Solving Sun-Earth-Moon Orbit Problem with Solar System Simulator

[Smoove]

Hi,

I am trying to create a program (3D simulator) that reproduces the Solar system's behaviour. In this idea, I am using Newton's first and second laws:

F = G * m1 * m2 / ( r * r)
a = F / m

I already managed to create a stable system containing the Sun, Earth, Jupiter and other planets... Everything seems to be working fine so far. However, an issue appears when I introduce the Moon: as we all know, it is supposed to be turning around the earth, but my simulator fails to reproduce that.

To illustrate the problem, allow me to present to you the following results:

Data:

Sun mass: 2 * 10^30
Earth mass: 6 * 10^24
Moon mass: 7.5 * 10^22
Distance Sun - Earth: 149000000 km
Distance Earth - Moon: 380000 km
Distance Sun - Moon: 149000000 + 380000 = 149380000 km

(note: the Moon is assumed to be on the opposite side of the Earth regarding the sun here, so the farest possible).

Newton's laws results:

F (Earth - Moon) = G * 6 * 10^24 * 7.5 * 10^22 / (380000 * 380000) = G * 3.116 * 10^36
F (Sun - Moon) = G * 2 * 10^30 * 7.5 * 10^22 / (149380000 * 149380000) = G * 6.722 * 10^36

a (Moon to Earth) = G * 3.116 * 10^36 / 7.5 * 10^22 = G * 4.15 * 10^13
a (Moon to Sun) = G * 6.722 * 10^36 / 7.5 * 10^22 = G * 8.96 * 10^13 (twice more!)

In other words, the Moon accelerates more toward the Sun than toward the Earth. Or another way to put it: the Earth cannot maintains the Moon's orbit.

The visible result in my simulator is that the Moon is orbiting around the Sun like any other planet.

I am obviously missing something important here, but I just can't figure out what it is!

Any lead would be much appreciate!

Many thanks,

Philippe

 

[D H]

In other words, the Moon accelerates more toward the Sun than toward the Earth.

Correct.

Or another way to put it: the Earth cannot maintains the Moon's orbit.

Incorrect.

The visible result in my simulator is that the Moon is orbiting around the Sun like any other planet.

Correct.

I answered your question on another forum. Why did you not reply to that response, Smoove? For readers here at PF, here is what I wrote:


In a sense, the Moon does indeed orbit the Sun. One obvious sense is that the specific energy of the Moon in a Sun-centered frame, $$v^2/2 - GM_{\mbox{sun}}/r$$, is negative. Another sense is that suppose we could magically make the Earth vanish (easy to do in a simulation, not so easy in real-life). The result: The Moon's orbit about the Sun would remain more or less the same (more or less meaning within about 10%).

On the other hand, that the Moon's acceleration toward the Earth is about 1/2 that of the Moon toward the Sun means that the nearby presence of the Earth is a bit more than a perturbative effect. Effects should be less than an order of magnitude smaller than the primary effect to be considered 'perturbative'. The presence of the Earth is considerably more than just a perturbation of the Moon's orbit about the Sun.

So, does the Moon orbit the Earth? Certainly. First off, the specific energy of the Moon in an Earth-centered frame is once again negative. The Moon is gravitationally bound to the Earth. Just as before, if we could magically make the Sun vanish, the Moon's orbit about the Earth would remain more or less unchanged.

From the perspective of an Earth-centered frame, the effect of the Sun's gravitational influence on the Moon's is the inertial frame acceleration of the Moon toward the Sun less the inertial frame acceleration of the Earth toward the Sun. This is greatest when the Earth is a perihelion, the Moon is at apogee, and the Moon is new. Even then, it is only about 1.4% of the Moon's gravitational acceleration toward the Earth. This is a small perturbative effect.

Yet another way to look at it is in terms of spheres of influence. There are two competing definitions of a gravitational sphere of influence of a planet on some object, Laplace's sphere of influence and the Hill sphere. With either definition, the Moon's orbit about the Earth is well inside Earth's gravitational sphere of influence. The Moon orbits the Earth (and the Sun, and the Milky Way, and the local group, and ...).

 

[K^2]

What are your initial velocities for Earth and the Moon? And do you use proper vector addition when finding distance from Earth to Moon once Moon begins to lead Earth in orbit around the Sun?

 

[tiny-tim]

Welcome to PF!

Hi Philippe! Welcome to PF!

(try using the X² and X₂ icons just above the Reply box )

… In other words, the Moon accelerates more toward the Sun than toward the Earth. Or another way to put it: the Earth cannot maintains the Moon's orbit.

The visible result in my simulator is that the Moon is orbiting around the Sun like any other planet.

There's a simpler way to check the ratio of the forces …

https://www.physicsforums.com/library.php?do=view_item&itemid=27" is ω²r, = (2π/T)²r …

so the ratio of forces is the ratio of r/T² …

that's 93million/.24million times (month/year)², or approximately 2.2

This isn't surprising … even if the Moon managed to get away from the Earth, it'd still be stuck with orbiting the Sun … clearly the Sun attracts it more strongly.

Someone looking from a long way off ("vertically") would see the Earth and the Moon each following two out-of-phase wavy orbits which change places 13 times a year … it would not look as if one was orbiting the other!

You need to reproduce the waviness of those orbits.

 

[Smoove]

Thank you for your answer D H. So if I understand you well, my results are correct and logical, I am only missing an appropriate start context or initial conditions (as K^2 suggest with his questions) . Is it a good conclusion to your answer?

K^2, it wouldn't be relevant to give you the values I am using, but you are right to point it out, as I "tried" many unless I found an apparently right balance. But this obviously not a good approach...

On the other hand, I am confident about my vector calculation: my program applies every gravitational forces for each object, coming from every others. Or more simply, considerating the system [Sun - Earth - Moon] it will calculate and sum up two acceleration vectors for each object of the system.

So my question would then be: do you have a clue to find the correct initial context (velocities)? This seems complex to me, because we are not able to observe it anymore... It was a long time ago!

Thank you again for your answers

 

[Smoove]

Thank you tiny-tim! Yes, I see that you are all suggesting the same, which I hadn't realized until now: the Moon is actually really orbiting around the Sun!

Unfortunately it means lots of troubles for me to be able to recreate that!

 

[D H]

You can obtain state vectors of the sun, the planets, the moon, and other objects with respect to the solar system barycenter from http://ssd.jpl.nasa.gov/?horizons.

What behavior you expect to see of the Moon from the perspective of a Sun-centered or barycentric frame? Epicycles? You won't. Just because you don't doesn't mean that the Moon is not orbiting the Earth.

 

[Smoove]

Sorry I forgot to give you an important element:
what I have been observing in my simulator for now, is a Moon turning around the Sun in an apparently completely disconnected way from the earth: although is has started at the same point (or almost, since the Moon is really really close to the earth) and with same initial direction vector.

To make it clear again (I am not good at explanations ^^): at one point, the Moon is on one side of the Sun and the Earth on the other side!

Thank you for this link D H, I will take a close look at it. I think I need to explore this initial condition aspect more deeply.

 

[D H]

To make it clear again (I am not good at explanations ^^): at one point, the Moon is on one side of the Sun and the Earth on the other side!.

In that case something is wrong with one or more of

• your equations of motion,
• your numerical integration technique, or
• your initial conditions.

 

[Smoove]

Ok D H,

I came here with this question to make sure I was not missing any important physics property: I think I have got my answer now.

As you sum it up very well, it leaves me three directions to search in, in which you can unfortunately not bring me anymore help!

Many thanks again, to all of you!

 

[K^2]

Your initial condition should be Moon's velocity = velocity needed to orbit Sun in circular orbit +/- velocity needed to orbit Earth in circular orbit. It's not exactly right, but it is close enough for you to see Moon orbiting Earth. If that doesn't happen, it's a problem with simulation.

 

[Smoove]

Thank you K^2, this is very helpful. I am now looking forward to leave the office... I have some tests to do! :)

 

[granpa]

http://www.math.nus.edu.sg/aslaksen/teaching/convex.html

 

[K^2]

^ Of course. The Earth-Moon system orbits Sun at what, around 30km/s? Moon's relative speed is never close, so it never goes retrograde. But Smoove is getting a result where Moon gets ahead of the Earth by half of the revolution around the Sun, and that's clearly wrong.

 

[granpa]

how big is each 'step' in his simulation?

normally one does either moons or planets but not both.

 

[D H]

What do you mean by "normally", granpa: "Normally" in the sense of "normally in the toy simulations done by undergrads" or "normally" in the sense of "normally in professional-quality, high-fidelity simulations of the solar system"?

 

[granpa]

normally as in steps small enough to calculate moons slows down the simulation so much that it takes forever to calculate planets. It doesn't matter whether its professional or not. And since moons have little effect on planets its usual to ignore them.

I could be a few years behind the times though. Maybe modern computers are fast enough to do both now.

But my point was that maybe his step is small enough to do planets but not small enough to do moons too.

 

[K^2]

Yeah, there is more than enough power for that. I've ran simulations for small galaxies with something like 100,000 stars, all interacting gravitationally. A solar system is easy enough.

It might be difficult to make accurate predictions for several hundred years into the future, but I don't think that's the goal here.

 

[granpa]

a hundred years?
I was thinking billions.

what kind of statistics are you going to get after only hundreds of orbits?

 

[D H]

I've ran simulations for small galaxies with something like 100,000 stars, all interacting gravitationally. A solar system is easy enough.

As a toy problem, yes. As a professional problem, anything but. The amount of scientific and even pre-scientific brainpower that has gone into and continues to go into simulating the solar system is immense.

How states are propagated is but one small part of what goes into a high-fidelity, high-precision solar system models. You probably used a very simple integrator such as leapfrog for your galaxy sim. That leapfrog technique and the kinds of techniques used to propagate those solar system models are at opposite ends of the spectrum of numerical integration techniques. People are still working on new integration techniques and are still playing the "my integrator is better than yours" game.

a hundred years?
I was thinking billions.

Can't be done with the expectation that the results will have any meaning. The solar system appears to be chaotic. What can be done is to repeatedly run the solar system model to determine that nature of the chaos. Is the chaos strong or weak? For example, can Jupiter make Mercury's orbit cross that of Venus orbit, possibly tossing Mercury out of the solar system, or even worse?

 

[granpa]

wouldnt that be a meaning?

 

[K^2]

As a toy problem, yes. As a professional problem, anything but.

I didn't expect that OP wanted anything but a toy model. Predictive power of a sim like this would be zero without some really tricky numerical manipulations. I'm aware of that.

And yes, my galaxy sim was very much simplified. A simple finite-step analysis, with analytic solutions for when two stars came too close to each other. The later prevents star ejection due to divisions by small number that you get from cheap sims. I wouldn't trust it to predict actual positions of stars, but it took care of general evolution, collisions, and arm formations quite nicely, which is all I really wanted from it.

 

[D H]

wouldnt that be a meaning?

Not necessarily. Suppose you have a solar system sim that shows that the solar system exhibits strong chaos. Others will argue that that chaos is merely a result of your faulty simulation; the observed chaos isn't real. For example, whether the outer planets are chaotic at all (strong or weak) is still subject to debate.

 

[granpa]

only good thing about a galaxy sim is that you can use parallel processing.

the op still hasnt told us the size of his step.

 

[granpa]

if you want both planets and moons then
first run the planet sim and store the positions of all the planets over time then
use that information to run a moon sim.

 

[K^2]

That works for Solar system, since it's more or less stable by now, but not for a general system, where gas giants can disrupt each-other's moons.

 

[D H]

That works, for a toy system. It will not work for a professional-quality model of the solar system. The mass of the Moon is about 0.0123 that of the Earth. Ignoring how that affects the orbit of the Earth-Moon system about the Sun makes it impossible to develop the kinds of ephemerides needed to support the milliarcsecond-level pointing requirements of the precision astronomy work that is being done nowadays.

 

[granpa]

I somehow doubt that is what the op is looking for.

the usual reason for doing this sort of thing is to test orbit stability over millions or billions of years.

 

[K^2]

He's not going to do that with his approach.