Orbits with repeating ground tracks

Sven Grahn


Orbits with repeating ground tracks play a great role in space systems engineering. Ground tracks that repeat according to a certain pattern have important applications in remote sensing missions, reconnaissance missions, and to provide multiple opportunities for  rendezvous and docking with an orbiting vehicle.

For example, the orbit of Salyut-3  at the time of Soyuz-14 and -15 launches to it was such that it repeated every 63 revolutions, i.e. almost every four days. The nodal period at such occasion was 89.85 minutes. The same was true for Soyuz-21 when it was launched towards Salyut-5. Soyuz-23 and -24 were launched when the period was nearer 89.55 minutes, which corresponds to a ground track that repeats every 79 revolutions.

A general theory can be developed to calculate such orbits. The repeating pattern is determined by four parameters:

This theory involves the solution of a seven degree equation. The results for the inclination 51.6o are shown in the table below for repeating ground track patterns that repeat within 1-7 days.


Stabilized ground tracks used in the Salyut/Mir program

An examination of the Salyut/Mir program shows that these repeating pattern orbits have been widely used. The example of the Salyut 2, 3, and 5 missions has been given above. We can plot the orbital periods of various Salyut missions as a function of the number of days that the ground track repeats. The diagram used is the one shown below, which displays how the constraint of  a repeating ground track leads to a kind of "quantization" of the orbital period - "hyperfine splitting" to use a nuclear physics term!

However, these points in the diagram can be connected  with curves for periods with the same N and M parameters, but with varying values of Q. This is shown below where some of the orbital periods used by Salyut missions have been marked with red rings.

The figure below shows the orbital period of Salyut-4 during 1975. The launches of Soyuz craft destined for Salyut are marked, including the launch failure called the "April 5 anomaly". All Soyuz launches except the unmanned Soyuz-20 took place at a period very close to 91.35 minutes.


Stabilized ground tracks in the US-P (EORSAT) and US-A (RORSAT) programs

The Soviet/russian satellite systems to monitor foreign fleet movements split into two branches, the passive US-P program still in use and the radar equipped satellites designated US-A that carried nuclear reactors. These mission types both showed extensive maneuvering by the spacecraft to maintain both stabilized ground tracks, and in the case of US-A, also to maintain the inter-satellite distance between an active pair of satellites.

The US-A spacecraft maintained an orbit at i = 65o and a nodal period of 89.65 min, which corresponds to an average altitude of 255 km. This turns out to be an orbit with the parameters (N,M,Q)=(16, -1, 7) which corresponds to a ground track that repeats every 111 orbits, i.e. weekly. the graph below shows how Kosmos-1249, a US-A spacecraft. launched in March 1981,  was maneuvered extensively to keep its nodal period extremely close to 89.65 minutes.

The US-P spacecraft are still in use and they have operated in pairs in widely separated orbital planes. However, they have always maintained orbits with a well defined repeating pattern. These spacecraft use an orbit at i = 65o and a nodal period of 93.30 min, which corresponds to an average altitude of 434 km. This turns out to be an orbit with the parameters (N,M,Q)=(15, 1, 4) which corresponds to a ground track that repeats every 61 orbits.

A simple Visual Basic program to design orbits with different repeating patterns can be found here.

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