Some questions regarding proportions in a two-body problem

[Ostsol]

I am programming an n-body simulation, but to test my program, I decided to start with two bodies. Specifically, I decided to model the Earth-Moon system. The issue I have is that Kepler's laws involve only the motion of one of the two bodies. From my understanding, the other is assumed to be static or of such mass that its motion is insignificant. That leaves me working with Newtonian gravitation and motion.

My problem is the starting conditions. Wikipedia tells me the mass of the Earth and Moon, and the orbital radius and velocity of the Moon. Simply inputting these parameters does not create a stable simulation, though. The entire system slowly moves. I was obvious that I was not accounting for the Earth's motion in my starting conditions. It was not obvious as to how I should determine the Earth's velocity.

Because my maths are effectively pre-calculus (I took single-variable calculus over a decade ago and have forgotten most of it), it is difficult to understand the maths behind celestial mechanics. Eventually, though, after playing with some numbers I found that the velocity of the Earth should be of a proportion to the Moon's that is inverse to the proportion of their masses. Basically:

[itex]v_e{}[/itex]/[itex]v_m{}[/itex]=[itex]m_m{}[/itex]/[itex]m_e{}[/itex]

Where v and m are velocity and mass, respectively; and the subscripts m and e are the Moon and the Earth, respectively. The Earth's velocity obviously points in the opposite direction as the Moon's. This creates a stables simulation regardless of the values input and the distances involved. As long as these proportions remain the same, the bodies trace the same orbital paths each time.

Now I'm sure that this proportion may be derived from the relevant formulae by those who have sufficient competence in the maths involved, but that's not me. I was unable to find this information on Wikipedia or in various Google searches, though perhaps that's not surprising. My questions therefore are: is this proportion documented somewhere and are there any others that I should be aware of?

Thank you.

-Daniel

 

[TurtleMeister]

The math for doing an n-body simulation is pretty simple. All you need are Newton's laws. It's the algorithm that you have to be careful with. I did my first program using the information at this site:

http://www.cs.princeton.edu/courses/archive/spring11/cos126/assignments/nbody.html

Although it's for Java, you can still use the info to write the program in any language you like. Also, the instructions at the link site are for x and y coordinates only. But it's pretty easy to add the Z coordinates once you understand how it works.

 

[Ostsol]

Yes, the simulation was relatively easy to create. I even incorporated RK4 integration based on some information I found elsewhere in these forums. The problem is figuring out the starting parameters for bodies of arbitrary mass and orbital radii.

 

[TurtleMeister]

I seed my simulation with the ephemerides generated at the Horizons web site. I then calculate the center of mass for the bodies I am simulating and set that coordinate at 0,0,0 adjusting the coordinates and velocities of the other bodies accordingly.

 

[PJC01]

I appreciate your efforts to understand and accurately model the Earth-Moon system in your simulation. The issue you have encountered is a common one in celestial mechanics and can often be challenging to solve without a strong understanding of the relevant mathematical principles.

Firstly, I would like to clarify that Kepler's laws do apply to the motion of both bodies in a two-body problem. However, as you correctly pointed out, the laws only describe the motion of one body in relation to the other and assume that the other body is either static or has negligible mass. In reality, both bodies are in motion and their motions affect each other.

In terms of determining the Earth's velocity in your simulation, you have correctly identified that it should be proportional to the Moon's velocity and inversely proportional to their masses. This is known as the "principle of equal areas" and is a fundamental concept in celestial mechanics. The idea is that in any given time interval, the line connecting the two bodies will sweep out equal areas, regardless of their masses. This principle is derived from Newton's laws of motion and can be mathematically proven, but it is beyond the scope of pre-calculus mathematics.

I would also like to point out that the Earth's velocity will not be exactly opposite to the Moon's velocity, as there are other factors at play such as the Earth's rotation and the gravitational influence of other celestial bodies. However, for the purposes of your simulation, your approach seems to be working well and providing stable results.

In terms of other proportions and equations that may be helpful in your simulation, I would suggest looking into the equations of motion for a two-body problem, which involve the gravitational constant, the masses of the two bodies, and their initial positions and velocities. Additionally, you may want to consider incorporating the effects of other celestial bodies, such as the Sun, in your simulation to make it more accurate.

Overall, I commend your efforts to understand and accurately model the Earth-Moon system in your simulation. While your approach may not be mathematically rigorous, it is producing stable results and can serve as a useful tool for further exploration and understanding of celestial mechanics. I encourage you to continue your research and learning in this field, as it is a fascinating and important area of study. Best of luck with your simulation!