# SGP4 gravity model accuracy

From a simulation, I obtain the instantaneous radius vector of the SFERA 2 satellite. If I compare that radius vector with the one obtained from the SGP4 propagator, I get very big differences. The following graph shows what I mean: How the radius vector is calculated

SGP4: starting from the TLE 17247.56557513, I propagate that TLE for 1 orbit centered on the TLE epoch to find the perigee and the apogee and I do the same for the next 539 TLEs (the SGP4 plot shows 540 points obtained from 540 TLEs).

The other 3 plots are obtained from a numerical propagator based on the DOPRI853 integrator. The initial TLE is the same used for the SGP4 plot (17247.56557513). The blue plot is obtained when I use the GRACE Gravity Model Version 3 Combined truncated to the order and degree 25, for the orange plot I use a spherical Earth and for the green plot I add the J2 perturbation.

Is it possible that the blue plot shows the true shape of the radius vector, while the SGP4 plot show just an approximate shape? In other words, is it possible that the SPG4 propagator is so wrong?

• SGP4 model show 1~3km error with each day. I think grace is better takes much more harmonics into consideration – Prakhar Apr 29 '18 at 15:48
• Not sure I'm understanding your method right, but if you're plugging in orbital elements straight from the TLE into your numerical integrator, don't do that – Chris Apr 29 '18 at 16:04
• @Chris I convert the TLE elements to the osculating initial state with the SGP4, then I convert the TEME osculating state to J2000 osculating state. – Cristiano Apr 29 '18 at 16:59
• @Prakhar. But if we take the blue plot as the most accurate, the SGP4 error is incredibly big (up to 40 km!) – Cristiano Apr 29 '18 at 17:03
• @Cristiano well, that's the way to do it. Carry on – Chris Apr 29 '18 at 19:27

## 1 Answer

Is it possible that the blue plot shows the true shape of the radius vector, while the SGP4 plot show just an approximate shape?

Just because SGP4 is indeed an approximation (a fairly low accuracy approximation) of the true orbit does not mean that the blue plot is better. I very strongly suspect that the blue plot is even worse, for a number of reasons. One is while SGP4 does lose accuracy at a fair clip, you are using each individual TLE to propagate over rather short time intervals, less than a day if I read the question correctly. This eliminates much of the error that results from using SGP4. In contrast, you are using an initial state and propagating for almost ten months. The error in that initial state will grow at least quadratically over time, and worse than that if there are losses in the system.

Another reason is that there obviously are losses in the system. This is a decaying orbit. An atmospheric drag model that yields two significant digits on the drag is a very good model. One minor solar event makes the upper atmosphere swell. The predicted drag versus the actual drag ten months out has zero significant digits.

Finally, you are using an integrator, Dormand-Prince, that is notoriously bad for orbit propagation. Bad as it is, plain old vanilla RK4 often does better job at propagating orbits than does Dormand-Prince. As a sign of this, your orbit gets more eccentric as time passes. Atmospheric drag should in general act to circularize an orbit.

• There's a lesson or two to be learned here. One is that a supposedly better integrator can in fact be a lousy integrator. This is particularly so for orbital mechanics. Another is that adding more precision can result in reduced accuracy. There's no reason to use a 25x25 earth gravity model in low earth orbit. The errors that result from not knowing atmospheric drag overwhelm the errors that result from using a lower fidelity gravity model. Even worse, going to higher degrees does not always improve accuracy. Sometimes it degrades accuracy. ... – David Hammen Apr 30 '18 at 17:03
• ... This is particularly so when using an adaptive integrator such as Dormand-Prince (matlab rk45) that can be artificially pulled this way and that by Runge's phenomenon (which is analogous to Gibb's phenomenon). – David Hammen Apr 30 '18 at 17:05