Adding Perturbing Accelerations to J2

[synchhh]

Homework Statement

I'm numerically integrating Gaussian Variational Equations on MATLAB. I have a script which, given initial values for a, e, i, Ω (RAAN), ω (argument at perigee), and θ (true anomaly), will compute the r and v vectors through numerical integration. This is done with J₂ and thrust (given a value T_r, T_v, T_h, radial thrust, along-V thrust, and along-h thrust per unit mass, respectively.) I'm then plotting the orbital elements obtained via GVE's and the orbital elements obtained via conversion of state vector on the same plot to ensure that they are equivalent. This needs to be done with J2 and additional perturbing accelerations.

Homework Equations

Note that there is a typo in Eq. (12.88). The final line reads p_s but should read p_w. Also, the equation I used for da/dt is as follows:

$$\frac{da}{dt} = \frac{2a^2esin(\theta)}{h}p_r + \frac{2a^3\sqrt{1-e^2}}{hr}p_s$$

The Attempt at a Solution

My scripts work when I run them without including the perturbing accelerations, but including J2. The orbital elements found from conversion of state vector are equivalent to those found from integrating GVE's. I was given a function that will apply the thrust to the state vector version. The only thrust that comes into play is the thrust acting along V. The author multiplied T_v by v, and that was all. I'm not exactly sure how to apply this thrust to the GVE's, since I am given orbital elements, and as such, the v vector is not explicitly defined. Also, I'm not sure at what point the v vector comes into play in this equation set? Regardless, I've tried taking the norm of the thrust elements and adding them to J2, and I've also tried adding each component to p_r, p_s, and p_w. I think a major problem is that I don't exactly understand what the number (0.00108263) associated with J2 is. I think that it somehow represents the effect of the oblateness of earth, but I'm not sure exactly how. How does it relate to perturbing accelerations? How could one combine the effects of J2 and perturbing accelerations? If necessary, I can provide the MATLAB files associated with this problem. Thanks for reading!

 

[synchhh]

Not sure how to edit my post, but I think that the script that factors in the perturbing acceleration into the orbital elements found through conversion of state vector actually just uses one perturbing acceleration at a time. This is due to the problem statement, found below (#2).