Rate of precession
The rate of precession depends on the inclination of the orbital plane to the equatorial plane, as well as the orbital eccentricity.
For a satellite in a prograde orbit around Earth, the precession is westward (nodal regression), that is, the node and satellite move in opposite directions.[1] A good approximation of the precession rate is
where
- ωp is the precession rate (in rad/s),
- RE is the body's equatorial radius (6378137 m for Earth),
- a is the semi-major axis of the satellite's orbit,
- e is the eccentricity of the satellite's orbit,
- ω is the angular velocity of the satellite's motion (2π radians divided by its period in seconds),
- i is its inclination,
- J2 is the body's "second dynamic form factor" [2](−√5C20[3] = 1.08262668×10−3 for Earth).
This last quantity is related to the oblateness as follows:
where
- εE is the central body's oblateness,
- RE is central body's equatorial radius (6378137 m for Earth),
- ωE is the central body's rotation rate (7.292115×10−5 rad/s for Earth),
- GME is the product of the universal constant of gravitation and the central body's mass (3.986004418×1014 m3/s2 for Earth).
The nodal progression of low Earth orbits is typically a few degrees per day to the west (negative). For a satellite in a circular (e = 0) 800 km altitude orbit at 56° inclination about Earth:
The orbital period is 6052.4 s, so the angular velocity is 0.001038 rad/s. The precession is therefore
This is equivalent to −3.683° per day, so the orbit plane will make one complete turn (in inertial space) in 98 days.
The apparent motion of the sun is approximately +1° per day (360° per year / 365.2422 days per tropical year ≈ 0.9856473° per day), so apparent motion of the sun relative to the orbit plane is about 2.8° per day, resulting in a complete cycle in about 127 days. For retrograde orbits ω is negative, so the precession becomes positive. (Alternatively, ω can be thought of as positive but the inclination is greater than 90°, so the cosine of the inclination is negative.) In this case it is possible to make the precession approximately match the apparent motion of the sun, resulting in a heliosynchronous orbit.
The used in this equation is the dimensionless coefficient from the geopotential model or gravity field model for the body.