2a:T7ac,When a small object moves near two large orbiting bodies (like the Sun and Jupiter, or the Earth and Moon), it feels two main forces: the gravitational pull of both bodies and the centrifugal pseudoforce arising from the rotating frame of reference. In 1772, mathematician Joseph-Louis Lagrange discovered five specific locations where these forces balance each other perfectly, allowing a small mass (assumed massless) to orbit at the same rotational speed as the secondary body. Known as Lagrange points (${tex`L_1`} through ${tex`L_5`}), these regions act as natural gravitational parking spots in space. While the collinear points (${tex`L_1`}, ${tex`L_2`}, ${tex`L_3`}) are unstable saddle points, the triangular points (${tex`L_4`} and ${tex`L_5`}) act like shallow valleys that can hold objects in stable, looping paths. This visualization displays the effective potential contours and zero-velocity boundaries in the **Circular Restricted Three-Body Problem (CR3BP)**. The *x* and *y* coordinates represent dimensionless spatial positions in a rotating coordinate frame centered at the system's center of mass (barycenter), where the **primary** and **secondary** bodies are held stationary along the *x*-axis. Select a preset below or click anywhere on the chart to drop a test mass from rest (${tex`v_0 = 0`}) and explore its orbit. ✦︎ This notebook was built with the assistance of Gemini AI. ---2b:T179b,lagrange_points = { const width = 700; const height = width; const margin = { top: 20, right: 20, bottom: 50, left: 50 }; const innerWidth = width - margin.left - margin.right; const innerHeight = height - margin.top - margin.bottom; const xScale = d3.scaleLinear().domain([-bound, bound]).range([0, innerWidth]); const yScale = d3.scaleLinear().domain([-bound, bound]).range([innerHeight, 0]); const svg = d3 .create("svg") .attr("width", width) .attr("height", height) .attr("viewBox", [0, 0, width, height]) .style("max-width", "100%") .style("height", "auto") .style("background", "#ffffff") .style("cursor", "crosshair"); const defs = svg.append("defs"); defs.append("clipPath") .attr("id", "chart-clip") .append("rect") .attr("width", innerWidth) .attr("height", innerHeight); const applyTextOutline = (selection) => { selection .attr("stroke", "#ffffff") .attr("stroke-width", "3.5px") .attr("stroke-linejoin", "round") .style("paint-order", "stroke fill"); }; const g = svg .append("g") .attr("transform", `translate(${margin.left},${margin.top})`); const xAxis = d3.axisBottom(xScale).ticks(10); const yAxis = d3.axisLeft(yScale).ticks(10); const xAxisG = g.append("g") .attr("transform", `translate(0,${innerHeight})`) .call(xAxis); xAxisG.append("text") .attr("x", innerWidth / 2) .attr("y", 40) .attr("fill", "#000") .attr("font-size", "14px") .attr("font-weight", "bold") .text("x") .call(applyTextOutline); const yAxisG = g.append("g") .call(yAxis); yAxisG.append("text") .attr("transform", "rotate(-90)") .attr("x", -innerHeight / 2) .attr("y", -40) .attr("fill", "#000") .attr("font-size", "14px") .attr("font-weight", "bold") .attr("text-anchor", "middle") .text("y") .call(applyTextOutline); xAxisG.selectAll(".tick text").call(applyTextOutline); yAxisG.selectAll(".tick text").call(applyTextOutline); const plotArea = g.append("g") .attr("clip-path", "url(#chart-clip)"); const values = new Float64Array(nx ** 2); const step = (2 * bound) / (nx - 1); const xSpan = d3.range(nx).map((i) => -bound + i * step); const ySpan = d3.range(nx).map((j) => -bound + j * step); for (let j = 0; j < nx; ++j) { for (let i = 0; i < nx; ++i) { const x = xSpan[i]; const y = ySpan[j]; const r1 = Math.hypot(x + mu, y); const r2 = Math.hypot(x - 1 + mu, y); const omega = 0.5 * (x * x + y * y) + (1 - mu) / r1 + mu / r2 + 0.5 * mu * (1 - mu); values[j * nx + i] = omega; } } const contours = d3 .contours() .size([nx, nx]) .thresholds(d3.range(1.4, 2.2, 0.03)); const contourScaleX = d3 .scaleLinear() .domain([0, nx - 1]) .range([xScale(-bound), xScale(bound)]); const contourScaleY = d3 .scaleLinear() .domain([0, nx - 1]) .range([yScale(-bound), yScale(bound)]); const geoPath = d3.geoPath( d3.geoTransform({ point: function (x, y) { this.stream.point(contourScaleX(x), contourScaleY(y)); } }) ); plotArea.append("g") .attr("class", "contours") .selectAll("path") .data(contours(values)) .join("path") .attr("d", geoPath) .attr("fill", "none") .attr("stroke", "#4682b4") .attr("stroke-width", 0.7) .attr("opacity", 0.65); if (trajectory && trajectory.length > 0) { const lineGenerator = d3 .line() .x((d) => xScale(d.x)) .y((d) => yScale(d.y)); plotArea.append("path") .datum(trajectory) .attr("fill", "none") .attr("stroke", "black") .attr("stroke-width", 0.8) .attr("d", lineGenerator); } // Primary Body g.append("circle") .attr("cx", xScale(-mu)) .attr("cy", yScale(0)) .attr("r", 12) .attr("fill", "#ffa500") .attr("stroke", "#cc8400") .attr("stroke-width", 1.5); // Secondary Body g.append("circle") .attr("cx", xScale(1 - mu)) .attr("cy", yScale(0)) .attr("r", 6) .attr("fill", "#4682b4") .attr("stroke", "#2b5278") .attr("stroke-width", 1.5); const gamma = Math.cbrt(mu / 3); const L1_x = 1 - mu - gamma; const L2_x = 1 - mu + gamma; const L3_x = -(1 + (5 / 12) * mu); const lagrangePoints = [ { name: "L1", x: L1_x, y: 0, dx: -15, dy: -8 }, { name: "L2", x: L2_x, y: 0, dx: 10, dy: 4 }, { name: "L3", x: L3_x, y: 0, dx: 8, dy: -8 }, { name: "L4", x: 0.5 - mu, y: Math.sqrt(3) / 2, dx: 8, dy: -8 }, { name: "L5", x: 0.5 - mu, y: -Math.sqrt(3) / 2, dx: 8, dy: 14 } ]; lagrangePoints.forEach((pt) => { g.append("circle") .attr("cx", xScale(pt.x)) .attr("cy", yScale(pt.y)) .attr("r", 4) .attr("fill", "red"); g.append("text") .attr("x", xScale(pt.x) + pt.dx) .attr("y", yScale(pt.y) + pt.dy) .attr("fill", "#000000") .attr("font-size", "13px") .attr("font-weight", "bold") .attr("font-family", "sans-serif") .text(pt.name) .call(applyTextOutline); }); if (particleStart) { g.append("circle") .attr("cx", xScale(particleStart[0])) .attr("cy", yScale(particleStart[1])) .attr("r", 4) .attr("fill", "black"); } g.append("rect") .attr("width", innerWidth) .attr("height", innerHeight) .attr("fill", "none") .attr("stroke", "#000000") .attr("stroke-width", 1.5); svg.on("click", (event) => { const [coordsX, coordsY] = d3.pointer(event, g.node()); const clickX = xScale.invert(coordsX); const clickY = yScale.invert(coordsY); mutable customStart = [clickX, clickY]; if (system?.name !== "Custom Orbit") { const selectEl = viewof system.querySelector("select"); if (selectEl) { selectEl.selectedIndex = 0; selectEl.dispatchEvent(new Event("input", { bubbles: true })); selectEl.dispatchEvent(new Event("change", { bubbles: true })); } } }); return svg.node(); }2c:T495,--- ## Lucy Mission and the Jupiter Trojan Asteroids Launched on October 16, 2021, NASA's **Lucy Mission** is the first space mission dedicated to studying the **Jupiter Trojan asteroids**. The mission is named after the famous Lucy hominin fossil because these asteroids are thought to be planetary "fossils" that may preserve primordial material from the early Solar System. These asteroids librate in stable orbits around the ${tex`L_4`} and ${tex`L_5`} Lagrange points of the Sun-Jupiter system. 2d:T411,viewof display_inspect = { const el = htl.html`

Appendix

⇣ Check to inspect the notebook script

`; const checkbox = el.querySelector("input"); const spacer = el.querySelector("div"); function update() { const show = checkbox.checked; document.querySelectorAll(".observablehq--inspect").forEach(node => { if (el.contains(node)) return; node.style.display = show ? "" : "none"; }); spacer.style.height = show ? "0" : "20vh"; } checkbox.addEventListener("change", update); const observer = new MutationObserver(update); observer.observe(document.body, { childList: true, subtree: true }); update(); invalidation.then(() => observer.disconnect()); return el; }2e:T454,systems = new Map([ [ "Custom Orbit (Free Exploration)", { name: "Custom Orbit", mu: null, start: null } ], [ "Sun–Jupiter: Horseshoe Orbit", { name: "Sun–Jupiter (Horseshoe)", mu: 0.0009537, start: [-1.005, 0.0] } ], [ "Sun–Jupiter: Trojan Asteroid (Greek camp, L4)", { name: "Sun–Jupiter Greek camp, L4", mu: 0.0009537, start: [0.998 * (0.5 - 0.0009537), 0.998 * (Math.sqrt(3) / 2)] } ], [ "Sun–Jupiter: Trojan Asteroid (Trojan Camp, L5)", { name: "Sun–Jupiter Trojan Camp, L5", mu: 0.0009537, start: [0.998 * (0.5 - 0.0009537), -0.998 * (Math.sqrt(3) / 2)] } ], [ "Sun–Earth: Earth Trojan Asteroid (2020 XL5)", { name: "Sun–Earth (2020 XL5)", mu: 0.000003, start: [ -0.000003 + Math.cos(Math.PI / 3 - 0.2), Math.sin(Math.PI / 3 - 0.2) ] } ], [ "Earth–Moon: Kordylewski Dust Cloud (L4)", { name: "Earth–Moon (L4 Kordylewski Cloud)", mu: 0.01215, start: [0.992 * (0.5 - 0.01215), 0.992 * (Math.sqrt(3) / 2)] } ] ]);2f:[] 28:{"title":"Lagrange Points and Planet Trojans","cells":[{"id":0,"value":"# Lagrange Points and Planet Trojans\n## Circular Restricted Three-Body Problem (CR3BP)","pinned":false,"hidden":false,"mode":"md"},{"id":452,"value":"$2a","pinned":false,"hidden":false,"mode":"md"},{"id":398,"value":"**Click anywhere on the chart** to set the initial position of a test particle with zero initial velocity. \n**Adjust the Mass Ratio (${tex`\\mu`}) slider** to change the mass fraction of the secondary body relative to the total system mass.","pinned":false,"hidden":false,"mode":"md"},{"id":677,"value":"viewof system = {\n const input = Inputs.select(systems, {\n label: \"Preset System\",\n value: systems.get(\"Sun–Jupiter: Horseshoe Orbit\")\n });\n input.querySelector(\"select\").style.width = \"310px\";\n \n return input;\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":7,"value":"viewof mu = {\n const previousValue = this ? (this.querySelector(\"input\")?.valueAsNumber ?? this.value) : 0.0009537;\n const initialValue = (system?.mu !== null && system?.mu !== undefined) ? system.mu : previousValue;\n\n const range = Inputs.range([0.000001, 0.05], {\n value: initialValue,\n step: 0.000001,\n label: html`Mass Ratio (${tex`\\mu`})`\n });\n\n range.addEventListener(\"input\", (e) => {\n if (e.isTrusted) {\n mutable userMovedMu = true;\n }\n });\n\n return range;\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":22,"value":"$2b","pinned":false,"hidden":false,"mode":"ojs"},{"id":823,"value":"$2c","pinned":false,"hidden":false,"mode":"md"},{"id":10,"value":"$2d","pinned":false,"hidden":false,"mode":"ojs"},{"id":680,"value":"$2e","pinned":false,"hidden":false,"mode":"ojs"},{"id":566,"value":"mutable customStart = [0.1968,0.981]","pinned":false,"hidden":false,"mode":"ojs"},{"id":793,"value":"bound = 1.3","pinned":false,"hidden":false,"mode":"ojs"},{"id":740,"value":"nx = 200;","pinned":false,"hidden":false,"mode":"ojs"},{"id":620,"value":"// Side-effect cell: Clean reactive coordinator\n{\n if (userMovedMu === true && system?.name !== \"Custom Orbit\") {\n const selectEl = viewof system.querySelector(\"select\");\n if (selectEl) {\n selectEl.selectedIndex = 0;\n selectEl.dispatchEvent(new Event(\"input\", { bubbles: true }));\n selectEl.dispatchEvent(new Event(\"change\", { bubbles: true }));\n }\n }\n\n if (system?.name !== \"Custom Orbit\") {\n mutable userMovedMu = false;\n }\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":615,"value":"mutable userMovedMu = false","pinned":false,"hidden":false,"mode":"ojs"},{"id":13,"value":"sun = [-mu, 0]","pinned":false,"hidden":false,"mode":"ojs"},{"id":16,"value":"planet = [1 - mu, 0]","pinned":false,"hidden":false,"mode":"ojs"},{"id":70,"value":"Lpoints = {\n const rawR = Math.cbrt(mu / 3); \n \n return [\n {\n name: \"L1\", \n x: 1 - mu - rawR + (1/3) * Math.pow(rawR, 2) - (1/27) * Math.pow(rawR, 3), \n y: 0\n },\n {\n name: \"L2\", \n x: 1 - mu + rawR + (1/3) * Math.pow(rawR, 2) - (1/27) * Math.pow(rawR, 3), \n y: 0\n },\n {\n name: \"L3\", \n x: -1 - mu + (7/12) * mu, \n y: 0\n },\n {\n name: \"L4\", \n x: 0.5 - mu, \n y: Math.sqrt(3) / 2\n },\n {\n name: \"L5\", \n x: 0.5 - mu, \n y: -Math.sqrt(3) / 2\n }\n ];\n}\n","pinned":false,"hidden":false,"mode":"ojs"},{"id":234,"value":"particleStart = {\n if (system && system.name !== \"Custom Orbit\" && Array.isArray(system.start)) {\n return system.start;\n }\n return customStart;\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":237,"value":"trajectory = {\n if (!particleStart || mu == null) return [];\n\n let state = [\n particleStart[0],\n particleStart[1],\n 0, // vx\n 0 // vy\n ];\n\n const dt = 0.005;\n const steps = 500000;\n const stride = 5;\n\n const result = [];\n\n for (let i = 0; i < steps; i++) {\n if (i % stride === 0) {\n result.push({\n x: state[0],\n y: state[1],\n t: i * dt\n });\n }\n\n if (Math.abs(state[0]) > 1.35 || Math.abs(state[1]) > 1.35) {\n break;\n }\n\n state = rk4(state, dt);\n }\n\n return result;\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":19,"value":"C = (x, y) => {\n const r1 = Math.hypot(x + mu, y);\n const r2 = Math.hypot(x - 1 + mu, y);\n\n return x*x + y*y\n + 2*(1 - mu)/r1\n + 2*mu/r2;\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":57,"value":"jacobi = (x, y) => {\n const r1 = Math.hypot(x + mu, y)\n const r2 = Math.hypot(x - 1 + mu, y)\n\n return x * x + y * y \n + 2 * (1 - mu) / r1 \n + 2 * mu / r2 \n + mu * (1 - mu)\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":228,"value":"acceleration = (x, y, vx, vy) => {\n const r1 = Math.hypot(x + mu, y)\n const r2 = Math.hypot(x - 1 + mu, y)\n\n return [\n 2*vy\n + x\n - (1 - mu)*(x + mu)/(r1**3)\n - mu*(x - 1 + mu)/(r2**3),\n\n -2*vx\n + y\n - (1 - mu)*y/(r1**3)\n - mu*y/(r2**3)\n ]\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":231,"value":"rk4 = (state, dt) => {\n\n const f = ([x, y, vx, vy]) => {\n const [ax, ay] = acceleration(x, y, vx, vy)\n return [vx, vy, ax, ay]\n }\n\n const add = (a, b, s) =>\n a.map((v, i) => v + s*b[i])\n\n const k1 = f(state)\n const k2 = f(add(state, k1, dt/2))\n const k3 = f(add(state, k2, dt/2))\n const k4 = f(add(state, k3, dt))\n\n return state.map(\n (v, i) =>\n v + dt/6 * (\n k1[i] + 2*k2[i] + 2*k3[i] + k4[i]\n )\n )\n}","pinned":false,"hidden":false,"mode":"ojs"}],"files":"$W2f","stdlib":"1","titling":"auto"}