# Effect of small variable transverse force on satellite orbital precession

Consider a small satellite which moves in a 2D elliptical orbit around a much larger body (e.g. the Sun) under the influence of Newtonian gravitational acceleration $$Ar = G.M/d^2$$

Next imagine that a small, time-varying force acts upon the satellite which produces acceleration of the satellite in the direction transverse to the instantaneous radial direction.

The general pattern of variable transverse acceleration (At) which I am interested in is given by the following equation:-

$$At = (K/c^2)*Vr*Vt*Ar$$

I have modelled such effects in a reiterative computer model and would like to validate those measurements. The principal effect is the introduction of a rotation of the aspides axis of the orbit ("precession of the line of aspides").

QUESTION

Is there an analytical technique I can use to produce a formula for the rate of orbital rotation?

TERMS & ASSUMPTIONS

• K is a constant (e.g. +/- 3.0).
• Ar is the radial acceleration of the satellite towards the Sun due to Newtonian gravitational influence of the Sun.
• At is the time-variable acceleration of the satellite in the direction transverse to the Sun:satellite radial direction (positive when directed forward along the future orbital path of the satellite, negative when directed backwards)
• c is speed of light.
• Vr is satellite radial velocity relative to the Sun (positive when satellite is moving further away from the Sun)
• Vt is satellite transverse velocity relative to the Sun (always forward)
• G is Newton's Universal Gravitational Constant.
• M is mass of the Sun.
• d is distance of satellite from centre of the Sun (assume at least 10 million km).
• Assume satellite mass and diameter are extremely small relative to the Sun (e.g. 20kg and 2m respectively).
• Satellite is spherical with uniform density.
• No other bodies are in the system.
• All motions and accelerations are confined to a two-dimensional plane.
• Transverse acceleration is produced by a very small nuclear-powered mass-reaction rocket attached at the centre of the satellite.
• Consider the mass ejected by the rocket to be insignificant. Thus kinetic energy is added to and taken away from the system at different times, but there is no significant loss in mass.
• Assume a purely Newtonian System (no General Relativity please).

I have seen various treatments of small radial accelerations which use Lagrangian and Hamiltonian principles but I do not think these are extendable to transverse accelerations (e.g. http://www.mathpages.com/home/kmath527/kmath527.htm).

UPDATE

I have accepted Walter's excellent answer at https://astronomy.stackexchange.com/questions/632/determining-effect-of-small-variable-force-on-planetary-perihelion-precession. (He used peturbation analysis to produce a precise formula for the rotation) But alternative approaches are welcome.

• Not quite an answer: Try using the Gauss/Lagrange/Delauney/Hill planetary equations. – David Hammen Jul 23 '14 at 2:53
• If $K$ is merely a (dimensionless) constant number, then your equation for $A_t$ does not have the correct physical dimension of an acceleration. In your question on Astronomy, you have $A_t=Kv_rv_tA_r/c^2$, i.e. an additional proportionality to the radial acceleration $A_r=-GM/r^2$. – Walter Sep 1 '14 at 8:28
• @Walter. You are right, my mistake. I have amended the formula for At to include Ar on the RHS. K is indeed dimensionless. – steveOw Sep 1 '14 at 16:22
• I've answered this question here. – Walter Sep 3 '14 at 8:04
• Crossposted from astronomy.stackexchange.com/q/632/476 and math.stackexchange.com/q/866836 – Qmechanic Sep 14 '14 at 11:24