# Re-creating Galileo's Observations of Jupiter's 4 Large Moons

Posted 3 years ago
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 Awhile back, Stephen Wolfram suggested that I try to re-create, using modern calculations instead of observations, Galileo's observations of Jupiter's 4 largest moons. I thought I'd share the results here for fun, especially since its nice of have something about Jupiter given the arrival of the Juno spacecraft at Jupiter this past weekend. These moons are also known today as the Galilean Satellites. At the time, the prevailing view of the cosmos was that Earth was the center of the universe and everything moved around it. So, it came as a surprise when Galileo observed Jupiter through his primitive telescope that Jupiter has 4 "stars" that appeared to be orbiting it. In his diagrams, Jupiter was represented as a simple circle, and the new satellites as star-like figures.Using Astronomical Algorithms, by Jean Meeus, as a source, I used the low accuracy method (since we just need the results to look okay to the eye) to compute the positions of Jupiter's 4 big moons in 2D. Higher accuracy methods are also provided in the source, but those are used for things like transits, eclipses, etc. and is more than what is needed here. JovianMoonCoordinates[jd_] := Module[{A, B, d, DE, Delta, DS, du1, du2, du3, du4, G, H, J, KK, lambda, M, NN, psi, r, R, r1, r2, r3, r4, u1, u2, u3, u4, V, w1, w2}, d = jd - 2451545; V = 3.0148817498 + 0.000019475780 d; M = 6.2400582213 + 0.017201970 d; NN = 0.349414916249264 + 0.0014501120450072 d + 0.00574213323906134 Sin[V]; J = 1.15392443495605 + 0.0157519089131849 d - 0.005742133239061344 Sin[V]; A = 0.033423055175691 Sin[M] + 0.00034906585039886 Sin[2 M]; B = 0.096953039948 Sin[NN] + 0.0029321531433504 Sin[2 NN]; KK = J + A - B; R = 1.00014 - .01671 Cos[M] - .00014 Cos[2 M]; r = 5.20872 - .25208 Cos[NN] - .00611 Cos[2 NN]; Delta = Sqrt[r^2 + R^2 - 2 r R Cos[KK]]; psi = ArcSin[R/Delta Sin[KK]]; w1 = 3.68229565585763 + 15.3207952882387 d - 0.088559510336640 Delta + psi - B; w2 = 3.26777995850898 + 15.18762666631161 d - 0.087789749516252 Delta + psi - B; lambda = 0.599520598060052 + 0.001450211528774608 d + 0.005742133239061344 Sin[V] + B; DS = 0.0544542726622230 Sin[0.7470009198535 + lambda]; DE = DS - 0.02268928027592 (r - Delta)/Delta Sin[ lambda - 1.754055898254301] - 0.03874630939427 Sin[psi] Cos[lambda + 0.383972435438752]; u1 = 2.858969742485099 + 3.5501020551357 d - 0.0205208211279524 Delta + psi - B; u2 = 6.25550438524295 + 1.767872509298387 d - 0.01021891623871900 Delta + psi - B; u3 = 0.0997909453120277 + 0.876757737252356 d - 0.0050679637991465 Delta + psi - B; u4 = 3.923660728774436 + 0.3750360006026911 d - 0.00216783815377278 Delta + psi - B; G = 5.780181416754821 + 0.878083559165341 d - 0.00507562750962625 Delta; H = 1.526290430869041 + 0.37645409807322 d - 0.002176035248978202 Delta; du1 = 0.00825540736193317 Sin[2 (u1 - u2)]; du2 = 0.0185877565337396 Sin[2 (u2 - u3)]; du3 = 0.002879793265790643 Sin[G]; du4 = 0.01471312559431219 Sin[H]; r1 = 5.9057 - .0244 Cos[2 (u1 - u2)]; r2 = 9.3966 - .0882 Cos[2 (u2 - u3)]; r3 = 14.9883 - .0216 Cos[G]; r4 = 26.3627 - .1939 Cos[H]; u1 += du1; u2 += du2; u3 += du3; u4 += du4; MapThread[{# Sin[#2], -# Cos[#2] Sin[DE]} &, {{r1, r2, r3, r4}, {u1, u2, u3, u4}}, 1]] I created a star symbol to represent the moons using the following: star[{x_, y_}] := Line[{{{-.75/2 + x, -.75/2 + y}, {.75/2 + x, .75/2 + y}}, {{-.75/2 + x, .75/2 + y}, {.75/2 + x, -.75/2 + y}}, {{x, 1/2 + y}, {x, -1/2 + y}}}] Finally, the pieces are assembled to create an individual "frame" here: JupiterSatellites[date_] := Module[{jd = JulianDate[date]}, Graphics[{GrayLevel[.6], {{Black, star[#]}} & /@ JovianMoonCoordinates[jd]}, ImageSize -> 400, PlotRange -> {{-25, 25}, {-1, 1}}, PlotRangePadding -> 3, Epilog -> {White, EdgeForm[Black], Thickness[.003], Disk[{0, 0}, {1, 1/1.071}]}, PlotLabel -> Block[{\$DateStringFormat = {"MonthNameShort", " ", "DayShort", " ", "Year", " ", "Hour12Short", ":", "Minute", "AMPM"}}, DateString[date]]]] A sequence of frames is easily created with a simple Table command: frames = Table[ JupiterSatellites[ DateObject[{1610, 1, 8, h, 0, 0}, TimeZone -> 1]], {h, 0, 120, 3}]; And you can view this list of frames as a stacked diagram as follows: Grid[List/@frames, Dividers -> All] You can also export the frames as an animation to see the positions change in time.
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Posted 3 years ago
 Very nice. I never learned Latin. Can you correlate each of Galileo's drawings with a corresponding calculation or provide a translation?Thanks!
Posted 3 years ago
 I'm sure there are other translations out there, but this one turned up via a quick Google search:http://people.reed.edu/~wieting/mathematics537/sideriusnuncius.pdfIf you scroll a little over half way through the document, you will see a number of his sketches. His description of time is a bit vague so I estimated about 6Pm Italy time (e.g. TimeZone -> 1); JupiterSatellites[DateObject[{1610, 1, 13, 18, 0, 0}, TimeZone -> 1]] This gives a view almost identical to what he described for Jan 13, 1610. Most of his observations refer to the "first hour of the night" and given its January, I'm assuming 6 PM local time would be a reasonable guess. Keep in mind that Galileo's telescope was very crude so its resolution was poor. If 2 moons were very close, he may not have been able to resolve them so this may appear to result in a discrepancy between what he describes and what the calculation show. But all the test cases I've done seem to be quite close.
Posted 3 years ago
 Very impressive!! I see a lot of constants in the JovianMoonCoordinates function. Is the accuracy of these necessary to ensure these coordinates still work centuries later?
Posted 3 years ago
 The source doesn't specify a range of dates for which the results are valid, at least not in an obvious way. Often, additional references are given for some of the sub-computations. Its possible that if you do enough source spelunking, you might find an originating source that provides that level of analysis.The first 17 or so variables in the first block of code are defined in chapter 43, while the remaining are in the following chapter 44.
Posted 3 years ago
 Thanks for explaining
Posted 3 years ago
 Here's a scan of the original 1610 publication: https://archive.org/details/Sidereusnuncius00GaliJovian diagrams begin here at the bottom of the right-hand page: https://archive.org/stream/Sidereusnuncius00Gali#page/n39/mode/2up
Posted 3 years ago
 If one is looking for yet another Saturday morning project, what about doing some image processing on cropped regions of the original prints to extract the typeset positions of the satellites, and then seeing what FindFormula is able to derive. Of course the amount of noise in the diagrams is atrocious.
Posted 3 years ago
 - another post of yours has been selected for the Staff Picks group, congratulations !We are happy to see you at the top of the "Featured Contributor" board. Thank you for your wonderful contributions, and please keep them coming!
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