Mean to Osculating conversion for non-J2 averaged elements

Question

What is the appropriate technique to convert mean elements (true anomaly averaged) into osculating elements when the model has J2, SRP, drag and additional forces modeled. Brouwer's transformation is only valid for J2-averaged mean elements.

Answer 1

There is not just one conversion, because the word "mean" can mean many different things, depending on the context.

Specific formulae that may or may not help you include those in Kozai 1959, Brouwer 1959, and Walter 1967, and I'll have more to say about them farther down, but the most important thing right now is to realize there are lots of possibilities. In ordinary English, mean can mean malicious, stingy, inferior, superior, humble, tool, method, riches, intermediate, to intend, to signify, or to imply. In mathematics, where it means average, there are differences between arithmetic mean, geometric mean, harmonic mean, and others. In orbital mechanics, there is the mean anomaly, as opposed to eccentric anomaly, true anomaly, and other anomalies (which aren't considered anomalous anymore, we just kept a very old name for these angles), but there is also a difference between the osculating mean anomaly and a mean mean anomaly, and there is more than one way to construct a mean mean anomaly.

When discussing orbits, mean usually means some kind of average (though not necessarily the one you think), but that means it also usually means fictitious. For example, mean anomaly is the true anomaly that an object with the same semimajor axis orbiting the same central body would have if that orbit were circular — but of course the orbit isn't circular, so there isn't actually anything where the mean anomaly says something would be. Similarly, "mean solar time" (the convention of pretending that all days have the same length) is based on the location of the fictitious mean sun, which is where the sun would appear to be if the earth's orbit were circular, even though it's not. If we kept time by where the sun actually is relative to earth (reckoning a day from noon to noon, as defined locally by "the sun is as high in the sky as it's going to get today"), the length of each day would vary by as much as 16 minutes from the nominal 24 hours (see Equation of Time).

This is related to the situation in probability, where the "expected" value (the arithmetic mean) when rolling a fair die is three and a half, which is impossible to roll, but is the most accurate way to locate the middle of the distribution. There are also "mean of date" coordinate frames, in which the equator, equinox, or ecliptic differ from their "true of date" values, which we use because sometimes (but not always) it is easier to work with the pretend orientations these things would have if every orbit were circular and not inclined, precessing, or otherwise deviated from the simplest conceivable relationship.

That last definition — easier to work with — is the one most relevant to the question of why "mean elements" are called that. If two-body mechanics were exact (the two items of interest are point masses, and there is nothing else at all in the observable universe), then five of the six Keplerian elements ($a$, $e$, $i$, $\Omega$ and $\omega$), everything except the anomaly, would be constant in time. Since the bodies aren't point masses, and there are lots more than just two of them in the solar system, then those orbit parameters change with time, as described by Lagrange's planetary equations. Mean elements are an attempt to find alternate coordinates in which to express the problem, that actually would be constant in time if certain perturbations of arbitrary complexity were exactly true, or at least much closer to constant than their osculating (instantaneous) namesakes are.

Walter, "Conversion of Osculating Orbital Elements into Mean Elements", The Astronomical Journal 72(8) 994-997, 1967, says of Brouwer, "Solution of the Problem of Artificial Satellite Theory without Drag", The Astronomical Journal 64 378-396, 1959, and Kozai, "The Motion of a Close Earth Satellite", The Astronomical Journal 64 367-377, 1959, (emphasis mine),

Both of them start out from a set of mean elements, however, which are different and generally not known. In this context, by mean elements we understand osculating elements from which short-periodic and long-periodic perturbations of the earth's potential have been subtracted. In practice, the osculating elements are the ones that are usually available as, for instance, in the case of a stepwise numerical integration or in the case of a set of elements obtained from an orbit injection maneuver. Thus one must resort to osculating elements as a starting point for the generation of mean elements.

The key is to know exactly which perturbations have been subtracted, so that you can add them back in. Walter's paper describes an iterative method for making those calculations, assuming that you already know which of Kozai's, Brouwer's, or somebody else's (perhaps Lyddane's, or Cefola's, or Delaunay's original) mean element theory you are trying to convert to or from. I am not going to write the dozens of pages of equations it takes to fully define these coordinates, or the necessary transformations among them, since the linked papers are free to access and the most likely result of me trying to translate them all into MathJax is a bunch of typos that just confuse you further. Go to the original sources, stare at them in horror, and then see if you can take the following shortcut. First, though, I'll convert just one expression, to give those who don't click through a taste of the mathematics one can easily drown in.

Kozai's mean mean anomaly, minus the osculating mean anomaly, so the linear part (nearly) cancels out, equals:

$$\frac{3 J_2^2 R^2}{2 a^2 e \sqrt{1-e^2}}\left[ \left( \frac{3}{2}\sin^2 i - 1 \right) \left\{ \left( 1- \frac{e^2}{4} \right) \sin\nu + \frac{e}{2}\sin 2\nu + \frac{e^2}{12}\sin 3\nu \right\} \right.$$ $$+\ \left. \sin^2 i \left\{ \left( \frac{1}{4}+\frac{5e^2}{16} \right) \sin (\nu + 2\omega) - \frac{e^2}{16}\sin (\nu - 2\omega) -\left( \frac{7}{12} -\frac{e^2}{48}\right) \sin(3\nu+2\omega) \right. \right.$$ $$\left. \left.- \frac{3e}{8}\sin(4\nu+2\omega) - \frac{e^2}{16}\sin(5\nu+2\omega) \right\} \right] $$

where $\nu$ is osculating true anomaly, which hasn't yet been averaged over by integration, because the result of that is even harder to write. David Hammen likened these papers to a blizzard of mathematics, which seems exactly right to me.

What you should actually do is NOT attempt to code these yourself! Instead, if you are interested in Kozai's or Brouwer's mean elements, go download the conversion functions that are part of the well-tested (and free to use, with a free registered account, but not open source) code library that comes with the official version of SGP4. If you are interested in Cefola's Semianalytic Satellite Theory, grab OreKit (both free to use and also open source), and use the well-tested conversion functions that come with that.

Using the US Space Force SGP4 version 8.2.1, released November 2021, with the input TLE from this question,

1 25544U 98067A   21356.62544795  .00006800  00000-0  13125-3 0  9998
2 25544  51.6428 130.9420 0004657 342.5227  11.5462 15.49048823317794

any version of mean anomaly you plot is going to be dominated by the linear trend caused by the fact that the ISS completes just under fifteen and a half orbits per day, which adds up to 22,300 degrees in the four days of the plots shown in one of the answers to that question. Subtracting this linear trend (3.873 degrees per minute), I find

mean mean anomaly looks like

osculating mean anomaly looks like

The Brouwer and Kozai theories have different definitions of mean motion, which is the time derivative of mean anomaly, and yet when I tried to use the SGP4 library functions to convert Brouwer mean anomaly into Kozai mean anomaly, it told me they were equal, which is somewhat bothersome, but probably just means I've never actually asked it to do this particular computation before, so I'm probably not asking it in quite the right way.