Kepler’s Impossible Equation

this video is sponsored by kiwi Co more on them later this is Kepler's impossible equation Kepler famously discovered that the planets sweep out equal areas and equal times on elliptical orbits but to actually use this fact we have to compute the angle Theta that the planet makes to the body it's orbiting inscribing Kepler's ellipse into a circle we can show that the area swept out by the planet plus this extra triangle is proportional to the area that would have been swept out on a circular orbit with some algebraic manipulation we arrive at a pair of equations one that connects time to the central angle e and one that connects e to Theta now to do much useful astronomy we need to be able to solve for e in the first equation so we can predict where the planet will be at some future time T Kepler wasn't able to solve this equation when he discovered it in 16005 and we've literally been trying ever since here's an attempt from 2024 why is such a simple looking equation so difficult to solve how do astronom actually used Kepler's laws without being able to solve for the angle e over the next 250 years the search for a solution to Kepler's equation would be a key part of the expansion of the frontiers of mathematics not once but twice and ultimately lead to an elegant theory of convergence for infinite series that today is a Cornerstone of modern analysis these are the rudolphine tables published by Kepler in 1627 they predict the position of the planets about 30 times more accurately than previous approaches how did Kepler get around his unsolvable equation to make these predictions an interesting property of equations like Kepler's is that while there's no obvious way to solve for E it's very easy to check if a given e value is any good all we have to do is plug some estimate of e let's call it e hat into Kepler's equation and see how close the resulting value is to the value of M that we're looking for Kepler used this fact to create a simple guessing and checking algorithm to estimate where the planet Mercury will be on a given date let's say December 25th 2024 we first comput the time elapsed since Mercury was at its last parah helion this is its closest point to the sun and the one closest to December 25th happens on December 6th 2024 giving a difference of 19 days from here we can compute the m in Kepler's equation this is the angle the planet would have swept out if its motion was uniform around the Sun Mercury will cover a full 360° or 2 pi radians in 88 Earth days so the angle m is equal to our lapse time of 19 days ided 88 * 2 pi giving an M value of 1. 36 radians or 77. 7 de of course as Kepler showed the planets do not actually follow uniform circular motion and Kepler's equation tells us how the actual central angle e varies with M when the planet's motion follows Kepler's equal area law while traveling on an ellipse with eccentricity of little e the eccentricity just measures how squished the ellipse is an ellipse with an centricity of zero is a circle and once eccentricity reaches one the ellipse breaks and turns into a parabola Kepler had a good estimate for the eccentricity of Mercury's orbit so from here he just needed to find an angle e that would return a value close enough to M when he plugged it into his equation he started with an initial guess for e one simple approach is just to set e hat equal to M from here we can plug e hat into Kepler's equation and compute m equal 66. 0 de which is a bit smaller than the true value of M to get closer to we need to increase the value of e hat but by how much should we increase it Kepler found a simple approach that works remarkably well compute the error between M and our estimate of M hat and add this error to eat to get a new value of eat so we're increasing our guess for E by exactly the error between the true and estimated value for M after just two iterations of this process our error shrinks to just 0. 4 de well within the measurement error of the observations Kepler was using Kepler doesn't give an explanation for how he landed on this exact approach why does adjusting our guess by exactly our error at each step work so well interestingly if we modify Kepler's method to adjust our guess for E by twice the error at each step it doesn't work at all and if we only adjust by half the error at each step it converges much more slowly to see why Kepler's approach works so well it's helpful to plot the value of M across a range of e values in the case of a purely circular orbit our eccentricity little e is just zero and M just equals e so our plot is just a straight line with a slope of one as eccentricity increases and our orbit becomes less circular we're effectively subtracting a scaled sine wave from our straight line causing it to bend downward between 0 and 180° and bend upward between 180 and 360° higher eccentricities make the relationship between time and the

planet's position less linear with the planet moving more quickly close to the Sun and more slowly when it's further away from the Sun solving Kepler's equation for m equal 77. 7 de is equivalent visually to figuring out where our curve intersects the horizontal line at m equal 77. 7 our initial guess for E 77. 7 de shows up here on our plot and following Kepler's approach we plug this eat into Kepler's equation effectively Computing this point on our curve we then computed the error by subtracting our estimate of M from the true value of M this is just the distance between the point on our curve and the line from here Kepler's approach tells us to add this error value to eat effectively moving us to the right on our plot by this error value and making nice progress towards our true solution where our line and curve intersect subsequent steps get us closer and closer to the true intersection value now the reason this works so well for Kepler's equation is because the EM curve is reasonably close to a straight line with a slope of one if our eccentricity was Zero making our orbit circular C's approach would work perfectly in just one step since our curve in this case is just a straight line with a slope of one adjusting e by the error term gets us exactly to the right solution since the base and height of our triangle are equal so Kepler's method works well when eccentricity is low and the EM curve is close to a straight line of the six known planets when Kepler published the rudol in tables mercury has the highest eccentricity at 0. 21 the worst case position on this orbit where m is around 140° requires four steps of Kepler's method to achieve sufficient accuracy which is not ideal when you're performing computation after computation to create something as extensive as the root offing tables but still is a viable approach six decades later around 1670 Isaac Newton would pick up or Kepler left off making a significant Improvement to Kepler's method now known as the Newton rafson method and used on a huge variety of problems Newton used a geometric construction to estimate the slope of the curve at eh hat and used this information to make smarter updates to e using Newton's method we're able to estimate e in the Mercury case with sufficient accuracy in just two steps a decade later in 1680 an incredibly bright object appeared in the sky that would challenge both Kepler and Newton's numerical approaches the great Comet of 1680 reportedly bright enough to see in the daytime and with an incredibly long tale was seen by many as an omen of the end of times here's a sermon about it from the Reverend increase ma given on January 20th 1680 in Boston back in England Newton's colleague Edmund Haley began to hypothesize that comets actually orbited the Sun as the planets did and that their arrivals could be predicted comparing observations of the Motions of various historical comets he conjectured that the Comets seen in 1681 and 1682 were actually the same comet Kepler had seen 75 years earlier in 16007 as he worked to publish his laws but for the math to work out the comet's orbit would have to have an incredibly High eccentricity we can explore how Kepler and Newton's numerical methods handle these Higher eccentricities by constructing a 2d grid of test cases where the y direction captures the eccentricity of the orbit and the X Direction captures the position of the object on its orbit running Kepler's method in each grid cell for six iterations we can visualize the final area using a heat map at an eccentricity of 0. 97 this is approximately the eccentricity of the orbit of haly's comet Kepler's method gives a massive worst case error of 20. 6 De Newton's method far is better but still yields significant error when the comet is close to the Sun visualizing our iterations on our em curves we can see that the high slope caused by the comet's highly eccentric orbit causes Kepler's method to jump around the correct solution and the high curvature causes Newton's method to overshoot the correct answer requiring more iterations from here there's a range of numerical approaches we can take to address these issues Haley would successfully use a different formulation of the problem that was more tractable for high eccentricities however at the end of the day all of these methods are just refinements of Kepler's guess and check idea and can be in precise messy require significant computation and it can be difficult to know in a given technique will work well with the digital computer still three centuries away astronomy would benefit greatly from more direct solution to Kepler's equation after all why can't we just solve for E The Next Century would bring the development of new sophisticated mathematical tools that would allow e to be expressed as the sum of various infinite series and would ultimately lead to an elegant theory of convergence for these series that today is a Cornerstone of modern analysis while they didn't push the frontiers of mathematics forward there were also a number of physical devices designed and built to solve Kepler's equation there's a set of purely geometric solutions that give

approximate answers to Kepler's equation in the early 1900s Arthur Rambo turned this construction into a physical device where the value of M could be set on a dial and an approximation of e could be found by taking readings of where this string intersects the dial there's nothing quite like building something physical like this for developing understanding which is why I was more than happy to partner again with this video sponsor kiwo now is a great time to have a fresh look at kiwo as they've just announced a big change to their lineup kids now Join one of five clubs where each crate is part of a learning series meticulously designed to build up specific skills and knowledge over the long run this is such an important part of learning and Mastery growing up I was obsessed with electronics and was lucky enough to be able to progress from taking apart old electronics to building a computer from Parts with my dad to soldering my own audio amplifiers like this one it sounded terrible but was a ton of fun and such a cool learning experience to progress from taking things apart to reading books to making my own circuits skills and experiences like these can really compound until you've arrived somewhere you could have never imagined with clubs kiwo is boxing up these long-term learning arcs I'm really excited to see my kids progress through the panda and Sprout clubs when my daughter turned six in a few years she can join the kiwo labs Club aside from being a great name kiwi Co Labs helps kids progress from science to engineering and includes brand new rebuildable stem projects like games and robots sign up for kiwo club today using Code Welch labs to receive 50% off your first Club crate when you join a club for kids age three and older or visit kiwico. com wlabs huge thanks to kiwo for sponsoring this video now back to Kepler's impossible equation in the 18th century mathematicians began to tame Infinity like imaginary numbers infinitely large and small numbers open up new ways of attacking problems for example we can use a tailor expansion to break apart the sign term in Kepler's equation into an infinite sum of simple polinomial terms each term contains higher and higher order derivatives of the sign function which tell us how much of each polinomial to add together to recreate the sign curve in the late 1700s Joseph Louis lrange while trying to solve Kepler's equation and other similar problems remarkably found a way to create Taylor series for inverse functions let's temporarily switch Kepler's equation to a more familiar F ofx notation and see how lra's result applies so we now have y = f ofx defined as x - e * the sin of X solving Kepler's equation with these new symbols means solving for x or equivalently finding the inverse function of F the lrange inversion theorem says that we can express X or F inverse as a fairly complicated looking infinite sum the thing to note here is that the sum does not involve F inverse itself which of course we don't know it instead involves the derivatives of f which do know applying lr's inverse theorem to Kepler's equation we're able to make some simplifications and arrive at this infinite sum where the coefficient of each term is equal to a derivative of s to the power of our term's index n * e to the power of n this formula has a similar shape to the tailor expansion we saw earlier for S of X but interestingly our powers are of the eccentricity of the ellipse e not X or Y switching back to E and M for X and Y we now have a fully solved expression for E so did lrange solve Kepler's equation let's try out lr's solution and see let's take the example of computing the position of Mercury that we considered earlier we'll use 0. 21 for the eccentricity little e and let's say again that we're trying to find the position of Mercury on December 25th 2024 which leads to an M value of 77. 7 De from here all we should have to do is plug in little E and M and see what lran solution returns now lran solution includes an infinite sum and of course we can't compute every term let's start with the first few terms and see how we do the first term evaluates to the sign of M * e Computing our second term requires us to take the derivative of sin squar resulting in the S of M * the cosine of M * e^ 2 the calculus really starts to pile up on the third term requiring us to compute the second derivative of sin cubed of M stopping at three terms for now we can plug in our M and E values for mercury and directly compute eal 89. 8 de we can measure the error in the same way we did with Kepler in Newton's method by plugging our estimate of e back into Kepler's equation and Computing the error in terms of M giving an impressive error of only 0. 02 de so how does lr's infinite series compar to Kepler and Newton's numerical methods plotting our errors across a full orbit of M values from Mercury we see that Lan's method outperforms Kepler's approach but not Newton's method converges

remarkably quickly to very low errors for low eccentricity orbits like this but what about high eccentricity orbits where we saw Newton and Kepler's methods struggle moving to the higher eccentricity value for Haley's comment of 0. 97 we see that Newton's method still generally outperforms the lrange inversion except for the M region around 6° where the comet is close to the Sun of course we're only using three terms from the gr's infinite series let's compute three more terms and see how this impacts our results as we can see moving to six terms the performance of our lren version actually gets worse why would more terms of our series make our estimate of e worse is the lren version Theory even correct returning to the lower eccentricity orbit of mercury for a moment we see the performance improves as we move from 3 to four to 5 to six terms of lrange series as expected why would lres series work for one ellipse but not for another for what values of our eccentricity e does adding more terms of our lrange Series start to make things worse in 1798 Pierre Simon the pl began publishing his massive and highly influential five volume treaty on astronomy mechanik Celeste where he immortalized lr's solution to Kepler's equation llas was aware that the lrange inv version didn't always work Computing that the series would diverge for eccentricity values larger than 0. 661 96 but lean CL did not rigorously justify this claim the early 1800s brought a formalization and unification of calculus led by the mathematician Augustine Louie Koshi studying Lan's solution to Kepler's equation LED Koshi to an elegant and precise method for determining when a series like the lren version would converge reportedly when Koshi read his paper on convergence of series at the French Academy of Sciences in 1831 llas was in the audience and ran home immediately after the meeting to see if he had published any Divergent series in mechanique Celeste so how does koshy's method work let's consider the power series for the function 1 over 1 + x^2 using a tailor series we can expand this function as 1 - x^2 + x 4th - x 6 Plus X 8th and so on plotting the function and its tailor expansion as we add more and more terms we can see that the expansion starts to break down around X = Plus or minus1 for X values between minus one in one additional terms make our tailor series more accurate while outside of this region more terms actually make the expansion worse just as we saw for the lrange inv version applied to Haley's Comet but what's special here about plus and minus one why does the expansion stop working at these values Koshi showed that there is a simple and general answer but to see it we have to expand X to be a complex number with real and imaginary components we can visualize the value of our function when X is a complex number as the height of a surface above the complex plane our original curve is directly above the real axis and now fits into a broader surface note these big Singularity spikes we see at xal plus or minus I our function is not defined here since i^ squ is minus one resulting in zero in our denominator remarkably it turns out that the location of these spikes has everything to do with why the tailor series of our real function diverges at plus or minus one in this example our tailor expansion is taken around the origin and the distance between the origin on our complex plane and these Singularity spikes is one what Koshi rigorously proved is that the radius of convergence is equal to the distance from the center of our expansion to the nearest Singularity on the complex plane this importantly works for points other than the origin if we had taken the tailor expansion of our function around xal 1/2 for example our tailor expansion looks a bit different and continues to match our function Beyond xal 1 but where exactly does it diverge what Koshi is telling us here is that the radius of convergence is exactly the distance between x = 1/2 and our nearest Singularity on the complex plane in this case xal I this distance comes out to the square root of 1^ 2 + 12 2ar so the square < TK of 5 over 4 about 1. 18 in general we can write the radius of convergence of our function as R = the < TK of 1 + k s where K is the center of our expansion Koshi applied a similar but necessarily more complex approach to the power series for the lrange inversion and showed that the series would converge for E values less than the square root of r^ 2us one where R is defined as a fairly complicated function of exponentials solving for R and then for E Koshi showed that the series will converge for E values less than 0. 6627 434 this constant is known today as the llao limit so the lrange version will only converge

for orbits with eccentricities below value at this point it's fair to wonder what astronomers are really getting from all this mathematics after all Newton's and even Kepler's method work well for lower eccentricity orbits and are simpler to use by 1860 work on Kepler's equation was mostly bifurcated and to astronomers using numerical or hybrid methods to get answers of sufficient accuracy as quickly as possible and mathematicians using trans Al equations like Kepler's to push the boundaries of mathematics as happen so often with mathematically driven Pursuits the search for a solution to Kepler's equation led to the creation of entirely new mathematical tools that have broad application outside of the initial problem statement in the decades after the gange applied his inverse methods to Kepler's equation the mathematician Frederick wilham Bessel took a fresh look at the problem and developed what we now call Bessel functions like lr's solution Bessel functions allow us to solve Kepler's equation using an infinite series but using sign functions instead of powers of e like the lrange series The Bessel series was more complex to work with and less accurate than competing numerical methods but Bessel functions themselves have turned out to have an absurdly broad range of applications from Acoustics to electromagnetism such as predicting the defraction behavior of light through helical objects like DNA today koshy's Radiance of convergence and more broadly the powerful idea of work with functions in the complex plane are key parts of modern analysis it's incredible to see how far the search for solutions to Kepler's simple equation has taken us in the fourth Century since Kepler encountered it and there remains new angles to be explored here's 17 different papers on solutions to Kepler's equation published in just the Last 5 Years including new numerical and analytical approaches we'll have to wait and see where the hunt for the solution to Kepler's impossible equation takes us next if you enjoy Welch lab's videos I really think you'll like my book on imaginary numbers it's coming out later this year way back in 2016 I made a massive 13-part YouTube series on imaginary numbers it's such an incredible topic I released an early version of this book back then and I'm now in the process of revising correcting and significantly expanding it my goal is to create the best book out there on imaginary numbers highquality hardcover printed books will start shipping later this year you can pre-order a copy today at the link in the description below and your order includes a free PDF copy of the 2016 version that you can download today you'll find all of this and more at the Welch Labs store