Nodal precession

Nodal precession is the precession of an orbital plane around the rotation axis of an astronomical body such as the Earth. This precession is due to the non spherical nature of a spinning body which creates a non spherical gravitational field.

Around a spherical body, an orbital plane would remain fixed in space around the central body. However, because most bodies are non spherical due to their rotation, this causes an equatorial bulge. Equatorial bulges have a gravitational effect that causes orbits to precess around the rotational axis of the central body.

For a satellite in a retrograde orbit about the Earth, the longitude of the ascending node increases and the node precesses eastward. This nodal progression allows sun-synchronous orbits to maintain a constant angle relative to the Sun.

Equation

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 about the Earth, the precession is westward (nodal regression), the node and satellite move in opposite directions[1]. A good approximation of the precession rate is

\omega_p = -\frac{3}{2}\frac{r^2}{(a(1-e^2))^2} J_2 \omega \cos i

where

\omega_p \, is the precession rate (rad/s)
r \, is the Earth's equatorial radius (6 378 137 m)
a \, is the semi-major axis of the satellite orbit
e \, is the eccentricity of the satellite orbit
\omega \, is its angular frequency (2\pi radians divided by its period)
i \, its inclination
J_2 \, is the Earth's second dynamic form factor \scriptstyle (-\sqrt{5}C_{20}=1.08262668\times10^{-3}).

This last quantity is related to the oblateness as follows:

J_2 = \frac{2 \varepsilon_E}{3} - \frac{a^3 \omega_E^2}{3 G M_E}

where

\varepsilon_E \, is the Earth's oblateness
a \, is Earth's equatorial radius
\omega_E \, is the Earth's rotation rate (7.292115×10−5 rad/s)
G M_E \, is the product of the universal constant of gravitation and the Earth's mass (3.986004418×1014 m3/s2)

References

  1. ^ Brown, Charles. Elements of spacecraft design. pp. 106. http://books.google.com/books?id=mTSSMhcmVbkC&lpg=PA95&ots=hkARYnIy9h&dq=ascending%20node%20regression&pg=PA121#v=onepage&q=ascending%20node%20regression&f=false. 

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