To first approximation, a satellite orbiting the earth moves in an elliptical orbit. That’s what would get from solving the two-body problem: two point masses orbiting their common center of mass, subject to no forces other than their gravitational attraction to each other.

But the earth is not a point mass. Neither is a satellite, though that’s much less important. The fact that the earth is not exactly a sphere but rather an oblate spheroid is the root cause of the J2 effect.

The J2 effect is the largest perturbation of a satellite orbit from a simple elliptical orbit, at least for satellites in low earth orbit (LEO) and medium earth orbit (MEO). The J2 effect is significant for satellites in higher orbits, though third body effects are larger.

Legendre showed that the gravitational potential of an axially symmetric planet is given by

V(r, phi) = frac{Gm}{r} left( 1 - sum_{k=2}^infty J_k  left( frac{r_{eq}}{r}right)^k P_k(cos phi) right)

Here (r, φ) are spherical coordinates. There’s no θ term because we assume the planet, and hence its gravitational potential, is axially symmetric, i.e. independent of θ. The term req is the equatorial radius of the planet. The Pk are Legendre polynomials.

For a moderately oblate planet, like the one we live on, the J2 coefficient is much larger than the others, and neglecting the rest of the coefficients gives a good approximation [1].

Here are the first few coefficients for Earth [2].

begin{align*} J_2 &= phantom{-}0.00108263 \ J_3 &= -0.00000254 \ J_4 &= -0.00000161 end{align*}

Note that J2 is three orders of magnitude smaller than 1, and so the J2 effect is small. And yet it matters a great deal. The longitude of the point at which a satellite crosses the equatorial plane may vary a few degrees per day. The rate of precession is approximately proportional to J2.

The value of J2 for Mars is about twice that of Earth (0.001960454). The largest J2 in our solar system is Neptune, which is about three times that of Earth (0.003411).

There are many factors left out of the assumptions of the two body problem. The J2 effect doesn’t account for everything that has been left out, but it’s the first refinement.

More orbital mechanics posts

More Legendre posts

[1] Legendre discovered/invented what we now call the Legendre polynomials in the course of calculating the gravitational potential above. I assume the convention of using J for the coefficients goes back to Legendre.

[2] Richard H. Battin. An Introduction to the Mathematics and Methods of Astrodynamics, Revised Edition, 1999.

The post The biggest perturbation of satellite orbits first appeared on John D. Cook.

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