Position of a body moving by Newtons Univ. Law. of Grav., at a point in time

[amn]

Hi all,

I am trying to code a simulation that pins two bodies against each other and animates their motion. I am using Sun and Earth as example, with preset positions and velocities, and Newtons Law of Universal Gravitation formula. It all works quite alright, up until I have decided to link my animation to a real clock source, which immediately presented the problem of obtaining acceleration, velocity and position vectors not just iteratively as I was used to, but at an arbitrary point in the future. We're talking delta-T here. I am not sure how to approach this problem.

Formally, say I have masses, position, acceleration, velocity vectors for both bodies and want to calculate position after a time interval T in the future.

Will there be definite integration involved?

 

[Andrew Mason]

Hi all,

I am trying to code a simulation that pins two bodies against each other and animates their motion. I am using Sun and Earth as example, with preset positions and velocities, and Newtons Law of Universal Gravitation formula. It all works quite alright, up until I have decided to link my animation to a real clock source, which immediately presented the problem of obtaining acceleration, velocity and position vectors not just iteratively as I was used to, but at an arbitrary point in the future. We're talking delta-T here. I am not sure how to approach this problem.

Formally, say I have masses, position, acceleration, velocity vectors for both bodies and want to calculate position after a time interval T in the future.

Will there be definite integration involved?

The orbit can be described in polar co-ordinates of radius and angle by the following equation:

$$r = a(1 - e^2)/(1 + e\cos(\theta))$$

where r is the distance from the focus of the ellipse (ie. the position of the sun), a is the semi-major axis of the orbit, e is the eccentricity of the orbit and $\theta$ is the angle through which the planet has moved since perigee (minimum r).

It is easy to find the angle as a function of t for a circular orbit (e = 0 => r = a for all $\theta$). It just becomes:

$$(r,\theta) = 2\pi at/T$$

where T is the period of the orbit and t is the time after perigee.

It is much more difficult for elliptical orbits (e>0). To determine $\theta$ at a given time one has to use conservation of angular momentum. L = mvr where L is constant (same for all time). The planet sweeps out equal areas in the orbital plane in equal times. It is a difficult calculation.

AM