32:T495,

Science is primarily a discipline based on observation. Many fundamental discoveries owe much to the quality of the underlying observations. A fascinating story that illustrates this perfectly is that of Kepler’s laws describing the motion of the planets around the Sun (☉). If German astronomer Johannes Kepler (1571 – 1630) succeeded to make his extraordinary discoveries, it is undeniably because he had at his disposal the extremely accurate data from Danish astronomer Tycho Brahe (1546 – 1601), especially his meticulous observations of Mars (♂) recorded over a whole decade. Johannes Kepler, who had been Tycho Brahe’s assistant, knew that these data were absolutely reliable, and definitely the most accurate at the time.
[Source: C. Yamahata, 2020]

33:T936,

Tycho Brahe's observations of the planet Mars (♂)
[ Motibus Martis (♂), 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 Mars (♂), as well as other planets (namely, Mercury ☿, Venus ♀, 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 Mars, δ, expressed in degrees [°] as a function of time, over the period extending from 1582 to 1600.

It was these very data that allowed Johannes Kepler to establish his laws of planetary motion. In particular, the elliptical orbits of the planets were inferred from Kepler's calculation of the orbit of Mars.

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

34:T609,{ 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: [1582.2, 1600.6], 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 Mars (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(Mars_theoretical_orbit, {x: "Date", y: "Declination", strokeWidth: 2, stroke: "steelblue" } ), Plot.dot(Mars_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; }35:T726,{ 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 3430° [modulo 360°] transform: RA_360 ? f => f/6 : f => f/360, domain: RA_360 ? [360/6,0] : [3430/360,0], 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 Mars, period 1582 – 1600
(ephemeris from NASA horizon's database).

`, marks: [ Plot.frame({stroke: "black"}), Plot.ruleY([0]), Plot.line(RA_360 ? Mars_theoretical_orbit_360 : Mars_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( Mars_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; }36: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).

37:T6d0,- *Tychonis Brahe Dani Opera Omnia*, [Tomus V](https://archive.org/details/operaomniaedidit05brahuoft/page/n9/mode/2up). Edited by J.L.E. Dreyer, Libraria Gyldendaliana: 1923. → In *Astronomiæ instauratæ mechanica* (pp. 1-162), Tycho Brahe describes the various instruments he had devised to carry out his astronomical observations. These observations were mainly made at [Uraniborg](https://archive.org/details/operaomniaedidit05brahuoft/page/138/mode/2up), his astronomical observatory built on [Hven island](https://archive.org/details/operaomniaedidit05brahuoft/page/150/mode/2up). Note that this section of the book was first published in 1598. - *Tychonis Brahe Dani Opera Omnia*, [Tomus X](https://archive.org/details/operaomniaedidit10brahuoft/page/n9/mode/2up). Edited by J.L.E. Dreyer, Libraria Gyldendaliana: 1923. → Collection of observation data of the planet Mars for the period **1582 - 1585**. - *Tychonis Brahe Dani Opera Omnia*, [Tomus XI](https://archive.org/details/operaomniaedidit11brahuoft/page/n9/mode/2up). Edited by J.L.E. Dreyer, Libraria Gyldendaliana: 1924. → Collection of observation data of the planet Mars for the period **1586 - 1589**. - *Tychonis Brahe Dani Opera Omnia*, [Tomus XII](https://archive.org/details/operaomniaedidit12brahuoft/page/n9/mode/2up). Edited by J.L.E. Dreyer, Libraria Gyldendaliana: 1925. → Collection of observation data of the planet Mars for the period **1590 - 1595**. - *Tychonis Brahe Dani Opera Omnia*, [Tomus XIII](https://archive.org/details/operaomniaedidit13brahuoft/page/n13/mode/2up). Edited by J.L.E. Dreyer, Libraria Gyldendaliana: 1926. → Collection of observation data of the planet Mars for the period **1596 - 1600**.38:Tf31,- ["Visualizing Tycho Brahe's Mars data"](http://www.pafko.com/tycho/) by Wayne Pafko (2000). [ URL last visited in July 2022, http://www.pafko.com/tycho/ ] ★★★ → A source of inspiration that convinced me to prepare this notebook. Wayne gathered most of the data I have presented here. He carried out this work in March 2000. Most notably, in an Excel file (see *data pane*), he provides the references for each data (Tome, pages, [Source: Tychonis Brahe Dani Opera Omnia]). I have only made a few corrections to his data. - ["Star map — Visualizing the apparent retrograde motion of Mars"](https://observablehq.com/@christophe-yamahata/star-map-visualizing-the-retrograde-motion-of-mars) ★★★ → It is a continuation of this notebook. Using the same data, the apparent retrograde motion of Mars can be visualized on a star map. - ["Understanding What Kepler Did — Part I"](http://www.ams.org/publicoutreach/feature-column/fc-2018-11) by Bill Casselman, *AMS Feature Column*, November 2018. ★★ See also: [Part II: Earth's motion](http://www.ams.org/publicoutreach/feature-column/fc-2019-02) (Feb. 2019) | [Part III: Predecessors](http://www.ams.org/publicoutreach/feature-column/fc-2019-06) (June 2019). → Once you have collected all these data (*i.e.*, Brahe's work), the next step is to build an appropriate theory (*i.e.*, Kepler's work). This page is about Kepler's contribution, from a mathematical perspective. - Data from the NASA HORIZONS project can be retrieved via its [data request page](https://ssd.jpl.nasa.gov/horizons/app.html). ★★★ The database is useful to obtain physical data used to calculate orbits. Interestingly, for the planet Mars, data are available only for dates after January 2^nd, 1600. Any request for dates starting prior to 1600 will return: *No ephemeris for target "Mars" prior to A.D. 1600-JAN-01 23:58:09.5423 UT*. - [Tycho Brahe timeline](https://astronomy.com/magazine/2003/09/tycho-brahe-timeline), Maggie McKee, in *Astronomy*, December 2003. → This chronology is uselful, especially to understand how and where the data were collected for the period 1582-1600. - On Observable, see the notebook: ["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. - Other Observable notebooks: - [Tycho Brahe's observations of Mercury (☿)](https://observablehq.com/@christophe-yamahata/visualizing-tycho-brahes-astronomical-observations-mercury) → A fork of this notebook, featuring data collected by Brahe for the planet Mercury. - - [Tycho Brahe's observations of Venus (♀)](https://observablehq.com/@christophe-yamahata/visualizing-tycho-brahes-astronomical-observations-venus) → A fork of this notebook, featuring data collected by Brahe for the planet Venus. - [Tycho Brahe's observations of Jupiter (♃)](https://observablehq.com/@christophe-yamahata/visualizing-tycho-brahes-astronomical-observations-jupiter) → A fork of this notebook, featuring data collected by Brahe for the planet Jupiter. - [Tycho Brahe's observations of Saturn (♄)](https://observablehq.com/@christophe-yamahata/visualizing-tycho-brahes-astronomical-observations-saturn) → A fork of this notebook, featuring data collected by Brahe for the planet Saturn. - ▶️ [YouTube video](https://www.youtube.com/watch?v=pQEkIdJK8ac): A 1-minute walkthrough of the Mars observation records (1582–1600).39: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; }3a:[{"id":"7a147161df583b5f2d38be07431ed50ba8d4c73010ddd5009eaa0062b404bf3b0729662d9b0233f39ef8c39d19659c3fca59326e548451c564f9c1e9126abe22","href":"https://static.observableusercontent.com/files/7a147161df583b5f2d38be07431ed50ba8d4c73010ddd5009eaa0062b404bf3b0729662d9b0233f39ef8c39d19659c3fca59326e548451c564f9c1e9126abe22","name":"Mars_theoretical_orbit_1582-1600@1.csv","type":"text/csv","lastModified":1658175211981.555,"size":243815},{"id":"cb3fe4a105f904a5bd2f488721445b67f2873bd8b901802594145e6fda6a62ebf46aee7884f8c4860e1606f29d53a5547718d86d1e905ffe8472d1a6d3cdee08","href":"https://static.observableusercontent.com/files/cb3fe4a105f904a5bd2f488721445b67f2873bd8b901802594145e6fda6a62ebf46aee7884f8c4860e1606f29d53a5547718d86d1e905ffe8472d1a6d3cdee08","name":"Tycho_Brahe_Mars_1582-1600.PNG","type":"image/png","lastModified":1658267963389.132,"size":1263494},{"id":"4e3d8de72b794da966430476e84e8a3dad0859854d8ba4cbcb25e54bd06dcfffabbd6998749d8ea6fb1c8bedbf1f14ae1d6f69c7103366fe8b30eebec3f1185e","href":"https://static.observableusercontent.com/files/4e3d8de72b794da966430476e84e8a3dad0859854d8ba4cbcb25e54bd06dcfffabbd6998749d8ea6fb1c8bedbf1f14ae1d6f69c7103366fe8b30eebec3f1185e","name":"Mars_data_1582-1600.csv","type":"text/csv","lastModified":1658155785100.955,"size":17418},{"id":"2e205806709d3f85c8db97225b9862d2104259d8624d9d41d12d8c7a5e2e2629b3c2654fdf00003638b066ad748f77098bab06a3e25dd71ab763ed5cfcf68169","href":"https://static.observableusercontent.com/files/2e205806709d3f85c8db97225b9862d2104259d8624d9d41d12d8c7a5e2e2629b3c2654fdf00003638b066ad748f77098bab06a3e25dd71ab763ed5cfcf68169","name":"Mars_observations_(original_data_by_Wayne_Pafko_2000).xls","type":"application/x-msi","lastModified":1775254537812.451,"size":368128},{"id":"9c91f89b993ce752c10b2147650ab95060d2505719991aa3919a009192739204ba92d517ea43392c7cdd38edf151052472204e58f27ecd363835bb3928813846","href":"https://static.observableusercontent.com/files/9c91f89b993ce752c10b2147650ab95060d2505719991aa3919a009192739204ba92d517ea43392c7cdd38edf151052472204e58f27ecd363835bb3928813846","name":"Mars_observations_Brahe.xlsx","type":"application/vnd.openxmlformats-officedocument.spreadsheetml.sheet","lastModified":1775314009164.939,"size":16720285}] 30:{"title":"♂ Visualizing Tycho Brahe's astronomical observations of Mars","cells":[{"id":62,"value":"# ♂ Visualizing Tycho Brahe's astronomical observations of Mars","pinned":false,"hidden":false,"mode":"md"},{"id":71,"value":"$32","pinned":false,"hidden":false,"mode":"html"},{"id":2105,"value":"

\n
Illustration: Tycho Brahe's observations of the planet Mars (period 1582-1600) [C. Yamahata]\n
\n
","pinned":false,"hidden":false,"mode":"html"},{"id":0,"value":"$33","pinned":false,"hidden":false,"mode":"html"},{"id":200,"value":"$34","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: false})","pinned":false,"hidden":false,"mode":"ojs"},{"id":805,"value":"$35","pinned":false,"hidden":false,"mode":"ojs"},{"id":2142,"value":"

Figure 2 shows the declination of Mars (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 Mars (Mars' loops) that has puzzled astronomers for centuries.

","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(Mars_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 = Mars_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":"$36","pinned":false,"hidden":false,"mode":"html"},{"id":2246,"value":"
\n

Download

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

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

References

","pinned":false,"hidden":false,"mode":"html"},{"id":76,"value":"$37","pinned":false,"hidden":false,"mode":"md"},{"id":111,"value":"
\n

Useful links

","pinned":false,"hidden":false,"mode":"html"},{"id":114,"value":"$38","pinned":false,"hidden":false,"mode":"md"},{"id":197,"value":"
\n

Appendix

","pinned":false,"hidden":false,"mode":"html"},{"id":1187,"value":"$39","pinned":false,"hidden":false,"mode":"ojs"},{"id":1511,"value":"Mars_theoretical_orbit_360 = Mars_theoretical_orbit\n .filter(d => d.Right_Ascension_360 = d.Right_Ascension - 360*Math.floor(d.Right_Ascension/360))\n .map(d => d.Right_Ascension_360 <= 0.5 || d.Right_Ascension_360 >= 359.5 ? NaN : d);","pinned":false,"hidden":false,"mode":"ojs"},{"id":754,"value":"Mars_theoretical_orbit = FileAttachment(\"Mars_theoretical_orbit_1582-1600@1.csv\").csv({typed: true})","pinned":false,"hidden":false,"mode":"ojs"},{"id":196,"value":"Mars_data_Tycho_Brahe = FileAttachment(\"Mars_data_1582-1600.csv\").csv({typed: true})","pinned":false,"hidden":false,"mode":"ojs"},{"id":1074,"value":"selection_coordinates = Mars_theoretical_orbit.filter(d => d.Date <= Date_data+0.005 && d.Date >= Date_data-0.005)[0];","pinned":false,"hidden":false,"mode":"ojs"}],"files":"$W3a","stdlib":"1","titling":"auto"}

Tycho Brahe's observations of the planet Mars (♂) [ Motibus Martis (♂), ex observationibus Tychonis Brahe ]