32:T554,

This is a continuation of my previous notebook on Tycho Brahe's observations of Mars (♂).

Tycho Brahe's astronomical observations of Mars focused on measuring its declination for an extended period of nearly twenty years (1582 – 1600). It it these meticulous measurements that later allowed Johannes Kepler to explain the trajectory of the planets in the solar system.

Tycho Brahe carried out most of his observations on the island of Hven [ Lat. ${φo > 0 ? '+'+φo.toFixed(2)+'° (North)' : (φo < 0 ? φo.toFixed(2)+'° (South)' : φo.toFixed(2)+'°')}, Lon. ${λo > 0 ? '+'+λo.toFixed(2)+'° (East)' : (λo < 0 ? λo.toFixed(2)+'° (West)' : λo.toFixed(2)+'°')} ]. It is therefore the place chosen in this notebook to display the celestial map (i.e., the night sky as Tycho Brahe could observe it at the end of the 16^th century).

33:T18a4,chart = { const svg = d3.create("svg") .attr("viewBox", [0, 0, width, height]) .attr("width", Math.min(width,Math.max(document.documentElement.clientWidth || 0, window.innerWidth || 0)-10) ) .attr("font-family", "sans-serif") .attr("font-size", 16) .attr("text-anchor", "middle") .attr("fill", "textColor") .style("margin", "0 -5px") .style("display", "block"); svg.append("clipPath") // define a clip path .attr("id", "disk-clip") // clipPath an ID .append("circle") .attr('cx', width/2) .attr('cy', height/2) .attr('r',width/2-60); svg.append("circle") .attr('cx', width/2) .attr('cy', height/2) .attr('r',width/2) .attr("fill", darkGrey) .attr("fill-opacity", 0.2); svg.append("path") .attr("d", path(horizon)) .attr("fill", darkSky); if(visible.includes("celestial_north")) { // North celestial pole let polaris = svg.append("circle") .attr("r", 4.2) .attr("fill", "darkblue") .attr("stroke", asterismColor) .attr("fill-opacity", 0.4) .attr("transform", d => { let coords = [0, 90]; let projectedCoords = projection(invertAzimuth(equatorial2horizontal(julianCentury(date), λo, φo)(coords))); return `translate(${projectedCoords})` }) } // Big dipper asterism let asterismPath = svg.append("path") .attr("clip-path", "url(#disk-clip)") .attr("fill", "none") .attr("stroke", asterismColor) .attr("stroke-opacity", 0.6); if(visible.includes("graticule")) { svg.append("path") .attr("d", path(graticule)) .attr("fill", "none") .attr("stroke", lightGrey) .attr("stroke-opacity", 0.25); } let eclipticPath = svg.append("path") .attr("clip-path", "url(#disk-clip)") .attr("fill", "none") .attr("stroke", eclipticColor) .attr("stroke-dasharray", "2,4") .attr("stroke-opacity", 0.4); let MarsPath = svg.append("path") .attr("clip-path", "url(#disk-clip)") .attr("fill", "none") .attr("stroke", eclipticColor) .attr("stroke-opacity", 0.6); let MarsPos = svg.append("circle") .attr("r", planetRadius) .attr("fill", planetColor); // The equator doesn’t change (much), so just render it once. const equator = circle_ .map(equatorial2horizontal(julianCentury(date), λo, φo)) .map(invertAzimuth); if(visible.includes("equator")) { let equatorPath = svg.append("path") .attr("clip-path", "url(#disk-clip)") .attr("fill", "none") .attr("stroke", equatorColor) .attr("stroke-opacity", 0.8) .attr("d", path({type: "LineString", coordinates: equator})); }; let planetSVG = svg.append("g") .attr("clip-path", "url(#disk-clip)") .selectAll("g"); let starSVG = svg.append("g") .selectAll("g"); svg.append("g") .call(xAxis); if(visible.includes("graticule")) { svg.append("g") .call(yAxis) .attr("fill", lightGrey) .attr("fill-opacity", 0.7); } return Object.assign(svg.node(), { update(date) { const T = julianCentury(date); starSVG = starSVG.data(stars) .join( enter => enter.append("g") .call(g => g.append("circle") .attr("r", d => d.radius) .attr("fill", starColor)), update => update, exit => exit.remove() ) .attr("transform", d => { let coords = [d.ra, d.dec]; let projectedCoords = projection(invertAzimuth(equatorial2horizontal(T, λo, φo)(coords))); return `translate(${projectedCoords})` }) // Compute the ecliptic. Note: D3 will apply adaptive resampling! if(visible.includes("ecliptic")) { let ecliptic = circle_ .map(ecliptic2equatorial(T)) .map(equatorial2horizontal(T, λo, φo)) .map(invertAzimuth); eclipticPath .attr("d", path({type: "LineString", coordinates: ecliptic})); }; // Compute the orbit of Mars if(visible.includes("planetPath")) { let Mars_orbit = Mars_path_RA_Dec .map(equatorial2horizontal(T, λo, φo)) .map(invertAzimuth); MarsPath .attr("d", path({type: "LineString", coordinates: Mars_orbit})); } if(visible.includes("celestial_north")) { // Big dipper asterism let asterism = big_dipper .map(equatorial2horizontal(T, λo, φo)) .map(invertAzimuth); asterismPath.attr("d", path({type: "LineString", coordinates: asterism})); } // Compute the planetary positions. Note: azimuth is already North-oriented! let Mars_pos = Mars_coordinates_RA_Dec .map(equatorial2horizontal(T, λo, φo)) .map(invertAzimuth); const planets = Mars_pos .map(Mars_pos => ({ name: "Mars", altitude: Mars_pos[1], azimuth: Mars_pos[0] })).slice(0, 1) const planets_alt = Object.entries(ephemeris.getAllPlanets(dateHour, λo, φo, 0).observed) .map(([name, {raw: {position: {altaz: {topocentric: {altitude, azimuth}}}}}]) => ({ name: name.slice(0, 1).toUpperCase() + name.slice(1), altitude, azimuth })) .filter( planets_alt => planetList.includes(planets_alt.name)); planetSVG = planetSVG.data(planets.length >0 ? planets : planets_alt) .join( enter => enter.append("g") .call(g => g.append("circle") .attr("r", planetRadius) .attr("fill", planetColor)) .call(g => g.append("text") .attr("fill", planetColor) .attr("dx", "-1em") .attr("dy", "1em") .attr("font-size", 16) .attr("font-face", "Open Sans") .text(d => visible.includes("planetNames") ? (d.name == "Mars" ? `♂` : d.name) : ``)), update => update, exit => exit.remove() ) .attr("transform", d => `translate(${projection([d.azimuth, d.altitude])})`); } }); }34:T641,{ let dotplot = Plot.plot({ height: 350, width: 700, grid: true, marginLeft: 40, marginRight: 40, marginTop: 50, marginBottom: 75, insetTop: 0, insetBottom: 0, style: { fontSize: "13px", background: "#fff", color: "#000" }, x: { label: " ← Right ascension (RA), α ", tickPadding: 6, labelOffset: 64, // Customized scale from 0° to 360° transform: f => f/6, domain: [360/6,0] , tickFormat: x => `${Math.round(6*x.toFixed(3))}°`, ticks: 6, tickSize: 9, tickRotate: 0 }, y: { label: "↑ Declination, δ", labelOffset: 35, tickFormat: x => x<=0 ?`${x}°`: `+${x}°`, ticks: 10, tickSize: 6, domain: [-30, 30] }, caption: html`
Theoretical astronomical coordinates of Mars, period 1582 – 1600
(ephemeris from NASA horizon's database).

`, marks: [ Plot.frame({stroke: "black"}), Plot.ruleY([0]), Plot.line(Mars_theoretical_orbit_360, {x: "Right_Ascension_360", y: "Declination", strokeWidth: 0.8, stroke: "steelblue"} ), Mars_coordinates_RA_Dec.length > 0 ? Plot.dot( Mars_coordinates_RA_Dec, {x: Mars_coordinates_RA_Dec[0][0], y: Mars_coordinates_RA_Dec[0][1], fill: "black", r: 4.5}) : Plot.dot( Mars_altaz, {x: Mars_altaz[0].RA, y: Mars_altaz[0].Dec, fill: "black", r: 4.5}), ] }) // Label in bold/large font let select_labels = d3.select(dotplot) .selectAll("svg > g > text") .style("font-weight", "bold") .style("font-size", "16px"); return dotplot; }35:T745,

Declination and Right Ascension

Altitude and azimuth coordinates are dependent on the observer's location on Earth. It is therefore much preferable to describe the position of a cellestial object using the declination (Dec, δ) and right ascension (RA, α). Indeed, thanks to the equatorial coordinate system, one can locate a point on the celestial sphere (i.e., its position relative the background stars). In particular, by definition, the celestial north pole has the coordinates: δ = +90°, α = 0° (or 0^h). For example, polaris (alpha Ursa Minoris) has the coordinates: δ = +89° 15′ 50.8″, α = 02^h 31^m 49.09^s (observation epoch J2000). We can therefore easily understand why it was the declination of Mars over the years that Tycho Brahe reported in his notebooks …

The graph above (♂ Dec vs. RA) shows the declination of Mars as a function of its right ascension, as calculated using ephemeris data extracted from the NASA Horizons database. This graph highlights the apparent retrograde motion of Mars (Mars drawing either a loop or an open zigzag) that has puzzled astronomers for centuries.

36:T465,

Endnote

To prepare this notebook and its sister page, I have used the ephemeris of Mars that I could calculate using the parameters found on NASA Horizons database. The results are in perfect agreement with the observations made by Tycho Brahe for this astronomical epoch of interest (1592 – 1600).

The readers who want to fork this page can proceed in a similar way or simply retrieve the ephemeris from Stellarium ( > Astronomical Calculations Window [F10] > Ephemeris ). For much greater flexibility, one can instead use the ephemeris JavaScript library (v. 2.0.0 is used sparingly in this notebook). Note however that this library suffers from some minor inaccuracies — at least for the epoch of interest — and that is why I have preferred to use a CSV file with Dec and RA data for the star map and for the graph.

37:T432,* On Observable's website * Visualizing [Tycho Brahe's astronomical observations of Mars](https://observablehq.com/@christophe-yamahata/visualizing-tycho-brahe-s-astronomical-observations-mars) → My first notebook on this topic. * ["We need more tiny knowledge projects—here’s one"](https://observablehq.com/@jsomers/we-need-more-tiny-knowledge-projects-heres-one) → On the hope of James Somers to find on Observable some dynamic and interactive visualizations showing Mars' retrograde motion. * Star maps * [@mbostock/star-map/2](https://observablehq.com/@mbostock/star-map/2) * [@aotten/star-map](https://observablehq.com/@aotten/star-map) * [@nsrivast/star-map](https://observablehq.com/@nsrivast/star-map) * [Stellarium](https://stellarium.org) → Use Stellarium to check how this observable celestial map compares with that computed with the open source planetarium software (see the illustration below). * ▶️ [YouTube video](https://www.youtube.com/watch?v=pQEkIdJK8ac): A 1-minute walkthrough of the Mars observation records (1582–1600).38:[{"id":"b72a112dc112a45fd0512dc049a9fc9576e35356e0fa6bc6cca995c1036c4317cfa3cb9db9679eadcca9b1a85b46bdbf8c70eb8907d5208ac30f33f25da5ebd3","href":"https://static.observableusercontent.com/files/b72a112dc112a45fd0512dc049a9fc9576e35356e0fa6bc6cca995c1036c4317cfa3cb9db9679eadcca9b1a85b46bdbf8c70eb8907d5208ac30f33f25da5ebd3","name":"stars_(from_star-map_Mike_Bostock).csv","type":"text/csv","lastModified":1714058156764.424,"size":60896},{"id":"a1053ffe25ff79a39968a52028df663d5ff9e0d9dd420e9b79ccf10da4899bcf97b79c3e54d8d3f36b993e437d6a66f1541f59fb3b0f7e7ff254a5d1573310fe","href":"https://static.observableusercontent.com/files/a1053ffe25ff79a39968a52028df663d5ff9e0d9dd420e9b79ccf10da4899bcf97b79c3e54d8d3f36b993e437d6a66f1541f59fb3b0f7e7ff254a5d1573310fe","name":"ephemeris-2.0.0.js","type":"application/javascript","lastModified":1658444690402.741,"size":230067},{"id":"70b8357f6d7630f60808aee7d0f0d52832e42ee8c29f0ff10354fb800223db3b98c8ecee79b68b33e609806ae9da56cc97a6b7cd1d5f62736756f8f4dd9ac8ab","href":"https://static.observableusercontent.com/files/70b8357f6d7630f60808aee7d0f0d52832e42ee8c29f0ff10354fb800223db3b98c8ecee79b68b33e609806ae9da56cc97a6b7cd1d5f62736756f8f4dd9ac8ab","name":"stars.csv","type":"text/csv","lastModified":1658444690402.741,"size":60896},{"id":"7a147161df583b5f2d38be07431ed50ba8d4c73010ddd5009eaa0062b404bf3b0729662d9b0233f39ef8c39d19659c3fca59326e548451c564f9c1e9126abe22","href":"https://static.observableusercontent.com/files/7a147161df583b5f2d38be07431ed50ba8d4c73010ddd5009eaa0062b404bf3b0729662d9b0233f39ef8c39d19659c3fca59326e548451c564f9c1e9126abe22","name":"Mars_theoretical_orbit_1582-1600.csv","type":"text/csv","lastModified":1658936038361.586,"size":243815},{"id":"df49bc99cb14d87928f405266aff20f9f93ad3cd4adc8f07f78db59a8c03bd2e65dd48417db20d39364153d474545fdd95dc76049dde29a370b0dcc29f5aafa2","href":"https://static.observableusercontent.com/files/df49bc99cb14d87928f405266aff20f9f93ad3cd4adc8f07f78db59a8c03bd2e65dd48417db20d39364153d474545fdd95dc76049dde29a370b0dcc29f5aafa2","name":"Stellarium_Hven_1591-10-05.PNG","type":"image/png","lastModified":1659005953810.422,"size":437414},{"id":"b98f3721f458a012d6f858b3af407feca5c5ce98ead9b4c0d4b02edcf5b76bff7df865cb9d90d26de049ce388e8d3d6be1e4d532615ea13f7ea33668610f023d","href":"https://static.observableusercontent.com/files/b98f3721f458a012d6f858b3af407feca5c5ce98ead9b4c0d4b02edcf5b76bff7df865cb9d90d26de049ce388e8d3d6be1e4d532615ea13f7ea33668610f023d","name":"esa_constellation_umbrella_inside.PNG","type":"image/png","lastModified":1659083608097.809,"size":841500}] 30:{"title":"Star map — Visualizing the apparent retrograde motion of Mars (1582-1600)","cells":[{"id":2327,"value":"# Star map — Visualizing the apparent retrograde motion of Mars (1582-1600)","pinned":false,"hidden":false,"mode":"md"},{"id":0,"value":"$32","pinned":false,"hidden":false,"mode":"html"},{"id":2489,"value":"
\n

Celestial map\n
(island of Hven, end of the 16^thcentury)

\nThe following star map shows the night sky as it could be observed by Tycho Brahe from the island of Hven.\nThis map uses stereographic projection: The cellestial hemisphere is projected on a plane. The celestial sphere presents itself to us like a dome covering our heads or, if you prefer, like the inner fabric of an umbrella. When this one is flipped over, the cardinal directions are reversed.\nAltitude and azimuth are the coordinates displayed on this map (horizontal coordinate system). Altitude (0° – 90°) expresses angular elevation above the horizon. Azimuth (0° – 360°) is the horizontal direction of the astronomical object from due North, relative to the observer.\n \n

","pinned":false,"hidden":false,"mode":"html"},{"id":2529,"value":"viewof observation_period = interval([1582, 1600], {\n step: 1,\n value: [1589, 1596],\n format: ([start, end]) => `${start == 1582 ? 'November' : 'January'} ${start} … ${end == 1600 ? 'April' : 'December'} ${end}`,\n label: 'Observation period (years between 1582 and 1600)'\n})","pinned":false,"hidden":false,"mode":"ojs"},{"id":1630,"value":" viewof date = Scrubber(d3.utcDays(Date.parse(dateStart), Date.parse(dateEnd)), {\n autoplay: true, loopDelay: 5000, delay: 0, loop: true,\n format: d3.utcFormat(\"%b %d, %Y\")})","pinned":false,"hidden":false,"mode":"ojs"},{"id":2334,"value":"viewof visible = checkbox({\n title: \"Set visibility\",\n options: [\n { value: \"planetNames\", label: \"♂ symbol\" },\n { value: \"planetPath\", label: \"Planet's path\" },\n { value: \"ecliptic\", label: \"Ecliptic plane\" },\n { value: \"equator\", label: \"Equatorial plane\" },\n { value: \"graticule\", label: \"Graticule (azimuthal grid)\" },\n { value: \"celestial_north\", label: \"Big dipper asterism & North celestial pole\" }\n ],\n value: [\"planetNames\", \"planetPath\", \"ecliptic\", \"equator\", \"graticule\", \"celestial_north\"]\n})","pinned":false,"hidden":false,"mode":"ojs"},{"id":17,"value":"$33","pinned":false,"hidden":false,"mode":"ojs"},{"id":2553,"value":"viewof hour_angle = slider({\n title: \"Observation angle (hour angle)\",\n description: \"Slide to change the observation angle (i.e., hour of observation)\",\n min: -180,\n max: 180,\n step: 30,\n value: 0,\n format: v => html`${v > 0? '+'+v : v}° (${v > 0? '+'+v/15 : v/15}^h)`,\n})","pinned":false,"hidden":false,"mode":"ojs"},{"id":4194,"value":"$34","pinned":false,"hidden":false,"mode":"ojs"},{"id":4078,"value":"Inputs.table(Mars_altaz, {\n columns: [\n \"date\",\n \"altitude\",\n \"azimuth\",\n \"Dec\",\n \"RA\"\n ],\n\n header: {\n date: \"Date of observation of ♂\",\n altitude: \"Altitude (°)\",\n azimuth: \"Azimuth (°)\",\n Dec: \"Declination (°)\",\n RA: \"Right Ascension (°)\"\n },\n \n align: {\n date: \"left\",\n altitude: \"center\",\n azimuth: \"center\",\n Dec: \"center\",\n RA: \"center\"\n },\n \n width: {\n date: 120,\n altitude: 80,\n azimuth: 80,\n Dec: 110,\n RA: 110\n },\n\n maxWidth: 800,\n maxHeight: 50,\n height: 50\n})","pinned":false,"hidden":false,"mode":"ojs"},{"id":4297,"value":"$35","pinned":false,"hidden":false,"mode":"html"},{"id":4982,"value":"$36","pinned":false,"hidden":false,"mode":"html"},{"id":2512,"value":"
\n

Useful links

","pinned":false,"hidden":false,"mode":"html"},{"id":2514,"value":"$37","pinned":false,"hidden":false,"mode":"md"},{"id":4916,"value":"
\n

Stellarium software – This screen capture show the night sky as it could be observed from the island of Hven On October 5^th, 1591.

\n","pinned":false,"hidden":false,"mode":"html"},{"id":5027,"value":"umbrella = html`\n
\n

A constellation umbrella. \n
[ Source: esa space shop ]

\n`","pinned":false,"hidden":false,"mode":"ojs"},{"id":2495,"value":"

\n
\n

Appendices

","pinned":false,"hidden":false,"mode":"html"},{"id":2500,"value":"### Parameters / Constants","pinned":false,"hidden":false,"mode":"md"},{"id":2502,"value":"Hven_island = [56.0881, 12.6966];","pinned":false,"hidden":false,"mode":"ojs"},{"id":2523,"value":"φo = Hven_island[0] // observer’s latitude (degrees)","pinned":true,"hidden":false,"mode":"ojs"},{"id":2527,"value":"λo = Hven_island[1] // Observer's longitude (degrees)","pinned":true,"hidden":false,"mode":"ojs"},{"id":2046,"value":"J2000 = Date.UTC(2000, 0, 1, 12)","pinned":false,"hidden":false,"mode":"ojs"},{"id":2918,"value":"julianCentury = date => {\n var Year = d3.utcFormat(\"%Y\");\n var Month = d3.utcFormat(\"%m\");\n var Day = d3.utcFormat(\"%d\");\n var Hour = hour_angle/15;\n return (Date.UTC(Year(date), Month(date)-1, Day(date), Hour) - Date.UTC(2000, 0, 1, 12) ) / 315576e7 ;\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":3108,"value":"dateHour = new Date(Date.parse(date)+0.5*86400000*hour_angle/180)","pinned":false,"hidden":false,"mode":"ojs"},{"id":2605,"value":"dateStart = {\n if(observation_period[0]==1582){\n const dateStart = observation_period[0]+'-11-01' // Something wrong with ephemeris-2.0.0.js before that date\n return dateStart\n }else{\n const dateStart = observation_period[0]+'-01-01'\n return dateStart\n }\n}\n","pinned":false,"hidden":false,"mode":"ojs"},{"id":2609,"value":"dateEnd = observation_period[1] == 1600 ? observation_period[1]+'-04-23' : observation_period[1]+'-12-31';\n","pinned":false,"hidden":false,"mode":"ojs"},{"id":3980,"value":"date_year = {\n var Year = d3.utcFormat(\"%Y\");\n var Month = d3.utcFormat(\"%m\");\n var Day = d3.utcFormat(\"%d\");\n return (1*Year(date) + Month(date)/12 + Day(date)/365.2524);\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":2114,"value":"---\n### Stars","pinned":false,"hidden":false,"mode":"md"},{"id":2200,"value":"nStars = 500","pinned":false,"hidden":false,"mode":"ojs"},{"id":2118,"value":"starRadius = d3.scaleLinear([6, -1], [0, 2])","pinned":false,"hidden":false,"mode":"ojs"},{"id":2760,"value":"starColor = \"#fafafa\"","pinned":false,"hidden":false,"mode":"ojs"},{"id":2772,"value":"darkSky = \"#000028\"","pinned":false,"hidden":false,"mode":"ojs"},{"id":2116,"value":"// Edit 25.04.2024 \n// > await FileAttachment(\"stars.csv\").txt()\n// replaced with > await FileAttachment(\"stars_(from_star-map_Mike_Bostock).csv\").text()\n// CSV file on that notebook: https://observablehq.com/@d3/star-map\nstars = (d3.csvParse(await FileAttachment(\"stars_(from_star-map_Mike_Bostock).csv\").text(), d => {\n d3.autoType(d);\n return {\n magnitude: d.magnitude,\n radius: starRadius(d.magnitude),\n ra: hms(d.RA_hour, d.RA_min, d.RA_sec),\n dec: dms(d.dec_deg, d.dec_min, d.dec_sec),\n };\n})).sort((a, b) => d3.ascending(a.magnitude, b.magnitude)).slice(0, nStars)","pinned":false,"hidden":false,"mode":"ojs"},{"id":4437,"value":"// δ UMa / RA 12h 15m 25.56063s / Dec +57° 01′ 57.4156″\n\n// α UMa / RA 11h 03m 43.67152s / Dec\t+61° 45′ 03.7249″\n// β UMa / RA 11h 01m 50.47654s / Dec\t+56° 22′ 56.7339″\n// γ UMa / RA 11h 53m 49.84732s / Dec\t+53° 41′ 41.1350″\n\n// ε UMa / RA 12h 54m 01.74959s / Dec\t+55° 57′ 35.3627″\n// ζ UMa / RA 13h 23m 55.54048s / Dec\t+54° 55′ 31.2671″\n// η UMa / RA \t13h 47m 32.43776s / Dec\t+49° 18′ 47.7602″\n\nbig_dipper = [[183.857,57.033],[165.932, 61.751],[165.460, 56.382],[178.458,53.695],[183.857,57.033],[193.507,55.960],[200.981,54.925],[207.089,49.313]] ","pinned":false,"hidden":false,"mode":"ojs"},{"id":2222,"value":"### Planet Mars","pinned":false,"hidden":false,"mode":"md"},{"id":2229,"value":"planetList = [\"Mars\"]","pinned":false,"hidden":false,"mode":"ojs"},{"id":2460,"value":"planetRadius = 3","pinned":false,"hidden":false,"mode":"ojs"},{"id":2753,"value":"planetColor = \"#ffbb22\"","pinned":false,"hidden":false,"mode":"ojs"},{"id":3234,"value":"eclipticColor = planetColor","pinned":false,"hidden":false,"mode":"ojs"},{"id":4074,"value":"Mars_altaz = Object.entries(ephemeris.getAllPlanets(dateHour, λo, φo, 0).observed)\n .map(([name, {raw: {position: {altaz: {topocentric: {altitude, azimuth, dRA, dec}}}}}]) => ({\n name: name.slice(0, 1).toUpperCase() + name.slice(1),\n date: dateHour,\n altitude, \n azimuth,\n RA: 15*(dRA.hours+dRA.minutes/60+(dRA.seconds+dRA.milliseconds/1000)/3600),\n Dec: dec*180/(Math.PI),\n }))\n .filter( planet => planet.name == \"Mars\");","pinned":false,"hidden":false,"mode":"ojs"},{"id":4190,"value":"Mars_theoretical_orbit = FileAttachment(\"Mars_theoretical_orbit_1582-1600.csv\").csv({typed: true})","pinned":false,"hidden":false,"mode":"ojs"},{"id":4188,"value":"Mars_theoretical_orbit_360 = Mars_theoretical_orbit\n .filter(d => d.Right_Ascension_360 = d.Right_Ascension - 360*Math.floor(d.Right_Ascension/360))\n .map(d => d.Right_Ascension_360 <= 0.5 || d.Right_Ascension_360 >= 359.5 ? NaN : d);","pinned":false,"hidden":false,"mode":"ojs"},{"id":4477,"value":"Mars_path = Mars_theoretical_orbit\n .filter(d => d.Date < date_year && d.Date > date_year-0.8)\n .filter(d => d.Right_Ascension_360 = d.Right_Ascension - 360*Math.floor(d.Right_Ascension/360));","pinned":false,"hidden":false,"mode":"ojs"},{"id":4514,"value":"Mars_path_RA_Dec = Mars_path\n .map(currentElement => [currentElement.Right_Ascension_360,currentElement.Declination])","pinned":false,"hidden":false,"mode":"ojs"},{"id":4554,"value":"Mars_coordinates_RA_Dec = Mars_path\n .filter(d => d.Date < date_year+0.001 && d.Date > date_year-0.01)\n .filter(d => d.Right_Ascension_360 = d.Right_Ascension - 360*Math.floor(d.Right_Ascension/360))\n .map(currentElement => [currentElement.Right_Ascension_360,currentElement.Declination])","pinned":false,"hidden":false,"mode":"ojs"},{"id":695,"value":"---\n### Plotting","pinned":false,"hidden":false,"mode":"md"},{"id":13,"value":"width = 600","pinned":false,"hidden":false,"mode":"ojs"},{"id":11,"value":"height = width","pinned":false,"hidden":false,"mode":"ojs"},{"id":1639,"value":"chart.update(date);","pinned":true,"hidden":false,"mode":"ojs"},{"id":43,"value":"graticule = d3.geoGraticule()\n .stepMinor([10, 15])\n .extentMajor([[-180, 0], [180, 89]])\n .extentMinor([[-180, 0], [180, 89]])()","pinned":false,"hidden":false,"mode":"ojs"},{"id":3199,"value":"textColor = \"#B0A0A0\"","pinned":false,"hidden":false,"mode":"ojs"},{"id":3397,"value":"lightGrey = \"#B0B0B0\"","pinned":false,"hidden":false,"mode":"ojs"},{"id":3808,"value":"darkGrey = \"F0F0F0\"","pinned":false,"hidden":false,"mode":"ojs"},{"id":3615,"value":"equatorColor = \"#FF0044\"","pinned":false,"hidden":false,"mode":"ojs"},{"id":4462,"value":"asterismColor = \"orange\"","pinned":true,"hidden":false,"mode":"ojs"},{"id":33,"value":"path = d3.geoPath(projection)","pinned":false,"hidden":false,"mode":"ojs"},{"id":8,"value":" projection = d3.geoStereographic()\n .scale((width-115) / 2)\n .clipExtent([[0, 0], [width, height]]) \n .rotate([180, -90]) // use North-oriented azimuth\n .translate([width / 2, height / 2])\n .precision(0.1) ","pinned":false,"hidden":false,"mode":"ojs"},{"id":98,"value":"xAxis = g => g\n .call(g => g.append(\"g\")\n .attr(\"stroke\", \"currentColor\")\n .selectAll(\"line\")\n .data(d3.range(360))\n .join(\"line\")\n .datum(d => [\n projection([d, 0]),\n projection([d, d % 10 ? -1 : -2])\n ])\n .attr(\"x1\", ([[x1]]) => x1)\n .attr(\"x2\", ([, [x2]]) => x2)\n .attr(\"y1\", ([[, y1]]) => y1)\n .attr(\"y2\", ([, [, y2]]) => y2))\n .call(g => g.append(\"g\")\n .selectAll(\"text\")\n .data(d3.range(0, 360, 10))\n .join(\"text\")\n .attr(\"dx\", d => d === 0 ? 0 : (d === 90 ? -12 : (d === 180 ? 0 : (d === 270 ? 12 : (d > 180 ? \"0.4em\" : (d < 90 ? \"-0.3em\" :\"-0.6em\"))))))\n .attr(\"dy\", d => d === 0 ? \"-0.1em\" : (d === 180 ? \"0.9em\" : \"0.4em\"))\n .text(d => d === 0 ? \"N\" : d === 90 ? \"E\" : d === 180 ? \"S\" : d === 270 ? \"W\" : `${d}°`)\n .attr(\"font-size\", d => d % 90 ? 12 : 26)\n .attr(\"font-weight\", d => d % 90 ? null : \"bold\")\n .datum(d => projection([d, -4]))\n .attr(\"x\", ([x]) => x)\n .attr(\"y\", ([, y]) => y))","pinned":false,"hidden":false,"mode":"ojs"},{"id":1954,"value":"yAxis = g => g\n .call(g => g.append(\"g\")\n .selectAll(\"text\")\n .data(d3.range(15, 90, 15))\n .join(\"text\")\n .attr(\"dy\", \"0.3em\")\n .attr(\"font-size\", 12)\n .text(d => `${d}°`)\n .datum(d => projection([-45, d]))\n .attr(\"x\", ([x]) => x)\n .attr(\"y\", ([, y]) => y))","pinned":false,"hidden":false,"mode":"ojs"},{"id":1729,"value":"circle_ = d3.range(10).map(i => [i / 9 * 360, 0]) // will be resampled by D3","pinned":false,"hidden":false,"mode":"ojs"},{"id":41,"value":"horizon = d3.geoCircle().radius(90).center([0, 90])()","pinned":false,"hidden":false,"mode":"ojs"},{"id":2447,"value":"---\n## Imports","pinned":false,"hidden":false,"mode":"md"},{"id":3976,"value":"Mars_orbit = FileAttachment(\"Mars_theoretical_orbit_1582-1600.csv\").csv()","pinned":false,"hidden":false,"mode":"ojs"},{"id":4066,"value":"Mars_orbit_360 = Mars_orbit.filter(d => d.Right_Ascension_360 = d.Right_Ascension - 360*Math.floor(d.Right_Ascension/360))\n .map(d => d.Right_Ascension_360 <= 0.5 || d.Right_Ascension_360 >= 359.5 ? NaN : d);","pinned":false,"hidden":false,"mode":"ojs"},{"id":1580,"value":"ephemeris = require(await FileAttachment(\"ephemeris-2.0.0.js\").url())","pinned":false,"hidden":false,"mode":"ojs"},{"id":1989,"value":"import { ecliptic2equatorial, equatorial2horizontal, invertAzimuth, hms, dms } from \"@mbostock/star-map/2\"","pinned":false,"hidden":false,"mode":"ojs"},{"id":1915,"value":"import { worldMapCoordinates, slider, checkbox, select, date as dateSelector } from \"@jashkenas/inputs\"","pinned":false,"hidden":false,"mode":"ojs"},{"id":1627,"value":"import {Scrubber} from \"@mbostock/scrubber\"","pinned":false,"hidden":false,"mode":"ojs"},{"id":2531,"value":"import {interval} from \"@mootari/range-slider\"","pinned":false,"hidden":false,"mode":"ojs"},{"id":4,"value":"d3 = require(\"d3@6\")","pinned":false,"hidden":false,"mode":"ojs"}],"files":"$W38","stdlib":"1","titling":"auto"}

Declination and Right Ascension

Endnote

Celestial map\n(island of Hven, end of the 16thcentury)

Useful links

Appendices

Celestial map\n
(island of Hven, end of the 16^thcentury)