.18584890514327, 64.04543330061168, - 62.906378006867236, + 62.90637800686723, 61.76886461871357, 60.633070316296966, 59.49916789625737, @@ -4843,13 +4872,13 @@ 50.51537433165906, 49.4055700214355, 48.29915303948659, - 47.196250679952996, + 47.19625067995299, 46.09698639508329, 45.00147985042811, 43.90984698103842, 42.822200048564426, 41.73864769915611, - 40.659295022069124, + 40.65929502206912, 39.58424360888411, 38.51359161325067, 37.44743381107045, @@ -4864,9 +4893,9 @@ 28.0682966073275, 27.051593833636367, 26.040222280919227, - 25.034240944965134, + 25.034240944965138, 24.03370623357131, - 23.038672025795986, + 23.038672025795982, 22.04918973076067, 21.065308345966073, 20.087074515087004, @@ -4874,13 +4903,13 @@ 18.14772466351731, 17.18669067329083, 16.23146840937568, - 15.282093592915535, + 15.282093592915537, 14.338599925437183, - 13.401019142232375, + 13.401019142232377, 12.469381065023656, 11.543713653898973, 10.624043058500265, - 9.710393668454003, + 9.710393668454005, 8.802788163032497, 7.90124756003609, 7.0057912638885735, @@ -4895,7 +4924,7 @@ -0.7770589862605897, -1.6109271278598745, -2.438591044211953, - -3.2600472324099514, + -3.260047232409951, -4.07529325090774, -4.884327682675407, -5.687150099139228, @@ -4903,8 +4932,8 @@ -7.274161903221896, -8.05835506228829, -8.836343682210716, - -9.608131762789247, - -10.373724092012063, + -9.608131762789249, + -10.373724092012065, -11.13312621528671, -11.886344405394054, -12.63338563315537, @@ -4920,7 +4949,7 @@ -19.76604239460465, -20.445769451116448, -21.119447895393495, - -21.787091409815474, + -21.78709140981547, -22.448714033367786, -23.10433014177393, -23.7539544281732, @@ -4929,13 +4958,13 @@ -25.6670276570587, -26.292837288467272, -26.912732685310083, - -27.526730068616054, + -27.526730068616057, -28.134845855932603, -28.73709664597936, -29.33349920375182, -29.924070446065056, -30.508827427527365, - -31.087787326933842, + -31.08778732693384, -31.660967434070567, -32.228385136919016, -32.79005790925213, @@ -4962,7 +4991,7 @@ -43.28938782012509, -43.72913278219019, -44.16353498865093, - -44.592611114174396, + -44.5926111141744, -45.01637771118136, -45.43485120472071, -45.848047887531386, @@ -4990,8 +5019,8 @@ -53.63436863448987, -53.93033039907178, -54.22135660993337, - -54.507459088838196, - -54.788649446235596, + -54.50745908883819, + -54.7886494462356, -55.06493907960334, -55.33633917186078, -55.60286068985008, @@ -5009,7 +5038,7 @@ -58.424281220143016, -58.62827854777447, -58.8275274379631, - -59.022035472642884, + -59.02203547264288, -59.21181000062085, -59.39685813685814, -59.577186761781185, @@ -5062,8 +5091,8 @@ 10.47428431602497, 11.058052029997349, 11.636115240039452, - 12.208407062140969, - 12.774864035761645, + 12.208407062140967, + 12.774864035761643, 13.335426082354932, 13.890036461720907, 14.43864172632235, @@ -5073,11 +5102,11 @@ 16.57205521587976, 17.089945009757066, 17.601575367481082, - 18.106914467148446, + 18.10691446714845, 18.605933356153987, 19.098605892872698, 19.584908687526948, - 20.064821042346118, + 20.06482104234612, 20.538324891122453, 21.005404738264073, 21.46604759744194, @@ -5103,7 +5132,7 @@ 29.326734539633794, 29.652990523289727, 29.97303472049666, - 30.286895165109506, + 30.286895165109502, 30.594600998283312, 30.89618241858934, 31.191670633086503, @@ -5138,7 +5167,7 @@ 37.276549591398904, 37.40702961000814, 37.532663175698026, - 37.653493848891884, + 37.65349384889189, 37.769565129919535, 37.8809204415446, 37.98760311220559, @@ -5154,7 +5183,7 @@ 38.80686921629049, 38.86497077465629, 38.91890707934848, - 38.968719002845596, + 38.9687190028456, 39.01444715458567, 39.056131872991266, 39.09381321795773, @@ -5172,7 +5201,7 @@ 39.24790031612475, 39.23704546323847, 39.22271560768073, - 39.204946003313516, + 39.20494600331352, 39.18377155737067, 39.15922682883948, 39.13134602708756, @@ -5200,7 +5229,7 @@ 37.73873050150281, 37.64275358985173, 37.54415661891918, - 37.442965154439236, + 37.44296515443923, 37.33920443612646, 37.23289938003622, 37.1240745809846, @@ -5239,10 +5268,10 @@ 32.26249760689152, 32.08052068366355, 31.896703806194456, - 31.711061867389766, + 31.71106186738977, 31.523609544367197, 31.33436130128091, - 31.143331392130786, + 31.143331392130783, 30.950533863555545, 30.755982557609517, 30.559691114522693, @@ -5251,16 +5280,16 @@ 29.96050939482845, 29.757389864620173, 29.552595466435253, - 29.346138686533706, + 29.346138686533703, 29.138031828173535, 28.9282870142257, 28.716916189770107, 28.50393112467255, - 28.289343416142678, + 28.28934341614268, 28.073164491272625, 27.855405609556744, 27.636077865392163, - 27.415192190560262, + 27.415192190560266, 27.192759356689255, 26.968789977697877, 26.743294512220302, @@ -5276,7 +5305,7 @@ 24.40659973445945, 24.164965548485146, 23.921919525280416, - 23.677470317253874, + 23.67747031725387, 23.431626453622005, 23.184396342579937, 22.935788273457593, @@ -5288,13 +5317,13 @@ 21.41565250157486, 21.15762128901484, 20.89827440820598, - 20.637619173559354, + 20.63761917355935, 20.375662801486644, 20.11241241242736, 19.847875032867087, - 19.582057597347674, + 19.58205759734767, 19.314966950469604, - 19.046609848887282, + 19.04660984888728, 18.776992963297655, 18.506122880423057, 18.234006104988374, @@ -5329,7 +5358,7 @@ }, "eye": { "x": 1.2812950198053348, - "y": 0.43084834204699235, + "y": 0.4308483420469923, "z": 1.2105576229066528 }, "projection": { @@ -6315,7 +6344,7 @@ " marker={\"symbol\": \"circle-open\", \"size\": [0, 20]},\n", " text=[\"\", \"periapsis\"],\n", " textfont=dict(size=20, family=\"sans-serif\", color=\"black\"),\n", - " textposition=\"middle left\", \n", + " textposition=\"middle left\",\n", ")\n", "\n", "nodeline = go.Scatter3d(\n", @@ -6361,7 +6390,7 @@ " y=y_new[52:312],\n", " z=z_new[52:312],\n", " mode=\"lines+text\",\n", - " line={\"width\": 8, \"color\":\"black\"},\n", + " line={\"width\": 8, \"color\": \"black\"},\n", " text=label,\n", " textfont=dict(size=20, family=\"sans-serif\", color=\"black\"),\n", " textposition=\"middle left\",\n", @@ -6428,7 +6457,7 @@ "# fig.add_trace(raanarc)\n", "fig.add_trace(omegaarc)\n", "\n", - "fig.show()\n" + "fig.show()" ] }, { diff --git a/drawing-scripts/definition-of-inclination.ipynb b/drawing-scripts/definition-of-inclination.ipynb index 66a3209c..a1336a85 100644 --- a/drawing-scripts/definition-of-inclination.ipynb +++ b/drawing-scripts/definition-of-inclination.ipynb @@ -17,8 +17,8 @@ "metadata": {}, "outputs": [], "source": [ - "import plotly.graph_objects as go\n", "import numpy as np\n", + "import plotly.graph_objects as go\n", "from scipy.spatial.transform import Rotation as R" ] }, @@ -62,7 +62,7 @@ " fig.add_trace(line)\n", " fig.add_trace(cone)\n", " else:\n", - " return line, cone\n" + " return line, cone" ] }, { @@ -98,9 +98,9 @@ "raan = 0\n", "omega = 0\n", "\n", - "rot = R.from_euler('zxy', [0, 0, inclination], degrees=True)\n", + "rot = R.from_euler(\"zxy\", [0, 0, inclination], degrees=True)\n", "\n", - "theta = np.arange(0, 2*np.pi, step=0.01)\n", + "theta = np.arange(0, 2 * np.pi, step=0.01)\n", "phi = 0\n", "# https://math.stackexchange.com/a/819533\n", "r = a * (1 - e**2) / (1 - e * np.cos(theta - phi))\n", @@ -128,8 +128,27 @@ " )\n", ")\n", "inclination_range = np.arange(0, np.radians(inclination), step=0.01)\n", - "arc = vector_scale / 2 * np.vstack((np.sin(inclination_range), np.zeros_like(inclination_range), np.cos(inclination_range))).T\n", - "arc_2 = -r_p * np.vstack((np.cos(inclination_range), np.zeros_like(inclination_range), -np.sin(inclination_range))).T" + "arc = (\n", + " vector_scale\n", + " / 2\n", + " * np.vstack(\n", + " (\n", + " np.sin(inclination_range),\n", + " np.zeros_like(inclination_range),\n", + " np.cos(inclination_range),\n", + " )\n", + " ).T\n", + ")\n", + "arc_2 = (\n", + " -r_p\n", + " * np.vstack(\n", + " (\n", + " np.cos(inclination_range),\n", + " np.zeros_like(inclination_range),\n", + " -np.sin(inclination_range),\n", + " )\n", + " ).T\n", + ")" ] }, { @@ -160,55 +179,55 @@ 121.08200615642058, 120.99122863517958, 120.88037065865358, - 120.74948744780421, + 120.7494874478042, 120.59864410488856, - 120.42791553465139, - 120.23738635387667, + 120.4279155346514, + 120.23738635387669, 120.02715078950114, 119.79731256551854, 119.54798477892736, - 119.27928976499719, + 119.2792897649972, 118.99135895215224, - 118.68433270679041, + 118.6843327067904, 118.35836016837732, - 118.01359907517289, + 118.01359907517288, 117.65021558096645, 117.2683840632116, - 116.86828692296797, + 116.86828692296795, 116.4501143770704, 116.01406424295818, - 115.56034171660801, - 115.08915914402347, - 114.60073578674255, - 114.09529758183007, + 115.560341716608, + 115.08915914402348, + 114.60073578674256, + 114.09529758183008, 113.5730768968282, - 113.03431228014135, + 113.03431228014136, 112.4792482073336, - 111.90813482381863, + 111.90813482381864, 111.32122768442105, 110.71878749028582, - 110.10107982361039, - 109.46837488066883, - 108.82094720359197, + 110.1010798236104, + 109.46837488066885, + 108.82094720359196, 108.15907541136102, - 107.48304193046341, + 107.4830419304634, 106.79313272565278, 106.0896370312417, - 105.37284708334833, + 105.37284708334832, 104.64305785350442, - 103.90056678401993, - 103.14567352548673, + 103.90056678401992, + 103.14567352548671, 102.37867967678967, 101.5998885279781, 100.80960480633746, - 100.00813442598277, + 100.00813442598276, 99.1957842412826, 98.37286180440265, - 97.53967512724479, + 97.5396751272448, 96.69653244803813, 95.8437420028229, 94.98161180205123, - 94.11044941251083, + 94.11044941251085, 93.2305617447616, 92.34225484625787, 91.44583370031246, @@ -259,7 +278,7 @@ 47.163303642020125, 46.175570243148634, 45.190579727545654, - 44.208468311460564, + 44.20846831146056, 43.22936831572624, 42.25340820807521, 41.28071264717805, @@ -279,12 +298,12 @@ 28.066910623492237, 27.155702515110253, 26.249276331750007, - 25.347699852613886, + 25.34769985261389, 24.451038013798815, 23.559352965953607, 22.672704131877747, 21.791148264006253, - 20.914739501727734, + 20.914739501727738, 20.043529428485908, 19.177567128617383, 18.316899243881178, @@ -293,11 +312,11 @@ 15.767093036396632, 14.928022336070818, 14.09444457289479, - 13.266392898052137, - 12.443898404012893, + 13.266392898052136, + 12.443898404012891, 11.626990177290702, 10.81569535059913, - 10.010039154384963, + 10.010039154384964, 9.21004496771864, 8.415734368523442, 7.627127183126938, @@ -327,13 +346,13 @@ -9.563087289533325, -10.206860938497158, -10.844885585672644, - -11.477172665329961, + -11.47717266532996, -12.103734277198086, -12.724583158298472, -13.339732655451431, -13.949196698446208, -14.552989773865267, - -15.151126899553367, + -15.151126899553368, -15.743623599721642, -16.330495880676914, -16.911760207166175, @@ -362,7 +381,7 @@ -28.77607302721975, -29.2285182737847, -29.6758583685586, - -30.118115165579134, + -30.118115165579137, -30.555310533980222, -30.98746634746953, -31.41460447413844, @@ -430,11 +449,11 @@ -48.93992525119252, -49.089870856758985, -49.235902626396744, - -49.378030703792604, + -49.37803070379261, -49.51626497668947, -49.65061507625514, -49.78109037648485, - -49.907699993636044, + -49.90769999363605, -50.03045278569428, -50.14935735186844, -50.2644220321143, @@ -463,10 +482,10 @@ -51.86637763401251, -51.89110686766426, -51.91211951361841, - -51.929417071942396, + -51.9294170719424, -51.94300077770722, -51.95287160094797, - -51.959030246633084, + -51.95903024663309, -51.961477154642, -51.96021249975136, -51.95523619162945, @@ -496,7 +515,7 @@ -50.71742489907726, -50.61885339697407, -50.51647503763928, - -50.410282605552396, + -50.4102826055524, -50.30026862417709, -50.18642535632955, -50.06874480456997, @@ -507,7 +526,7 @@ -49.42248471333739, -49.281597993218114, -49.13681075821631, - -48.988112946100564, + -48.98811294610056, -48.83549423914582, -48.67894406482494, -48.51845159653804, @@ -515,7 +534,7 @@ -48.185595205958776, -48.0132083672329, -47.836833403425885, - -47.656458229963484, + -47.65645822996349, -47.47207051346934, -47.283657672809795, -47.091206880191685, @@ -535,7 +554,7 @@ -43.96515122996103, -43.710428157150965, -43.45142947880948, - -43.188138404650836, + -43.18813840465083, -42.92053792933214, -42.648610835048046, -42.372339694235926, @@ -568,13 +587,13 @@ -33.20211785582683, -32.796398191363785, -32.38575498121607, - 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-12.927649705082635, - -13.438623515911933, + -12.41962643826582, + -12.927649705082636, + -13.438623515911932, -13.952515588855528, - -14.469292110832479, + -14.46929211083248, -14.98891770441749, -15.51135539460404, -16.036566575522908, @@ -3829,7 +3848,7 @@ -20.33290448489509, -20.881054300600553, -21.431494569194932, - -21.984166572899774, + -21.984166572899777, -22.539009574122353, -23.095960785276525, -23.654955339190803, @@ -3842,10 +3861,10 @@ -27.619232233805164, -28.191979566134535, -28.766088342948517, - -29.341471447722622, + -29.341471447722625, -29.918039415917168, -30.495700417596375, - -31.074360241548234, + -31.07436024154823, -31.653922280987235, -32.23428752092159, -32.81535452726938, @@ -3854,7 +3873,7 @@ -34.56171534107273, -35.1445264145715, -35.727495549910905, - -36.310506678578044, + -36.31050667857805, -36.89344129293319, -37.47617845240837, -38.05859479223162, @@ -3884,7 +3903,7 @@ -51.579270475896614, -52.1070080589834, -52.63046615033712, - -53.149470519493924, + -53.14947051949392, -53.663846065903215, -54.17341692688251, -54.67800658911705, @@ -3897,16 +3916,16 @@ -58.05575156808746, -58.51409855455153, -58.96586649223321, - -59.410879212900724, + -59.41087921290072, -59.8489613318309, -60.27993839378673, -60.703637020903145, - -61.119885062287196, + -61.1198850622872, -61.52851174513204, -61.929347827134364, -62.322225749997195, -62.706979793792485, - -63.083446231949516, + -63.08344623194952, -63.45146348662981, -63.81087228424085, -64.16151581083733, @@ -4232,19 +4251,19 @@ 6.481707130984743, 6.977155732211824, 7.471906623679961, - 7.9659103307122985, + 7.965910330712299, 8.459117453349801, 8.951478671291209, 9.44294474882503, - 9.933466539753061, - 10.422994992304977, + 9.93346653975306, + 10.422994992304975, 10.911481154043466, 11.39887617675942, 11.88513132135673, - 12.370197962726147, + 12.370197962726149, 12.854027594607755, - 13.336571834441557, - 13.817782428205689, + 13.336571834441555, + 13.817782428205687, 14.297611255241776, 14.776010333066978, 15.252931822172174, @@ -4257,8 +4276,8 @@ 18.546023470649136, 19.00942075615807, 19.470917115432528, - 19.930466399221146, - 20.388022652978506, + 19.930466399221142, + 20.388022652978503, 20.84354012146054, 21.29697325329998, 21.748276705561512, @@ -4331,7 +4350,7 @@ 49.99000033332889, 49.977501687449376, 49.9600053330489, - 49.937513019748316, + 49.93751301974832, 49.91002699676021, 49.87755001266398, 49.84008531513097, @@ -4353,7 +4372,7 @@ 48.56689874260148, 48.44562108553224, 48.31949890672566, - 48.188544818294524, + 48.18854481829453, 48.05277191553854, 47.91219377563486, 47.7668244562803, @@ -4463,12 +4482,12 @@ -59.70024991668155, -59.637365877401805, -59.56851815123198, - -59.493713622887284, + -59.49371362288728, -59.41295977275823, -59.32626467616254, -59.23363700253761, -59.135086014573645, - -59.030621567287284, + -59.03062156728728, -58.92025410703622, -58.803994670474495, -58.6818548834489, @@ -4480,8 +4499,8 @@ -57.82625378195343, -57.663326298646254, -57.494632530761834, - -57.320189347536356, - -57.140014193142804, + -57.32018934753635, + -57.1400141931428, -56.95412508494645, -56.76254061170322, -56.56527993170077, @@ -4576,7 +4595,7 @@ 7.778048557181691, 8.37258687865419, 8.966287948415953, - 9.559092396854759, + 9.55909239685476, 10.150940944019762, 10.741774405549451, 11.331533698590034, @@ -4585,7 +4604,7 @@ 13.09377738485216, 13.678651412111304, 14.262157585628074, - 14.844237555271377, + 14.844237555271375, 15.424833113529306, 16.00388620132987, 16.581338913846825, @@ -4594,7 +4613,7 @@ 18.30351818660661, 18.873993636967064, 19.442581703692102, - 20.009225528448866, + 20.009225528448862, 20.573868447327083, 21.136453996505395, 21.69692591789772, @@ -4612,7 +4631,7 @@ 28.23755329026948, 28.76553231625218, 29.29063481297445, - 29.812808270624206 + 29.81280827062421 ] }, { @@ -4665,7 +4684,7 @@ "scene": { "aspectmode": "auto", "aspectratio": { - "x": 1.1291564382163535, + "x": 1.1291564382163537, "y": 1.020736111755783, "z": 0.8676256949924155 }, @@ -5663,7 +5682,7 @@ " marker={\"symbol\": \"circle-open\", \"size\": [0, 20]},\n", " text=[\"\", \"periapsis\"],\n", " textfont=dict(size=20, family=\"sans-serif\", color=\"black\"),\n", - " textposition=\"top left\", \n", + " textposition=\"top left\",\n", ")\n", "\n", "layout = dict(\n", @@ -5723,7 +5742,7 @@ "fig.add_trace(arcline_2)\n", "fig.add_trace(periline)\n", "\n", - "fig.show()\n" + "fig.show()" ] }, { @@ -5745,6 +5764,7 @@ ], "source": [ "from plotly.io import to_html\n", + "\n", "to_html(fig, full_html=False)" ] }, diff --git a/drawing-scripts/definition-of-mean-eccentric-anomaly.ipynb b/drawing-scripts/definition-of-mean-eccentric-anomaly.ipynb index fcf8c151..e29a2a77 100644 --- a/drawing-scripts/definition-of-mean-eccentric-anomaly.ipynb +++ b/drawing-scripts/definition-of-mean-eccentric-anomaly.ipynb @@ -26,10 +26,11 @@ } ], "source": [ - "import numpy as np\n", - "import matplotlib.pyplot as plt\n", "import matplotlib.animation as animation\n", + "import matplotlib.pyplot as plt\n", + "import numpy as np\n", "from scipy.optimize import newton\n", + "\n", "plt.ioff()" ] }, @@ -76,19 +77,25 @@ "r_a = p / (1 - e)\n", "r_p = 2 * a - r_a\n", "M_e = np.linspace(0, 2 * np.pi, 300)\n", + "\n", + "\n", "def kepler(E, M_e, e):\n", " \"\"\"Kepler's equation, to be used in a Newton solver.\"\"\"\n", " return E - e * np.sin(E) - M_e\n", "\n", + "\n", "def d_kepler_d_E(E, M_e, e):\n", " \"\"\"The derivative of Kepler's equation, to be used in a Newton solver.\n", - " \n", + "\n", " Note that the argument M_e is unused, but must be present so the function\n", " arguments are consistent with the kepler function.\n", " \"\"\"\n", " return 1 - e * np.cos(E)\n", "\n", - "E = newton(func=kepler, fprime=d_kepler_d_E, x0=np.ones_like(M_e) * np.pi, args=(M_e, e))\n", + "\n", + "E = newton(\n", + " func=kepler, fprime=d_kepler_d_E, x0=np.ones_like(M_e) * np.pi, args=(M_e, e)\n", + ")\n", "nu = (2 * np.arctan(np.sqrt((1 + e) / (1 - e)) * np.tan(E / 2))) % (2 * np.pi)\n", "r = p / (1 + e * np.cos(nu))\n", "x = a * e + r * np.cos(nu)\n", @@ -119,7 +126,18 @@ " true_anomaly_arc.set_data([], [])\n", " mean_anomaly_arc.set_data([], [])\n", " eccentric_anomaly_arc.set_data([], [])\n", - " return (ellipse, circle, spacecraft, mean_anomaly, eccentric_anomaly, ecc_line, true_anomaly_arc, mean_anomaly_arc, eccentric_anomaly_arc)\n", + " return (\n", + " ellipse,\n", + " circle,\n", + " spacecraft,\n", + " mean_anomaly,\n", + " eccentric_anomaly,\n", + " ecc_line,\n", + " true_anomaly_arc,\n", + " mean_anomaly_arc,\n", + " eccentric_anomaly_arc,\n", + " )\n", + "\n", "\n", "def animate(t):\n", " E = newton(func=kepler, fprime=d_kepler_d_E, x0=np.pi, args=(t, e))\n", @@ -133,13 +151,21 @@ "\n", " nu_arc = np.linspace(0, nu, 50)\n", " true_anomaly_arc.set_data(a * e + 0.2 * np.cos(nu_arc), 0.2 * np.sin(nu_arc))\n", - " \n", + "\n", " Me_arc = np.linspace(0, M_e, 50)\n", " mean_anomaly_arc.set_data(0.1 * np.cos(Me_arc), 0.1 * np.sin(Me_arc))\n", - " \n", + "\n", " ecc_arc = np.linspace(0, E, 50)\n", " eccentric_anomaly_arc.set_data(0.4 * np.cos(ecc_arc), 0.4 * np.sin(ecc_arc))\n", - " return (ecc_line, spacecraft, mean_anomaly, eccentric_anomaly, true_anomaly_arc, mean_anomaly_arc, eccentric_anomaly_arc)" + " return (\n", + " ecc_line,\n", + " spacecraft,\n", + " mean_anomaly,\n", + " eccentric_anomaly,\n", + " true_anomaly_arc,\n", + " mean_anomaly_arc,\n", + " eccentric_anomaly_arc,\n", + " )" ] }, { @@ -272199,6 +272225,7 @@ ], "source": [ "from IPython.display import HTML\n", + "\n", "js = anim.to_jshtml()\n", "HTML(js)" ] diff --git a/drawing-scripts/definition-of-raan.ipynb b/drawing-scripts/definition-of-raan.ipynb index 5617926e..99284cc6 100644 --- a/drawing-scripts/definition-of-raan.ipynb +++ b/drawing-scripts/definition-of-raan.ipynb @@ -17,8 +17,8 @@ "metadata": {}, "outputs": [], "source": [ - "import plotly.graph_objects as go\n", "import numpy as np\n", + "import plotly.graph_objects as go\n", "from scipy.spatial.transform import Rotation as R" ] }, @@ -62,7 +62,7 @@ " fig.add_trace(line)\n", " fig.add_trace(cone)\n", " else:\n", - " return line, cone\n" + " return line, cone" ] }, { @@ -99,9 +99,9 @@ "raan = 30\n", "omega = 0\n", "\n", - "rot = R.from_euler('ZY', [raan, inclination], degrees=True)\n", + "rot = R.from_euler(\"ZY\", [raan, inclination], degrees=True)\n", "\n", - "theta = np.arange(0, 2*np.pi, step=0.01)\n", + "theta = np.arange(0, 2 * np.pi, step=0.01)\n", "phi = 0\n", "# https://math.stackexchange.com/a/819533\n", "r = a * (1 - e**2) / (1 - e * np.cos(theta - phi))\n", @@ -129,21 +129,50 @@ " )\n", ")\n", "inclination_range = np.arange(0, np.radians(inclination), step=0.01)\n", - "arc = vector_scale / 2 * np.vstack((np.sin(inclination_range), np.zeros_like(inclination_range), np.cos(inclination_range))).T\n", + "arc = (\n", + " vector_scale\n", + " / 2\n", + " * np.vstack(\n", + " (\n", + " np.sin(inclination_range),\n", + " np.zeros_like(inclination_range),\n", + " np.cos(inclination_range),\n", + " )\n", + " ).T\n", + ")\n", "arc_rot = R.from_euler(\"Z\", [raan], degrees=True)\n", "arc = arc_rot.apply(arc)\n", - "arc_2 = -r_p * np.vstack((np.cos(inclination_range), np.zeros_like(inclination_range), -np.sin(inclination_range))).T\n", + "arc_2 = (\n", + " -r_p\n", + " * np.vstack(\n", + " (\n", + " np.cos(inclination_range),\n", + " np.zeros_like(inclination_range),\n", + " -np.sin(inclination_range),\n", + " )\n", + " ).T\n", + ")\n", "arc_2 = arc_rot.apply(arc_2)\n", "N_vec = np.cross([0, 0, 1], rot_vec[1, :])\n", "N = np.linalg.norm(N_vec)\n", "u_N = N_vec / N\n", - "node_line = np.vstack((np.linspace(-100, 100, 2), u_N[1] / u_N[0] * np.linspace(-100, 100, 2), np.zeros(2))).T\n", + "node_line = np.vstack(\n", + " (\n", + " np.linspace(-100, 100, 2),\n", + " u_N[1] / u_N[0] * np.linspace(-100, 100, 2),\n", + " np.zeros(2),\n", + " )\n", + ").T\n", "\n", "nodes = np.array(((0, p, 0), (0, -p, 0)))\n", "nodes = rot.apply(nodes)\n", "\n", "raan_range = np.arange(0, np.radians(90 + raan), step=0.01)\n", - "raan_arc = vector_scale / 2 * np.vstack((np.cos(raan_range), np.sin(raan_range), np.zeros_like(raan_range))).T" + "raan_arc = (\n", + " vector_scale\n", + " / 2\n", + " * np.vstack((np.cos(raan_range), np.sin(raan_range), np.zeros_like(raan_range))).T\n", + ")" ] }, { @@ -167,24 +196,24 @@ "mode": "lines", "type": "scatter3d", "x": [ - 105.00000000000003, + 105.00000000000004, 104.29128536314914, 103.56528578696982, 102.82222417564698, - 102.06233182007969, + 102.06233182007968, 101.28584820190578, - 100.49302078783643, + 100.49302078783644, 99.68410481467166, 98.85936306538694, 98.01906563670094, 97.16348969855196, - 96.29291924592691, - 95.40764484350187, - 94.50796336356701, + 96.29291924592692, + 95.40764484350188, + 94.507963363567, 93.5941777177202, - 92.66659658282535, - 91.72553412173947, - 90.77130969932075, + 92.66659658282536, + 91.72553412173949, + 90.77130969932077, 89.8042475942357, 88.82467670708823, 87.8329302653955, @@ -219,7 +248,7 @@ 55.35854977939462, 54.16582063287363, 52.971797874709765, - 51.776803938271556, + 51.77680393827155, 50.581157106269195, 49.38517135837881, 48.189156227016376, @@ -241,7 +270,7 @@ 29.28246046534393, 28.128459520989644, 26.979123599058155, - 25.834660802951234, + 25.83466080295123, 24.69527351635537, 23.561158410238654, 22.43250645528731, @@ -251,10 +280,10 @@ 17.976151180082248, 16.877481544768614, 15.785302509490236, - 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"x": 0.9307095489136137, + "x": 0.9307095489136136, "y": 1.4796662430239833, "z": 0.726142848162594 }, @@ -6134,7 +6163,7 @@ " marker={\"symbol\": \"circle-open\", \"size\": [0, 20]},\n", " text=[\"\", \"periapsis\"],\n", " textfont=dict(size=20, family=\"sans-serif\", color=\"black\"),\n", - " textposition=\"top left\", \n", + " textposition=\"top left\",\n", ")\n", "\n", "nodeline = go.Scatter3d(\n", @@ -6232,7 +6261,7 @@ "fig.add_trace(nodesmarks)\n", "fig.add_trace(raanarc)\n", "\n", - "fig.show()\n" + "fig.show()" ] }, { diff --git a/intro/reference-frames.md b/intro/reference-frames.md index 945cf866..0ab1a8d0 100644 --- a/intro/reference-frames.md +++ b/intro/reference-frames.md @@ -12,6 +12,7 @@ The two types of reference frames are: Profound. (sec:inertial-reference-frame)= + ### Inertial Reference Frame An **inertial** reference frame is one that is not _accelerating_. It may be moving at constant velocity, but there can be absolutely no acceleration, _including rotation!_ diff --git a/orbital-maneuvers/nonimpulsive-maneuver-example.ipynb b/orbital-maneuvers/nonimpulsive-maneuver-example.ipynb index 26bcb19a..fd9c240a 100644 --- a/orbital-maneuvers/nonimpulsive-maneuver-example.ipynb +++ b/orbital-maneuvers/nonimpulsive-maneuver-example.ipynb @@ -36,14 +36,14 @@ "metadata": {}, "outputs": [], "source": [ - "import numpy as np\n", - "from scipy.integrate import solve_ivp\n", "# %matplotlib notebook\n", "import matplotlib.pyplot as plt\n", + "import numpy as np\n", "from matplotlib.patches import Circle\n", + "from scipy.integrate import solve_ivp\n", "\n", "R_E = 6378 # km\n", - "mu = 3.986E5 # km**3/s**2\n", + "mu = 3.986e5 # km**3/s**2\n", "\n", "r_LEO = R_E + 300 # km\n", "v_LEO = np.sqrt(mu / r_LEO) # km/s\n", @@ -52,12 +52,12 @@ "\n", "r_0 = np.array((r_LEO, 0, 0)) # km\n", "v_0 = np.array((0, v_LEO, 0)) # km/s\n", - "m_0 = np.array((1000)) # kg\n", + "m_0 = np.array(1000) # kg\n", "Y_0 = np.hstack((r_0, v_0, m_0))\n", "\n", - "T = 2.5E-3 # kN\n", + "T = 2.5e-3 # kN\n", "I_sp = 10_000 # s\n", - "g_0 = 9.807E-3 # km/s**2" + "g_0 = 9.807e-3 # km/s**2" ] }, { @@ -75,7 +75,7 @@ "source": [ "def nonimpulsive_maneuver(t, Y, mu, T, I_sp, g_0, r_2):\n", " \"\"\"Residual function for non-impulsive maneuvers.\n", - " \n", + "\n", " t: Current simulation time\n", " Y: State vector [x y z xdot ydot zdot m], km, km/s, kg\n", " mu: Gravitational parameter, km**3/s**2\n", @@ -89,7 +89,7 @@ " dY_dt = np.zeros(len(Y))\n", " dY_dt[0:3] = Y[3:6]\n", " dY_dt[3:6] = -mu * Y[0:3] / r**3 + T * Y[3:6] / (m * v)\n", - " dY_dt[-1] = - T / (I_sp * g_0)\n", + " dY_dt[-1] = -T / (I_sp * g_0)\n", " return dY_dt" ] }, @@ -112,7 +112,7 @@ "source": [ "def reached_destination(t, Y, mu, T, I_sp, g_0, r_2):\n", " \"\"\"Determine if the spacecraft reaches the destination radius.\n", - " \n", + "\n", " Returns the difference between the current orbital radius and the\n", " destination radius.\n", " \"\"\"\n", @@ -152,12 +152,13 @@ "source": [ "def mass(t, Y, mu, T, I_sp, g_0, r_2):\n", " \"\"\"Return the current mass of the spacecraft.\n", - " \n", + "\n", " The mass is stored in the last element of the solution vector.\n", " If this becomes zero, the integration should terminate.\n", " \"\"\"\n", " return Y[-1]\n", "\n", + "\n", "mass.terminal = True" ] }, @@ -180,10 +181,11 @@ "source": [ "def orbit(t, Y, mu, T, I_sp, g_0, r_2):\n", " \"\"\"Trigger when the y-component of the position crosses zero.\n", - " \n", + "\n", " Useful to count the number of orbits.\"\"\"\n", " return Y[1]\n", "\n", + "\n", "# Only trigger when going from negative to positive\n", "orbit.direction = 1" ] @@ -222,17 +224,17 @@ ], "source": [ "t_end = 2_000_000 # s\n", - "t_eval = np.linspace(0, t_end, int(1E6))\n", + "t_eval = np.linspace(0, t_end, int(1e6))\n", "sol = solve_ivp(\n", " nonimpulsive_maneuver,\n", " t_span=(0, t_end),\n", " y0=Y_0,\n", " t_eval=t_eval,\n", " events=(reached_destination, mass, orbit),\n", - " rtol=1E-12,\n", - " atol=1E-15,\n", + " rtol=1e-12,\n", + " atol=1e-15,\n", " method=\"DOP853\",\n", - " args=(mu, T, I_sp, g_0, r_2)\n", + " args=(mu, T, I_sp, g_0, r_2),\n", ")\n", "print(sol.status)" ] @@ -281,9 +283,9 @@ "outputs": [], "source": [ "r_vec = sol.y[0:3].T\n", - "r = np.sqrt(r_vec[:, 0]**2 + r_vec[:, 1]**2 + r_vec[:, 2]**2)\n", + "r = np.sqrt(r_vec[:, 0] ** 2 + r_vec[:, 1] ** 2 + r_vec[:, 2] ** 2)\n", "v_vec = sol.y[3:6].T\n", - "v = np.sqrt(v_vec[:, 0]**2 + v_vec[:, 1]**2 + v_vec[:, 2]**2)\n", + "v = np.sqrt(v_vec[:, 0] ** 2 + v_vec[:, 1] ** 2 + v_vec[:, 2] ** 2)\n", "m = sol.y[-1]" ] }, @@ -325,7 +327,7 @@ "ax.add_patch(Circle((0, 0), r_2, ec=\"C1\", fc=\"none\", lw=2, ls=\"--\"))\n", "ax.plot(r_vec[:, 0], r_vec[:, 1], color=\"C2\")\n", "orbit_crossings = sol.y_events[2][:, 0]\n", - "ax.plot(orbit_crossings, np.zeros(orbit_crossings.shape), 'ko', fillstyle='none')" + "ax.plot(orbit_crossings, np.zeros(orbit_crossings.shape), \"ko\", fillstyle=\"none\")" ] }, { @@ -425,7 +427,7 @@ } ], "source": [ - "h_t = np.sqrt(2 * mu * r_LEO*r_2/(r_LEO + r_2))\n", + "h_t = np.sqrt(2 * mu * r_LEO * r_2 / (r_LEO + r_2))\n", "delta_v_1 = h_t / r_LEO - v_LEO\n", "delta_v_2 = np.sqrt(mu / r_2) - h_t / r_2\n", "delta_v_t = delta_v_1 + delta_v_2\n", @@ -434,8 +436,10 @@ "delta_m = m_0 * (1 - np.exp(-delta_v_t / (300 * g_0)))\n", "print(f\"The Δm for a Hohmann transfer is: {delta_m:.4F} kg\")\n", "\n", - "transit_t = np.pi / np.sqrt(mu) * ((r_LEO + r_2) / 2)**(3/2)\n", - "print(f\"The time of flight for a Hohmann transfer is: {transit_t / (3600 * 24):.4F} days\")" + "transit_t = np.pi / np.sqrt(mu) * ((r_LEO + r_2) / 2) ** (3 / 2)\n", + "print(\n", + " f\"The time of flight for a Hohmann transfer is: {transit_t / (3600 * 24):.4F} days\"\n", + ")" ] }, { diff --git a/orbital-maneuvers/plane-change-maneuvers.md b/orbital-maneuvers/plane-change-maneuvers.md index 95d73916..d543b6b1 100644 --- a/orbital-maneuvers/plane-change-maneuvers.md +++ b/orbital-maneuvers/plane-change-maneuvers.md @@ -153,7 +153,7 @@ glue("plane-change-pure-rotation", fig, display=False) The required velocity increment as a function of dihedral angle for pure rotation maneuvers. ::: -As we can see from {numref}`fig:plane-change-pure-rotation`, plane change of about 24° requires the same $\Delta v$ as is needed to increase the speed to the escape velocity from a circular orbit, $\Delta v / v \approx$ 0.414. +As we can see from {numref}`fig:plane-change-pure-rotation`, plane change of about 24° requires the same $\Delta v$ as is needed to increase the speed to the escape velocity from a circular orbit, $\Delta v / v \approx$ 0.414. Similarly, a plane change of 60° requires a $\Delta v$ equal to the current spacecraft velocity! In LEO, the velocity is approximately 7.5 km/s. 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"poliastro>=0.18.dev0", - "scipy>=1.12.0", - "skyfield>=1.47", - "sphinx-design>=0.5.0", - "sympy>=1.12", - "wheel>=0.42.0", + "astropy>=6.0.0", + "docutils>=0.20.1", + "ipympl>=0.9.3", + "jupyter-book>=1.0.0", + "matplotlib>=3.8.3", + "myst-nb-bokeh>=2024.1.0", + "myst-nb>=1.0.0", + "numpy>=1.26.4", + "pint>=0.23", + "plotly>=5.19.0", + "poliastro>=0.18.dev0", + "scipy>=1.12.0", + "skyfield>=1.48", + "sphinx-design>=0.5.0", + "sympy>=1.12", + "wheel>=0.42.0", ] -# Numba 0.58.1 requires Python < 3.12 -requires-python = ">=3.10,<3.12" +requires-python = ">=3.10,<3.13" name = "orbital-mechanics-notes" -version = "2024.01.0" -description = "Orbital Mechanics Notes built with JupyterBook" -authors = [ - {name = "Bryan Weber", email = "bryan.w.weber@gmail.com"}, -] -license = {text = "CC-BY-SA-4.0"} +authors = [{ name = "Bryan Weber", email = "bryan.w.weber@gmail.com" }] +license = { text = "CC-BY-SA-4.0" } [tool.pdm] distribution = false [tool.pdm.dev-dependencies] -dev = [ - "doit>=0.36.0", - "pyyaml>=6.0.1", - "mypy>=1.8.0", - "ruff>=0.1.14", -] +dev = ["doit>=0.36.0", "pyyaml>=6.0.1", "ruff>=0.3.2", "requests>=2.31.0"] + [tool.pdm.resolution.overrides] # Unfortunately, poliastro is no longer maintained. I updated some dependencies on my # fork to be able to upgrade some other things like astropy. Nothing seems to be broken @@ -46,6 +37,8 @@ dev = [ [tool.ruff] extend-include = ["*.ipynb"] + +[tool.ruff.lint] select = [ "E", # pycodestyle "W", # pycodestyle @@ -55,3 +48,4 @@ select = [ "NPY", # NumPy rules "RUF", # Ruff rules ] +ignore = ["E402"] diff --git a/raw_svg/render_math_svg.py b/raw_svg/render_math_svg.py index 65b7ae10..65eeab19 100644 --- a/raw_svg/render_math_svg.py +++ b/raw_svg/render_math_svg.py @@ -1,20 +1,19 @@ -from __future__ import annotations -import requests -from xml.etree import ElementTree as ET -from pathlib import Path import json -from typing import Optional, Union, Dict +import re import sys from datetime import datetime -import re +from pathlib import Path +from xml.etree import ElementTree as ET + +import requests HERE = Path(__file__).parent OUTPUT = HERE.parent / "images" # HOST = "https://math.vercel.app" HOST = "http://localhost:3000" -PARAMS: dict[str, Optional[Union[str, bytes]]] = {"from": None} +PARAMS: dict[str, str | bytes | None] = {"from": None} CACHE = HERE / "mathjax_cache.json" -CACHE_T = Dict[Optional[Union[str, bytes]], str] +CACHE_T = dict[str | bytes | None, str] SESSION = requests.Session() NAMESPACES = { "": "http://www.w3.org/2000/svg", @@ -60,7 +59,7 @@ -""" +""" # noqa: E501 # Match sizes from height and width attributes SIZE_REGEX = re.compile(r"([\d.]+)?(\w*)?") # Match a scaling factor in the class attribute @@ -148,7 +147,7 @@ def replace_text(search_elem: ET.Element, cache: CACHE_T) -> None: search_elem.remove(elem) -def main(svg_file: Optional[Path] = None) -> None: +def main(svg_file: Path | None = None) -> None: if svg_file is None: for svg_f in HERE.glob("*.svg"): main(svg_f) diff --git a/reference/planetary-ephemeris.md b/reference/planetary-ephemeris.md index f16f193f..2810ae7a 100644 --- a/reference/planetary-ephemeris.md +++ b/reference/planetary-ephemeris.md @@ -359,10 +359,10 @@ The default interface of HORIZONS is shown in {numref}`fig:horizons-default` The default web interface for HORIZONS. ::: -Each of the options can be changed by clicking the *Edit* buttons. For our purposes, we can change the following: +Each of the options can be changed by clicking the _Edit_ buttons. For our purposes, we can change the following: 1. _Ephemeris Type_: Either _Vector Table_ or _Osculating Orbital Elements_ is suitable, although the latter is more direct for this example -2. _Target Body_: This option opens a pop-up where we can search for the body of interest. In the drop-down menu under *Choose a method for specifying the target body*, you can choose *Select from a list of major bodies*, then choose *Mercury* +2. _Target Body_: This option opens a pop-up where we can search for the body of interest. In the drop-down menu under _Choose a method for specifying the target body_, you can choose _Select from a list of major bodies_, then choose *Mercury* 3. _Center_: The default selection here is _Solar System Barycenter_, the center of gravity of the entire solar system. This is usually a little bit outside the sun, depending on the relative locations of the planets, especially Jupiter. In our case, we want the center of the Sun as the focus of the orbit, so click _Edit_ and then type `@sun` into the search box. 4. _Time Span_: This can be used to generate a range of dates, or to input specific dates. We will choose _Specify a list of times_ for this example, and then input the date of interest, in JDT, {glue:text}`planetary-ephemeris-JDT`. 5. _Table Settings_: Here, we want to change the units to _km and seconds_. Another useful option is the _Reference plane_. The default of _ecliptic x-y plane derived from reference plane_ is appropriate for this example. You may also want to set the CSV output option, depending on how you will use the data. diff --git a/reference/planetary-parameters.md b/reference/planetary-parameters.md index e063e756..dc0d44e1 100644 --- a/reference/planetary-parameters.md +++ b/reference/planetary-parameters.md @@ -13,6 +13,7 @@ kernelspec: --- (sec:planetary-parameters)= + # Planetary Parameters ## Planetary Mass Parameters diff --git a/reference/scripts/planet-orbital-elements.py b/reference/scripts/planet-orbital-elements.py index 8460e6fd..45aa6e06 100644 --- a/reference/scripts/planet-orbital-elements.py +++ b/reference/scripts/planet-orbital-elements.py @@ -1,17 +1,16 @@ -from __future__ import annotations -from pathlib import Path import re from datetime import datetime +from pathlib import Path -from dateutil.rrule import rrule, DAILY +import numpy as np from dateutil import tz -from skyfield.iokit import load_file, Loader +from dateutil.rrule import DAILY, rrule +from IPython.display import display +from myst_nb.ext.glue import GLUE_PREFIX from skyfield.elementslib import OsculatingElements, osculating_elements_of from skyfield.framelib import build_ecliptic_matrix +from skyfield.iokit import Loader, load_file from skyfield.units import Distance, Velocity -import numpy as np -from myst_nb.ext.glue import GLUE_PREFIX -from IPython.display import display class CelestialObject: @@ -33,7 +32,7 @@ def format_units(self, obj: float): mantissa, exponent = initial_format.split("E") sign = exponent[0] if exponent[0] == "-" else "" value = int(exponent[1:]) - return f"{mantissa} × 10{sign}{value}" + return f"{mantissa} × 10{sign}{value}" # noqa: RUF001 def glue(self): metadata = {"scrapbook": {"mime_prefix": GLUE_PREFIX}} diff --git a/reference/scripts/planet-radii.py b/reference/scripts/planet-radii.py index 932d3acc..1866e32e 100644 --- a/reference/scripts/planet-radii.py +++ b/reference/scripts/planet-radii.py @@ -1,6 +1,7 @@ import csv -from myst_nb.ext.glue import GLUE_PREFIX + from IPython.display import display +from myst_nb.ext.glue import GLUE_PREFIX # Data source: https://link.springer.com/article/10.1007/s10569-017-9805-5/tables/5 planet_data = """\ @@ -16,7 +17,7 @@ Saturn,58232 ± 6,60268 ± 4,54364 ± 10,102.9,8,205 Uranus,25362 ± 7,25559 ± 4,24973 ± 20,16.8,28,0 Neptune,24622 ± 19,24764 ± 15,24341 ± 30,8.0,14,0 -""" +""" # noqa: E501 # Data source: https://link.springer.com/article/10.1007/s10569-017-9805-5/tables/6 moon_data = """\ @@ -87,7 +88,7 @@ Neptune,VIII,Proteus,208 ± 8,218,208,201,7.9,18.0,13.0 Neptune,I,Triton,1352.6 ± 2.4,,,,,, Neptune,II,Nereid,170 ± 25,,,,,, -""" +""" # noqa: E501 # Data source: https://link.springer.com/article/10.1007/s10569-017-9805-5/tables/7 dwarf_data = """\ @@ -113,7 +114,7 @@ 67P/Churyumov–Gerasimenko,1.65,2.40,1.55,1.20,(f) 81P/Wild 2,1.975,2.7,1.9,1.5, 103P/Hartley 2,0.58,0.34,1.16,1.16,(g) -""" +""" # noqa: E501, RUF001 def glue(row: dict[str, str], name: str, object: str) -> None: diff --git a/reference/scripts/standard-gravitational-parameter.py b/reference/scripts/standard-gravitational-parameter.py index e52884aa..0dd44151 100644 --- a/reference/scripts/standard-gravitational-parameter.py +++ b/reference/scripts/standard-gravitational-parameter.py @@ -1,5 +1,4 @@ from dataclasses import dataclass -from typing import Optional import pint from IPython.display import display @@ -17,8 +16,8 @@ class CelestialObject: GM_in_au: pint.Quantity GM_ratio: pint.Quantity GM_in_km: pint.Quantity - mass: Optional[pint.Quantity] = None - mass_fraction: Optional[pint.Quantity] = None + mass: pint.Quantity | None = None + mass_fraction: pint.Quantity | None = None def compute_mass(self): self.mass = (self.GM_in_km / self.G).to("kg") @@ -35,7 +34,7 @@ def format_units(self, obj): sign = exponent[0] if exponent[0] == "-" else "" value = int(exponent[1:]) - return f"{mantissa} × 10{sign}{value}{units}" + return f"{mantissa} × 10{sign}{value}{units}" # noqa: RUF001 def glue(self): assert self.mass is not None, "Mass has not been computed" diff --git a/references.bib b/references.bib index d81518a6..cec4a346 100644 --- a/references.bib +++ b/references.bib @@ -59,7 +59,7 @@ @book{Bate2020 title = {Fundamentals of Astrodynamics}, author = {Bate, Roger R. and Mueller, Donald D. and White, Jerry E. and Saylor, William W.}, year = {2020}, - edition = {Second edition}, + edition = {second}, publisher = {{Dover Publications, Inc}}, address = {{Mineola, New York}}, abstract = {"Developed at the U.S. Air Force Academy, this teaching text is widely known and used throughout the astrodynamics and aerospace engineering communities. Completely revised and updated, this second edition takes into account new developments of the past four decades, especially regarding information technology. Central emphasis is placed on the use of the universal variable formulation, although classical methods are also discussed. The development of the basic two-body and n-body equations of motion serves as a foundation for all that follows. Subsequent topics include orbit determination and the classical orbital elements, coordinate transformations, and differential correction. The Kepler and Gauss problems are treated in detail, and two-body mechanics are applied to the ballistic missile problem. Perturbations, integration schemes and error, and analytic formulations of several common perturbations are introduced. Example problems and exercises appear throughout the text, along with photographs, diagrams, and drawings. Four helpful appendixes conclude the book"--}, @@ -109,7 +109,7 @@ @book{Curtis2020 title = {Orbital Mechanics for Engineering Students}, author = {Curtis, Howard D}, year = {2020}, - edition = {Fourth}, + edition = {fourth}, publisher = {{Elsevier}}, isbn = {978-0-08-102133-0}, langid = {english}, @@ -135,7 +135,7 @@ @book{Gradshtein2007 title = {Table of Integrals, Series, and Products}, author = {Gradshte{\u \i}n, I. S. and Ryzhik, I. M. and Jeffrey, Alan}, year = {2007}, - edition = {7th ed}, + edition = {seventh}, publisher = {{Academic Press}}, address = {{Amsterdam ; Boston}}, isbn = {978-0-12-373637-6}, @@ -281,7 +281,7 @@ @book{Prussing2013 title = {Orbital Mechanics}, author = {Prussing, John E. and Conway, Bruce A.}, year = {2013}, - edition = {Second edition}, + edition = {Second}, publisher = {{Oxford University Press}}, address = {{New York}}, isbn = {978-0-19-983770-0}, @@ -373,7 +373,7 @@ @book{Seidelmann1992 title = {Explanatory Supplement to the {{Astronomical}} Almanac}, editor = {Seidelmann, P. Kenneth and United States Naval Observatory and Great Britain}, year = {1992}, - edition = {Rev. ed.}, + edition = {Rev.}, publisher = {{University Science Books}}, address = {{Mill Valley, Calif}}, isbn = {978-0-935702-68-2}, @@ -437,7 +437,7 @@ @book{Vallado2013 author = {Vallado, David A. and McClain, Wayne D.}, year = {2013}, series = {Space Technology Library}, - edition = {4. ed}, + edition = {fourth}, number = {21}, publisher = {{Microcosm Press}}, address = {{Hawthorne, Calif}}, @@ -456,5 +456,3 @@ @book{Zwillinger2003 langid = {english}, annotation = {OCLC: 493531816} } - - diff --git a/the-n-body-problem/Equations-of-Motion-CR3BP.ipynb b/the-n-body-problem/Equations-of-Motion-CR3BP.ipynb index b05278c0..fe248c56 100644 --- a/the-n-body-problem/Equations-of-Motion-CR3BP.ipynb +++ b/the-n-body-problem/Equations-of-Motion-CR3BP.ipynb @@ -2,6 +2,7 @@ "cells": [ { "cell_type": "markdown", + "metadata": {}, "source": [ "# Example: CR3BP Equations of Motion\n", "\n", @@ -33,45 +34,46 @@ ":::\n", "\n", "We need to provide $\\pi_2$, the mass ratio, as the parameter of the system, along with the initial position and velocity. Let's solve this system of equations numerically to find the position of $m$ as a function of non-dimensional time in the rotating frame of reference. First, we import the appropriate libraries:" - ], - "metadata": {} + ] }, { "cell_type": "code", "execution_count": null, + "metadata": {}, + "outputs": [], "source": [ - "from scipy.integrate import solve_ivp\n", + "import matplotlib.pyplot as plt\n", "import numpy as np\n", - "import matplotlib.pyplot as plt" - ], - "outputs": [], - "metadata": {} + "from scipy.integrate import solve_ivp" + ] }, { "cell_type": "code", "execution_count": null, + "metadata": {}, + "outputs": [], "source": [ "# These masses represent the Earth-Moon system\n", - "m_1 = 5.974E24 # kg\n", - "m_2 = 7.348E22 # kg\n", - "pi_2 = m_2/(m_1 + m_2)" - ], - "outputs": [], - "metadata": {} + "m_1 = 5.974e24 # kg\n", + "m_2 = 7.348e22 # kg\n", + "pi_2 = m_2 / (m_1 + m_2)" + ] }, { "cell_type": "markdown", + "metadata": {}, "source": [ "The system of equations is very sensitive to the initial conditions that we pick. In this case, we are choosing the initial position at the $x^*$ position of the secondary mass, offset slightly in the $y^*$ axis. The initial velocity is directed along the positive $y^*$ axis and the negative $x^*$ axis." - ], - "metadata": {} + ] }, { "cell_type": "code", "execution_count": null, + "metadata": {}, + "outputs": [], "source": [ "x_0 = 1 - pi_2\n", - "y_0 = .0455\n", + "y_0 = 0.0455\n", "z_0 = 0\n", "vx_0 = -0.5\n", "vy_0 = 0.5\n", @@ -81,27 +83,27 @@ "r_0 = np.array((x_0, y_0, z_0))\n", "v_0 = np.array((vx_0, vy_0, vz_0))\n", "Y_0 = np.hstack((r_0, v_0))" - ], - "outputs": [], - "metadata": {} + ] }, { "cell_type": "markdown", + "metadata": {}, "source": [ "Now we need to define the function that solves the non-dimensional equations of motion, much like the function in the relative motion problem." - ], - "metadata": {} + ] }, { "cell_type": "code", "execution_count": null, + "metadata": {}, + "outputs": [], "source": [ "def nondim_cr3bp(t, Y):\n", " \"\"\"Solve the CR3BP in nondimensional coordinates.\n", - " \n", + "\n", " The state vector is Y, with the first three components as the\n", " position of $m$, and the second three components its velocity.\n", - " \n", + "\n", " The solution is parameterized on $\\\\pi_2$, the mass ratio.\n", " \"\"\"\n", " # Get the position and velocity from the solution vector\n", @@ -114,24 +116,29 @@ "\n", " sigma = np.sqrt(np.sum(np.square([x + pi_2, y, z])))\n", " psi = np.sqrt(np.sum(np.square([x - 1 + pi_2, y, z])))\n", - " Ydot[3] = 2 * ydot + x - (1 - pi_2) * (x + pi_2) / sigma**3 - pi_2 * (x - 1 + pi_2) / psi**3\n", + " Ydot[3] = (\n", + " 2 * ydot\n", + " + x\n", + " - (1 - pi_2) * (x + pi_2) / sigma**3\n", + " - pi_2 * (x - 1 + pi_2) / psi**3\n", + " )\n", " Ydot[4] = -2 * xdot + y - (1 - pi_2) * y / sigma**3 - pi_2 * y / psi**3\n", - " Ydot[5] = -(1 - pi_2)/sigma**3 * z - pi_2/psi**3 * z\n", + " Ydot[5] = -(1 - pi_2) / sigma**3 * z - pi_2 / psi**3 * z\n", " return Ydot" - ], - "outputs": [], - "metadata": {} + ] }, { "cell_type": "markdown", + "metadata": {}, "source": [ "Now we can solve the problem with `solve_ivp()`. We choose an end time of 25 non-dimensional units, pretty much arbitrarily to see what we can get." - ], - "metadata": {} + ] }, { "cell_type": "code", "execution_count": null, + "metadata": {}, + "outputs": [], "source": [ "t_0 = 0 # nondimensional time\n", "t_f = 20 # nondimensional time\n", @@ -141,69 +148,68 @@ "Y = sol.y.T\n", "r = Y[:, :3] # nondimensional distance\n", "v = Y[:, 3:] # nondimensional velocity" - ], - "outputs": [], - "metadata": {} + ] }, { "cell_type": "markdown", + "metadata": {}, "source": [ "Finally, to plot the trajectory, we will define some values for the circular orbit of $m_2$ around the barycenter and then plot the trajectory of $m$ in red." - ], - "metadata": {} + ] }, { "cell_type": "code", "execution_count": null, - "source": [ - "from myst_nb import glue" - ], - "outputs": [], "metadata": { "tags": [ "remove-cell" ] - } + }, + "outputs": [], + "source": [ + "from myst_nb import glue" + ] }, { "cell_type": "code", "execution_count": null, + "metadata": { + "tags": [ + "remove-output" + ] + }, + "outputs": [], "source": [ "x_2 = (1 - pi_2) * np.cos(np.linspace(0, np.pi, 100))\n", "y_2 = (1 - pi_2) * np.sin(np.linspace(0, np.pi, 100))\n", - "fig, ax = plt.subplots(figsize=(5,5), dpi=96)\n", + "fig, ax = plt.subplots(figsize=(5, 5), dpi=96)\n", "\n", "# Plot the orbits\n", - "ax.plot(r[:, 0], r[:, 1], 'r', label=\"Trajectory\")\n", - "ax.axhline(0, color='k')\n", + "ax.plot(r[:, 0], r[:, 1], \"r\", label=\"Trajectory\")\n", + "ax.axhline(0, color=\"k\")\n", "ax.plot(np.hstack((x_2, x_2[::-1])), np.hstack((y_2, -y_2[::-1])))\n", - "ax.plot(-pi_2, 0, 'bo', label=\"$m_1$\")\n", - "ax.plot(1 - pi_2, 0, 'go', label=\"$m_2$\")\n", - "ax.plot(x_0, y_0, 'ro')\n", + "ax.plot(-pi_2, 0, \"bo\", label=\"$m_1$\")\n", + "ax.plot(1 - pi_2, 0, \"go\", label=\"$m_2$\")\n", + "ax.plot(x_0, y_0, \"ro\")\n", "ax.set_aspect(\"equal\")" - ], - "outputs": [], - "metadata": { - "tags": [ - "remove-output" - ] - } + ] }, { "cell_type": "code", "execution_count": null, - "source": [ - "glue(\"cr3bp-example-trajectory\", fig, display=False)" - ], - "outputs": [], "metadata": { "tags": [ "remove-cell" ] - } + }, + "outputs": [], + "source": [ + "glue(\"cr3bp-example-trajectory\", fig, display=False)" + ] }, { "cell_type": "markdown", + "metadata": {}, "source": [ ":::{glue:figure} cr3bp-example-trajectory\n", ":name: fig:cr3bp-example-trajectory\n", @@ -211,14 +217,12 @@ "The trajectory of $m$ in the rotating frame of reference, in non-dimensional coordinates.\n", ":::\n", "\n", - "As shown in {numref}`fig:cr3bp-example-trajectory`, we can get some very interesting behavior of $m$ in the rotating frame of reference. The behavior of $m$ depends very strongly on the initial conditions that we set.\n", - "\n", - "**See what interesting patterns you can get, and post your efforts on Piazza!**" - ], - "metadata": {} + "As shown in {numref}`fig:cr3bp-example-trajectory`, we can get some very interesting behavior of $m$ in the rotating frame of reference. The behavior of $m$ depends very strongly on the initial conditions that we set." + ] }, { "cell_type": "markdown", + "metadata": {}, "source": [ "## Calculating the Jacobi Constant\n", "\n", @@ -229,12 +233,13 @@ ":::\n", "\n", "This value should be constant over the entire trajectory of the tertiary mass." - ], - "metadata": {} + ] }, { "cell_type": "code", "execution_count": null, + "metadata": {}, + "outputs": [], "source": [ "speed_sq = np.sum(np.square(v), axis=1)\n", "\n", @@ -244,51 +249,55 @@ "psi = np.sqrt(np.sum(np.square(r), axis=1))\n", "r[:, 0] = r[:, 0] + 1.0 - pi_2\n", "\n", - "J = 0.5 * speed_sq - (1 - pi_2) / sigma - pi_2 / psi - 0.5 * ((1 - pi_2) * sigma**2 + pi_2 * psi**2)" - ], - "outputs": [], - "metadata": {} + "J = (\n", + " 0.5 * speed_sq\n", + " - (1 - pi_2) / sigma\n", + " - pi_2 / psi\n", + " - 0.5 * ((1 - pi_2) * sigma**2 + pi_2 * psi**2)\n", + ")" + ] }, { "cell_type": "markdown", + "metadata": {}, "source": [ "Now that we have calculated $J$ as a function of time, let's plot it! Hopefully it's constant..." - ], - "metadata": {} + ] }, { "cell_type": "code", "execution_count": null, + "metadata": { + "tags": [ + "remove-output" + ] + }, + "outputs": [], "source": [ - "fig, ax = plt.subplots(figsize=(8,6), dpi=96)\n", + "fig, ax = plt.subplots(figsize=(8, 6), dpi=96)\n", "ax.plot(sol.t, J, label=\"Jacobi Constant\")\n", "ax.axhline(J[0], color=\"C1\", label=\"Initial Jacobi Constant\")\n", "ax.legend(loc=\"center left\")\n", "ax.set_xlabel(\"$\\\\tau$\")\n", "ax.set_ylabel(\"$J$\")" - ], - "outputs": [], - "metadata": { - "tags": [ - "remove-output" - ] - } + ] }, { "cell_type": "code", "execution_count": null, - "source": [ - "glue(\"runge-kutta-jacobi-constant\", fig, display=False)" - ], - "outputs": [], "metadata": { "tags": [ "remove-cell" ] - } + }, + "outputs": [], + "source": [ + "glue(\"runge-kutta-jacobi-constant\", fig, display=False)" + ] }, { "cell_type": "markdown", + "metadata": {}, "source": [ ":::{glue:figure} runge-kutta-jacobi-constant\n", ":name: fig:runge-kutta-jacobi-constant\n", @@ -303,12 +312,17 @@ "According to Rubinsztejn {cite}`Rubinsztejn2019`, the problem may be with the integrator we're using, which doesn't guarantee that energy is conserved because of the algorithm it uses. We can change the precision requirements for the solver. The default integrator for `solve_ivp()` is a [5th order Runge-Kutta integrator](https://docs.scipy.org/doc/scipy/reference/generated/scipy.integrate.RK45.html#scipy.integrate.RK45), with absolute and relative tolerances of $10^{-6}$ and $10^{-3}$, respectively.\n", "\n", "We will decrease the absolute and relative tolerances, to ensure that the solver is producing a high-accuracy solution." - ], - "metadata": {} + ] }, { "cell_type": "code", "execution_count": null, + "metadata": { + "tags": [ + "remove-output" + ] + }, + "outputs": [], "source": [ "sol_hp = solve_ivp(nondim_cr3bp, [t_0, t_f], Y_0, atol=1e-9, rtol=1e-6, t_eval=t_points)\n", "Y_hp = sol_hp.y.T\n", @@ -323,9 +337,14 @@ "psi_hp = np.sqrt(np.sum(np.square(r_hp), axis=1))\n", "r_hp[:, 0] = r_hp[:, 0] + 1 - pi_2\n", "\n", - "J_hp = 0.5 * speed_sq_hp - (1 - pi_2) / sigma_hp - pi_2 / psi_hp - 0.5 * ((1 - pi_2) * sigma_hp**2 + pi_2 * psi_hp**2)\n", + "J_hp = (\n", + " 0.5 * speed_sq_hp\n", + " - (1 - pi_2) / sigma_hp\n", + " - pi_2 / psi_hp\n", + " - 0.5 * ((1 - pi_2) * sigma_hp**2 + pi_2 * psi_hp**2)\n", + ")\n", "\n", - "fig, ax = plt.subplots(figsize=(6,4), dpi=96)\n", + "fig, ax = plt.subplots(figsize=(6, 4), dpi=96)\n", "ax.plot(sol_hp.t, J_hp, label=\"Jacobi Constant\")\n", "ax.axhline(J_hp[0], color=\"C1\", label=\"Initial Jacobi Constant\")\n", "ax.set_ylim(-1.4988, -1.49886)\n", @@ -333,29 +352,24 @@ "ax.set_xlabel(\"$\\\\tau$\")\n", "ax.set_ylabel(\"$J$\")\n", "ax.ticklabel_format(style=\"plain\", useOffset=False)" - ], - "outputs": [], - "metadata": { - "tags": [ - "remove-output" - ] - } + ] }, { "cell_type": "code", "execution_count": null, - "source": [ - "glue(\"runge-kutta-small-tolerance\", fig, display=False)" - ], - "outputs": [], "metadata": { "tags": [ "remove-cell" ] - } + }, + "outputs": [], + "source": [ + "glue(\"runge-kutta-small-tolerance\", fig, display=False)" + ] }, { "cell_type": "markdown", + "metadata": {}, "source": [ ":::{glue:figure} runge-kutta-small-tolerance\n", ":name: fig:runge-kutta-small-tolerance\n", @@ -366,35 +380,40 @@ "As shown in {numref}`fig:runge-kutta-small-tolerance`, reducing the tolerances significantly improves the behavior of the solver. The change in $J$ is now on the order of $1\\times 10^{-5}$, which is acceptable for reasonable lengths of integration time.\n", "\n", "{numref}`fig:trajectory-comparison-cr3bp` shows the trajectory of $m$ for both of the solution cases here. The trajectories clearly diverge quite quickly after the simulation starts." - ], - "metadata": {} + ] }, { "cell_type": "code", "execution_count": null, + "metadata": { + "tags": [ + "remove-output" + ] + }, + "outputs": [], "source": [ "fig, ax = plt.subplots(figsize=(5, 5), dpi=96)\n", "\n", "# Plot the orbits\n", - "ax.plot(r_hp[:, 0], r[:, 1], 'r', label=\"Reduced Tolerances Trajectory\")\n", - "ax.axhline(0, color='k')\n", + "ax.plot(r_hp[:, 0], r[:, 1], \"r\", label=\"Reduced Tolerances Trajectory\")\n", + "ax.axhline(0, color=\"k\")\n", "ax.plot(np.hstack((x_2, x_2[::-1])), np.hstack((y_2, -y_2[::-1])))\n", - "ax.plot(-pi_2, 0, 'bo')\n", - "ax.plot(1 - pi_2, 0, 'go')\n", - "ax.plot(x_0, y_0, 'ro')\n", - "ax.plot(r[:, 0], r[:, 1], 'C2', label=\"Original Trajectory\")\n", + "ax.plot(-pi_2, 0, \"bo\")\n", + "ax.plot(1 - pi_2, 0, \"go\")\n", + "ax.plot(x_0, y_0, \"ro\")\n", + "ax.plot(r[:, 0], r[:, 1], \"C2\", label=\"Original Trajectory\")\n", "ax.set_aspect(\"equal\")" - ], - "outputs": [], - "metadata": { - "tags": [ - "remove-output" - ] - } + ] }, { "cell_type": "code", "execution_count": null, + "metadata": { + "tags": [ + "remove-cell" + ] + }, + "outputs": [], "source": [ "box = ax.get_position()\n", "ax.set_position([box.x0, box.y0, box.width * 0.8, box.height])\n", @@ -402,16 +421,11 @@ "# Put a legend to the right of the current axis\n", "ax.legend(loc=\"upper left\", bbox_to_anchor=(1, 1))\n", "glue(\"trajectory-comparison-cr3bp\", fig, display=False)" - ], - "outputs": [], - "metadata": { - "tags": [ - "remove-cell" - ] - } + ] }, { "cell_type": "markdown", + "metadata": {}, "source": [ ":::{glue:figure} trajectory-comparison-cr3bp\n", ":name: fig:trajectory-comparison-cr3bp\n", @@ -422,50 +436,50 @@ "## Cause of Error\n", "\n", "Now, let's examine {numref}`fig:runge-kutta-jacobi-constant` and {numref}`fig:runge-kutta-small-tolerance` again. There appears to be a structure to the changes in $J$. This gives us a hint that the problem might be related to the physical situation that the tertiary mass finds itself in. Let's plot the distance from the larger mass, given by $\\sigma$, on the same plot as the Jacobi constant." - ], - "metadata": {} + ] }, { "cell_type": "code", "execution_count": null, + "metadata": { + "tags": [ + "remove-output" + ] + }, + "outputs": [], "source": [ "fig, J_ax = plt.subplots(figsize=(6, 4), dpi=96)\n", "Jacobi_plot = J_ax.plot(sol_hp.t, J_hp, label=\"Jacobi Constant\")\n", - "initial_Jacobi = J_ax.axhline(J_hp[0], color='C1', label=\"Initial Jacobi Constant\")\n", + "initial_Jacobi = J_ax.axhline(J_hp[0], color=\"C1\", label=\"Initial Jacobi Constant\")\n", "s_ax = J_ax.twinx()\n", - "sigma_plot = s_ax.plot(sol_hp.t, sigma_hp, 'C2', label=\"$\\sigma$\")\n", + "sigma_plot = s_ax.plot(sol_hp.t, sigma_hp, \"C2\", label=r\"$\\sigma$\")\n", "s_ax.set_ylim(0, 1.15)\n", "J_ax.ticklabel_format(style=\"plain\", useOffset=False)\n", "J_ax.set_xlabel(\"$\\\\tau$\")\n", "J_ax.set_ylabel(\"$J$\")\n", "s_ax.set_ylabel(\"$\\\\sigma$\")" - ], - "outputs": [], - "metadata": { - "tags": [ - "remove-output" - ] - } + ] }, { "cell_type": "code", "execution_count": null, - "source": [ - "lns = Jacobi_plot + [initial_Jacobi] + sigma_plot\n", - "labs = [l.get_label() for l in lns]\n", - "J_ax.legend(lns, labs, loc=\"best\")\n", - "\n", - "glue(\"jacobi-constant-with-sigma\", fig, display=False)" - ], - "outputs": [], "metadata": { "tags": [ "remove-cell" ] - } + }, + "outputs": [], + "source": [ + "lns = [*Jacobi_plot, initial_Jacobi, *sigma_plot]\n", + "labs = [ll.get_label() for ll in lns]\n", + "J_ax.legend(lns, labs, loc=\"best\")\n", + "\n", + "glue(\"jacobi-constant-with-sigma\", fig, display=False)" + ] }, { "cell_type": "markdown", + "metadata": {}, "source": [ ":::{glue:figure} jacobi-constant-with-sigma\n", ":name: fig:jacobi-constant-with-sigma\n", @@ -476,12 +490,17 @@ "As you can see in {numref}`fig:jacobi-constant-with-sigma`, the error in the Jacobi constant spikes when the tertiary mass gets closer to the primary mass, $m_1$. This makes a certain amount of sense, because the acceleration terms depend inversely on the cube of $\\sigma$. As the value of $\\sigma$ gets smaller, the error grows.\n", "\n", "This error can be avoided by using a different class of numerical integrators, called [**symplectic integrators**](https://en.wikipedia.org/wiki/Symplectic_integrator). We won't have time to discuss those further right now, though. If you're interested in taking a crack at an implementation, you can find sample code here: , and you will be interested in the **Structure-Preserving Algorithms**." - ], - "metadata": {} + ] }, { "cell_type": "code", "execution_count": null, + "metadata": { + "tags": [ + "remove-cell" + ] + }, + "outputs": [], "source": [ "# Plot the inertial motion\n", "from scipy.spatial.transform import Rotation as R\n", @@ -489,38 +508,32 @@ "theta = sol_hp.t % (2 * np.pi)\n", "rot = R.from_euler(\"z\", theta)\n", "r_abs = rot.apply(r_hp)" - ], - "outputs": [], - "metadata": { - "tags": [ - "remove-cell" - ] - } + ] }, { "cell_type": "code", "execution_count": null, - "source": [ - "fig, ax = plt.subplots(figsize=(5, 5), dpi=96)\n", - "ax.plot(r_abs[:, 0], r_abs[:, 1], 'r')\n", - "ax.plot(np.hstack((x_2, x_2[::-1])), np.hstack((y_2, -y_2[::-1])))\n", - "ax.plot(-pi_2, 0, 'bo')\n", - "ax.plot(1 - pi_2, 0, 'go')\n", - "ax.plot(x_0, y_0, 'ro')" - ], - "outputs": [], "metadata": { "tags": [ "remove-cell" ] - } + }, + "outputs": [], + "source": [ + "fig, ax = plt.subplots(figsize=(5, 5), dpi=96)\n", + "ax.plot(r_abs[:, 0], r_abs[:, 1], \"r\")\n", + "ax.plot(np.hstack((x_2, x_2[::-1])), np.hstack((y_2, -y_2[::-1])))\n", + "ax.plot(-pi_2, 0, \"bo\")\n", + "ax.plot(1 - pi_2, 0, \"go\")\n", + "ax.plot(x_0, y_0, \"ro\")" + ] }, { "cell_type": "code", "execution_count": null, - "source": [], + "metadata": {}, "outputs": [], - "metadata": {} + "source": [] } ], "metadata": { @@ -539,9 +552,9 @@ "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", - "version": "3.9.6" + "version": "3.12.2" } }, "nbformat": 4, "nbformat_minor": 4 -} \ No newline at end of file +} diff --git a/the-n-body-problem/Lagrange-points-example.ipynb b/the-n-body-problem/Lagrange-points-example.ipynb index 2601e1c0..5dd785b0 100644 --- a/the-n-body-problem/Lagrange-points-example.ipynb +++ b/the-n-body-problem/Lagrange-points-example.ipynb @@ -2,6 +2,7 @@ "cells": [ { "cell_type": "markdown", + "metadata": {}, "source": [ "# Example: Plotting Lagrange Points\n", "\n", @@ -43,42 +44,43 @@ " * MATLAB (`fzero`): `[-1, -1.5]`\n", "\n", "For some reason, the `fzero()` in MATLAB seems much more sensitive to the initial guess value, and if you only provide a single value for the initial guess, it chooses a positive value as the second part of the interval. Thus, if the root is below your initial guess the MATLAB solver will not be able to find it." - ], - "metadata": {} + ] }, { "cell_type": "markdown", + "metadata": {}, "source": [ "## Python\n", "\n", "First, we will demonstrate the Python solver. Like for the `solve_ivp` function, we need to define a function that returns a value given the single input `xstar`. Python is flexible enough to allow us to define `pi_2` as another parameter." - ], - "metadata": {} + ] }, { "cell_type": "code", "execution_count": 1, + "metadata": {}, + "outputs": [], "source": [ "# %matplotlib notebook\n", - "from scipy.optimize import newton\n", - "import numpy as np\n", + "import matplotlib.path as mpath\n", "import matplotlib.pyplot as plt\n", - "import matplotlib.path as mpath" - ], - "outputs": [], - "metadata": {} + "import numpy as np\n", + "from scipy.optimize import newton" + ] }, { "cell_type": "code", "execution_count": 7, - "source": [ - "from myst_nb import glue" - ], + "metadata": { + "tags": [ + "remove-cell" + ] + }, "outputs": [ { - "output_type": "error", "ename": "ModuleNotFoundError", "evalue": "No module named 'myst_nb'", + "output_type": "error", "traceback": [ "\u001b[0;31m---------------------------------------------------------------------------\u001b[0m", "\u001b[0;31mModuleNotFoundError\u001b[0m Traceback (most recent call last)", @@ -87,49 +89,51 @@ ] } ], - "metadata": { - "tags": [ - "remove-cell" - ] - } + "source": [ + "from myst_nb import glue" + ] }, { "cell_type": "code", "execution_count": 2, + "metadata": {}, + "outputs": [], "source": [ "# This code defines a nice shape for the center of mass of the system.\n", "circle = mpath.Path.unit_circle()\n", "wedge_1 = mpath.Path.wedge(90, 180)\n", "wedge_2 = mpath.Path.wedge(270, 0)\n", "\n", - "verts = np.concatenate([circle.vertices, wedge_1.vertices[::-1, ...], wedge_2.vertices[::-1, ...]])\n", + "verts = np.concatenate(\n", + " [circle.vertices, wedge_1.vertices[::-1, ...], wedge_2.vertices[::-1, ...]]\n", + ")\n", "codes = np.concatenate([circle.codes, wedge_1.codes, wedge_2.codes])\n", "center_of_mass = mpath.Path(verts, codes)" - ], - "outputs": [], - "metadata": {} + ] }, { "cell_type": "code", "execution_count": 3, + "metadata": {}, + "outputs": [], "source": [ "# These masses represent the Earth-Moon system\n", - "m_1 = 5.974E24 # kg\n", - "m_2 = 7.348E22 # kg\n", - "pi_2 = m_2/(m_1 + m_2)\n", + "m_1 = 5.974e24 # kg\n", + "m_2 = 7.348e22 # kg\n", + "pi_2 = m_2 / (m_1 + m_2)\n", "\n", "# These give us the coordinates of the orbits of m_2 and m_1\n", "x_2 = (1 - pi_2) * np.cos(np.linspace(0, np.pi, 100))\n", "y_2 = (1 - pi_2) * np.sin(np.linspace(0, np.pi, 100))\n", "x_1 = (-pi_2) * np.cos(np.linspace(0, np.pi, 100))\n", "y_1 = (-pi_2) * np.sin(np.linspace(0, np.pi, 100))" - ], - "outputs": [], - "metadata": {} + ] }, { "cell_type": "code", "execution_count": 4, + "metadata": {}, + "outputs": [], "source": [ "def collinear_lagrange(xstar, pi_2):\n", " \"\"\"Calculate the resultant of the collinear Lagrange point equation.\n", @@ -139,18 +143,18 @@ " scipy.optimize.newton (or another Newton solver) to find a value for xstar\n", " that satsifies the equation, for a given value of pi_2.\n", "\n", - " The solver will try different values of xstar until the return value is equal to zero.\n", + " The solver will try different values of xstar until the return value is equal to\n", + " zero.\n", " \"\"\"\n", " first_term = xstar\n", - " second_term = (1 - pi_2) / np.abs(xstar + pi_2)**3 * (xstar + pi_2)\n", - " third_term = pi_2 / np.abs(xstar - 1 + pi_2)**3 * (xstar - 1 + pi_2)\n", + " second_term = (1 - pi_2) / np.abs(xstar + pi_2) ** 3 * (xstar + pi_2)\n", + " third_term = pi_2 / np.abs(xstar - 1 + pi_2) ** 3 * (xstar - 1 + pi_2)\n", " return first_term - second_term - third_term" - ], - "outputs": [], - "metadata": {} + ] }, { "cell_type": "markdown", + "metadata": {}, "source": [ "Then we need to pass this to the Newton solver. The function signature is:\n", "\n", @@ -159,75 +163,55 @@ "```\n", "\n", "where `func` is the function to be solved, `x0` is the initial guess, and `args` is a tuple of additional arguments to pass to `func`." - ], - "metadata": {} + ] }, { "cell_type": "code", "execution_count": 8, - "source": [ - "L_2 = newton(func=collinear_lagrange, x0=1, args=(pi_2,))\n", - "L_1 = newton(func=collinear_lagrange, x0=0, args=(pi_2,))\n", - "L_3 = newton(func=collinear_lagrange, x0=-1, args=(pi_2,))\n", - "print(f\"{L_1=}, {L_2=}, {L_3=}\")" - ], + "metadata": {}, "outputs": [ { - "output_type": "stream", "name": "stdout", + "output_type": "stream", "text": [ "L_1=0.8369154703225321, L_2=1.1556818961296604, L_3=-1.0050626166357435\n" ] } ], - "metadata": {} + "source": [ + "L_2 = newton(func=collinear_lagrange, x0=1, args=(pi_2,))\n", + "L_1 = newton(func=collinear_lagrange, x0=0, args=(pi_2,))\n", + "L_3 = newton(func=collinear_lagrange, x0=-1, args=(pi_2,))\n", + "print(f\"{L_1=}, {L_2=}, {L_3=}\")" + ] }, { "cell_type": "markdown", + "metadata": {}, "source": [ "Remember, these are in nondimensional coordinates. We can then plot the Lagrange points relative to $m_1$ and $m_2$ in the rotating frame of reference." - ], - "metadata": {} + ] }, { "cell_type": "code", "execution_count": 14, - "source": [ - "fig, ax = plt.subplots(figsize=(5,5), dpi=96)\n", - "ax.set_xlabel(\"$x^*$\")\n", - "ax.set_ylabel(\"$y^*$\")\n", - "\n", - "# Plot the orbits\n", - "ax.axhline(0, color='k')\n", - "ax.plot(np.hstack((x_2, x_2[::-1])), np.hstack((y_2, -y_2[::-1])))\n", - "ax.plot(np.hstack((x_1, x_1[::-1])), np.hstack((y_1, -y_1[::-1])))\n", - "ax.plot([-pi_2, 0.5 - pi_2, 1 - pi_2, 0.5 - pi_2, -pi_2], [0, np.sqrt(3)/2, 0, -np.sqrt(3)/2, 0], 'k', ls=\"--\", lw=1)\n", - "\n", - "# Plot the Lagrange Points and masses\n", - "ax.plot(L_1, 0, 'rv', label=\"$L_1$\")\n", - "ax.plot(L_2, 0, 'r^', label=\"$L_2$\")\n", - "ax.plot(L_3, 0, 'rp', label=\"$L_3$\")\n", - "ax.plot(0.5 - pi_2, np.sqrt(3)/2, 'rX', label=\"$L_4$\")\n", - "ax.plot(0.5 - pi_2, -np.sqrt(3)/2, 'rs', label=\"$L_5$\")\n", - "ax.plot(0, 0, 'k', marker=center_of_mass, markersize=10)\n", - "ax.plot(-pi_2, 0, 'bo', label=\"$m_1$\")\n", - "ax.plot(1 - pi_2, 0, 'go', label=\"$m_2$\")\n", - "ax.legend()\n", - "ax.set_aspect(\"equal\")" - ], + "metadata": { + "tags": [ + "remove-output" + ] + }, "outputs": [ { - "output_type": "execute_result", "data": { "text/plain": [ "" ] }, + "execution_count": 14, "metadata": {}, - "execution_count": 14 + "output_type": "execute_result" }, { - "output_type": "display_data", "data": { "image/png": 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", "text/plain": [ @@ -236,18 +220,49 @@ }, "metadata": { "needs_background": "light" - } + }, + "output_type": "display_data" } ], - "metadata": { - "tags": [ - "remove-output" - ] - } + "source": [ + "fig, ax = plt.subplots(figsize=(5, 5), dpi=96)\n", + "ax.set_xlabel(\"$x^*$\")\n", + "ax.set_ylabel(\"$y^*$\")\n", + "\n", + "# Plot the orbits\n", + "ax.axhline(0, color=\"k\")\n", + "ax.plot(np.hstack((x_2, x_2[::-1])), np.hstack((y_2, -y_2[::-1])))\n", + "ax.plot(np.hstack((x_1, x_1[::-1])), np.hstack((y_1, -y_1[::-1])))\n", + "ax.plot(\n", + " [-pi_2, 0.5 - pi_2, 1 - pi_2, 0.5 - pi_2, -pi_2],\n", + " [0, np.sqrt(3) / 2, 0, -np.sqrt(3) / 2, 0],\n", + " \"k\",\n", + " ls=\"--\",\n", + " lw=1,\n", + ")\n", + "\n", + "# Plot the Lagrange Points and masses\n", + "ax.plot(L_1, 0, \"rv\", label=\"$L_1$\")\n", + "ax.plot(L_2, 0, \"r^\", label=\"$L_2$\")\n", + "ax.plot(L_3, 0, \"rp\", label=\"$L_3$\")\n", + "ax.plot(0.5 - pi_2, np.sqrt(3) / 2, \"rX\", label=\"$L_4$\")\n", + "ax.plot(0.5 - pi_2, -np.sqrt(3) / 2, \"rs\", label=\"$L_5$\")\n", + "ax.plot(0, 0, \"k\", marker=center_of_mass, markersize=10)\n", + "ax.plot(-pi_2, 0, \"bo\", label=\"$m_1$\")\n", + "ax.plot(1 - pi_2, 0, \"go\", label=\"$m_2$\")\n", + "ax.legend()\n", + "ax.set_aspect(\"equal\")" + ] }, { "cell_type": "code", "execution_count": null, + "metadata": { + "tags": [ + "remove-input" + ] + }, + "outputs": [], "source": [ "box = ax.get_position()\n", "ax.set_position([box.x0, box.y0, box.width * 0.8, box.height])\n", @@ -255,27 +270,22 @@ "# Put a legend to the right of the current axis\n", "ax.legend(loc=\"upper left\", bbox_to_anchor=(1, 1))\n", "glue(\"lagrange-points-earth-moon\", fig, display=False)" - ], - "outputs": [], - "metadata": { - "tags": [ - "remove-input" - ] - } + ] }, { "cell_type": "markdown", + "metadata": {}, "source": [ ":::{glue:figure} lagrange-points-earth-moon\n", ":name: fig:lagrange-points-earth-moon\n", "\n", "The location of the Lagrange points in the Earth-Moon system, shown in non-dimensional coordinates.\n", ":::" - ], - "metadata": {} + ] }, { "cell_type": "markdown", + "metadata": {}, "source": [ "## Matlab\n", "\n", @@ -288,15 +298,14 @@ "Notice that to find $L_1$ in Matlab, we had to use the initial guess range from 0.01 to 0.97. Matlab is more sensitive to the initial guess, so you need to make sure that the root is within the initial guess you choose. The `collinear_lagrange` function is discontinuous around 0 and 1.0, so you need to choose your limits carefully.\n", "\n", "It may help to plot the function for a given problem. To do so, within the script, add the line `fplot(@collinear_lagrange)` to show a plot of the function." - ], - "metadata": {} + ] }, { "cell_type": "code", "execution_count": null, - "source": [], + "metadata": {}, "outputs": [], - "metadata": {} + "source": [] } ], "metadata": { @@ -323,4 +332,4 @@ }, "nbformat": 4, "nbformat_minor": 4 -} \ No newline at end of file +} diff --git a/the-n-body-problem/circular-restricted-three-body-problem.md b/the-n-body-problem/circular-restricted-three-body-problem.md index 89820c11..0c54141e 100644 --- a/the-n-body-problem/circular-restricted-three-body-problem.md +++ b/the-n-body-problem/circular-restricted-three-body-problem.md @@ -255,9 +255,9 @@ The characteristic length is the circular orbit radius, $r_{12}$. Using this, we :::{math} :label: eq:non-dim-r-vectors-cr3bp \begin{aligned} - \vector{\rho} &= \frac{\vector{r}}{r_{12}} = x^*\uvec{\imath} + y^*\uvec{\jmath} + z^*\uvec{k} \\ - \vector{\sigma} &= \frac{\vector{r}_1}{r_{12}} = \left(x^* + \pi_2\right)\uvec{\imath} + y^*\uvec{\jmath} + z^*\uvec{k} \\ - \vector{\psi} &= \frac{\vector{r}_2}{r_{12}} = \left(x^* - 1 + \pi_2\right)\uvec{\imath} + y^*\uvec{\jmath} + z^*\uvec{k} + \vector{\rho} &= \frac{\vector{r}}{r_{12}} = x^_\uvec{\imath} + y^_\uvec{\jmath} + z^_\uvec{k} \\ + \vector{\sigma} &= \frac{\vector{r}_1}{r_{12}} = \left(x^_ + \pi_2\right)\uvec{\imath} + y^_\uvec{\jmath} + z^_\uvec{k} \\ + \vector{\psi} &= \frac{\vector{r}_2}{r_{12}} = \left(x^* - 1 + \pi_2\right)\uvec{\imath} + y^_\uvec{\jmath} + z^_\uvec{k} \end{aligned} ::: @@ -281,7 +281,7 @@ where $\tau = t/t_C$. Making the terms on the right hand side of Eq. {eq}`eq:fiv :::{math} :label: eq:non-dim-five-term-accel-cr3bp -\ddot{\vector{\rho}} = \left(\ddot{x}^* - 2\dot{y}^* - x^*\right)\uvec{\imath} + \left(\ddot{y}^* + 2\dot{x}^* - y^*\right)\uvec{\jmath} + \ddot{z}^*\uvec{k} +\ddot{\vector{\rho}} = \left(\ddot{x}^* - 2\dot{y}^* - x^_\right)\uvec{\imath} + \left(\ddot{y}^_ + 2\dot{x}^* - y^_\right)\uvec{\jmath} + \ddot{z}^_\uvec{k} ::: Now we have the non-dimensional inertial acceleration, we need to make Eq. {eq}`eq:vector-eom-cr3bp`, the equation of motion, non-dimensional. After a bunch of algebra, not shown here, we end up with: diff --git a/the-n-body-problem/jacobi-constant.md b/the-n-body-problem/jacobi-constant.md index 03bb5911..c88d3631 100644 --- a/the-n-body-problem/jacobi-constant.md +++ b/the-n-body-problem/jacobi-constant.md @@ -187,7 +187,7 @@ The forbidden regions in the Earth-Moon system, for varying values of the Jacobi At a given instant, assume that the position and and velocity of the satellite are known, such that $J$ can be calculated. {numref}`fig:jacobi-forbidden-regions` tells us what regions of space a satellite can access purely by exchanging kinetic and potential energy. -One example of a use for these data is to determine whether or not a spacecraft can complete a [**free return trajectory**](https://en.wikipedia.org/wiki/Free-return_trajectory) from the moon back to Earth. A free return trajectory requires no additional propulsion. Depending on the exact trajectory, the return can result in the spacecraft entering Earth's atmosphere for a safe landing, as was done in the _Apollo 13_ mission. +One example of a use for these data is to determine whether or not a spacecraft can complete a [**free return trajectory**](https://en.wikipedia.org/wiki/Free-return_trajectory) from the moon back to Earth. A free return trajectory requires no additional propulsion. Depending on the exact trajectory, the return can result in the spacecraft entering Earth's atmosphere for a safe landing, as was done in the *Apollo 13* mission. Imagine a spacecraft in orbit around the moon. The spacecraft has $J =$ -1.797, corresponding to the upper left axis in {numref}`fig:jacobi-forbidden-regions`. With this value of $J$, the spacecraft cannot leave orbit around the moon, because it does not have enough energy to enter any of the gray areas. diff --git a/the-n-body-problem/lagrange-points.md b/the-n-body-problem/lagrange-points.md index 5b735535..5fcf930e 100644 --- a/the-n-body-problem/lagrange-points.md +++ b/the-n-body-problem/lagrange-points.md @@ -80,7 +80,7 @@ Then we can use the definition of $\psi$ to find $r_2$ in dimensional coordinate :::{math} :label: eq:r_2-equilateral-lagrange -\psi = \frac{\mag{\vector{r}_2}}{r_{12}} \Rightarrow r_2 = r_{12} +\psi = \frac{\mag{\vector{r}*2}}{r*{12}} \Rightarrow r_2 = r_{12} ::: Plugging the result of Eq. {eq}`eq:psi-equilateral-lagrange` back into Eq. {eq}`eq:y-star-equilateral-lagrange` equation, we find: @@ -94,7 +94,7 @@ Then we can use the definition of $\sigma$ to find $r_1$ in dimensional coordina :::{math} :label: eq:r_1-equilateral-lagrange -\sigma = \frac{\mag{\vector{r}_1}}{r_{12}} \Rightarrow r_1 = r_{12} +\sigma = \frac{\mag{\vector{r}*1}}{r*{12}} \Rightarrow r_1 = r_{12} ::: Since if $a = b$ and $b = c$, then $a = c$, we find: @@ -104,7 +104,7 @@ Since if $a = b$ and $b = c$, then $a = c$, we find: r_1 = r_2 = r_{12}\qquad \text{and}\qquad \psi = \sigma ::: -for the _equilateral Lagrange points_. Eq. {eq}`eq:equilateral-lagrange-distances` shows that the distance from $m_1$ to $m$ is the same as the distance from $m_2$ to $m$ and from $m_1$ to $m_2$. This defines an equilateral triangle, giving these Lagrange points their name. +for the *equilateral Lagrange points*. Eq. {eq}`eq:equilateral-lagrange-distances` shows that the distance from $m_1$ to $m$ is the same as the distance from $m_2$ to $m$ and from $m_1$ to $m_2$. This defines an equilateral triangle, giving these Lagrange points their name. From the definition of $\vector{\sigma}$ and $\vector{\psi}$, we can take their magnitudes and equate them to solve for the values of $x^*$ and $y^{*}$ at the equilibrium points: @@ -335,7 +335,7 @@ For the Earth-Moon system, the value of $\pi_2$ is approximately 0.012. ## Lagrange Point Stability -Although all the Lagrange points are equilibrium points, they may not be _stable_ equilibrium points. Stability is the ability of the system to return to its initial position after being perturbed. +Although all the Lagrange points are equilibrium points, they may not be *stable* equilibrium points. Stability is the ability of the system to return to its initial position after being perturbed. To analyze the stability of the Lagrange points, we will use the potential energy function for the CR3BP. diff --git a/the-n-body-problem/scripts/two-body-inertial-numerical-solution.py b/the-n-body-problem/scripts/two-body-inertial-numerical-solution.py index 66128a8f..29b66c2e 100644 --- a/the-n-body-problem/scripts/two-body-inertial-numerical-solution.py +++ b/the-n-body-problem/scripts/two-body-inertial-numerical-solution.py @@ -21,19 +21,19 @@ r = np.sqrt((X_2 - X_1) ** 2 + (Y_2 - Y_1) ** 2 + (Z_2 - Z_1) ** 2) -ddotX_1 = G * m_2 * (X_2 - X_1) / r ** 3 -ddotY_1 = G * m_2 * (Y_2 - Y_1) / r ** 3 -ddotZ_1 = G * m_2 * (Z_2 - Z_1) / r ** 3 -ddotX_2 = -G * m_1 * (X_2 - X_1) / r ** 3 -ddotY_2 = -G * m_1 * (Y_2 - Y_1) / r ** 3 -ddotZ_2 = -G * m_1 * (Z_2 - Z_1) / r ** 3 +ddotX_1 = G * m_2 * (X_2 - X_1) / r**3 +ddotY_1 = G * m_2 * (Y_2 - Y_1) / r**3 +ddotZ_1 = G * m_2 * (Z_2 - Z_1) / r**3 +ddotX_2 = -G * m_1 * (X_2 - X_1) / r**3 +ddotY_2 = -G * m_1 * (Y_2 - Y_1) / r**3 +ddotZ_2 = -G * m_1 * (Z_2 - Z_1) / r**3 # [section-3] R_1 = y_0[:3] R_2 = y_0[3:6] r = np.sqrt(np.sum(np.square(R_2 - R_1))) -ddot = G * (R_2 - R_1) / r ** 3 +ddot = G * (R_2 - R_1) / r**3 ddotR_1_0 = m_2 * ddot ddotR_2_0 = -m_1 * ddot @@ -46,9 +46,8 @@ R_2_1 = dotR_2_0 * Delta_t + R_2_0 # [section-5] -from scipy.integrate import solve_ivp import matplotlib.pyplot as plt -from mpl_toolkits.mplot3d import Axes3D +from scipy.integrate import solve_ivp # [section-6] @@ -76,7 +75,7 @@ def absolute_motion(t, y): # Calculate the acceleration terms and fill them in to the rest # of the derivative array r = np.sqrt(np.sum(np.square(R_2 - R_1))) - ddot = G * (R_2 - R_1) / r ** 3 + ddot = G * (R_2 - R_1) / r**3 ddotR_1 = m_2 * ddot ddotR_2 = -m_1 * ddot diff --git a/the-n-body-problem/scripts/two-body-relative-numerical-solution.py b/the-n-body-problem/scripts/two-body-relative-numerical-solution.py index 219ffeff..b8c95850 100644 --- a/the-n-body-problem/scripts/two-body-relative-numerical-solution.py +++ b/the-n-body-problem/scripts/two-body-relative-numerical-solution.py @@ -1,8 +1,8 @@ # [section-1] -from scipy.integrate import solve_ivp -import numpy as np import matplotlib.pyplot as plt +import numpy as np from mpl_toolkits.mplot3d import Axes3D # noqa: F401, needed for 3d plots +from scipy.integrate import solve_ivp # [section-2] G = 6.67430e-20 # km**3/(kg * s**2) @@ -34,7 +34,7 @@ def relative_motion(t, Y): # Calculate the accelerations r = np.sqrt(np.sum(np.square(r_vec))) - a_vec = -mu * r_vec / r ** 3 + a_vec = -mu * r_vec / r**3 Ydot[3:] = a_vec return Ydot diff --git a/the-n-body-problem/scripts/two-body-relative.npz b/the-n-body-problem/scripts/two-body-relative.npz index cfce8d89..24cf6c1a 100644 Binary files a/the-n-body-problem/scripts/two-body-relative.npz and b/the-n-body-problem/scripts/two-body-relative.npz differ diff --git a/the-orbit-equation/hyperbolic_trajectory.py b/the-orbit-equation/hyperbolic_trajectory.py index 324fb4f8..6edd2377 100644 --- a/the-orbit-equation/hyperbolic_trajectory.py +++ b/the-orbit-equation/hyperbolic_trajectory.py @@ -1,6 +1,6 @@ -import numpy as np -import matplotlib.pyplot as plt import matplotlib.animation as animation +import matplotlib.pyplot as plt +import numpy as np plt.ioff() @@ -30,13 +30,13 @@ def plot_figure(): e = 1.1 theta_inf = np.arccos(-1 / e) a = 1 - b = a * np.sqrt(e ** 2 - 1) + b = a * np.sqrt(e**2 - 1) r_p = a * (e - 1) (m1,) = ax.plot(-a - r_p, 0, "ko") x = np.linspace(a, 2 * a, 1000) - y = b / a * np.sqrt(x ** 2 - a ** 2) + y = b / a * np.sqrt(x**2 - a**2) (r_pos,) = ax.plot(x, y, color="C0") (r_neg,) = ax.plot(x, -y, color="C0") (l_neg,) = ax.plot(-x, -y, color="C1") @@ -58,7 +58,7 @@ def init(): return (r_pos, r_neg, l_neg, l_pos, ann_2, ann_3, ann_4, m1, point, ann) def animate(t): - r = a * (e ** 2 - 1) / (1 + e * np.cos(t)) + r = a * (e**2 - 1) / (1 + e * np.cos(t)) point.set_data(-a - r_p + r * np.cos(t), r * np.sin(t)) ann.set_text(rf"$\nu$ = {np.degrees(t):.2F}°") return (point, ann) diff --git a/the-orbit-equation/orbital-nomenclature.md b/the-orbit-equation/orbital-nomenclature.md index 35e16c61..b02aecc5 100644 --- a/the-orbit-equation/orbital-nomenclature.md +++ b/the-orbit-equation/orbital-nomenclature.md @@ -40,7 +40,7 @@ For circular orbits, we usually pick the apse line to be along the $x$-axis and A parabolic trajectory, $e = 1$ ::: -The parabolic trajectory has eccentricity $e = 1$. The parabolic trajectory extends to infinity towards the left of the domain. It is the limiting case where an object on a parabolic trajectory will never return to the primary mass. Thus, it is called a _trajectory_ rather than _orbit_ since it is not periodic. +The parabolic trajectory has eccentricity $e = 1$. The parabolic trajectory extends to infinity towards the left of the domain. It is the limiting case where an object on a parabolic trajectory will never return to the primary mass. Thus, it is called a *trajectory* rather than *orbit* since it is not periodic. The parabolic trajectory has a periapsis, a point of closest approach, but no apoapsis, since the trajectory extends to infinity. :::: diff --git a/time-since-periapsis-and-keplers-equation/time-since-periapsis.md b/time-since-periapsis-and-keplers-equation/time-since-periapsis.md index 7346013f..bf8622c0 100644 --- a/time-since-periapsis-and-keplers-equation/time-since-periapsis.md +++ b/time-since-periapsis-and-keplers-equation/time-since-periapsis.md @@ -217,6 +217,7 @@ The true, mean, and eccentric anomalies for an elliptical orbit. The true anomal ::: (sec:time-since-periapsis-solution-procedures)= + ## Solution Procedures There are now two cases that we want to consider: