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The classic orbit elements, $(a,e,i,\Omega,\omega,\nu)$, have annnoying singularities in their non-Keplerian perturbed dynamics at $i=0\pm \pi$ and $e=0$. So Arsenault, then Braucke & Cefola, then Walker, came up with versions of the so-called "equinoctial elements" that are free of these singularities (except at $i=\pi$). The set developed by Walker in 1985, called the "modified equinoctial elements" (MEEs), seem to be the most popular and are defined algebraically in relation to the classic elements as

$$ p = \tfrac{h^2}{\mu} = a(1-e^2) \quad,\quad e_1 = e\cos(\Omega+\omega) \quad,\quad e_2 = e\sin(\Omega+\omega) \\ q_1 = \tan \tfrac{i}{2}\cos{\Omega} \quad,\quad q_2 = \tan \tfrac{i}{2}\sin{\Omega} \quad,\quad l = \Omega + \omega +\nu $$

Where I have used the labeling convention of the first paper attached at the bottom of this post.

question 1: Why do the MEEs use the semilatus rectum, $p$, instead of the classic semi-major axis, $a$? This was clearly intentional as they actively replaced $a$ with $p$. In the 1985 paper they say that this makes the MEE "applicable to all orbits". Is the semi-major axis not defined for all orbits?

question 2: If $p$ really is better than $a$, is there any reason why the angular momentum magnitude, $h=||\vec{\mathbf{r}}\times\dot{\vec{\mathbf{r}}}||=\sqrt{\mu p}$, could not be used in place of $p$? Both cause issues for rectilinear motion but, other than that, it seems that $h$ would be just as good as $p$ (and is slightly more intuitive).


Recent conference paper explaining the meaning of the equinoctial elements: On Equinoctial Elements and Rodrigues Parameters

Walker's 1985 paper

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    $\begingroup$ I'm upvoting this question solely because it has so many cool words in the title $\endgroup$ Aug 24, 2022 at 13:22

2 Answers 2

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Partial Answer:

For the extreme special case of the parabolic trajectory, where Orbital Eccentricity = $e=1$, the semi-major axis hits a singularity at $a=\pm\infty$, whereas the semi-latus rectum remains defined.

In the case of the parabola, the semi-latus rectum is equal to twice the periapsis distance, allowing differentiation amongst different parabolic orbits where the other Keplerian parameters are the same.

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  • $\begingroup$ Ah ok. I was thinking Of the semi-major axis as defined in terms of specific energy so didn’t see how it’d be singular, but your answer makes sense from the geometry. The angular momentum magnitude, h, is always defined isn’t it? I feel like swapping p for h makes the dynamics slightly simpler $\endgroup$
    – J Peterson
    Aug 23, 2022 at 19:30
  • $\begingroup$ @JPeterson I think it might be geometric convention. Certainly if you're perturbing an orbit, and using the general polar equation for conic sections to graph it, $r=\frac{a(1-e^2)}{1 + e \cos \theta}$ is troubled by the parabolic discontinuity with increasing eccentricity, whereas $r=\frac{p}{1 + e \cos \theta}$ passes through it smoothly. $\endgroup$
    – notovny
    Aug 23, 2022 at 21:35
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Another partial answer: there's nothing wrong with using specific angular momentum, but beware coordinates that use $h$ to mean something else. Also, being "intuitive" is not one of the criteria used to choose these elements. The goal is to find combinations that behave well when you start applying the calculus-based machinery of classical mechanics.

The conference paper you linked, Peterson, Junkins, and Majji (2022), "On Equinoctial Elements and Rodrigues Parameters", AAS 22-832, says on page 16:

We note that, in place of the semilatus rectum, $p$, some sources instead use the mean motion, $n$, the specific angular momentum magnitude, $h$, the semi-major axis, $a$, or the specific energy, $\mathcal{E} [-\mu/2a]$. What we have labeled $e_1$ , $e_2$ , $q_1$ , and $q_2$ above, other sources often label using some permutation of the letters $f$, $g$, $h$, $p$, and $k$, with different sources using different labeling conventions. However, these letters are already in common use for various other parameters in celestial mechanics (e.g., this work and many others use $h$ and $p$ for the specific angular momentum and semilatus rectum, respectively, and $f$ is often used for the true anomaly).

There are many different ways to do it, but they all depend on each other, so the ones you didn't choose pop up all over the place in the equations. For example, your other source, Walker, Ireland, and Owens (1985), "A Set of Modified Equinoctial Orbit Elements", Celestial Mechanics 36(4), 409-419, chooses $p$ as a coordinate, but 10 of the 13 equations on page 412 have $\sqrt{\mu p}$ in them, which equals $h$. Similarly, Peterson et al. chose $h$ as a coordinate, and many of their equations contain $h^2/\mu$, which equals $p$. Which of them you prefer is partly just a matter of taste.

Broucke and Cefola (1972), "On the Equinoctial Orbit Elements", Celestial Mechanics 5(3) 303-310 describe the criteria they used for choosing elements as

For the particular form of the equinoctial elements studied, all the partial derivatives and the Poisson and Lagrange brackets exist for orbits with an inclination of 90$^\circ$ as well as for orbits with zero eccentricity and/or zero inclination. ... For a small eccentricity application as the planet Venus, the equinoctial elements give another improvement of a factor of two in running time, compared with the classical orbit elements, thanks to the removal of the singularity in the classical partial derivatives at $e=0$.

Writing them all out takes quite a while, but the main idea is to look at things in the denominator that might be zero, and invent alternative parameterizations that alter the offending element into something else that doesn't divide by zero.

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  • $\begingroup$ "Also, being intuitive is not one of the criteria used to choose these elements." This is certainly true but if there is an opportunity to make things more intuitive, without loss of utility, then I prefer to take it. Messy algebraic relations in celestial mechanics seem to often have a semi-intutive geometric interpretation behind them. $\endgroup$
    – J Peterson
    Aug 30, 2022 at 18:21
  • $\begingroup$ @JPeterson when possible, sure --- but it is hard to tell what has more utility, until you've spent a long time buried in the equations. $\endgroup$
    – Ryan C
    Aug 30, 2022 at 18:59

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