# How much variation in orbit altitude is caused by gravitational variation, in LEO?

Imagine a hypothetical spacecraft in LEO, at 200km. Ignore air resistance for a moment.

It's actual height will vary, due to both gravitational anomalies and the shape of the Earth. Do we know how much variation there would be? Don't really care about the precise orbit, am interested in the order of magnitude.

• Very cool question! I guess the orbit should have a substantial inclination to make both effects large? – uhoh Apr 24 '20 at 10:29
• I didnt mind either high or low inclination, but now you've said that I want both! :) – user2702772 Apr 24 '20 at 10:45
• Hmmmm... first question needs to be "How much variation in the G field is there at a nominal altitude of X km ? After that, some ugly equations involving current velocity and current net forces. – Carl Witthoft Apr 24 '20 at 11:13
• I don't want to specify a specific height, in case someone had an odd height handy, say, 157.89Km. And my physics isn't good enough for the ugly. – user2702772 Apr 24 '20 at 11:18

Here's the result of my simulation:

The Earth's gravity field is modeled with the SGG-UGM-1 gravity model (computed using EGM2008 derived gravity anomaly and GOCE observation data) truncated to the degree and order 15. No atmosphere.

• In a message, I said "I get the same result as yzokras", but it's not true: my 0 deg plot is similar to yzokras 90 deg. Also the perturbation frequency shown in the yzokras plot is twice as high as mine. – Cristiano Apr 25 '20 at 11:50
• Your plots look good! I fixed my calculations, now we have similar results in the 90 deg inclination case. Which software did you use for this simulation? It would be nice to see some source code here. – yzokras Apr 25 '20 at 14:39
• I wrote my own C++ code based on the DOPRI853 integrator. – Cristiano Apr 25 '20 at 14:40
• @Cristiano I've just asked How to get analytical expressions for acceleration components due to zonal harmonics of a gravitational field? I want to integrate using $J_2$, $J_3$ and $J_4$ by myself without calling an external Geopotential function, but I'm stuck on the algebra! @@ – uhoh Apr 27 '20 at 12:49
• @uhoh I uploaded the code I use for J2...J5: cristianopi.altervista.org/J2J5.cpp (double checked). – Cristiano Apr 27 '20 at 17:40

Here is my take using the Orekit Python wrapper, a numerical propagator and the Eigen6s gravity field model limited to 8*8 spherical harmonics here. Using more harmonics has no visible effect on the plots anyways.

I chose to include a second definition of altitude (fourth plot), the altitude above a sphere whose radius is equal to Earth's equatorial radius. I find this quantity better to analyze the impact of orbit perturbations, because this pseudo-altitude should stay constant when there are no perturbations.

In the 90° inclination case, the peak-to-peak variation of the spherical altitude is around 8 kilometers. One can also see that this perturbations are periodic, because after one orbit the eccentricity goes back to 0 and the semi-major axis to its initial value.

https://nbviewer.jupyter.org/github/GorgiAstro/some-orbit-stuff/blob/c7ac315f246e7accace7a2bb154a57c782d589f2/gravity-perturbations.ipynb

• We should double check why we get different graphs. If uhoh posts his integration, it could help. – Cristiano Apr 25 '20 at 14:14
• I realized my calculation of the spherical radius was wrong, I was subtracting 6378.1km to the semi-major axis... Now this is fixed, and the results are similar with @Cristiano – yzokras Apr 25 '20 at 14:37
• There's two answers that are both worthy of acceptance. I've accepted the other, because it allowed me to understand the impact of varied inclination at a glance. Your answer has prompted other questions which will asked tomorrow. But I wanted you to know your answer was good. – user2702772 Apr 25 '20 at 19:37
• I've just asked How to get analytical expressions for acceleration components due to zonal harmonics of a gravitational field? I want to integrate using $J_2$, $J_3$ and $J_4$ by myself without calling an external Geopotential function, but I'm stuck on the algebra. – uhoh Apr 27 '20 at 12:50