Solving Kepler's Equation with Newton's Method

[@alastairmackay1758]

What a fantastic video! Scoured the internet in search of an explanation to solving Kepler's equation and this is by far the most clear. Thank you so much!!

[@csvaughen]

thank you - so glad it was helpful - thanks to the Kerbals for inspiring me too

[@alastairmackay1758]

@@csvaughen hahaha of course! hey just out of curiosity, do you have any idea how to model the apparent path of the sun in the sky as a function of time? i'm doing a project and i'm looking to model both the position of the earth in its orbit around the sun as well as its rotation on its own axis as a function of time, if that makes any sense at all! sorry if that's a bit confusing, i'm way out of my depth at this point!

[@csvaughen]

@@alastairmackay1758 no, dang, sorry, can't help... wish I could, that sounds awesome

[@unflexian]

hey, welch labs just posted a video on this equation and in the comments HarvesteR replied with details of how it was solved in KSP, i thought you would find that interesting

[@csvaughen]

whoa! yes - I'm definitely checking that out! thanks

[@jakobtrnovsek323]

@Christopher Scott Vaughen, would you be able to upload the geogebra file? I would like to see what equation you used to graph the elliptical orbit.

[@csvaughen]

yes, I have it here sites.google.com/view/kspmath scroll down and you'll find both the text version of how it works, scroll down more and you'll find solutions, scroll down further and you'll find the geogebra file

[@jakobtrnovsek323]

@@csvaughen thank you very much.

[@CesGu]

Thanks a lot!!

[@carmenguachalla2349]

Very nice video. Thanks to share your knowledge

[@ftmhaidar5181]

hi, thank you for video! i have few Qs: why do we have to set M= E-esinE to zero? and how would you find E from newton's law method?

[@csvaughen]

Kepler's equation begins with M=E-esinE, it is then convenient to re-write as M - E + esinE = 0 to solve for E (or even as M - x + esinx =0) because it's natural to find where a curve crosses the x-axis as a point that solves the equation. Newton's method is then used as a numerical method to find the solution, by iterations, each step improving the approximation... you can google and search for Newton's method for a more complete explanation of this process on its own

[@ftmhaidar5181]

@@csvaughen thank you for your response! i have tried search to find how can kepler can be solved by newton law however, there was not any that it was specifically related to f=ma to M=E-esinE.

[@csvaughen]

@@ftmhaidar5181 wait, newton's method is this: kzbin.info/www/bejne/Z6axaZZ4fLujnNU And it is NOT related to F=ma (law of motion) , the equation F=ma is also from Newton but totally different

[@Game_Masters]

Why don't they teach us this is universities and schools ? This is really important stuff. I feel like most universities go over the details and miss the essence of math and physics.

[@csvaughen]

Thanks!!! well, probably you'd eventually get to this in school doing aerospace or astrophysics, for example check out "Orbital Dynamics for Engineering Students" by Howard Curtis

[@voidisyinyangvoidisyinyang885]

OK at 37 minutes - it finally hit me how ingenious this rocket science stuff is. Fascinating - how the ratios are transferred to get the changes in speed in contrast to the circle average.

[@csvaughen]

click the show more in the video description and also check out sites.google.com/view/kspmath, thanks for checking out the video and the feedback

[@simadavoodi]

I really appreciate this, I wonder is there any way to ask you question

[@csvaughen]

yes, for sure, what's the questions? also I have a workbook on this material at sites.google.com/view/kspmath

[@simadavoodi]

@@csvaughen

[@simadavoodi]

@@csvaughen thanks . so actually I need to know about ground tracks, sth like calculating RAAN ,inclination ,or also how to portrait the satellite ground track by identifying latitude limits , direction,.....

[@csvaughen]

@@simadavoodi ok, that's an excellent question, sorry but I don't know enough to really help much with that, I've been wanting to study that more myself, I did something close to that, on launch latitude, azimuth and inclination, but maybe you already saw I have that video (here: kzbin.info/www/bejne/paWYiqCenbWGqbM) it is related but I didn't get into plotting the ground track...

[@simadavoodi]

@@csvaughen thank you very much, yup actually I'm studying space and astronautical engineering at Sapienza, so I was wonder is there any online course or sth as help, but u know I've already learned a lot from your video, thank you again, good luck

[@zemlidrakona2915]

Thanks! Now I can have my planet orbit a sun!.

[@voidisyinyangvoidisyinyang885]

Why do you think Kepler was against the closed form solution of the Golden Ratio? Because he understood the iterations as male and female numbers with infinity not contained by geometric symmetry. Therefore the ratio can not simply just be inverted for the zero answer - hence chaos.

[@csvaughen]

I don't know what you are referring to but don't think that matters particularly, at least for modern math and science. It might be interesting for historical reasons, but Kepler's understanding of math and science were just different than our's is now. Math and science have evolved a lot and continue to evolve.

[@sebastiancorreazapata8367]

When solving Kepler’s equation using Newton’s method, Does M remain constant throughout every iteration?

[@csvaughen]

Yes, M is a constant in the equation. You can see an example at sites.google.com/view/kspmath, look for chapter 1, and its written out there, and also on that page, scroll down and look for the excel file, here is a link to just the excel file: drive.google.com/file/d/1T_XFVKbq-2tulmyc4JouJC6y7tUkkcou/view

[@maggiemakgill]

Of course Kepler didn't have access to Newton's method. He was dead (1630) before it was published. Calculus was incomplete in his time (NO FToC!) and a lot of series methods didn't exist, no Taylor series or Newton's method etc. And so there wasn't easy ways of generating algorithms to solve equations. There were available trig calculations and was a series method for sine etc. (because of their importance in navigation!) but I have always wondered how KEPLER solved Kepler's equations (was there a team of monks somewhere? Sometimes that was the solution to something laborious in this time, their "supercomputer" was a cloister full of monks and nuns!). An iterative method is supposily mentioned in his book, (he might have developed something like Newton's method from other arguments, say geometric, the "Taylor series" for sine existed pre-Taylor and pre-calculus and was developed in India in the ... 12th century?). I might just have to read Astronomia nova and Epitome Astronomiae Copernicanae! but I don't know latin :( -BWT The first volume was put on the Index of Prohibited Books on 28 February 1619! so it might have been hard to get those monks! But at least those guys were around filling in the trig tables!

[@csvaughen]

right, good question, how did Kepler solve that equation? I don't know... just figured Kepler found some particular solutions by observation but no general method for solving and I thought it was beautiful that Newton (who came along after Kepler) gave the general understanding to orbital mechanics (using calculus) that proved Kepler's laws and oh yeah, by the way, a general method for solving that "kepler" equation too

[@maggiemakgill]

@@csvaughen No Kepler MADE predictions. GOOD ONES. (The success of them likely motivated Newton and meant that his theory of gravity HAD to be consistent with Kepler when he finished his analysis about how planets would move under gravity it HAD to say "ellipses", parabolas and hyperbolas were allow to be present at well, but ellipses HAD to be there!) Kepler made good predictions about how the planets would be seen to move in the future based on past positions and his equations. He predicted various transits of mercury and venus, including the events of 1631, now Venus wasn't observable from Europe (the first observed transit of Venus there was the 1639 transit of Venus), but the transit of Mercury was visible in Europe and it was observed by Pierre Gassendi (a Catholic priest, scientist, and philosopher clearly ignoring the church ban!) and it was when Kepler said it was going to be (sadly, he didn't live to see it, as he died of an illness on 15 November 1630). This means that Kepler HAD to calculate not just what equation the planets were following, but the orbital elements themselves (semi-major axis, eccentricity, inclination, the longitude of the ascending node, the argument of periapsis, and the true anomaly and orbital speed at some point in time), from where they were in the sky in the past, then how they would move in space around the sun, and how they would look on the sky in the future! NOT EASY IN A TIME WITHOUT CALCULUS! To do it accurately he needed to treat the orbits as ellipses and NOT circles and include inclination. Going back and forth between sky coordinates and the path around the sun is tricky. You can MEASURE the angular location of a planet along the ecliptic plane, essentially how far ahead or behind the sun a planet is, and its angular distance above or below the ecliptic. Even ignoring inclination, you measure the angular distance between the sun and Venus at say .. 13 degrees east today and say 17 degrees east a few days from now. You CANNOT turn that into the change in angle around the sun (true anomaly) without the orbital distance or at least the ratios with the earth's orbital distance. And at first ... Kepler didn't have that. You'd need to do a fit to find the orbit and then make predictions to check. You're going to need lots and lots of planetary locations and do a ton of trig. Kepler did have lots of observations of the locations of the planets in the sky at various dates. He had his observations but he also had Tycho Brahe's observations, who was an uninteresting theorist but a great observational astronomer who had very carefully built tables of planetary locations. This is EXACTLY what Kelper needed to work this out. Kepler started out working for Brahe and Kepler replaced Brahe after his death (NOT OF AN EXPLODING BLADDER, honestly that's a crazy myth!) and got all of Brahe's data! It is sometimes said that Brahe's greatest discovery was ... Johannes Kepler. Still, the MATH itself would have been a huge undertaking. The existing theoretical framework, earth-centered orbits with epicycles would have meant a very different type of calculation. I've read comments on Kepler's books on the topic which says he had an iterative method, but they never go into the details.

[@csvaughen]

@@maggiemakgill wow, thank you for this addition here, absolutely beautiful, the science and history, all so fascinating and so much more to learn... another thing that is interesting to explore that you hint at a little is the history of regression analysis, essential in so many other applications, and how that originated with the study of the motion of the planets

[@voidisyinyangvoidisyinyang885]

Zero comments? It's been proven Newton got his inverse square law from Pythagorean science (actually from Archytas). Zero, as negative infinity, was opposed by Aristotle.

[@csvaughen]

let's please just stick with consensus of modern science on this page, no conspiracy theories or pseudo-science please

[@voidisyinyangvoidisyinyang885]

@@csvaughen So you think Fields Medal math professor Alain Connes' lecture on noncommutative phase music as the unified field theory is pseudo-science? kzbin.info/www/bejne/mHrdmqisYrx_g68 I'm all ears - please share any details why you think this is the case. The fact is that in music theory there is the "phantom tonic" as the Perfect Fourth - meaning that for the root tonic, there will never be a 4/3 overtone because the denominator is not an octave multiple. So therefore the symmetric commutative math, thus far used by "consensus of modern science" is wrong - as Alain Connes emphasizes! Do you really relegate him to conspiracy level? haha.