Orbit Perturbations¶
→¨r=−μr3→r+→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)=μr∞∑k=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=−2.33936×10−3J2J4=−1.49601×10−3J2J5=−0.20995×10−3J2J6=0.49941×10−3J2J7=0.32547×10−3J2
The perturbing acceleration is →aP=−∇Φ.
The gravitational perturbation due to J2 is
→aP=32J2μR2r4[xr(5z2r2−1)ˆI+yr(5z2r2−1)ˆJ+zr(5z2r2−3)ˆK]
The gravitational perturbation due to J3 is
→aP=12J3μR3r5[5xr(7z3r3−3zr)ˆI+5yr(7z3r3−3zr)ˆJ+(35z4r4−30z2r2+3)ˆK]
Then, solving the equation of motion yields the perturbed orbit:
→¨r=−μr3→r+→aP