Orbits with repeating ground tracks
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.
N = the integer number of orbits
per day upon which the pattern is based
M = the number of revolutions
from the base revolution on the initial day that denotes the revolution
with which the ground track shall be matched on the day when the repeating
pattern shall occur.
Q = ground track is synchronized
after this number of days.
i = the inclination.
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
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.
ground tracks in the US-P (EORSAT) and US-A (RORSAT) programs
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.
to Space Tracking Notes