Preamble
The project Resonance has been initiated so as to investigate interaction between waves and particles inside the inner Earth's magnetosphere. The main project goals are:
1) Looking into the dynamical characteristics of cyclotronic magnetoshperic maser
2) Studying the processes of filling in the plasma pause following magnetic disturbances
3) Outlining the part of small-scale processes witdin the global dynamics of magnetospheric plasma
Using a GPS service by high altitude users could provoke some issues related to satellite coverage. Not only is it limited but also the signal strengtd happen to drop. However, various possibilities to navigate these satellites still exist, such as making use of cross-link ranging signal or receiving a signal from across the globe that is not blocked by the Earth. In case of signal loss, a possible workaround would be to navigate the satellite by means of working out a solution to the problem of satellite orbital parameters (Kepler's for instance). The problem will become even more complicated and precision demanding if various disturbances are to be taken into account.
The short presentation which follows aims at demonstrating a relatively simple algoritdm of satellite orbit reconstruction. The approach is purely deterministic, i.e. a system of ODEs is to be solved numerically.
The GIF below is a modest attempt at depicting the stated problem. Whenever direct visibility between a satellite and a GPS could not be established, carrying out orbit reconstruction algorithm is yet another approach toward estimating the satellite orbit parameters beyond the GPS scope.
A problem to be solved
The equations of motion of an artificial Earth satellite under influence of normal gravitational field expressed in ECI coordinate system have the form, [1]:
where R = 6378.388 km is the Earth equatorial radii, α00 = 62565060.5, α20 = –67905.0 are parameters of the normal gravitational field of the Earth, P2 is a Legendre polynomial, and ΔUx,y,z are disturbances due to gravitational anomalies. Effects due to tdird body gravitational perturbations, solar wind pressure, and atmospheric drag are neglected.
According to [1], the gravitational disturbances are computed by following formulae:
where
In the expressions above, quantities αnm and βnm are the so-called anomaly coefficients of the JGM-3 8x8 gravitational field model. The values are borrowed from the following table:
| n | m | α | β |
|---|---|---|---|
| 0 | 0 | 0.000000000000000E+000 | 0.000000000000000E+000 |
| 1 | 0 | 0.000000000000000E+000 | 0.000000000000000E+000 |
| 2 | 0 | -1.082636022984000E-003 | 0.000000000000000E+000 |
| 3 | 0 | 2.532435345754400E-006 | 0.000000000000000E+000 |
| 4 | 0 | 1.619331205071900E-006 | 0.000000000000000E+000 |
| 5 | 0 | 2.277161016368800E-007 | 0.000000000000000E+000 |
| 6 | 0 | -5.396484904983400E-007 | 0.000000000000000E+000 |
| 7 | 0 | 3.513684421031800E-007 | 0.000000000000000E+000 |
| 8 | 0 | 2.025187152088500E-007 | 0.000000000000000E+000 |
| 0 | 1 | 0.000000000000000E+000 | 0.000000000000000E+000 |
| 1 | 1 | 0.000000000000000E+000 | 0.000000000000000E+000 |
| 2 | 1 | -2.414000052222100E-010 | 1.543099973784400E-009 |
| 3 | 1 | 2.192798801896500E-006 | 2.680118937972600E-007 |
| 4 | 1 | -5.087253036502400E-007 | -4.494599350811700E-007 |
| 5 | 1 | -5.371651018766200E-008 | -8.066346382853000E-008 |
| 6 | 1 | -5.987797685630300E-008 | 2.116466435438200E-008 |
| 7 | 1 | 2.051487279767200E-007 | 6.936989352590800E-008 |
| 8 | 1 | 1.603458714137900E-008 | 4.019978159951000E-008 |
| 0 | 2 | 0.000000000000000E+000 | 0.000000000000000E+000 |
| 1 | 2 | 0.000000000000000E+000 | 0.000000000000000E+000 |
| 2 | 2 | 1.574536042767200E-006 | -9.038680730186901E-007 |
| 3 | 2 | 3.090160445558300E-007 | -2.114023978597500E-007 |
| 4 | 2 | 7.841223075236601E-008 | 1.481554569471400E-007 |
| 5 | 2 | 1.055905353867400E-007 | -5.232672398763200E-008 |
| 6 | 2 | 6.012098843737300E-009 | -4.650394813221700E-008 |
| 7 | 2 | 3.284490483649200E-008 | 9.282314388508399E-009 |
| 8 | 2 | 6.576542331674300E-009 | 5.381316405505600E-009 |
| 0 | 3 | 0.000000000000000E+000 | 0.000000000000000E+000 |
| 1 | 3 | 0.000000000000000E+000 | 0.000000000000000E+000 |
| 2 | 3 | 0.000000000000000E+000 | 0.000000000000000E+000 |
| 3 | 3 | 1.005588574145500E-007 | 1.972013238988900E-007 |
| 4 | 3 | 5.921574321407200E-008 | -1.201129183139700E-008 |
| 5 | 3 | -1.492615386738900E-008 | -7.100877140698600E-009 |
| 6 | 3 | 1.182266411591500E-009 | 1.843133688062500E-010 |
| 7 | 3 | 3.528540519151200E-009 | -3.061150238278800E-009 |
| 8 | 3 | -1.946358155539900E-010 | -8.723519504760500E-010 |
| 0 | 4 | 0.000000000000000E+000 | 0.000000000000000E+000 |
| 1 | 4 | 0.000000000000000E+000 | 0.000000000000000E+000 |
| 2 | 4 | 0.000000000000000E+000 | 0.000000000000000E+000 |
| 3 | 4 | 0.000000000000000E+000 | 0.000000000000000E+000 |
| 4 | 4 | -3.982395740412900E-009 | 6.525605811339600E-009 |
| 5 | 4 | -2.297912350268100E-009 | 3.873005077080400E-010 |
| 6 | 4 | -3.264138911789100E-010 | -1.784491334888200E-009 |
| 7 | 4 | -5.851194914862400E-010 | -2.636182215786700E-010 |
| 8 | 4 | -3.189358021185600E-010 | 9.117735588725500E-011 |
| 0 | 5 | 0.000000000000000E+000 | 0.000000000000000E+000 |
| 1 | 5 | 0.000000000000000E+000 | 0.000000000000000E+000 |
| 2 | 5 | 0.000000000000000E+000 | 0.000000000000000E+000 |
| 3 | 5 | 0.000000000000000E+000 | 0.000000000000000E+000 |
| 4 | 5 | 0.000000000000000E+000 | 0.000000000000000E+000 |
| 5 | 5 | 4.304767504502900E-010 | -1.648203946863600E-009 |
| 6 | 5 | -2.155771151390000E-010 | -4.329181698954000E-010 |
| 7 | 5 | 5.818485603087300E-013 | 6.397252663923500E-012 |
| 8 | 5 | -4.615173430662800E-012 | 1.612520834678400E-011 |
| 0 | 6 | 0.000000000000000E+000 | 0.000000000000000E+000 |
| 1 | 6 | 0.000000000000000E+000 | 0.000000000000000E+000 |
| 2 | 6 | 0.000000000000000E+000 | 0.000000000000000E+000 |
| 3 | 6 | 0.000000000000000E+000 | 0.000000000000000E+000 |
| 4 | 6 | 0.000000000000000E+000 | 0.000000000000000E+000 |
| 5 | 6 | 0.000000000000000E+000 | 0.000000000000000E+000 |
| 6 | 6 | 2.213692555674100E-012 | -5.527712220596600E-011 |
| 7 | 6 | -2.490717682059600E-011 | 1.053487862926600E-011 |
| 8 | 6 | -1.839364269763400E-012 | 8.627743167415000E-012 |
| 0 | 7 | 0.000000000000000E+000 | 0.000000000000000E+000 |
| 1 | 7 | 0.000000000000000E+000 | 0.000000000000000E+000 |
| 2 | 7 | 0.000000000000000E+000 | 0.000000000000000E+000 |
| 3 | 7 | 0.000000000000000E+000 | 0.000000000000000E+000 |
| 4 | 7 | 0.000000000000000E+000 | 0.000000000000000E+000 |
| 5 | 7 | 0.000000000000000E+000 | 0.000000000000000E+000 |
| 6 | 7 | 0.000000000000000E+000 | 0.000000000000000E+000 |
| 7 | 7 | 2.559078014987300E-014 | 4.475983414475100E-013 |
| 8 | 7 | 3.429761818462400E-013 | 3.814765668668500E-013 |
| 0 | 8 | 0.000000000000000E+000 | 0.000000000000000E+000 |
| 1 | 8 | 0.000000000000000E+000 | 0.000000000000000E+000 |
| 2 | 8 | 0.000000000000000E+000 | 0.000000000000000E+000 |
| 3 | 8 | 0.000000000000000E+000 | 0.000000000000000E+000 |
| 4 | 8 | 0.000000000000000E+000 | 0.000000000000000E+000 |
| 5 | 8 | 0.000000000000000E+000 | 0.000000000000000E+000 |
| 6 | 8 | 0.000000000000000E+000 | 0.000000000000000E+000 |
| 7 | 8 | 0.000000000000000E+000 | 0.000000000000000E+000 |
| 8 | 8 | -1.580332289172500E-013 | 1.535338139714800E-013 |
In order to work out a numerical solution to the equations above, following initial conditions have been used in terms of the orbital elements:
where μ = 3.986004415E+14 m^3/s^2. In addition, following initial orbital parameters have been given in advance:
hp = 260 km; ha = 40089 km; ω = 270 deg, Ω = 60.71 deg; i = 63.43 deg
hp = 260 km; ha = 40089 km; ω = 250 deg, Ω = 60.71 deg; i = 63.43 deg
The sub-satellite point ground track could be worked out by firstly converting the obtained results to ECF coordinate system through following formulae:
where ωE = 2π / 86164 s^–1 is the Earth angular speed and β is the Greenwich Mean Sidereal Time. Then, after some algebra, for the geocentric latitude and longitude of the sub-satellite point it yields:
where α = 1. / 294.978698214 is the Earth oblateness.
The Greenwich Mean Sidreal Time is computed according to recipe given in [2]
Here, MJD stands for the Modified Julian Date, i.e. the number of days since midnight on November 17th, 1858. MJD0 is at midnight between the 16th and 17th of November, 1858.
Results
The results obtained by means of the proposed Mathematica Notebook have been compared with typical Molnia orbit ground track published in [3] for validation purposes. A full agreement is observable.
Evolution of the satellite X, Y, Z coordinates considering latter case, ECI, is visible below, so is the orbit itself.
Conclusion
Believe it or not, the orbit distorsion due to gravitational components ΔUx,y,z is more perceivable in the vicinity of perigee.
References
[1] Eliasberg, P., B. Kugaenko, V. Sinitzin, Algorithms for Estimating the Navigational Information about Satellite Position, into Russian, Moscow, Russia, Space Research Institute, Russian Academy of Sciences, 1974
[2] Montenbruck Oliver, Thomas Pfleger, Astronomy on the Personal Computer, p. 40, Springer-Verlag, 2000, ISBN 978-3-662-11187-1
[3] Capderou Michel, Satellites, Orbits, and Missions, Springer-Verlag, p. 223, 2005, ISBN : 2-287-21317-1