32:T8d4,

Tycho Brahe's observations of the planet Venus (♀)
[ Motibus Veneris (♀), ex observationibus Tychonis Brahe ]

Tycho Brahe was convinced of the need to undertake measurements of planetary orbits over very long periods of time (i.e., over several years) in order to uncover the laws governing the motion of the planets. He thus meticulously measured the orbit of Venus (♀), as well as other planets (namely, Mercury ☿, Mars ♂, Jupiter ♃, Saturn ♄), over a period of about twenty years. His observations have been collected in notebooks that have been reproduced in Tychonis Brahe Dani Opera Omnia, a series of books from which all the data presented here could be extracted. More specifically, Tycho Brahe accurately measured the declination of Venus, δ, expressed in degrees [°] as a function of time, over the period extending from 1582 to 1600.

It was thanks to these data that Johannes Kepler could establish his laws of planetary motion.

Figure 1 shows the raw data (○) collected by Tycho Brahe from his observations of the declination of Venus (Observationes Veneris, declinatio ♀). In the same figure, the solid line curve was calculated using the ephemeris data retrieved from the NASA Horizons database.

33:T60d,{ let dotplot = Plot.plot({ height: 470, width: 850, grid: true, marginLeft: 50, marginRight: 40, marginTop: 50, marginBottom: 75, insetTop: 0, insetBottom: 0, style: { fontSize: "13px", background: "#fafafa", color: "#000" }, x: { label: " Year → ", tickPadding: 6, labelOffset: 64, tickFormat: "f", tickSize: 9, ticks: 20, domain: [1582.01, 1600.4], tickRotate: -35 }, y: { label: "↑ Declination, δ", labelOffset: 35, tickFormat: x => x<=0 ?`${x}°`: `+${x}°`, ticks: 10, tickSize: 6, domain: [-30, 30] }, caption: html`
Figure 1: Tycho Brahe's observations of the planet Venus (period 1582 – 1600).
⚪ Tycho Brahe's observations
— Theoretical declination (ephemeris from NASA horizons' database)

`, marks: [ Plot.frame({stroke: "black"}), Plot.ruleX([selection_coordinates.Date], {stroke: "blue", strokeDasharray: "4,3"}), Plot.ruleY([0]), Plot.line(Venus_theoretical_orbit, {x: "Date", y: "Declination", strokeWidth: 2, stroke: "steelblue" } ), Plot.dot(Venus_data_Tycho_Brahe, {x: "Date", y: "Declination", stroke:"#111", fill: "orange", strokeOpacity: 0.4, r: 3.4}), ] }) // 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; }34:T72a,{ let dotplot = Plot.plot({ height: 470, width: 850, grid: true, marginLeft: 50, 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 6480° [modulo 360°] transform: RA_360 ? f => f/6 : f => f/360, domain: RA_360 ? [360/6,0] : [6480/360,0], tickFormat: RA_360 ? x => `${Math.round(6*x.toFixed(3))}°` : x => `${Math.round(360*x.toFixed(3))}°`, ticks: RA_360 ? 6 : 10, 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`
Figure 2: Theoretical astronomical coordinates of Venus, period 1582 – 1600
(ephemeris from NASA horizon's database).

`, marks: [ Plot.frame({stroke: "black"}), Plot.ruleY([0]), Plot.line(RA_360 ? Venus_theoretical_orbit_360 : Venus_theoretical_orbit, {x: RA_360 ? "Right_Ascension_360" : "Right_Ascension", y: "Declination", strokeWidth: RA_360? 0.8 : 2, stroke: "steelblue"} ), RA_360 ? null : Plot.ruleX([selection_coordinates.Right_Ascension], {stroke: "blue", strokeDasharray: "4,3"}), RA_360 ? Plot.dot( Venus_theoretical_orbit_360, {x: selection_coordinates.Right_Ascension_360 , y: selection_coordinates.Declination, fill: "black", r: 4.5}) : null, ] }) // 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:T473,

Notes

If you intend to analyse the data from Opera Omnia, be aware of the following:

The Gregorian calendar was introduced by Pope Gregory XIII in October 1582. He decreed that the Julian date October 4, 1582, was to be followed by the Gregorian date October 15, 1582. However, this modern calendar was not adopted by Protestant countries until much later. As a consequence, if you want to compare any of the data collected by Tycho Brahe with those calculated by modern ephemeris, you need to account for a difference of 10 days between the Julian and Gregorian calendars.
For example, you can use Stellarium to observe the partial eclipse that could be observed from Hven island in 1584 ("Observatio eclipsis solaris anno 1584 die 30 aprilis", see Tychonis Brahe Dani Opera Omnia, Tomus X, pp. 299). The actual date is May 10, 1584 (JD 2299733.68).

36: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; }37:[{"id":"80303ae1297f7f18ae72ff03703c0814c67e25b7ad046f4d8a8ccfb251fa65039fe2aaaf1be9f72d71690c0a9741faa260462babbb2540e88b90eaa5de1f7e83","href":"https://static.observableusercontent.com/files/80303ae1297f7f18ae72ff03703c0814c67e25b7ad046f4d8a8ccfb251fa65039fe2aaaf1be9f72d71690c0a9741faa260462babbb2540e88b90eaa5de1f7e83","name":"Tycho_Brahe_Venus_1582-1600.PNG","type":"image/png","lastModified":1716053991810.944,"size":857364},{"id":"54ea4ed6549dce1e612ebd626b4bfa27b3a01a9ff8931c76a51be4d90ba86bb09de3a7d069b87b26dd604fa6c0f9886a9945b000625beac38d746f1a94451c78","href":"https://static.observableusercontent.com/files/54ea4ed6549dce1e612ebd626b4bfa27b3a01a9ff8931c76a51be4d90ba86bb09de3a7d069b87b26dd604fa6c0f9886a9945b000625beac38d746f1a94451c78","name":"Venus_data_1582-1600.csv","type":"text/csv","lastModified":1716054191136.48,"size":6309},{"id":"2b7c89b861d8bb2e37111534cc1544222f53e59e1f0d6ec0e5f18dddc30d8f04aad1c34873ae53ed4e507493558e2c0ec33cbd1a2b17570c4cd2e2da232d991b","href":"https://static.observableusercontent.com/files/2b7c89b861d8bb2e37111534cc1544222f53e59e1f0d6ec0e5f18dddc30d8f04aad1c34873ae53ed4e507493558e2c0ec33cbd1a2b17570c4cd2e2da232d991b","name":"Venus_theoretical_orbit_1582-1600@1.csv","type":"text/csv","lastModified":1716060862327.511,"size":245048},{"id":"e962593a56551bfb5539b83815d318432f110ab32e5c734b453aaa72eb22dfc4050d5697418081ef1aba735de4a1755a84982589f4733c90325e89153084b574","href":"https://static.observableusercontent.com/files/e962593a56551bfb5539b83815d318432f110ab32e5c734b453aaa72eb22dfc4050d5697418081ef1aba735de4a1755a84982589f4733c90325e89153084b574","name":"Venus_observations_Brahe.xlsx","type":"application/vnd.openxmlformats-officedocument.spreadsheetml.sheet","lastModified":1775312654595.665,"size":5029527}] 30:{"title":"♀ Visualizing Tycho Brahe's astronomical observations of Venus","cells":[{"id":62,"value":"# ♀ Visualizing Tycho Brahe's astronomical observations of Venus","pinned":false,"hidden":false,"mode":"md"},{"id":71,"value":"

This is a fork of a notebook about Tycho Brahe's observations of Mars (♂).

In this version, we focus on the observational data collected by Tycho Brahe for the planet Venus (♀).

\n","pinned":false,"hidden":false,"mode":"html"},{"id":2105,"value":"

\n
Illustration: Tycho Brahe's observations of the planet Venus (period 1582-1600) [C. Yamahata]\n
\n
","pinned":false,"hidden":false,"mode":"html"},{"id":0,"value":"$32","pinned":false,"hidden":false,"mode":"html"},{"id":200,"value":"$33","pinned":false,"hidden":false,"mode":"ojs"},{"id":1190,"value":"viewof Date_data = Scrubber(d3.range(1582.5, 1600.39, RA_360? 1/24 : 1/12), {autoplay: true, loop: RA_360? false : true, alternate: false, initial: RA_360? 183 : 0, format: f => `Observation date: ${(Math.ceil(12*(f+0.0001-Math.floor(f))) < 10)? \"0\":\"\"}${Math.ceil(12*(f+0.0001-Math.floor(f)))}/${Math.floor(f)}`})\n\n// viewof Date_data = Inputs.range([1582.397, 1600.38], {format: x => x.toFixed(0), value: 1591.5, step: 1/12, label: \"Observation date (Year): \"})","pinned":false,"hidden":false,"mode":"ojs"},{"id":1569,"value":"viewof RA_360 = Inputs.toggle({label: \"RA (α): Modulo 360°\", value: true})","pinned":false,"hidden":false,"mode":"ojs"},{"id":805,"value":"$34","pinned":false,"hidden":false,"mode":"ojs"},{"id":2142,"value":"

Figure 2 shows the declination of Venus (Dec, δ) as a function of its right ascension (RA, α), calculated using ephemeris data extracted from the NASA Horizons database. This graph highlights the apparent retrograde motion of Venus.

","pinned":false,"hidden":false,"mode":"html"},{"id":1033,"value":"
\n

Tycho Brahe's observations (around ${(Math.ceil(12(Date_data+0.0001-Math.floor(Date_data))) < 10)? \"0\":\"\"}${Math.ceil(12(Date_data+0.0001-Math.floor(Date_data)))}/${Math.floor(Date_data)}):

","pinned":false,"hidden":false,"mode":"html"},{"id":917,"value":"Inputs.table(Venus_data_Tycho_Brahe.filter(d => d.Date >= Date_data-0.1 && d.Date <= Date_data+0.1), {\n columns: [\n \"Date\",\n \"Declination\",\n ],\n\n header: {\n Date: \"Year\",\n Declination: \"Declination, δ (°)\",\n },\n \n align: {\n Date: \"center\",\n Declination: \"center\",\n },\n \n width: {\n Date: 80,\n Declination: 150,\n },\n\n maxWidth: 320,\n maxHeight: 140,\n height: 140\n})","pinned":false,"hidden":false,"mode":"ojs"},{"id":1039,"value":"
\n

Theoretical astronomical coordinates:

","pinned":false,"hidden":false,"mode":"html"},{"id":1292,"value":"{\n \n let filtered_data = Venus_theoretical_orbit.filter(d => d.Date <= Date_data+0.004 && d.Date >= Date_data-0.004),\n\n transformed_data = filtered_data.map(filtered_data => ({ \n 'Date': `${Math.ceil(12*(filtered_data.Date-Math.floor(filtered_data.Date))) < 10? \"0\":\"\"}`+Math.ceil(12*(filtered_data.Date-Math.floor(filtered_data.Date))) +\"/\"+ Math.floor(filtered_data.Date)+\" (\"+filtered_data.Date.toFixed(3)+\")\",\n 'Declination': filtered_data.Declination,\n 'Right_Ascension': filtered_data.Right_Ascension\n }))\n\n return Inputs.table(transformed_data, {\n columns: [\n \"Date\",\n \"Declination\",\n \"Right_Ascension\"\n ],\n \n header: {\n Date: \"Date\",\n Declination: \"Declination, δ (°)\",\n \"Right_Ascension\": \"Right Ascension, α (°) \"\n },\n \n align: {\n Date: \"center\",\n Declination: \"center\",\n \"Right_Ascension\": \"center\"\n },\n \n width: {\n Date: 120,\n Declination: 150,\n Right_Ascension: 150\n },\n\n maxWidth: 550,\n maxHeight: 120,\n height: 120\n})\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":341,"value":"$35","pinned":false,"hidden":false,"mode":"html"},{"id":2315,"value":"
\n

Download

\n📎 In the Files Pane, you will find an Excel (.XLSX) file containing Tycho Brahe’s observations of the planet Venus (♀), along with orbit calculations derived from the osculating orbital elements for the year 1600 (J1600).\n

","pinned":false,"hidden":false,"mode":"md"},{"id":74,"value":"
\n

References

","pinned":false,"hidden":false,"mode":"html"},{"id":76,"value":"- *Tychonis Brahe Dani Opera Omnia*, [Tomus X](https://archive.org/details/operaomniaedidit10brahuoft/page/n9/mode/2up).\n Edited by J.L.E. Dreyer, Libraria Gyldendaliana: 1923. → Collection of observation data for the period **1582 - 1585**.\n- *Tychonis Brahe Dani Opera Omnia*, [Tomus XI](https://archive.org/details/operaomniaedidit11brahuoft/page/n9/mode/2up).\n Edited by J.L.E. Dreyer, Libraria Gyldendaliana: 1924. → Collection of observation data for the period **1586 - 1589**.\n- *Tychonis Brahe Dani Opera Omnia*, [Tomus XII](https://archive.org/details/operaomniaedidit12brahuoft/page/n9/mode/2up).\n Edited by J.L.E. Dreyer, Libraria Gyldendaliana: 1925. → Collection of observation data for the period **1590 - 1595**.\n- *Tychonis Brahe Dani Opera Omnia*, [Tomus XIII](https://archive.org/details/operaomniaedidit13brahuoft/page/n13/mode/2up).\n Edited by J.L.E. Dreyer, Libraria Gyldendaliana: 1926. → Collection of observation data for the period **1596 - 1600**.","pinned":false,"hidden":false,"mode":"md"},{"id":111,"value":"
\n

Useful links

","pinned":false,"hidden":false,"mode":"html"},{"id":114,"value":"- [\"Visualizing Tycho Brahe's observations of Mars\"](https://observablehq.com/@christophe-yamahata/visualizing-tycho-brahe-s-astronomical-observations-mars?collection=@christophe-yamahata/tycho-brahe-and-mars)\n\n → Original notebook from which I have forked this one. In that notebook, Tycho Brahe's observations of Mars are utilized.\n\n\n- Data from the NASA HORIZONS project can be retrieved via its [data request page](https://ssd.jpl.nasa.gov/horizons/app.html).\n\n → This database is valuable for acquiring physical data used in orbit calculations (i.e., Osculating orbital elements).","pinned":false,"hidden":false,"mode":"md"},{"id":197,"value":"
\n

Appendix

","pinned":false,"hidden":false,"mode":"html"},{"id":1187,"value":"$36","pinned":false,"hidden":false,"mode":"ojs"},{"id":1511,"value":"Venus_theoretical_orbit_360 = Venus_theoretical_orbit\n .filter(d => d.Right_Ascension_360 = d.Right_Ascension - 360*Math.floor(d.Right_Ascension/360))\n .map(d => d.Right_Ascension_360 <= 0.6 || d.Right_Ascension_360 >= 359.4 ? NaN : d);","pinned":false,"hidden":false,"mode":"ojs"},{"id":754,"value":"Venus_theoretical_orbit = FileAttachment(\"Venus_theoretical_orbit_1582-1600@1.csv\").csv({typed: true})","pinned":false,"hidden":false,"mode":"ojs"},{"id":196,"value":"Venus_data_Tycho_Brahe = FileAttachment(\"Venus_data_1582-1600.csv\").csv({typed: true})","pinned":false,"hidden":false,"mode":"ojs"},{"id":1074,"value":"selection_coordinates = Venus_theoretical_orbit.filter(d => d.Date <= Date_data+0.2 && d.Date >= Date_data-0.2)[0];","pinned":false,"hidden":false,"mode":"ojs"}],"files":"$W37","stdlib":"1","titling":"auto"}

Tycho Brahe's observations of the planet Venus (♀) [ Motibus Veneris (♀), ex observationibus Tychonis Brahe ]