Applying secondary orbital perturbation effects

Question

I read about secondary effects which influence on orbit propagation like solar radiation, moon, etc. (https://en.wikipedia.org/wiki/Orbit_modeling). First I calculate position vector Rpqw to object (artificial satellite) using 6 orbit elements with following formulas:

u = v + ω 
r = a*(1-e*e)/(1+e*cos(v))
p = r * cos(v) 
q = r * sin(v)
w = 0.0
RotationMatrix = R3(-Ω)R1(-i)R3(- ω)
Rijk = RotationMatrix * Rpqw
Where:
U – latitude argument
r – distance to object
Rijk – coordinates of object in 3D space in ECI coordinate system (a.k.a. IJK - axis)
a -  semimajor axis
i - inclination
Ω - right ascension of the ascending node
ω  - argument of perigee
v - true anomaly

I want to know when I must apply effects like solar radiation pressure or atmospheric drag and others, after I calculate Rijk or during the process? And does the applying order matter?

Answer 1

Actually the process you described is correct in both cases, but it doesn’t point the problem you’re referring to, since it is only a way to change the reference frame, from a perifocal one to a geocentric equatorial one.

The difference between the unperturbed 2-Body problem and the perturbed one (containing as you stated solar radiation pressure, drag, three body effects) resides in the dynamics of the problem. The process you showed refers to a static condition Rpqw which is only the state of the dynamical system.

You can include the perturbation effects by adding the perturbing accelerations in the equations of motion for the direct integration. By integrating them, starting from proper initial conditions, you obtain the complete history of position and velocity vectors of the spacecraft.

Otherwise you may use Gauss’ Planetary Equations (1, 2, 3) to integrate the evolution of the orbital elements in time and use the istant values of these to get Rpqw and afterwards Rijk.

For reference you may look at the books of Vallado (Fundamentals of Astrodynamics and Applications) or Chobotov (Orbital Mechanics).