Converting Modified Keplerian Orbital Parameters to Classical Keplerian

Question

I'm in the process of comparing the data in the USSC's SATCAT DB with the ESA's DISCOS DB. Unfortunately, the Americans use modified Keplerian orbit descriptions that look like:

{
    "PERIOD": "96.19",
    "INCLINATION": "65.10",
    "APOGEE": "938",
    "PERIGEE": "214",
}

I'm planning on working with the classical keplerian descriptions that the ESA uses that look like:

{
    "raan": 0,
    "inc": 26.13,
    "sma": 24124000,
    "ecc": 0.72566885,
    "aPer": 0
}

With those parameters being the right ascension ascending node, inclination, semi-major axis, eccentricity and argument of periapsis.

Now, I think I can do the following:

eccentricity = eccentricity = (apogee - perigee) / (apogee + perigee)
semiMajorAxis = (period^2 * gravitational parameter of central body) / (4 * pi^2)
apoapsis = apogee + radius of central body
periapsis = perigee + radius of central body

But leaves me with no idea how to get the raan given there's seemingly nothing to do with this angle in the SATCAT's orbit model.

Am I missing something here? Can I actually derive this value from the four provided values (and maybe some known constants).

In my search I came across this paper from JPL that seems to describe some of these relationships.

However, I'm confused even further by this as the parameter I need is seemingly circularly defined:

The symbols are defined as:

And further we see:

Answer 1

See also my answer to the version posted on Space Exploration SE (https://space.stackexchange.com/questions/61473/get-fully-defined-orbit-from-space-track-org-satcat/61482#61482)

First, the way the catalog is organized, you have the wrong URL. SATCAT is just general knowledge of the things that stay constant, like country of origin and date of launch, and things that are close to constant, like a necessarily incomplete description of the orbit. If you want actual orbit data, which changes sometimes several times a day, you want the ELSET database, with "gp" in the URL. The resulting data format is usually called a "TLE", meaning Two-Line Element Set. The names of the fields look familiar, but in truth they are something very different, which brings us to the other problem.

Second, the numbers you get from the correct URL should NOT be used to just plug into equations you got from anywhere else. The quantities published on the US Space Force's space-track.org are "mean" elements, which are complicated functions of the osculating elements, in a way that is not clearly described. That is, we have the original papers from 1959 to 1979 that describe the history of how these complicated models evolved and were then simplified to reduce run times on old computers, but then the US Government stopped publishing the changes. The computer program that takes in the TLEs and outputs equinoctial and other kinds of elements is freely available in that anyone can use it, but the actual code it uses has not been seen by anyone outside that particular US military element in a very long time.

The concept of "general perturbations" is a beautiful exercise in Hamiltonian mechanics, in the classical "contact transformation" sense that the work of creating a useful model is by coming up with complicated ways that change the coordinates of the problem into something that makes the Hamiltonian as close to constant, and therefore trivial, as possible. If you are interested in the history and references of how this all works, I recommend starting with a few of my posts on space exploration for an annotated list of links:

https://space.stackexchange.com/questions/43283/confused-about-sgp4-implementation-published-by-celestrack/56096#56096

https://space.stackexchange.com/questions/23810/how-do-sdp4s-deep-space-corrections-to-sgp4-account-for-the-suns-and-moons/46346#46346

https://space.stackexchange.com/questions/31174/differences-between-sgp8-and-the-standard-sgp4-is-it-ever-used-in-practice/56113#56113