# Analyze rotation of satellite orbit due to transverse acceleration.

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

$$Ar=G.M/d^2$$

QUESTION:- Is there an analytical technique for determining the rate of orbital rotation (precession) which would be produced when an additional small, time-variable, force acts upon the satellite to produce acceleration of the satellite in the direction of the satellite's current transverse component of velocity .

I have modelled such effects in a reiterative computer model and would like to validate those measurements.

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

$$At=(K/c^2)∗Vr∗Vt*Ar$$

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).

(I have asked this question on Physics and Astronomy stackexchange with no response so far).

PROGRESS

The problem equates to a Lagrangian with an additional friction-like term as per Goldstein Equation 1.70 (thanks @Julio for the steer).

UPDATE

I have accepted Walter's answer at https://astronomy.stackexchange.com/questions/632/determining-effect-of-small-variable-force-on-planetary-perihelion-precession (he used peturbation analysis to derive a formula for the rotation). Alternative methods are welcome.

• I think, you can see the Classic Mechanics of Goldstein to have the idea that solve this – Julio Jul 14 '14 at 12:51
• @Julio Thanks for the tip. I see that my problem equates to a Lagrangian with an additional friction-like term as per Goldstein Equation 1.70. Not sure where to go next but I will read on. – steveOw Jul 15 '14 at 9:35
• I've answered this question here. – Walter Sep 3 '14 at 8:05
• Crossposted from astronomy.stackexchange.com/q/632/476 and to physics.stackexchange.com/q/123685/2451 – Qmechanic Sep 14 '14 at 11:25