31:T502,

To describe the elliptical orbits of the planets in the solar system, we use the six Keplerian elements:

Shape and size
- Eccentricity (e): Defines the shape of the ellipse.
- Semi-major Axis (a): Defines the longest radius of the ellipse.
Orientation of the orbital plane

Inclination (i): Defines the tilt of the orbital plane with respect to the reference plane (i.e., Earth's ecliptic plane).
Longitude of the Ascending Node (Ω): Defines the orientation of the orbital plane relative to a reference direction (i.e., the vernal point).

Orientation and position within the orbital plane

Argument of Periapsis (ω): Defines the orientation of the ellipse within the orbital plane.
True Anomaly (𝜈): Gives the position of the orbiting body along the ellipse at a specific epoch, expressed as an angle from the periapsis (also known as perihelion for objects orbiting the Sun).

32:Tc635,illustration = { document.addEventListener('fullscreenchange', isFullScreen); document.addEventListener('webkitfullscreenchange', isFullScreen); document.addEventListener('mozfullscreenchange', isFullScreen); document.addEventListener('MSFullscreenChange', isFullScreen); function isFullScreen() { return document.fullscreenElement ? true : false; } const width = isFullScreen()? "100%" : 900; const height = isFullScreen()? Math.ceil(width * screen.height / screen.width): 620; const svg = htl.svg``; return svg; }33:Taae,// Imported from [https://observablehq.com/@mbostock/scrubber] // import {Scrubber} from "@mbostock/scrubber" function Scrubber(values, { format = value => value, initial = 0, delay = null, autoplay = true, loop = true, loopDelay = null, alternate = false } = {}) { values = Array.from(values); const form = html``; let frame = null; let timer = null; let interval = null; let direction = 1; function start() { form.b.textContent = "Pause"; if (delay === null) frame = requestAnimationFrame(tick); else interval = setInterval(tick, delay); } function stop() { form.b.textContent = "Play"; if (frame !== null) cancelAnimationFrame(frame), frame = null; if (timer !== null) clearTimeout(timer), timer = null; if (interval !== null) clearInterval(interval), interval = null; } function running() { return frame !== null || timer !== null || interval !== null; } function tick() { if (form.i.valueAsNumber === (direction > 0 ? values.length - 1 : direction < 0 ? 0 : NaN)) { if (!loop) return stop(); if (alternate) direction = -direction; if (loopDelay !== null) { if (frame !== null) cancelAnimationFrame(frame), frame = null; if (interval !== null) clearInterval(interval), interval = null; timer = setTimeout(() => (step(), start()), loopDelay); return; } } if (delay === null) frame = requestAnimationFrame(tick); step(); } function step() { form.i.valueAsNumber = (form.i.valueAsNumber + direction + values.length) % values.length; form.i.dispatchEvent(new CustomEvent("input", {bubbles: true})); } form.i.oninput = event => { if (event && event.isTrusted && running()) stop(); form.value = values[form.i.valueAsNumber]; form.o.value = format(form.value, form.i.valueAsNumber, values); }; form.b.onclick = () => { if (running()) return stop(); direction = alternate && form.i.valueAsNumber === values.length - 1 ? -1 : 1; form.i.valueAsNumber = (form.i.valueAsNumber + direction) % values.length; form.i.dispatchEvent(new CustomEvent("input", {bubbles: true})); start(); }; form.i.oninput(); if (autoplay) start(); else stop(); Inputs.disposal(form).then(stop); return form; }34:T420,orbital_elements = { return [ { name: "Mean motion", symbol: "n", unit: "°/day", }, { name: "Semi-major axis", symbol: "a", unit: "AU", }, { name: "Eccentricity", symbol: "e", unit: "-", }, { name: "Inclination", symbol: "i", unit: "°", }, { name: "Longitude of the ascending node", symbol: "Ω", unit: "°", }, { name: "Argument of periapsis", symbol: "ω", unit: "°", }, { name: "Mean anomaly", symbol: "M", unit: "°", }, { name: "True anomaly", symbol: "𝜈", unit: "°", }, { name: "Mean longitude", symbol: "L", unit: "°", }, { name: "Longitude of periapsis", symbol: "ϖ", unit: "°", }, { name: "Obliquity of the ecliptic (Earth's tilt)", symbol: "ε", unit: "°", } ]; } 35:[{"id":"9b706e7a911221ab6e86515d8ec961068387dc13e824b2746869abf8e4429a2a0091d685b4894defdfd6845051fbb7ebe61cd8b0ccda7b87ee6161b573ed3440","href":"https://static.observableusercontent.com/files/9b706e7a911221ab6e86515d8ec961068387dc13e824b2746869abf8e4429a2a0091d685b4894defdfd6845051fbb7ebe61cd8b0ccda7b87ee6161b573ed3440","name":"astronomical_coordinates.svg","type":"image/svg+xml","lastModified":1717402205957.797,"size":352452},{"id":"0a424ff5fd07185150f7473f47e34dfdc2b5971292e5101a69a5c986f0c51b020ccba11da93e75e134fa38f564ab2f259eb39420c48c0b59bacd6f777bcbcbf5","href":"https://static.observableusercontent.com/files/0a424ff5fd07185150f7473f47e34dfdc2b5971292e5101a69a5c986f0c51b020ccba11da93e75e134fa38f564ab2f259eb39420c48c0b59bacd6f777bcbcbf5","name":"comet_Halley_diagram.jpg","type":"image/jpeg","lastModified":1717402205957.797,"size":9772},{"id":"0433e2ee1884764dcd41429480c16c966a8350c23c037b732c1466a98223410fb1daafbeba6b39a00da9cf382904c1ce6dbe287ed0f3aecdd49ceaf91819f6c5","href":"https://static.observableusercontent.com/files/0433e2ee1884764dcd41429480c16c966a8350c23c037b732c1466a98223410fb1daafbeba6b39a00da9cf382904c1ce6dbe287ed0f3aecdd49ceaf91819f6c5","name":"Earth.gif","type":"image/gif","lastModified":1717402205957.797,"size":302090},{"id":"b09a11b597234a184944c6166d69c2e5fe0a3d685183156e44bed1a885c2decfe4f5b88256118eba387c58acd14ddba9a6542cd29d09d538994677a39bd8057e","href":"https://static.observableusercontent.com/files/b09a11b597234a184944c6166d69c2e5fe0a3d685183156e44bed1a885c2decfe4f5b88256118eba387c58acd14ddba9a6542cd29d09d538994677a39bd8057e","name":"Sun.png","type":"image/png","lastModified":1717402205957.797,"size":19323},{"id":"0a3d0071b98093a9564817cf11cf4e84d416e6af08c912d461ab4a5202841cbb7bdbb40bfa6cd9e75320aaabfa275a3d7c50ac764af9d4cfd3eff8d627b51f41","href":"https://static.observableusercontent.com/files/0a3d0071b98093a9564817cf11cf4e84d416e6af08c912d461ab4a5202841cbb7bdbb40bfa6cd9e75320aaabfa275a3d7c50ac764af9d4cfd3eff8d627b51f41","name":"ellipse_drawn_with_pins_and_string.png","type":"image/png","lastModified":1723291268401.159,"size":588256}] 2f:{"title":"Orbital elements","cells":[{"id":0,"value":"# Orbital elements","pinned":false,"hidden":false,"mode":"md"},{"id":1120,"value":"

\n Orbital elements are the parameters required to describe a conic section orbit — most commonly an ellipse — in space. These parameters are typically used to express a celestial body's osculating orbit, which approximates its actual orbit at a specified time (the \"epoch\").\n

\n Orbits can be described using either (i) Keplerian elements or (ii) comet elements:\n

Keplerian elements — eccentricity (e), semi-major axis (a), true anomaly (𝜈), inclination (i), longitude of the ascending node (Ω), and argument of periapsis (ω).
Comet elements — eccentricity (e), periapsis distance, time of periapsis passage, inclination (i), longitude of the ascending node (Ω), and argument of periapsis (ω).

","pinned":false,"hidden":false,"mode":"html"},{"id":1212,"value":"$31","pinned":false,"hidden":false,"mode":"html"},{"id":970,"value":"

","pinned":false,"hidden":false,"mode":"html"},{"id":1435,"value":"

\n The following diagram highlights the four Keplerian elements that define trigonometric angles. As an illustrative example, we use the comet Halley, which has a pronounced eccentricity and inclination.\n

","pinned":false,"hidden":false,"mode":"html"},{"id":228,"value":"{\n // See also: https://observablehq.com/@mbostock/fullscreen-canvas\n return htl.html``\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":1259,"value":"$32","pinned":false,"hidden":false,"mode":"ojs"},{"id":313,"value":"viewof angle = Scrubber(angles, {autoplay: true, loop: true, alternate: false, initial: 50/angles_factor, format: f => `${Math.round(f,0)}°`})","pinned":false,"hidden":false,"mode":"ojs"},{"id":69,"value":"viewof stroke = Inputs.range([0.3,1.2], {step: 0.1, value: 0.5, label: \"Stroke size of Earth's orbit\"})","pinned":false,"hidden":false,"mode":"ojs"},{"id":742,"value":"viewof font_size = Inputs.range([9.5,12], {step: 0.5, value: 10, label: \"Font size\"})","pinned":false,"hidden":false,"mode":"ojs"},{"id":974,"value":"viewof opacity_ecliptic = Inputs.range([0,0.3], {step: 0.01, value: 0.02, label: \"Opacity of the ecliptic plane\"})","pinned":false,"hidden":false,"mode":"ojs"},{"id":981,"value":"viewof opacity_orbital = Inputs.range([0,0.3], {step: 0.01, value: 0.06, label: htl.html`Opacity of the
orbital plane`})","pinned":false,"hidden":false,"mode":"ojs"},{"id":20,"value":"viewof r_earth = Inputs.range([10,20], {value: 15, step: 1, label: \"Size of Earth\"})","pinned":false,"hidden":false,"mode":"ojs"},{"id":1453,"value":"
\n

Orbital elements in Observable notebooks

","pinned":false,"hidden":false,"mode":"html"},{"id":1456,"value":"

\n Orbital elements have been extensively utilized in numerous Observable notebooks authored by the writer of this text. Particularly noteworthy are those detailing the orbits of Mars, and other planets , allowing for a comparison of astronomical coordinates with observations made by Tycho Brahe at the close of the 16^th century.\n

","pinned":false,"hidden":false,"mode":"html"},{"id":1459,"value":"

\n \nUpon reviewing these notebooks, you will observe that the true anomaly (𝜈) is derived from the mean anomaly (M), a fictitious angle that greatly simplifies calculations. As a reliable approximation, the true anomaly (𝜈) can be obtained from the mean anomaly (M) and eccentricity (e) using the following relation:\n\n
\n ${tex`\\qquad v = M + \\frac{180}{\\pi} \\times [ (2e - \\frac{1}{4}e^3) \\times \\sin(M) \n + \\frac{5}{4}e^2 \\times \\sin(2M) + \\frac{13}{12} e^3 \\times \\sin(3M) ]`}\n

","pinned":false,"hidden":false,"mode":"html"},{"id":1482,"value":"

\n Furthermore, to complete observations from Earth, one must also account for the Earth's tilt (ε).\n

\n The table below lists the various orbital elements used to describe the orbits of the planets of the solar system. \n Parameters not mentioned above can be derived from the six primary elements. For instance, the mean motion (n) is derived from the orbital period, which can be calculated from the semi-major axis (a) using Kepler's third law:\n
${tex`\\qquad \\quad \\; n = \\sqrt{\\frac{G \\cdot M_{\\odot}}{a^3}}`}, \n
with ${tex`\\;\\;\\; G \\; = 6.674 \\times 10^{−11}`} N·m²·kg^-2, the gravitational constant,\n
and ${tex`\\;\\:\\, M_{\\odot} = 1.9884 \\times 10^{30}`} kg, the solar mass.\n
For Earth, with ${tex`a = 1 `} AU ${tex` = 1.496 \\times 10^{11}`} m, we find:\n
${tex`\\qquad n \\approx 0.98556`}°/day.\n

\n
","pinned":false,"hidden":false,"mode":"html"},{"id":1140,"value":"Inputs.table(orbital_elements,{\n columns: [\n \"symbol\",\n \"unit\",\n \"name\",\n ],\n header: {\n symbol: \"Symbol\",\n unit: \"Unit\",\n name: \"Description\",\n },\n align: {\n symbol: \"center\",\n unit: \"center\",\n name: \"left\",\n },\n width: {\n symbol: 40,\n unit: 50,\n name: 250,\n },\n maxWidth: 500,\n maxHeight: 280,\n height: 280\n})","pinned":false,"hidden":false,"mode":"ojs"},{"id":968,"value":"
\n

About the illustration in this notebook

\nThe SVG diagram embedded in this notebook was initially created using Adobe Illustrator. By carefully organizing the file (naming elements and layers, grouping, and ordering), it was then saved as an SVG file:\n

\nGenerator: Adobe Illustrator 16.0.0, SVG Export Plug-In . SVG Version: 6.00 Build 0)\n

\n

Since the SVG format is ASCII-encoded, it is easily editable in Observable. Indeed, in Observable, we can render SVG by typing \"svg\" followed by backticks: \n

\n svg`<svg></svg>` \n

\n

Most of the work in this notebook focused on making the illustration interactive.

You can download the original SVG file (Adobe Illustrator CS6) used to prepare the illustration.\n

\n

\n","pinned":false,"hidden":false,"mode":"html"},{"id":1053,"value":"## Useful links\n- [Orbital elements](https://en.wikipedia.org/wiki/Orbital_elements) – Comprehensive Wikipedia page about the orbital elements.\n- [NASA's FAQ: Orbital elements](https://ssd.jpl.nasa.gov/faq.html) – This webpage provides detailed information on orbital elements, including the diagram shown below. The interactive illustration in this notebook was inspired by this diagram.\n- [Comet Halley](https://en.wikipedia.org/wiki/Halley%27s_Comet) – The orbital elements of the comet 1P/Halley are available on this Wikipedia page.\n- [3D solar system](https://observablehq.com/@christophe-yamahata/3d-solar-system-model-using-three-js) – Observable notebook using Three.js to display an animated 3D model of the solar system.","pinned":false,"hidden":false,"mode":"md"},{"id":1062,"value":"

**Diagram of the orbit of comet 1P/Halley illustrating comet elements.** The point S represents the Sun, P represents perihelion, i is the inclination, lower case omega (ω) is the argument of perihelion, and upper case omega (Ω) is the longitude of the ascending node. The symbol ♈︎ to the far left (the rams horn - symbol of aries) indicates the direction of the vernal equinox. [Source: NASA/JPL]

","pinned":false,"hidden":false,"mode":"html"},{"id":1577,"value":"
\n

**Drawing an ellipse with two pins and a string.** As a bonus, this illustration demonstrates a simple technique for drawing an ellipse on paper using two pins and a string. The pins are placed at the focal points A and B of the ellipse.

[Source: Adapted from a vintage Gallaher's Cigarettes illustration. The text *\"Gallaher's Cigarettes\"* has been removed, and the letter *\"B\"* has been moved under the second pin.]

","pinned":false,"hidden":false,"mode":"html"},{"id":617,"value":"---\n## Appendix","pinned":false,"hidden":false,"mode":"md"},{"id":316,"value":"$33","pinned":false,"hidden":false,"mode":"ojs"},{"id":201,"value":"earth = await FileAttachment(\"Earth.gif\").url()","pinned":false,"hidden":false,"mode":"ojs"},{"id":78,"value":"sun = await FileAttachment(\"Sun.png\").url()","pinned":false,"hidden":false,"mode":"ojs"},{"id":963,"value":"astronomical_coordinates = await FileAttachment(\"astronomical_coordinates.svg\").url()","pinned":false,"hidden":false,"mode":"ojs"},{"id":1135,"value":"$34","pinned":false,"hidden":false,"mode":"ojs"},{"id":612,"value":"### Illustration parameters","pinned":false,"hidden":false,"mode":"md"},{"id":490,"value":"viewof tilt_angle = Inputs.range([-90,90], {step: 0.5, value: 23.5, label: \"Tilt angle\", placeholder: \"23.5°\", disabled: true})","pinned":false,"hidden":false,"mode":"ojs"},{"id":933,"value":"angles_factor = 1/50","pinned":false,"hidden":false,"mode":"ojs"},{"id":432,"value":"angles = Array.from({length: 360/angles_factor}, (_, i) => i*angles_factor)","pinned":false,"hidden":false,"mode":"ojs"},{"id":833,"value":"cx_sun = 330.8","pinned":false,"hidden":false,"mode":"ojs"},{"id":835,"value":"cy_sun = 220.25","pinned":false,"hidden":false,"mode":"ojs"},{"id":572,"value":"cx = 342.831","pinned":false,"hidden":false,"mode":"ojs"},{"id":575,"value":"cy=230.931","pinned":false,"hidden":false,"mode":"ojs"},{"id":578,"value":"rx = 246.258","pinned":false,"hidden":false,"mode":"ojs"},{"id":581,"value":"ry = 145.392","pinned":false,"hidden":false,"mode":"ojs"}],"files":"$W35","stdlib":"1","titling":"auto"}