[Space Geodesy] Perturbation Theory

Motivation

The perturbation theory is to solve initial value problem. In other words through perturbation theory we could recover the satellite location or orbit.

There are mainly two categories of perturbation theory:

• General perturbation theory: find the approximated solution analytically. By the end of general perturbation, we can get the derivative of keplerian orbital elements w.r.t time, thus we can recover the location of satellite by integration and provided the integration.
• Special perturbation theory: find the approximated solution numerically. By the end of special perturbation, r(t) is writen as
  r(t)=∑i=1qaq(t−t0)q$$r(t)=\sum _{i=1}^q{a}_q(t-t_0{)}^q$$
  . Notice that the coefficient in above formula should be provided in advance.

General Perturbation Theory

• Keplerian element is a function of the position and velocity vector:
  I=I(r⃗ (t),r⃗ ˙(t))$$I=I(\overrightarrow{r}(t),\dot{\overrightarrow{r}}(t))$$
• The simplified formula development:
  I˙=∇rI⋅r˙+∇vI⋅{f0+f1}$$\dot{I}={\mathrm{\nabla }}_{r}I\cdot \dot{\mathbf{\text{r}}}+{\mathrm{\nabla }}_{v}I\cdot \{{\mathbf{\text{f}}}_0+{\mathbf{\text{f}}}_1\}$$
  ∇rI⋅r˙+∇vI⋅f0=0$${\mathrm{\nabla }}_{r}I\cdot \dot{\mathbf{\text{r}}}+{\mathrm{\nabla }}_{v}I\cdot {\mathbf{\text{f}}}_0=0$$
  Thus:I˙=∇vI⋅f1$$Thus:\dot{I}={\mathrm{\nabla }}_{v}I\cdot {\mathbf{\text{f}}}_1$$
• Notice that the time derivative of keplerian element is a scalar product of its gradient in velocity space and the perturbing term(acceleration).
• Perturbing term f1$f_1$ can be decompsed in an orthogonal coordinate:
  f1=⎡⎣⎢⎢RSW⎤⎦⎥⎥$$f_1=\left[\begin{array}{c}R\\ S\\ W\end{array}\right]$$
• The following equation shows the derivative of keplerian element (Gaussian perturbation equation):
  
• Know the difference between Eccentric anomaly, true Anomaly, mean anomaly.
• Remember something:
  
  • Numerical eccentricity of orbit are normally near zero. This is why sometimes we replace perigee passing time by mean anomaly. And also to change a (semi-major axis), we need to impose force along the track direction (S ), rather than the radial direction (R ).
  • Notice that (a,e,tp)$(a,e,t_p)$ only related to R and S, while the orientation of the orbit (i,Ω,ω)$(i,\mathrm{\Omega },\omega )$ related to R , S and W.
  • To change the inclination of the orbit, first we should impose force orthogonal to the orbit (W ), secondly u=ω+v$u=\omega +v$ should equals to 0 or 180 degree.
  • To change the ascending node of the orbit, first we should impose force orthogonal to the orbit (W ), secondlyu=ω+v$u=\omega +v$ should equals to 90 or 270 degree.