Help in runge kutta problem to determine period of a comet

[Gvilla1000]

Hi everyone:

I have trouble in an assignment I was given. The instructions are to utilize Runge-Kutta order 2 and 4 to determine the period of the comet Halley (perihelion = 0.586 AU and aphelion = 35.1 AU).

The period of the orbit is given by Kepler's third law:
Period = Sm^3/2

where Sm is the Semimajor axis acquired from:
Sm= (P+A)/2

where P is the perihelion and A is the aphelion

My problem is...if this is such a straightforward answer...how can I use Runge-Kutta in a problem like this? what would be my function that I would need to use Runke Kutta on?

 

[AlephZero]

You could use Newton's laws of motion and the inverse square law of gravitiation. Or you could set up some equations using conservation of energy and angular momentum. You will also need the initial conditions, e.g. the speed of the planet at periohelion that gives you the correct aphelion distance.

It's hard to say what is the "best" way to set this up without knowing the bigger picture of how this course fits into your studies. A computer scientist who knows about numerical methods but not much physics, a mech engineer who knows about dynamics but not much astronomy, and an astronomer, would probably do it three different ways.

 

[jtbell]

If I had given this problem to my students, I would have meant it as follows:

1. Set up a system of linear differential equations to describe the motion of the comet in the Sun's gravitational field, starting with F = ma and Newton's law of gravitation.

2. Write a program to solve those equations using the specified methods, to generate the path of the comet given its initial position and velocity.

3. For the initial position, use the known perihelion for Halley's comet. Adjust the initial velocity so that the aphelion is the known value for Halley's comet.

4. Trace the comet through one complete orbit and find how much time it takes.

But your professor might have meant something else, so you had best ask him if this is the sort of thing he is looking for.