# In a 3D space defined by semimajor axis, eccentricity and inclination, what would be the shape of the space where SDP4 works better than SGP4?

@BillGray's answer to Why can't custom-made TLEs for the DSCOVR launch booster in orbit around Earth work with SDP4? is quite interesting and informative; I fail to do it justice by summarizing it as saying that SDP4 is an improvement for orbits of 12 and 24 hours since it tries to address perturbative effects of the Sun and Moon's gravity on high Earth orbits, but doesn't work well for Earth orbits much higher (such as some discarded rocket bodies for missions to Lagrange points and deep space.

Thus some of the discussed custom TLE's of very high orbits end up using SDP4 and some stick with SGP4 (via a flag in the TLE it seems).

A TLE creator's gotta do what a TLE creator's gotta do.

The following was also asked and somewhat answered by means of "read this" answers: How do SDP4's "Deep space" corrections to SGP4 account for the Sun's and Moon's gravity?

But in a comment under @BillGray's answer to the DSCOVR booster question, I wrote:

I'm surprised that SDP4's "deep space" math has the narrow range of applicability that it does. In a 3D space defined by $$a, \epsilon, i$$ I wonder what the shape of the space where SDP4 works better than SGP4 would look like.

So I thought I would just ask that as a new question.

Question: In a 3D space defined by semimajor axis, eccentricity and inclination $$(a, \epsilon, i)$$, what would be the shape of the space where SDP4 works better than SGP4?

• I think that this is one of the most interesting questions I've ever asked, why the anonymous, unhelpful, silent down vote? Without some feedback I'll never know!
– uhoh
May 26 at 2:27
• I'll chip in an upvote because it seems to be a perfectly legit question albeit one I don't really grok. May 26 at 2:42
• @OrganicMarble thanks! Maybe the dv was for generalized nongrokability
– uhoh
May 26 at 3:46

I have no formal, mathematical answer to this. When I have to fit TLEs to a new object, I try SDP4; if it fails, I drop back to SGP4. It's an entirely empirical "gee, SDP4 didn't work here" approach. I also haven't kept particularly careful track of when it happens, and I've only had SDP4 fail on maybe a dozen objects. So my statistics are not of high quality. With those warnings :

SGP4 almost alway works for objects that are not at escape velocity relative to the earth, even, for example, Gaia (at the earth-sun L2 point) and a few temporarily captured objects such as the Surveyor 2 Centaur in late 2020/early 2021. Exceptions involve extreme closeness to the moon, such as during the lunar flybys of the Chang'e 2, 3, and 4 boosters.

SDP4 is much less robust, and fails for higher and more eccentric orbits. Again, I've not done a careful survey of the matter, but inclination doesn't seem to matter much. I can get a decent fit for IMP-7, in a twelve-day, e=0.08 orbit, and for any object with a period of four days or less. But SDP4 usually fails miserably on objects with apogees beyond the moon.

• The fitting you mention here is to reported observations, correct? I'm wondering if a simple state vector propagation with say only $J_2$ could be compared directly to SGP4 and SDP4 outputs based on TLEs that are chosen to match the state vectors. I'm not proposing that you do that of course because it's work. But if it could be scripted then a "data cube" of state vectors propagated for $a, \epsilon, i$ points could be generated and the space of $\text{rms}{}_{SDP4} \le \text{rms}{}_{SGP4}$ discovered? (State vectors would use ephemeris Sun, Earth and Moon positions)
– uhoh
May 30 at 1:30
• Although, even for a single $\text{rms}$ value one might need to sample a variety epochs within a lunar month and combinations of $\Omega$ and $\omega$ which makes this more like a six than a three dimensional problem.
– uhoh
May 30 at 1:34
• I found the deep space equations, now looking for additional explanations/insight for how they work. Textbook or scholarly discussion of equations used by the SDP4 part of SGP4 TLE propagators beyond Space Track Report #1?
– uhoh
May 30 at 3:02
• I fit an orbit to observations, and then generate an ephemeris of state vectors and fit TLEs to that. You'll see that the TLEs will have comments such as "Worst residual: nn.nn km". That's relative to the state vector ephems for the particular time span (usually one day) to which they were fitted, and describe only the error of the TLEs relative to the computed state vector. Usually, the error in the state vector is less... but as described in the boilerplate at the top of the file, it may not be, if we didn't have many observations to work with. May 30 at 19:30
• I think the 'simple state vector propagation' would have to include more than J2. For these high-flying guys, J2 is less significant than lunar and solar perturbations. My sense is that SGP4/SDP4 actually does a good job of handling J2 and even J3 and J4, plus some lower harmonics... none of which (except J2) matter much for cislunar objects. SDP4 has primitive handling of lunisolar perturbations, good enough for 12 and 24-hour orbits, but insufficiently nuanced for week-long orbits. May 30 at 19:33