Orbit Perturbations

¨r=μr3r+ap,

where ap is the acceleration due to perturbations from thrust, atmospheric drag, gravitational perturbations, or any other force.

Atmospheric drag

Gravitational perturbations

The rotationally symmetric perturbation Φ(r,ϕ) is

Φ(r,ϕ)=μrk=2Jk(Rr)kPk(cosϕ),

where Jk are the zonal harmonics of the planet, R is the equatoriall radius (R/r<1), and Pk are Legendre polynomials of order k.

For the Earth, the six zonal harmonics are:

J2=0.00108263J3=2.33936×103J2J4=1.49601×103J2J5=0.20995×103J2J6=0.49941×103J2J7=0.32547×103J2

The perturbing acceleration is aP=Φ.

The gravitational perturbation due to J2 is

aP=32J2μR2r4[xr(5z2r21)ˆI+yr(5z2r21)ˆJ+zr(5z2r23)ˆK]

The gravitational perturbation due to J3 is

aP=12J3μR3r5[5xr(7z3r33zr)ˆI+5yr(7z3r33zr)ˆJ+(35z4r430z2r2+3)ˆK]

Then, solving the equation of motion yields the perturbed orbit:

¨r=μr3r+aP