2a:T805,md` In 2014, astronomer [Juna Kollmeier](https://en.wikipedia.org/wiki/Juna_Kollmeier)’s son, then aged 4, asked her: can moons have moons? In April, I, then aged 28, [tweeted](https://twitter.com/tophtucker/status/990454127992942598) pretty much the same thing. On Oct. 3, astronomers [Alex Teachey and David Kipping said](https://www.nasa.gov/press-release/astronomers-find-first-evidence-of-possible-moon-outside-our-solar-system) they suspected they’d detected a moon orbiting the exoplanet Kepler-1625b, which would be the first known moon outside our solar system. It’d be about the size of the planet Neptune, which is big, and has several moons, renewing interest in whether the [Kepler-1625b I](https://en.wikipedia.org/wiki/Kepler-1625b_I) exomoon might have any moons of its own, e.g. from [D.H. Forgan](https://arxiv.org/pdf/1810.02712v1.pdf) on Oct. 8. And so on Oct. 9, Kollmeier and Sean Raymond published a preprint of their paper named for the child’s question, which came to my attention via a graham cracker marshmellow confectionary brand’s [tweet](https://twitter.com/MoonPie/status/1050132123678588928), in which paper, I and presumably the son are relieved to hear, the authors write that “[i]t is natural to ask” this question. All you need is “the host moon must have a Hill sphere that is larger than its physical radius as well as its Roche limit.” OK! Sure. Except it’s tricky ’cause a planet is not a point and so satellites are subject to tidal forces which induce a destabilizing migration in many orbits. But nevertheless the paper ([arXiv](https://arxiv.org/abs/1810.03304), [pdf](https://arxiv.org/pdf/1810.03304.pdf)) finds that several moons in our solar system could support small stable long-lived submoons, as could Kepler-1625b I. I really like their **Figure 1**! So I’m just trying to reproduce it. The authors present this equation (adapted from [Barnes & O’Brien 2002](https://arxiv.org/pdf/astro-ph/0205035.pdf)) for the critical size of a moon that can support a sub-moon:`2b:T57f,key = md`

⚠️ Note that there is a \`fudge\` factor of ${fudge} in the numerator, because I am doing something wrong! So I eyeballed the diagrams and added that to get them to roughly match the authors’. 😳

Where: - ${tex`R_{moon}`} is the critical radius of a moon that can host a long-lived submoon - ${tex`M_{sub}`} is the (fixed) mass of the submoon in question - ${tex`a_{moon}`} is the moon’s orbital radius - ${tex`\rho_{moon}`} is the moon’s bulk density (assumed to be ${tex`2500\ kg\ m^{−3}`}) - ${tex`Q_{moon}`} is the moon’s [tidal quality factor](https://link.springer.com/referenceworkentry/10.1007%2F978-3-662-44185-5_1319) (assumed to be ${tex`100`}) - ${tex`k_{2,moon}`} is the moon’s [tidal Love number](https://en.wikipedia.org/wiki/Love_number) (assumed to be ${tex`0.25`} per [Moore & Schubert 2000](https://www.sciencedirect.com/science/article/pii/S0019103500964601)) - ${tex`M_p`} is the planet mass - ${tex`T`} is fixed at ${tex`4.6\ Gyr`} - ${tex`G`} is the gravitational constant - ${tex`f`} is the fraction of the moon’s [Hill sphere](https://en.wikipedia.org/wiki/Hill_sphere) out to which a submoon’s orbit is stable (assumed to be ${tex`0.4895`} per [Domingos et al. 2006](https://academic.oup.com/mnras/article/373/3/1227/1063626)) - ${tex`\pi`} is one of the few Aronofsky movies I still haven’t seen `2c:T5ba,viewof Rsub = { let val = 10 const x = d3.scaleLog() .domain([0.078125, 320]) .range([1, Math.min(300, width - 20)]) .base(10) const span = x.range()[1] - x.range()[0] const padding = 7.5 const div = d3.select(DOM.element("div")) const svg = div.append("svg") .attr("height", 15) .attr("width", span) .style("display", "block") .style("overflow", "visible") .style("margin-left", `${padding}px`) const input = div.append("input") .attr("min", x.range()[0]) .attr("max", x.range()[1]) .attr("value", x(val)) .style("width", `${span + padding * 2}px`) .style("display", "block") .attr("type", "range") .on("input", setValue) const label = svg.append("text") .attr("x", x(input.node().value)) .attr("y", 11) .attr("text-anchor", "middle") .text("hi") .attr("font-size", 10) .attr("font-family", "sans-serif") const ticks = svg.append("g") .attr("class", "axis") .attr("transform", "translate(0, 20)") .selectAll("line.tick") .data(x.ticks()) .enter() .append("line") .attr("x1", d => x(d) + 0.5) .attr("x2", d => x(d) + 0.5) .attr("y1", -6) render() function setValue(e) { val = x.invert(d3.event.target.value) render() } function render() { div.property("value", val).dispatch("input") label .attr("x", x(val)) .text(`${val.toPrecision(3)} km`) } return div.node() }2d:T12cf,{ const div = d3.select(this || DOM.element("div")) const planets = div.selectAll(".planet-system") .data(system.children.filter(d => d.children)) planets.exit().remove() planets.enter() .append("div") .attr("class", "planet-system") .each(initPlanet) .merge(planets) .each(renderPlanet) function initPlanet(planet, i) { const sel = d3.select(this) sel.append("h3").text(planet.data.name) const svg = sel.append("svg") .attr("class", "diagram") .attr("width", w + margin.left + margin.right) .attr("height", h + margin.top + margin.bottom) const orrery = sel.append("svg") .attr("class", "orrery") .style("transform", `translate( ${(w + margin.left + margin.right + width) / 2}px, ${margin.top + h/2}px)` ) const defs = svg.append("defs").html(` `) defs.append("clipPath") .attr("id", `clip${i}`) .append("rect") .attr("width", w) .attr("height", h) const g = svg.append("g") .attr("class", "inner") .attr("clip-path", `url(#clip${i})`) .attr("transform", `translate(${margin.left}, ${margin.top})`) .on("mousemove", function() { const p = d3.mouse(this) planet.data.mouse = {data: { distance: x.invert(p[0]) * planet.data.radius, radius: y.invert(p[1]), seed: planet.data.mouse ? planet.data.mouse.data.theta || 0 : 0, epoch: Date.now(), mouse: true, }} }) .on("mouseleave", () => { planet.data.mouse = undefined }) g.append("rect") .attr("width", w) .attr("height", h) .attr("fill", "rgba(255,255,255,0.9)") svg.append("text").text("Radius of moon (km)") .attr("y", 10) .style("font-weight", "bold") svg.append("text").text(`Orbital distance (${planet.data.name} radii)`) .attr("x", w + margin.left) .attr("y", h + margin.top + margin.bottom) .attr("text-anchor", "end") .style("font-weight", "bold") svg.append("g") .call(d3.axisBottom(x)) .attr("transform", `translate(${margin.left}, ${h + margin.top})`) svg.append("g") .call(d3.axisLeft( y.copy() .domain(y.domain().map(d => d / 1000)) )) .attr("transform", `translate(${margin.left}, ${margin.top})`) svg.append("line") .attr("x1", margin.left) .attr("x2", margin.left + w) .attr("y1", margin.top + 0.5) .attr("y2", margin.top + 0.5) svg.append("line") .attr("x1", margin.left + w + 0.5) .attr("x2", margin.left + w + 0.5) .attr("y1", margin.top) .attr("y2", margin.top + h) g.append("path") .attr("class", "hatched-area") .style("fill", "url(#diagonalHatch)") defs.append("clipPath") .attr("id", `moonmoon${i}`) .append("path") const moonmoonLabel = g.append("text") .attr("class", "moonmoon") .attr("clip-path", `url(#moonmoon${i})`) .attr("x", w * 0.9) .attr("y", h * 0.2) moonmoonLabel.append("tspan").text("MOONS OF").attr("dx", 0).attr("dy", "0em") moonmoonLabel.append("tspan").text("MOONS").attr("dx", -20).attr("dy", "1.5em") moonmoonLabel.append("tspan").text("STABLE").attr("dx", -20).attr("dy", "1.5em") g.selectAll("circle") .data(planet.children) .enter() .append("circle") .attr("cx", moon => x(moon.data.distance / planet.data.radius)) .attr("cy", moon => y(moon.data.radius)) } function renderPlanet(planet, i) { const sel = d3.select(this) const svg = sel.select("svg.diagram") const g = svg.select("g.inner") g.selectAll("circle") .attr("fill", moon => Rmoon(planet.data.mass, moon.data.distance) <= moon.data.radius ? "#f0f" : "#000") .attr("r", moon => r(moon.data.radius)) sel.selectAll(".orrery") .each(renderOrrery) const thresholdData = d3.range(1, 91) .map(d => d * planet.data.radius) .map(d => [d, Rmoon(planet.data.mass, d)]) const line = d3.line() .x(d => x(d[0] / planet.data.radius)) .y(d => y(d[1])) const area = d3.area() .x(d => x(d[0] / planet.data.radius)) .y0(y.range()[1]) .y1(d => Math.min(Math.max(y(d[1]), y.range()[1]), y.range()[0])) svg.select(".hatched-area") .attr("d", area(thresholdData)) svg.select(`#moonmoon${i}`).select("path") .attr("d", area(thresholdData)) } return div.node() }2e:T45f,md`#### Future work - Figure out where my fudge factor of ${fudge} is coming from; it’d help to see one fully-worked example by the authors; maybe I should email them. - Add Kepler-1625b system. - Add illustrations of the Roche limits and Hill spheres. - Add toggle to show both the sizes and distances of bodies on the same scale, though it gets really… empty. - I’d love to relax the ${tex`M \gg m`} assumptions (i.e. assuming the thing being orbited is way bigger than the thing orbiting), maybe treat Pluto–Charon more like a proper [two-body problem](https://en.wikipedia.org/wiki/Gravitational_two-body_problem). - Make it clear which planets the animated moons belong to!! They overflow and overlap and sometimes wander to confusing places. - Pattern fill and svg clip-path don’t work on my mobile Safari. Meanwhile I have an old draft of an eclipse notebook, which I think I was working on when I sent that April tweet. Haven’t looked at it in a while.` // And [this was sorta interesting](https://blogs.scientificamerican.com/observations/our-solar-system-is-even-stranger-than-we-thought/).2f:T58d5,planetsRaw = [ { "name": "Sun", "number": "", "parent": "", "distance": "", "period": "", "inclination": "", "eccentricity": "", "discoverer": "", "date": "", "radius": 695000, "mass": 1.99e+30 }, { "name": "Mercury", "number": "I", "parent": "Sun", "distance": 57910, "period": 87.97, "inclination": 7, "eccentricity": 0.21, "discoverer": "", "date": "", "radius": 2440, "mass": 3.3e+23 }, { "name": "Venus", "number": "II", "parent": "Sun", "distance": 108200, "period": 224.7, "inclination": 3.39, "eccentricity": 0.01, "discoverer": "", "date": "", "radius": 6052, "mass": 4.87e+24 }, { "name": "Earth", "number": "III", "parent": "Sun", "distance": 149600, "period": 365.26, "inclination": 0, "eccentricity": 0.02, "discoverer": "", "date": "", "radius": 6378, "mass": 5.97e+24 }, { "name": "Mars", "number": "IV", "parent": "Sun", "distance": 227940, "period": 686.98, "inclination": 1.85, "eccentricity": 0.09, "discoverer": "", "date": "", "radius": 3397, "mass": 6.42e+23 }, { "name": "Jupiter", "number": "V", "parent": "Sun", "distance": 778330, "period": 4332.71, "inclination": 1.31, "eccentricity": 0.05, "discoverer": "", "date": "", "radius": 71492, "mass": 1.9e+27 }, { "name": "Saturn", "number": "VI", "parent": "Sun", "distance": 1429400, "period": 10759.5, "inclination": 2.49, "eccentricity": 0.06, "discoverer": "", "date": "", "radius": 60268, "mass": 5.68e+26 }, { "name": "Uranus", "number": "VII", "parent": "Sun", "distance": 2870990, "period": 30685, "inclination": 0.77, "eccentricity": 0.05, "discoverer": "Herschel", "date": 1781, "radius": 25559, "mass": 8.68e+25 }, { "name": "Neptune", "number": "VIII", "parent": "Sun", "distance": 4504300, "period": 60190, "inclination": 1.77, "eccentricity": 0.01, "discoverer": "Adams", "date": 1846, "radius": 24766, "mass": 1.02e+26 }, { "name": "Pluto", "number": "IX", "parent": "Sun", "distance": 5913520, "period": 90550, "inclination": 17.15, "eccentricity": 0.25, "discoverer": "Tombaugh", "date": 1930, "radius": 1150, "mass": 1.27e+22 }, { "name": "Moon", "number": "I", "parent": "Earth", "distance": 384, "period": 27.32, "inclination": 5.14, "eccentricity": 0.05, "discoverer": "", "date": "", "radius": 1738, "mass": 7.35e+22 }, { "name": "Phobos", "number": "I", "parent": "Mars", "distance": 9, "period": 0.32, "inclination": 1, "eccentricity": 0.02, "discoverer": "Hall", "date": 1877, "radius": 11, "mass": 10800000000000000 }, { "name": "Deimos", "number": "II", "parent": "Mars", "distance": 23, "period": 1.26, "inclination": 1.8, "eccentricity": 0, "discoverer": "Hall", "date": 1877, "radius": 6, "mass": 1800000000000000 }, { "name": "Metis", "number": "XVI", "parent": "Jupiter", "distance": 128, "period": 0.29, "inclination": 0, "eccentricity": 0, "discoverer": "Synnott", "date": 1979, "radius": 20, "mass": 95600000000000000 }, { "name": "Adrastea", "number": "XV", "parent": "Jupiter", "distance": 129, "period": 0.3, "inclination": 0, "eccentricity": 0, "discoverer": "Jewitt", "date": 1979, "radius": 10, "mass": 19100000000000000 }, { "name": "Amalthea", "number": "V", "parent": "Jupiter", "distance": 181, "period": 0.5, "inclination": 0.4, "eccentricity": 0, "discoverer": "Barnard", "date": 1892, "radius": 94, "mass": 3500000000000000000 }, { "name": "Thebe", "number": "XIV", "parent": "Jupiter", "distance": 222, "period": 0.67, "inclination": 0.8, "eccentricity": 0.02, "discoverer": "Synnott", "date": 1979, "radius": 50, "mass": 777000000000000000 }, { "name": "Io", "number": "I", "parent": "Jupiter", "distance": 422, "period": 1.77, "inclination": 0.04, "eccentricity": 0, "discoverer": "Galileo", "date": 1610, "radius": 1821, "mass": 8.93e+22 }, { "name": "Europa", "number": "II", "parent": "Jupiter", "distance": 671, "period": 3.55, "inclination": 0.47, "eccentricity": 0.01, "discoverer": "Galileo", "date": 1610, "radius": 1565, "mass": 4.8e+22 }, { "name": "Ganymede", "number": "III", "parent": "Jupiter", "distance": 1070, "period": 7.15, "inclination": 0.19, "eccentricity": 0, "discoverer": "Galileo", "date": 1610, "radius": 2634, "mass": 1.48e+23 }, { "name": "Callisto", "number": "IV", "parent": "Jupiter", "distance": 1883, "period": 16.69, "inclination": 0.28, "eccentricity": 0.01, "discoverer": "Galileo", "date": 1610, "radius": 2403, "mass": 1.08e+23 }, { "name": "Themisto", "number": "XVIII", "parent": "Jupiter", "distance": 7507, "period": "", "inclination": "", "eccentricity": "", "discoverer": "Sheppard", "date": 2000, "radius": "", "mass": "" }, { "name": "Leda", "number": "XIII", "parent": "Jupiter", "distance": 11094, "period": 238.72, "inclination": 27, "eccentricity": 0.15, "discoverer": "Kowal", "date": 1974, "radius": 8, "mass": 5680000000000000 }, { "name": "Himalia", "number": "VI", "parent": "Jupiter", "distance": 11480, "period": 250.57, "inclination": 28, "eccentricity": 0.16, "discoverer": "Perrine", "date": 1904, "radius": 93, "mass": 9560000000000000000 }, { "name": "Lysithea", "number": "X", "parent": "Jupiter", "distance": 11720, "period": 259.22, "inclination": 29, "eccentricity": 0.11, "discoverer": "Nicholson", "date": 1938, "radius": 18, "mass": 77700000000000000 }, { "name": "Elara", "number": "VII", "parent": "Jupiter", "distance": 11737, "period": 259.65, "inclination": 28, "eccentricity": 0.21, "discoverer": "Perrine", "date": 1905, "radius": 38, "mass": 777000000000000000 }, { "name": "Ananke", "number": "XII", "parent": "Jupiter", "distance": 21200, "period": -631, "inclination": 147, "eccentricity": 0.17, "discoverer": "Nicholson", "date": 1951, "radius": 15, "mass": 38200000000000000 }, { "name": "Carme", "number": "XI", "parent": "Jupiter", "distance": 22600, "period": -692, "inclination": 163, "eccentricity": 0.21, "discoverer": "Nicholson", "date": 1938, "radius": 20, "mass": 95600000000000000 }, { "name": "Pasiphae", "number": "VIII", "parent": "Jupiter", "distance": 23500, "period": -735, "inclination": 147, "eccentricity": 0.38, "discoverer": "Melotte", "date": 1908, "radius": 25, "mass": 191000000000000000 }, { "name": "Sinope", "number": "IX", "parent": "Jupiter", "distance": 23700, "period": -758, "inclination": 153, "eccentricity": 0.28, "discoverer": "Nicholson", "date": 1914, "radius": 18, "mass": 77700000000000000 }, { "name": "Iocaste", "number": "XXIV", "parent": "Jupiter", "distance": 20216, "period": "", "inclination": "", "eccentricity": "", "discoverer": "Sheppard", "date": 2000, "radius": "", "mass": "" }, { "name": "Harpalyke", "number": "XXII", "parent": "Jupiter", "distance": 21132, "period": "", "inclination": "", "eccentricity": "", "discoverer": "Sheppard", "date": 2000, "radius": "", "mass": "" }, { "name": "Praxidike", "number": "XXVII", "parent": "Jupiter", "distance": 20964, "period": "", "inclination": "", "eccentricity": "", "discoverer": "Sheppard", "date": 2000, "radius": "", "mass": "" }, { "name": "Taygete", "number": "XX", "parent": "Jupiter", "distance": 23312, "period": "", "inclination": "", "eccentricity": "", "discoverer": "Sheppard", "date": 2000, "radius": "", "mass": "" }, { "name": "Chaldene", "number": "XXI", "parent": "Jupiter", "distance": 23387, "period": "", "inclination": "", "eccentricity": "", "discoverer": "Sheppard", "date": 2000, "radius": "", "mass": "" }, { "name": "Kalyke", "number": "XXIII", "parent": "Jupiter", "distance": 23745, "period": "", "inclination": "", "eccentricity": "", "discoverer": "Sheppard", "date": 2000, "radius": "", "mass": "" }, { "name": "Callirrhoe", "number": "XVII", "parent": "Jupiter", "distance": 24100, "period": "", "inclination": "", "eccentricity": "", "discoverer": "Sheppard", "date": 2000, "radius": "", "mass": "" }, { "name": "Megaclite", "number": "XIX", "parent": "Jupiter", "distance": 23911, "period": "", "inclination": "", "eccentricity": "", "discoverer": "Sheppard", "date": 2000, "radius": "", "mass": "" }, { "name": "Isonoe", "number": "XXVI", "parent": "Jupiter", "distance": 23078, "period": "", "inclination": "", "eccentricity": "", "discoverer": "Sheppard", "date": 2000, "radius": "", "mass": "" }, { "name": "Erinome", "number": "XXV", "parent": "Jupiter", "distance": 23168, "period": "", "inclination": "", "eccentricity": "", "discoverer": "Sheppard", "date": 2000, "radius": "", "mass": "" }, { "name": "Pan", "number": "XVIII", "parent": "Saturn", "distance": 134, "period": 0.58, "inclination": 0, "eccentricity": 0, "discoverer": "Showalter", "date": 1990, "radius": 10, "mass": "?" }, { "name": "Atlas", "number": "XV", "parent": "Saturn", "distance": 138, "period": 0.6, "inclination": 0, "eccentricity": 0, "discoverer": "Terrile", "date": 1980, "radius": 15, "mass": "?" }, { "name": "Prometheus", "number": "XVI", "parent": "Saturn", "distance": 139, "period": 0.61, "inclination": 0, "eccentricity": 0, "discoverer": "Collins", "date": 1980, "radius": 46, "mass": 270000000000000000 }, { "name": "Pandora", "number": "XVII", "parent": "Saturn", "distance": 142, "period": 0.63, "inclination": 0, "eccentricity": 0, "discoverer": "Collins", "date": 1980, "radius": 42, "mass": 220000000000000000 }, { "name": "Epimetheus", "number": "XI", "parent": "Saturn", "distance": 151, "period": 0.69, "inclination": 0.34, "eccentricity": 0.01, "discoverer": "Walker", "date": 1980, "radius": 57, "mass": 559000000000000000 }, { "name": "Janus", "number": "X", "parent": "Saturn", "distance": 151, "period": 0.69, "inclination": 0.14, "eccentricity": 0.01, "discoverer": "Dollfus", "date": 1966, "radius": 89, "mass": 1980000000000000000 }, { "name": "Mimas", "number": "I", "parent": "Saturn", "distance": 186, "period": 0.94, "inclination": 1.53, "eccentricity": 0.02, "discoverer": "Herschel", "date": 1789, "radius": 199, "mass": 37500000000000000000 }, { "name": "Enceladus", "number": "II", "parent": "Saturn", "distance": 238, "period": 1.37, "inclination": 0.02, "eccentricity": 0, "discoverer": "Herschel", "date": 1789, "radius": 249, "mass": 73000000000000000000 }, { "name": "Tethys", "number": "III", "parent": "Saturn", "distance": 295, "period": 1.89, "inclination": 1.09, "eccentricity": 0, "discoverer": "Cassini", "date": 1684, "radius": 530, "mass": 622000000000000000000 }, { "name": "Telesto", "number": "XIII", "parent": "Saturn", "distance": 295, "period": 1.89, "inclination": 0, "eccentricity": 0, "discoverer": "Smith", "date": 1980, "radius": 15, "mass": "?" }, { "name": "Calypso", "number": "XIV", "parent": "Saturn", "distance": 295, "period": 1.89, "inclination": 0, "eccentricity": 0, "discoverer": "Pascu", "date": 1980, "radius": 13, "mass": "?" }, { "name": "Dione", "number": "IV", "parent": "Saturn", "distance": 377, "period": 2.74, "inclination": 0.02, "eccentricity": 0, "discoverer": "Cassini", "date": 1684, "radius": 560, "mass": 1.05e+21 }, { "name": "Helene", "number": "XII", "parent": "Saturn", "distance": 377, "period": 2.74, "inclination": 0.2, "eccentricity": 0.01, "discoverer": "Laques", "date": 1980, "radius": 16, "mass": "?" }, { "name": "Rhea", "number": "V", "parent": "Saturn", "distance": 527, "period": 4.52, "inclination": 0.35, "eccentricity": 0, "discoverer": "Cassini", "date": 1672, "radius": 764, "mass": 2.31e+21 }, { "name": "Titan", "number": "VI", "parent": "Saturn", "distance": 1222, "period": 15.95, "inclination": 0.33, "eccentricity": 0.03, "discoverer": "Huygens", "date": 1655, "radius": 2575, "mass": 1.35e+23 }, { "name": "Hyperion", "number": "VII", "parent": "Saturn", "distance": 1481, "period": 21.28, "inclination": 0.43, "eccentricity": 0.1, "discoverer": "Bond", "date": 1848, "radius": 143, "mass": 17700000000000000000 }, { "name": "Iapetus", "number": "VIII", "parent": "Saturn", "distance": 3561, "period": 79.33, "inclination": 14.72, "eccentricity": 0.03, "discoverer": "Cassini", "date": 1671, "radius": 718, "mass": 1.59e+21 }, { "name": "Phoebe", "number": "IX", "parent": "Saturn", "distance": 12952, "period": -550.48, "inclination": 175.3, "eccentricity": 0.16, "discoverer": "Pickering", "date": 1898, "radius": 110, "mass": "" }, { "name": "Cordelia", "number": "VI", "parent": "Uranus", "distance": 50, "period": 0.34, "inclination": 0.14, "eccentricity": 0, "discoverer": "Voyager 2", "date": 1986, "radius": 13, "mass": "?" }, { "name": "Ophelia", "number": "VII", "parent": "Uranus", "distance": 54, "period": 0.38, "inclination": 0.09, "eccentricity": 0, "discoverer": "Voyager 2", "date": 1986, "radius": 16, "mass": "?" }, { "name": "Bianca", "number": "VIII", "parent": "Uranus", "distance": 59, "period": 0.43, "inclination": 0.16, "eccentricity": 0, "discoverer": "Voyager 2", "date": 1986, "radius": 22, "mass": "?" }, { "name": "Cressida", "number": "IX", "parent": "Uranus", "distance": 62, "period": 0.46, "inclination": 0.04, "eccentricity": 0, "discoverer": "Voyager 2", "date": 1986, "radius": 33, "mass": "?" }, { "name": "Desdemona", "number": "X", "parent": "Uranus", "distance": 63, "period": 0.47, "inclination": 0.16, "eccentricity": 0, "discoverer": "Voyager 2", "date": 1986, "radius": 29, "mass": "?" }, { "name": "Juliet", "number": "XI", "parent": "Uranus", "distance": 64, "period": 0.49, "inclination": 0.06, "eccentricity": 0, "discoverer": "Voyager 2", "date": 1986, "radius": 42, "mass": "?" }, { "name": "Portia", "number": "XII", "parent": "Uranus", "distance": 66, "period": 0.51, "inclination": 0.09, "eccentricity": 0, "discoverer": "Voyager 2", "date": 1986, "radius": 55, "mass": "?" }, { "name": "Rosalind", "number": "XIII", "parent": "Uranus", "distance": 70, "period": 0.56, "inclination": 0.28, "eccentricity": 0, "discoverer": "Voyager 2", "date": 1986, "radius": 27, "mass": "?" }, { "name": "Belinda", "number": "XIV", "parent": "Uranus", "distance": 75, "period": 0.62, "inclination": 0.03, "eccentricity": 0, "discoverer": "Voyager 2", "date": 1986, "radius": 34, "mass": "?" }, { "name": "Puck", "number": "XV", "parent": "Uranus", "distance": 86, "period": 0.76, "inclination": 0.31, "eccentricity": 0, "discoverer": "Voyager 2", "date": 1985, "radius": 77, "mass": "?" }, { "name": "Miranda", "number": "V", "parent": "Uranus", "distance": 130, "period": 1.41, "inclination": 4.22, "eccentricity": 0, "discoverer": "Kuiper", "date": 1948, "radius": 236, "mass": 65900000000000000000 }, { "name": "Ariel", "number": "I", "parent": "Uranus", "distance": 191, "period": 2.52, "inclination": 0, "eccentricity": 0, "discoverer": "Lassell", "date": 1851, "radius": 581, "mass": 1.35e+21 }, { "name": "Umbriel", "number": "II", "parent": "Uranus", "distance": 266, "period": 4.14, "inclination": 0, "eccentricity": 0, "discoverer": "Lassell", "date": 1851, "radius": 585, "mass": 1.17e+21 }, { "name": "Titania", "number": "III", "parent": "Uranus", "distance": 436, "period": 8.71, "inclination": 0, "eccentricity": 0, "discoverer": "Herschel", "date": 1787, "radius": 789, "mass": 3.53e+21 }, { "name": "Oberon", "number": "IV", "parent": "Uranus", "distance": 583, "period": 13.46, "inclination": 0, "eccentricity": 0, "discoverer": "Herschel", "date": 1787, "radius": 761, "mass": 3.01e+21 }, { "name": "Caliban", "number": "XVI", "parent": "Uranus", "distance": 7169, "period": -580, "inclination": 140, "eccentricity": 0.08, "discoverer": "Gladman", "date": 1997, "radius": 40, "mass": "" }, { "name": "Stephano", "number": "XX", "parent": "Uranus", "distance": 7948, "period": -674, "inclination": 143, "eccentricity": 0.24, "discoverer": "Gladman", "date": 1999, "radius": 15, "mass": "" }, { "name": "Sycorax", "number": "XVII", "parent": "Uranus", "distance": 12213, "period": -1289, "inclination": 153, "eccentricity": 0.51, "discoverer": "Nicholson", "date": 1997, "radius": 80, "mass": "" }, { "name": "Prospero", "number": "XVIII", "parent": "Uranus", "distance": 16568, "period": -2019, "inclination": 152, "eccentricity": 0.44, "discoverer": "Holman", "date": 1999, "radius": 20, "mass": "" }, { "name": "Setebos", "number": "XIX", "parent": "Uranus", "distance": 17681, "period": -2239, "inclination": 158, "eccentricity": 0.57, "discoverer": "Kavelaars", "date": 1999, "radius": 20, "mass": "" }, { "name": "Trinculo", "number": "XXI", "parent": "Uranus", "distance": "", "period": "", "inclination": "", "eccentricity": "", "discoverer": "Gladman", "date": 2001, "radius": "", "mass": "" }, { "name": "Naiad", "number": "III", "parent": "Neptune", "distance": 48, "period": 0.29, "inclination": 0, "eccentricity": 0, "discoverer": "Voyager 2", "date": 1989, "radius": 29, "mass": "?" }, { "name": "Thalassa", "number": "IV", "parent": "Neptune", "distance": 50, "period": 0.31, "inclination": 4.5, "eccentricity": 0, "discoverer": "Voyager 2", "date": 1989, "radius": 40, "mass": "?" }, { "name": "Despina", "number": "V", "parent": "Neptune", "distance": 53, "period": 0.33, "inclination": 0, "eccentricity": 0, "discoverer": "Voyager 2", "date": 1989, "radius": 74, "mass": "?" }, { "name": "Galatea", "number": "VI", "parent": "Neptune", "distance": 62, "period": 0.43, "inclination": 0, "eccentricity": 0, "discoverer": "Voyager 2", "date": 1989, "radius": 79, "mass": "?" }, { "name": "Larissa", "number": "VII", "parent": "Neptune", "distance": 74, "period": 0.55, "inclination": 0, "eccentricity": 0, "discoverer": "Reitsema", "date": 1989, "radius": 96, "mass": "?" }, { "name": "Proteus", "number": "VIII", "parent": "Neptune", "distance": 118, "period": 1.12, "inclination": 0, "eccentricity": 0, "discoverer": "Voyager 2", "date": 1989, "radius": 209, "mass": "?" }, { "name": "Triton", "number": "I", "parent": "Neptune", "distance": 355, "period": -5.88, "inclination": 157, "eccentricity": 0, "discoverer": "Lassell", "date": 1846, "radius": 1353, "mass": 2.15e+22 }, { "name": "Nereid", "number": "II", "parent": "Neptune", "distance": 5513, "period": 360.13, "inclination": 29, "eccentricity": 0.75, "discoverer": "Kuiper", "date": 1949, "radius": 170, "mass": "?" }, { "name": "Charon", "number": "I", "parent": "Pluto", "distance": 20, "period": 6.39, "inclination": 98.8, "eccentricity": 0, "discoverer": "Christy", "date": 1978, "radius": 603, "mass": 1.52e+21 }, { "name": "Nix", "number": "II", "parent": "Pluto", "distance": 49, "period": 24.86, "inclination": 0.2, "eccentricity": 0, "discoverer": "Stern", "date": 2005, "radius": 23, "mass": 2000000000000000000 }, { "name": "Hydra", "number": "III", "parent": "Pluto", "distance": 65, "period": 38.21, "inclination": 0.21, "eccentricity": 0.01, "discoverer": "Stern", "date": 2005, "radius": 30, "mass": 2000000000000000000 } ]30:T7f6,renderOrrery = function(planet) { const sel = d3.select(this) const planetOutline = sel.selectAll("circle.planet") .data([0]) planetOutline.enter() .append("circle") .attr("class", "planet") .style("stroke", planetColors[planet.data.name]) .style("fill", "none") .merge(planetOutline) .attr("r", r(planet.data.radius)) const kids = planet.data.mouse ? planet.children.concat([planet.data.mouse]) : planet.children const moon = sel.selectAll("g.moon") .data(kids) moon.exit().remove() const moonEnter = moon.enter() .append("g") .attr("class", "moon") moonEnter.append("circle") .attr("class", "moon-circle") moonEnter.merge(moon) .each(function(moon) { const sel = d3.select(this) const circle = sel.select("circle") const supercritical = Rmoon(planet.data.mass, moon.data.distance) <= moon.data.radius const distance = x(moon.data.distance / planet.data.radius) // if no mass for moon, calculate from assumed ρmoon density const moonMass = isNaN(moon.data.mass) ? ρmoon * (4 / 3) * π * moon.data.radius ** 3 : moon.data.mass const speed = getAngularSpeed(planet.data.mass, moonMass, moon.data.distance) * orbitalSpeedFactor const theta = (now - moon.data.epoch) * speed + moon.data.seed moon.data.theta = theta sel .attr("transform", `translate(${polarToCartesian(distance, theta)})`) .classed("mousemoon", moon.data.mouse) circle .attr("r", r(moon.data.radius)) .attr("fill", supercritical ? "#f0f" : "#000") const moonmoon = sel.selectAll("circle.moonmoon") .data(supercritical ? [0] : []) moonmoon.exit().remove() moonmoon.enter() .append("circle") .attr("class", "moonmoon") .merge(moonmoon) .attr("r", r(Rsub * 1000)) .attr("transform", `translate(${polarToCartesian( r(moon.data.radius) + 2, now / 100 )})`) }) }31:[] 29:{"title":"Re: “Can Moons Have Moons?”","cells":[{"id":0,"value":"md`# Re: “Can Moons Have Moons?”\n### [Kollmeier & Raymond (2018)](https://arxiv.org/pdf/1810.03304.pdf)`","pinned":false,"hidden":false,"mode":"ojs"},{"id":397,"value":"$2a","pinned":false,"hidden":false,"mode":"ojs"},{"id":8,"value":"tex.block`R_{moon} \\geq \\left[\\frac{39M_{sub}k_{2,moon}T\\sqrt{G}}{2(4\\pi\\rho_{moon})^{8/3}Q_{moon}}\\left(\\frac{3M_p}{(fa_{moon})^3}\\right)^{13/6}\\right]^{1/3}`","pinned":false,"hidden":false,"mode":"ojs"},{"id":415,"value":"md`Or, in code:`","pinned":false,"hidden":false,"mode":"ojs"},{"id":58,"value":"Rmoon = (Mp, amoon) => (\n (\n (\n (39 * Msub * k2moon * T * fudge * Math.sqrt(G)) / \n (2 * (4 * π * ρmoon) ** (8 / 3) * Qmoon)\n ) * \n (\n (3 * Mp) /\n (f * amoon) ** 3\n ) ** (13 / 6)\n ) ** (1 / 3)\n)","pinned":true,"hidden":false,"mode":"ojs"},{"id":36,"value":"$2b","pinned":false,"hidden":false,"mode":"ojs"},{"id":530,"value":"md`The submoon mass ${tex`M_{sub}`} is here derived from the submoon radius ${tex`R_{sub}`}, with an assumed density of ${tex`2000\\ kg\\ m^{−3}`}, “generally appropriate for smaller bodies”, per the paper. Fiddle with it and watch how smaller submoons have a larger potential region of stability in the parameter space:`","pinned":false,"hidden":false,"mode":"ojs"},{"id":145,"value":"$2c","pinned":false,"hidden":false,"mode":"ojs"},{"id":272,"value":"$2d","pinned":false,"hidden":false,"mode":"ojs"},{"id":959,"value":"viewof radiusScale = radio({\n title: 'Radius scale',\n description: 'Choose whether bodies should be scaled linearly by their radius or by the logarithm of their radius',\n options: [\n { label: 'Log radius', value: 'scaleLog' },\n { label: 'Linear radius', value: 'scaleLinear' },\n ],\n value: 'scaleLog'\n})","pinned":false,"hidden":false,"mode":"ojs"},{"id":1290,"value":"html`

\n${md`### ⚠️ Warnings\n- These were reproducing by eyeballing the diagrams in the paper and fiddling with an inexplicable fudge factor until they seemed to match. I do all the calculations and then just multiply by ${fudge} for no good reason except that it makes it work.\n- In the cute little orreries, the orbits are just circles, and I’m assuming that the mass of the central body is much larger than the orbiting satellites.\n- The distance at which the imagined sub-moon orbits its host is totally made-up.\n`}\n

`","pinned":false,"hidden":false,"mode":"ojs"},{"id":520,"value":"md`---`","pinned":false,"hidden":false,"mode":"ojs"},{"id":876,"value":"$2e","pinned":false,"hidden":false,"mode":"ojs"},{"id":528,"value":"md`#### Changelog \n\n- *Oct. 10* — Ugh this doesn’t match the paper... I think I’m messing up unit conversations... it’s a mess... especially meters and kilometers... and the csv I pulled planetary data from has orbital distance in thousands of km and radius in km... but I think I’m accounting for that... but between the assumed mass densities, the gravitational constant, and... yeah... gotta sleep, will clean up later, and hopefully get to match. \n\n- *Oct. 11* — Update! T was in gigayears and G was in seconds, lmao. Fudge factor reduced from 1e18 to 14! LOL! \n\n- *Oct. 28* — Re-published with more text, better organization, more accurate orbital velocities, lil imagined animated submoons, and mouseover ghost.\n\nSee also [Gizmodo](https://gizmodo.com/astronomers-wonder-can-moons-have-moons-1829639045?IR=T) and [Newsweek](https://www.newsweek.com/child-question-spark-research-moon-orbit-1165948).`","pinned":false,"hidden":false,"mode":"ojs"},{"id":1288,"value":"md`---`","pinned":false,"hidden":false,"mode":"ojs"},{"id":1180,"value":"md`#### Data\n`","pinned":false,"hidden":false,"mode":"ojs"},{"id":775,"value":"planetColors = ({\n \"Earth\": \"blue\",\n \"Mars\": \"red\",\n \"Jupiter\": \"orange\",\n \"Saturn\": \"gold\",\n \"Uranus\": \"blue\",\n \"Neptune\": \"green\",\n \"Pluto\": \"brown\"\n})","pinned":false,"hidden":false,"mode":"ojs"},{"id":139,"value":"$2f","pinned":false,"hidden":false,"mode":"ojs"},{"id":547,"value":"planets = planetsRaw.map(({name, parent, mass, radius, distance}) => (\n {\n name, \n parent, \n mass, \n radius: radius * 1e3, \n distance: distance * 1e6, \n seed: Math.random() * Math.PI * 2,\n epoch: 0,\n }\n))","pinned":false,"hidden":false,"mode":"ojs"},{"id":141,"value":"system = d3.stratify()\n .id(d => d.name)\n .parentId(d => d.parent)\n (planets)","pinned":false,"hidden":false,"mode":"ojs"},{"id":1202,"value":"md`#### Astronomical constants\nSee [above](#key) for explanations and assumptions.`","pinned":false,"hidden":false,"mode":"ojs"},{"id":5,"value":"G = 6.674e-11","pinned":false,"hidden":false,"mode":"ojs"},{"id":48,"value":"T = 4.6 * 3.154e16","pinned":false,"hidden":false,"mode":"ojs"},{"id":68,"value":"f = 0.4895","pinned":false,"hidden":false,"mode":"ojs"},{"id":70,"value":"π = Math.PI","pinned":false,"hidden":false,"mode":"ojs"},{"id":129,"value":"ρmoon = 2500","pinned":false,"hidden":false,"mode":"ojs"},{"id":132,"value":"Qmoon = 100","pinned":false,"hidden":false,"mode":"ojs"},{"id":134,"value":"k2moon = 0.25","pinned":false,"hidden":false,"mode":"ojs"},{"id":344,"value":"Msub = ((4 / 3) * π * (Rsub * 1000) ** 3) * 2000","pinned":false,"hidden":false,"mode":"ojs"},{"id":578,"value":"fudge = 14 // 1e18","pinned":false,"hidden":false,"mode":"ojs"},{"id":1182,"value":"md`#### Chart options`","pinned":false,"hidden":false,"mode":"ojs"},{"id":821,"value":"w = Math.min(400, width - margin.left - margin.right)","pinned":false,"hidden":false,"mode":"ojs"},{"id":823,"value":"h = w * 0.6","pinned":false,"hidden":false,"mode":"ojs"},{"id":826,"value":"margin = ({top: 15, right: 0, bottom: 30, left: 33})","pinned":false,"hidden":false,"mode":"ojs"},{"id":1104,"value":"orbitalSpeedFactor = 1/3000","pinned":false,"hidden":false,"mode":"ojs"},{"id":1184,"value":"md`#### Scales`","pinned":false,"hidden":false,"mode":"ojs"},{"id":830,"value":"r = d3[radiusScale]()\n .domain([1000, 5000000])\n .range([0.1, 5])","pinned":false,"hidden":false,"mode":"ojs"},{"id":835,"value":"x = d3.scaleLinear()\n .domain([0, 90])\n .range([0, w])","pinned":false,"hidden":false,"mode":"ojs"},{"id":839,"value":"y = d3.scaleLog()\n .domain([10000, 5000000])\n .range([h, 0])","pinned":false,"hidden":false,"mode":"ojs"},{"id":1187,"value":"md`#### Rendering`","pinned":false,"hidden":false,"mode":"ojs"},{"id":843,"value":"$30","pinned":false,"hidden":false,"mode":"ojs"},{"id":1197,"value":"md`#### Helpers`","pinned":false,"hidden":false,"mode":"ojs"},{"id":1080,"value":"// https://en.wikipedia.org/wiki/Orbital_speed#Mean_orbital_speed\ngetAngularSpeed = (M, m, r) => {\n const speed = Math.sqrt((M + m) / r)\n const circumference = 2 * π * r\n const angularSpeed = (speed / circumference) * 2 * π\n return angularSpeed\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":1108,"value":"polarToCartesian = (r, θ) => [\n r * Math.cos(θ),\n r * Math.sin(θ)\n]","pinned":false,"hidden":false,"mode":"ojs"},{"id":1239,"value":"md`#### Imports`","pinned":false,"hidden":false,"mode":"ojs"},{"id":957,"value":"import {radio} from \"@jashkenas/inputs\"","pinned":false,"hidden":false,"mode":"ojs"},{"id":143,"value":"d3 = require(\"d3\")","pinned":false,"hidden":false,"mode":"ojs"},{"id":1274,"value":"md`#### Styles`","pinned":false,"hidden":false,"mode":"ojs"},{"id":299,"value":"html``","pinned":false,"hidden":false,"mode":"ojs"},{"id":1386,"value":"width > 800 ? html`` : html``","pinned":false,"hidden":false,"mode":"ojs"}],"files":"$W31","stdlib":"1","titling":"auto"}