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The Wikipedia article Two Line Element Set says:

The SGP4 model was later extended with corrections for deep space objects, creating SDP4, which used the same TLE input data. Over the years a number of more advanced prediction models have been created, but these have not seen widespread use. This is due to the TLE not containing the additional information needed by some of these formats, which makes it difficult to find the elements needed to take advantages of the improved model. More subtly, the TLE data is massaged in a fashion to improve the results when used with the SGP series models, which may cause the predictions of other models to be less accurate than SGP when used with common TLEs. The only new model to see widespread use is SGP8/SDP8, which were designed to use the same data inputs and are relatively minor corrections to the SGP4 model.

Then the Wikipedia article Simplified Perturbations Models says:

SGP8/SDP8 introduced additional improvements for handling orbital decay.

Note that the current usage of the term SGP4 tends to refer to implementations that contain both SGP4 proper and SDP4, the switchover being at a period of 225 minutes.

The reference for the above quote is Revisiting Spacetrack Report #3: Rev 2 which says:

Spacetrack Report Number 3 officially introduced five orbital propagation models to the user community—SGP, SGP4, SDP4, SGP8 and SDP8—all “generally” compatible with the TLE data. At the time, SGP had just been replaced by SGP4/SDP4 (the latter having included deep-space perturbations). The SGP8/SDP8 model was developed to alleviate deficiencies of SGP4/SDP4 for the special cases of orbital decay and reentry. The approach provided a closed-form solution based on the general trends of orbital elements as they neared reentry, and was quite successful. However, there is no evidence to suggest that SGP8/SDP8 was implemented for operational TLE formation.

  1. Is this "closed-form solution based on the general trends of orbital elements as they neared reentry" the only significant difference between SGP8 and SGP4?
  2. What is this closed-form solution? Is there a reference for further reading?
  3. What does "...there is no evidence to suggest that SGP8/SDP8 was implemented for operational TLE formation..." mean exactly? A "forensic" examination of TLEs shows no sign that they are being generated for SPG8? Or no evidence that people are using SPG8? Something else?
  4. Is SGP8 ever used in practice?
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  • $\begingroup$ I know that this is a big question, but I think in this particular case it might be good to have the answer(s) collected in one place or future readers, and it is possible that if someone familliar with SGP8 know some of the answers, they may know all of them. $\endgroup$
    – uhoh
    Oct 8 '18 at 5:08
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About fifteen years ago, I very carefully implemented SGP8 and SDP8 (and the 'original' SGP) into my software (see https://github.com/Bill-Gray/sat_code ). It is hard to prove a negative -- i.e., that the 8 versions never saw any use -- but I've since seen SDP4/SGP4 get an incredible amount of use, and have seen the 8 variants get absolutely zero use and have never seen a reference to them being used.

I should note that I work almost exclusively with objects in multi-day to multi-month orbits, rather than those about to re-enter. Still, people using my code (of whom there are many) do use it for a lot of purposes; I think if they were using the 8 variants, I'd have heard about it.

But the convincer, to me, would be the 'Revisiting Spacetrack Report #3' comments. Those of us outside Space-Track/JSpOC/etc. have to do a certain amount of reverse-engineering; details on how SGP4/SDP4 are implemented internally, for example, have never been publicly released. The authors of 'Revisiting' are not only experts in this field; they also have some inside knowledge and access to historical data which I lack. If they can't come up with a case where the 8 versions were used, I'd be reasonably confident such use never occurred.

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    $\begingroup$ Thank you for your answer! As you mention, "it is hard to prove a negative" so in cases like these it may be necessary to fall back on "expert testimony". $\endgroup$
    – uhoh
    Apr 18 '19 at 2:13
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I think the statement you asked question 3 about describes Vallado, Kelso, and associates, nearly all of whom are former USAF Space Command people with experience inside the unit that uses SGP4 operationally, carefully comparing results from their reconstructed version of the code to results from the official code, in search of places they need to change their code to better match what's being done inside the latest SGP4. If they didn't see any sign of it, then I doubt anyone else would, either, so I can't help you on question 4.

You asked earlier whether SGP4 really changes anymore, and I have one thing to add on that topic now: yesterday I downloaded SGP4 code version 8.2, released just a couple of weeks ago (the link was broken for a while, but it's back in working order now). The release notes say they "Fixed some bugs hindering performance. SGP4 v8.2 is now over twice as fast as v8.1 (over four times as fast as v8.0)", so it appears they are indeed continuing to modify the official SGP4 code, even if only slightly.

Question 2, however, I can now answer. The reference you need is Hoots 1980, A Short Efficient Analytical Satellite Theory, Journal of Guidance, Control, and Dynamics 5.2 (194-199). This is the article I mentioned in this answer as being paywalled, although now that I have my hands on a copy, I note that this particular article states explicitly, "This paper is declared a work of the U.S. Government and therefore is in the public domain", so perhaps a complaint to the publisher (AIAA) is in order.

In the spirit of that statement about the public domain, let me quote some of the historical commentary:

One of the first theories of this type was developed by Hilton and Kuhlman <1> in 1966. (For a more recent reference see Hoots and Roehrich. <2>) This simplified general perturbations theory, hereafter called SGP, uses a simplification of the work of Kozai <3> for its gravitational model and it takes the drag effect on mean motion as linear in time. This assumption dictates a quadratic variation of mean anomaly with time. The drag effect on eccentricity is modeled in such a way that perigee height remains constant.

In 1970 Cranford (see Lane and Hoots <4>) developed a simplified general perturbations theory called SGP4. This model was obtained by simplification of the more extensive analytical theory of Lane and Cranford <5>, which uses the solution of Brouwer <6> for its gravitational model and a power density function for its atmospheric model.

The SGP8 theory is obtained by simplification of an extensive analytical theory of Hoots <7> which uses the same gravitational and atmospheric models as Lane and Cranford did but integrates the differential equations in a much different manner. The full theory is valid for all eccentricities between 0 and 0.1 and all inclinations not near 0 deg or critical inclination. Since the full theory contains several terms which only become important for larger eccentricities, it was felt many terms could be dropped without affecting predictions on most satellites. Furthermore, it can be shown that many terms included in the full theory are much smaller than the differences introduced by using the power density function atmosphere to model the real-world atmosphere. Thus, in an operational environment, many of the differences between the full theory and the simplified theory will be masked by differences between the model and the real world.

The descriptions of the changes, and the justifications for them, are:

The first type of term to be considered for simplification was the short-period drag periodics of Eqs. (3). It was found that these terms could be excluded from the theory with negligible effect. Second, we dropped terms of the form of drag coupled with drag which appear in the secular Eqs. (1) for mean anomaly and argument of perigee. The next type of term examined was of the form of drag coupled with gravity. Although terms of this form occur in the triple-primed differential equations for all six orbital elements, it was found that the direct effect on each of the elements was very small. However, since changes in the mean motion and eccentricity directly cause changes in total drag and since the mean motion is integrated a second time in the mean anomaly equation, it was found that the dominant part of the coupled terms must be retained in the mean motion and eccentricity differential equations, but all coupled terms can be neglected in the other four differential equations.

It is well known that the gravitational model of Vinti <9> allows algebraic combination of the second and fourth zonal harmonic terms. Since the Vinti potential differs from the true gravitational coefficients, we found that adopting the Vinti relationship $J_4=-J_2^2$ caused measurable degradation. However, if we use the Vinti potential in the long-period terms only and retain the values of $J_2$ and $J_4$ in the secular terms, we found a significant algebraic simplification in our equations with little loss of accuracy. In addition, adopting the Vinti potential removes from SGP8 any singularities at the critical inclination.

The final simplification concerned the purely gravitational terms. It was found that terms of size second order times $e^2$ could be neglected in the secular gravitational terms. Additionally, the gravitational periodics can be simplified considerably. It has been shown by Hoots <10> that the Lyddane-modified Brouwer geopotential transformation can be reformulated in terms of an alternate set of variables which allows a direct conversion from double-primed elements to Cartesian position and velocity while reducing the formula amount of the transformation by one-third. By using this alternate set of variables and retaining only the dominant periodic terms, we obtained a significant decrease in the number of terms in the transformation while sacrificing little in accuracy for most satellites.

References to "double-primed" and "triple-primed" equations and elements assume you have copies of Brouwer <6> and Lyddane <8> at hand and are following along in their notation.

The citations from the paper are:

<1> Hilton, C.G. and Kuhlman, J.R., "Mathematical Models for the Space Defense Center," Philco-Ford Corporation, Colorado Springs, Colo., U-3871, Nov. 1966.

<2> Hoots, F.R. and Roehrich, R.L., "Models for Propagation of NORAD Element Sets," Aerospace Defense Command, Peterson AFB, Colo., Project SPACETRACK Rept. No. 3, Dec. 1980.

<3> Kozai, Y., "The Motion of a Close Earth Satellite," Astronomical Journal Vol. 64, Nov. 1959, pp. 367-377.

<4> Lane, M.H. and Hoots, F.R., "General Perturbations Theories Derived from the 1965 Lane Drag Theory," Aerospace Defense Command, Peterson AFB, Colo., Project SPACETRACK Rept. No. 2, Dec. 1979.

<5> Lane, M.H. and Cranford, K.H., "An Improved Analytical Drag Theory for the Artificial Satellite Problem," Paper 69-925 presented at the AIAA/AAS Astrodynamics Conference, Aug. 20-22,1969.

<6> Brouwer, D., "Solution of the Problem of Artificial Satellite Theory Without Drag," Astronautical Journal, Vol. 64, Nov. 1959, pp. 378-397.

<7> Hoots, F.R., "Theory of the Motion of an Artificial Earth Satellite," Celestial Mechanics, Vol. 23, April 1981, pp. 307-363.

<8> Lyddane, R.H., "Small Eccentricities or Inclinations in the Brouwer Theory of the Artificial Satellite," Astronautical Journal, Vol. 68, Oct. 1963, pp. 555-558.

<9> Vinti, J.P., ''New Method of Solution for Unretarded Satellite Orbits," Journal of Research of the National Bureau of Standards, Vol.62B, 1959, pp. 105-116.

<10> Hoots, F.R., "Reformulation of the Brouwer Geopotential Theory for Improved Computational Efficiency," Celestial Mechanics, Vol. 24, Aug. 1981, pp. 367-375.

<11> Lerch, F.J. et al., "Gravity Model Improvement Using GEOS-3 (GEM 9&10)," Goddard Space Flight Center, Greenbelt, Md., GSFC Report No. X-921-77-246, Sept. 1977.

<12> Jacchia, L.G., "New Static Models of the Thermosphere and Exosphere with Empirical Temperature Profiles," Smithsonian Astrophysical Observatory, Boston, Mass., SP-313, May 1970.

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