Talk:Kepler's equation

Initial Comments

It would be helpful to many to give the answer in traditional Cartesian form, as functions of the initial conditions and of time giving the position in polar coordinates. 82.163.24.100 (talk) 19:57, 16 March 2010 (UTC)

Naming the article

This article assumes that the reader knows about mean anomaly and eccentric anomaly, and then he probably already knows what Kepler's equation is. Perhaps the name should be changed to Solving Kepler's equation ? Bo Jacoby (talk) 19:44, 3 June 2010 (UTC).

Parabolic Case?

We're missing the formulae for the parabolic case. It will go something like M = αG + βG³, with a suitable replacement, G, for {E,H} and suitably-defined coefficients α and β. Don't any of the references cited list this case? One derivation can be found by taking either of the limits E/√(1 - e) → G ← H/√(e - 1), as e → 1. This should also work for the rectilinear case. — Preceding unsigned comment added by 65.29.226.169 (talk) 02:44, 13 November 2022 (UTC)

Inverse Kepler equation

This paragraph is factually incorrect as it ignores different meaning of solution in Kepler and present times. Kepler himself was fully aware that his equation may be solved by recurrence E_{n+1}=M+esin(E_{n}). It is a matter of notation to write such a solution as an infinite series using standard formula for sin(a+b). So what he has meant is that no analytical solution may be obtained in a final number of steps. This asesement was true in his times and remains so today. alex@camk.edu.pl —Preceding unsigned comment added by 62.121.103.177 (talk) 21:08, 10 December 2010 (UTC)

The formula

$E={\begin{cases}\displaystyle \sum _{n=1}^{\infty }{\frac {M^{\frac {n}{3}}}{n!}}\lim _{\theta \to 0}\left({\frac {\mathrm {d} ^{\,n-1}}{\mathrm {d} \theta ^{\,n-1}}}\left({\frac {\theta }{\sqrt[{3}]{\theta -\sin(\theta )}}}^{n}\right)\right),&\epsilon =1\\\displaystyle \sum _{n=1}^{\infty }{\frac {M^{n}}{n!}}\lim _{\theta \to 0}\left({\frac {\mathrm {d} ^{\,n-1}}{\mathrm {d} \theta ^{\,n-1}}}\left({\frac {\theta }{\theta -\epsilon \cdot \sin(\theta )}}^{n}\right)\right),&\epsilon \neq 1\end{cases}}$

makes no sense because the exponent n is inside the parenthesis. I am not going to guess what was ment. How is the formula proved? Is the first formula a limiting case of the second formula?

The next formula

$E={\begin{cases}\displaystyle x+{\frac {1}{60}}x^{3}+{\frac {1}{1400}}x^{5}+{\frac {1}{25200}}x^{7}+{\frac {43}{17248000}}x^{9}+{\frac {1213}{7207200000}}x^{11}+{\frac {151439}{12713500800000}}x^{13}\cdots \ |\ x=(6M)^{\frac {1}{3}},&\epsilon =1\\\\\displaystyle {\frac {1}{1-\epsilon }}M-{\frac {\epsilon }{(1-\epsilon )^{4}}}{\frac {M^{3}}{3!}}+{\frac {(9\epsilon ^{2}+\epsilon )}{(1-\epsilon )^{7}}}{\frac {M^{5}}{5!}}-{\frac {(225\epsilon ^{3}+54\epsilon ^{2}+\epsilon )}{(1-\epsilon )^{10}}}{\frac {M^{7}}{7!}}+{\frac {(11025\epsilon ^{4}+4131\epsilon ^{3}+243\epsilon ^{2}+\epsilon )}{(1-\epsilon )^{13}}}{\frac {M^{9}}{9!}}\cdots ,&\epsilon \neq 1\end{cases}}$

is too long to be read on the screen without using the horizontal bar.

It is unclear if these formulas are supposed to be useful. Is convergence fast?

Kepler's equation M=E−e sin(E) is solved numerically by E₀=0 , E_i+1=M+e sin(E_i) , E=lim E_i. This is written in J like this:

Kepler =. (]-[*1 o.]):.(,.~(p.1&o.)^:_])

Forwards: e Kepler E gives M. Backwards: e (Kepler^:_1) M gives E. For example: 0.2 Kepler 1 gives 0.8317058 and 0.2 (Kepler^:_1) 0.8317058 gives 1.

Compared to this, the series of the article seem complicated. Still I think that an article on the methods of solving Kepler's equation is nice to have. Bo Jacoby (talk) 19:44, 3 June 2010 (UTC).

Thanks for all the input. The inverse Kepler equation above can be punched verbatim into TI calculator and evaluated, there isn't any ambiguity in it's interpretation. For example, when n=7, the seventh term of the top expression is:

${\frac {M^{\frac {7}{3}}}{7!}}\lim _{\theta \to 0}\left({\frac {\mathrm {d} ^{\,6}}{\mathrm {d} \theta ^{\,6}}}\left({\frac {\theta ^{7}}{(\theta -\sin(\theta ))^{7/3}}}\right)\right)$

Although it's not stated, the focus of this article is intended to be mathematical not physical. There were already many articles on the physical application of Kepler's equation. The solution provided is the unique taylor series, it is in a certain mathematical sense the simplest and most fundamental solution possible. It usually not the best solution to use in actual applications. Iteration methods require either a computer or manual labor, but power series can be evaluated on simple non-scientific calculators. Power series are simple formula, and they can be readily manipulated algebrically. For example, they can be integrated and differentiated, iteration techniques cannot. Norbeck (talk) 15:04, 8 June 2010 (UTC)

A PDF file of the cited work of Stumpff is available online from the NASA technical report server at (search for author's name "Stumpff"): http://ntrs.larc.nasa.gov/ . The expressions can be exactly reproduced using the free CAS Maxima. I doubt the convergence of the series in the elliptical case for any reasonably large value of M, because the coefficients do not form a decreasing sequence. For example, in case of e=0.375 I get the values of 1.6, -0.4096, 0.3670016, -0.43269239, 0.583820131, -0.852693109, and 1.312129996 for the coefficients corresponding to n = 1,3,5,7,8 9,11, and 13. Thus, these formulae seem pretty useless to me at the moment.Drgst (talk) 05:48, 22 November 2011 (UTC)

You are correct, the series at $\epsilon$ = .375 diverges for M >= 1. But the series does converge for some value of M whenever $\scriptstyle -1\leq \displaystyle \!\epsilon \scriptstyle \leq 1$. This series is a function of two variables and the convergence depends on both variables. In general, tailor series are rarely an appropriate tool for numerical evaluation. In practice the best way to solve Kepler's equation is usually with a solver. Similarly, the formal power series of the sin function, $\textstyle \sum _{n=0}^{\infty }{\frac {(-1)^{n}}{(2n+1)!}}x^{2n+1}$ is used to derive Euler's Identity $\scriptstyle e^{i\pi }+1=0$, while the CORDIC algorithm is used to make actual calculations. The power series provides an immediate description of the essence of the function. In this case the series representation tells you that it is a homogenous odd function with a nonlinear dominant term, among other things. NOrbeck (talk) 05:37, 23 November 2011 (UTC)

The emphasis in this section -- obtaining a series in M -- is completely misplaced. M can't be regarded as a small parameter! M is the time and the whole emphasis of astronomical investigations is predicting the behavior of celestial objects over long times. The "obvious" small parameter is e and treatment of this case is relegated to a short paragraph as the end. I recommend discarding the series in M and expanding the part treating the expansion in e. cffk (talk) 12:14, 3 June 2019 (UTC)

Alternate forms

The discussion in the subsection on Alternate forms seem to assume constant major semiaxis a, (rather than, say, constant semi latus rectum p), but does not say so explicitely. The parabolic orbit has eccentricity one, but the article says: When epsilon = 1, the orbit is completely flat, and it appears to be a straight line segment. Bo Jacoby (talk) 04:58, 4 June 2010 (UTC).

Yeah, this part gets confusing. The article needs work here. It is often incorrectly stated that orbits with orbital eccentricity=1 are necessarily parabolic, radial orbits also have an eccentricity of 1. Kepler's equation has nothing to do with parabolic orbits. The standard form of Kepler's equation applies only to elliptic orbits (0<epsilon<1). The article provides the complete solution to the standard form of Kepler's equation for all values of epsilon, even though when epsilon>=1 or epsilon<0 this solution has no physical application. When the eccentricity=1 and the angular momentum=0, the orbit is radial (not parabolic) and the radial "form" of Kepler's equation is used.

abuse of ellipsis

Ellipsis#In_mathematical_notation says: Normally dots should only be used where the pattern to be followed is clear. This is not the case in the series formulas in the present article. Also To indicate the omission of values in a repeated operation, an ellipsis raised to the center of the line is used between two operation symbols or following the last operation symbol. Plus sign is missing before '...' Bo Jacoby (talk) 05:28, 4 June 2010 (UTC).

I'm curious, how do you represent infinite series with unknown patterns? Without using an ellipsis, how do you distinguish infinite series from polynomials?

At any rate the pattern IS known, it is provided by the first two equations. The first equations are challenging to evaluate, so the power series expansions were included. The power series converge quickly only for small values of e, but they provide insight and they are easily differentiable to get the equations for velocity and acceleration as a function of time.

I recognize that it is common to include a plus/minus operator prior to the ellipsis. This is not a convention, and cannot be a convention. To include the next operator you must know the sign of the next omitted term, this is certainly not always the case. In the situation where the sign of the omited term is unknown, the convention would be impossible to comply with. In those situations the author would be forced to omit the operator, which would then be incorrectly interpreted as a either a typo or abuse of notation. Including an operator without an associated operand serves no purpose, except to make the expression larger. Norbeck (talk) 15:04, 8 June 2010 (UTC)

Numerical solution

I have added a short section on numerical approximation via Newton's method. Was there a reason that a numerical approximation was not discussed in earlier versions of this article? Jaxcp3 (talk) 16:40, 4 August 2012 (UTC)

Since the derivative can get quite close to zero, I recommend not using any derivative-based methods such as Newton-Raphson, secant, or regula falsi. I have written up an article here, for why I think bisection is the best method, and extremely stable (guaranteed convergence!). In practice, 17 or 18 iterations is sufficient to obtain 4-5 digits of accuracy.

Ackbeet (talk) 18:12, 28 December 2019 (UTC)

@Ackbeet: It would be best if you could be specific as to the conditions where Newton's method fails. By the way, 4-5 digits of accuracy after 17 iterations is terrible. Nowadays, I would expect close to full double precision accuracy and Newton's method frequently gives this in 4 or so iterations. cffk (talk) 19:44, 6 January 2020 (UTC)

Numerical approximation of inverse problem

In this article the Newton's method equation which is mentioned under the title: Numerical approximation of inverse problem reads

E_{n+1} = E_{n} - \frac{f'(E_{n})}{f(E_{n})} = E_{n} - \frac{ E_{n} - \epsilon \sin(E_{n}) - M(t) }{ 1 - \epsilon \cos(E_{n})}

   and should read:

E_{n+1} = E_{n} - \frac{f(E_{n})}{f'(E_{n})} = E_{n} - \frac{ E_{n} - \epsilon \sin(E_{n}) - M(t) }{ 1 - \epsilon \cos(E_{n})}

In the equation the function, f(E), and the derivative of the function, f'(E), are interchanged in the ratio. The expanded expression on the right side is correct. That ratio has the function as the numerator and the derivative of the function as the denominator. See: http://en.wikipedia.org/wiki/Newton%27s_method Dragonbyte (talk) 09:08, 23 August 2012 (UTC) dragonbyte

Link to video

I reverted the insertion of the link because I think it violates WP:V. Let's find a way to solve this. If it's linked to from another reliable source in the sense of WP:RS, then that would be fine I think. Martijn Meijering (talk) 20:43, 8 April 2014 (UTC)

The terms state that if the cite belongs to a recognizable expert it is fine. What is the problem?

By the way, I encourage you to take a look at the video to see what it is about. Have you?

Boazkaka (talk) 22:19, 8 April 2014 (UTC)

Yes, but only briefly. I didn't see any cite. Martijn Meijering (talk) 22:23, 8 April 2014 (UTC)

The whole point is that the argument should be presented in a clear way such that the viewers can judge for themselves. Look - I now realize that links to youtube from Wikipedia are hopeless as 3 editors already removed my links to different values without even trying to see if the explanation was helpful or not. This is very unfortunate since it makes a huge resource of information and pedagogical material (by numerous people) inaccessible . If you have ideas how to implement video to Wikipedia or make links that will stay there I will be very happy to hear. Otherwise I give up.

I think you guys have the wrong approach: you ask yourself- is this verifiable?

This is the wrong question - information on Wikipedia is never reliable. It could be very helpful though. You should ask yourself - is this helpful.

Best,

Boaz Katz Boazkaka (talk) 22:39, 8 April 2014 (UTC)

possesive

Just wondering, is it supposed to be Kepler's equation, or the Kepler equation? The actual reason for the question is another article about someone else's equation, and I will probably find an MOS to ask, but thought I would start here. Gah4 (talk) 07:03, 30 January 2019 (UTC)

The Explanatory Supplement to the Astronomical Almanac 3rd ed, 2013, edited by Urban and Seidelmann, give a section heading of "8.10.2 Solution of Kepler's Equation, M == E − e sin E" on page 340. Jc3s5h (talk) 14:28, 30 January 2019 (UTC)

From Bessel there is: Bessel beam, Bessel ellipsoid, Bessel function, Bessel's inequality, Bessel's correction, Bessel filter, Bessel (crater), Bessel transform, Bessel window. Seems not to be a consistent rule, some are posessive, some aren't. Any thoughts on on WP:MOS to ask? Gah4 (talk) 17:19, 30 January 2019 (UTC)

Usages that are inconsistent in reliable sources probably should not be covered by the MOS, unless the inconsistency creates confusion. The other justification for picking one choice when several equally good choices exist is to prevent editors from constantly editing articles just to change from one version to another, such as changing dates from the format 30 January 2019 to January 30, 2019. Since there are no editors constantly changing between "Kepler equation" and "Kepler's equation" this reason does not apply. Jc3s5h (talk) 17:24, 30 January 2019 (UTC)

What I thought is that law would go one way, function one way, and equation one way, but not the same for all of them. I am not sure now that is true, though. It does seem that laws are usually possessive, and functions usually not. Equations might go either way. That isn't so obvious to me. Gah4 (talk) 02:04, 31 January 2019 (UTC)

Derivation

You have not cited a reliable source, as required by the verifiability policy. Also, it is standard in mathematics to increase in the counterclockwise direction. Jc3s5h (talk) 16:00, 12 February 2020 (UTC)

Maths error

There is a maths script error in sub section "Radial Kepler equation" of section "Alternate form". Please correct it if possible. Hu741f4 (talk) 08:11, 21 January 2023 (UTC)