Kepler's Laws and dark matter

1. Oct 20, 2006

FeynmanMH42

I have been interested in astronomy since I was two years old (I knew the names of all the moons of Uranus, the fact that there were only 15 back then was irrelevant :P) but even now at nearly 16 something confuses me:
Planets orbit the Sun (or stars orbiting a galaxy, or moons orbiting a planet, whatever) because the force of gravity pulls on them - the closer they are to the Sun, the faster they have to move because they feel a stronger pull of gravity.
However gravity depends on both distance and mass, so if Jupiter was as close to the Sun as Mercury is (assuming it could hold on to its mass being so close to the Sun, which it obviously couldn't as it's liquid and gas) wouldn't it travel faster?
However doesn't Kepler's third law state that the velocity of a planet in orbit is proportional to its distance from the sun? (One is squared, one is cubed, I forget which.)
This makes no sense to me; so Mercury and Jupiter at the same distance would travel at the same speed and have the same period? What about mass?
Also, I always thought Dark Matter was needed to explain the rotation curves of galaxies because the extra mass around the galaxy created a stronger gravitational pull and caused the matter round the edges to move faster than it should if all there was was visible matter?
And if that's not true, then can someone explain how Dark Matter creates galaxy rotation curves?
Thanks.

2. Oct 20, 2006

neutrino

Kepler's third law states that the square of the period of revelution is proporitional to cube of the mean distance between the planet and the sun.

Last edited: Oct 20, 2006
3. Oct 20, 2006

FeynmanMH42

But if Jupiter and Mercury were at the same distance from the Sun, then their speeds would have to be the same for their periods to be the same, despite the fact they have very different masses and clearly feel different gravitational pulls.

4. Oct 20, 2006

neutrino

The 'constant speed' (that I assume you're referring to) is the tangential speed, in the case of a circular orbit. The gravitational force, on the other hand, is radial, i.e. along the direction from the planet to the sun.

5. Oct 20, 2006

Labguy

Yes, but consider two objects at any distance from the "main" body, like jupiter's moons or your example above. If you have a massive planet (or moon or small rock) orbiting at the same distance, their period will be the same. The small one won't go faster than the big one just because it has a different mass.

Because, you have two opposing forces to contend with. Centrifugal force tries to send the body away in a straight line. Centripetal force (gravity) keeps them in orbit, just like a string on a ball spinning around your head. It just so happens that the formula for both centrifugal and centripetal force are exactly the same; MV2/R.

So, the big body has more centrifugal force because of larger M, but the larger M also has more gravity (centripetal force) so they offset and any two bodies at the same distance travel with the same orbital period.

6. Oct 20, 2006

xAbsoluteZerox

Hopefully I can clear a couple things up:

p^2 = a^3 is only for bodies orbiting our sun where p must be in years and a must be in AU and it is only an approximation. A better formula is newton's form of kepler's 3rd law:

p^2 = [(4*pi^2)/(G*(m+M))]*a^3

this formula can be easily defived from F=ma. So what does kepler's law assume? It assumes that the mass of a planet orbiting the sun is MUCH less massive than the actual sun itself, and to Kepler's precision a long time ago, this is the case. This is easily proven by simply making the m+M term equal to the mass of the sun. When you do this, it turns back into kepler's equation (after unit conversions).

So - if jupiter and mercury were at the same radii from the sun and not interacting with eachother, jupiter would revolve faster. Kepler was not wrong for his time, he just didn't have the accuracy and precision in his instruents too see the small difference which newton later found and proved.

As for the dark matter: that is correct...with more mass, you DO orbit faster as i just proved above...and in the case of dark matter, the dark matter theory proposes a LOT of dark matter, even to a point where some galaxies would be composed of up to 90% dark matter and 10% baryonic matter (normal stuff). So yes, with that much dark matter, there would be a very significent change in rotation curves from just visible light.

**EDIT:
I do not agree with this because the forces are not in opposing directions but rather normal to eachother, which will make the object with the larger mass move faster than the less massive object, as confirmed by newton.

Last edited: Oct 20, 2006
7. Oct 20, 2006

neutrino

Why? What happens if I use the second and the metre?

8. Oct 20, 2006

xAbsoluteZerox

If you use the second and the metre you need to have a conversion term, which is basically what newton's formula comes up with. Try it out with the earth-sun system, seconds and metre's dont work in kepler's formula, you come up with nonsensical results.

9. Oct 20, 2006

neutrino

Oh, it's the equality sign! Sorry, I had $$\propto$$ in mind.

10. Oct 20, 2006

Staff: Mentor

It's pretty simple Feynman - consider f=ma: if you double the force of gravity and double the mass being accelerated, what happens to the acceleration?

11. Oct 20, 2006

FeynmanMH42

Thanks, I understand now. :)
I've been into astronomy for ages and that was the last nagging question I've found the answer to - thank you!

12. Oct 20, 2006

Labguy

So, are you saying that two asteroids at the same distance from the sun revolve with different periods because one is massive and one is a small rock? Are you saying that the Space Shuttle and the ISS (not docked) revolve with different periods because one has more mass? Does a golf ball released from the Shuttle bay (no push) suddenly revolve around Earth slower than the Shuttle?

I don't think so....

EDIT: And yes, Newton changed Kepler a bit to include orbits around the COM and included both masses. But, especially on small things (like 200 trillion tons) the decyclopedia says:

Last edited: Oct 20, 2006
13. Oct 21, 2006

Chronos

This is an interesting question and fundamental to planet formation. Planets form by gravitationally vacuming up less massive particles in their path. The most massive object in any orbit eventually wins the battle. This also explains why two separate planets cannot occupy the same orbit. The less massive body is eventually captured or devoured by the more massive companion.

14. Oct 21, 2006

FeynmanMH42

Okay, so can someone explain why we need Dark Matter please?
As far as I knew:
1) Galaxies' centres were supposed to rotate faster than the edges, but they don't.
2) Therefore there needs to be extra, invisible matter in a halo round the galaxy. This creates more mass around the edges, thus increasing the gravitational pull and making the mass move as fast as the mass in the inner parts of the galaxy.
If everything at the same distance from the centre of the galaxy has the same period and therefore moves at the same speed, then how come we need extra mass to explain why the edges rotate as quickly as the inside parts?

15. Oct 21, 2006

Labguy

These may not be technical enough, but try http://arxiv.org/PS_cache/astro-ph/pdf/0401/0401088.pdf" [Broken].

Last edited by a moderator: May 2, 2017
16. Oct 21, 2006

xAbsoluteZerox

Yes, that is what I am saying. If two objects have different masses and are their centers or mass are at the same radius from a massive body, they always will revolve at different speeds, always. Now in the case of the space shuttle and the ISS, there is a lot more to it than just this, you have to consider initial velocities and angles and a bunch of technicalities. And yes, if a golf ball is released from the space shuttle, it will revolve at a different rate than the spacecraft, guaranteed.

17. Oct 22, 2006

Labguy

In looking at the formulae for two-body systems, it appears that you are right and Labguy blew it, so I was likely wrong.... But, that's Ok, I think I was wrong once before a long time ago on simple, stupid stuff.... I hate and "don't do" planetary and solar system stuff.

Why doesn't anybody post to the AP forum about stellar evolution, supernovae, etc. anymore??..

18. Oct 22, 2006

SpaceTiger

Staff Emeritus
I find this argument strange, given that you already wrote out Kepler's Third Law. The period of an orbiting body is given by

$$P^2=\frac{4\pi^2 a^3}{G(M_p+M_s)}$$

In planet-sun systems or shuttle-earth systems, the primary mass is considerably larger than the secondary, so the period is effectively mass-independent.

In the simple case of a circular orbit, we can visualize this as the fictitious centrifugal force opposing the gravitational force. They are equal and opposite along the line connecting the two bodies. Both "forces" are proportional to the secondary mass, so it cancels in the equation and the velocity becomes mass-independent (again, assuming the secondary is much more massive than the primary).

Last edited: Oct 22, 2006
19. Oct 22, 2006

xAbsoluteZerox

Again, it comes down to precision. I hold to my statement that two bodies of different masses will always revolve at different rates if at the same radii. When you say that you assume the primary mass is considerably larger, I agree, but it is not infinately larger. The only way for two bodies of different masses to be revolving at the EXACT same rate, is if the primary mass they are revolving around is infinately massive, which doesn't exist. It is not exactly mass-independent, it is an assuption you make which limits your precision. However, after extreme lengths of time, even the smallest differences in mass will have an observable effect on their revolution periods, it is just impractical in most cases to use this much precision.

20. Oct 22, 2006

SpaceTiger

Staff Emeritus
While we're at it, why don't we throw in GR, drag forces, solar wind pressure, quantum effects, many-body effects, etc.? When you're working with two-body systems in Newtonian gravity, everything is an approximation. You're earlier statement:

...is incorrect. The forces are not normal to one another and this is not the reason that Mercury and Jupiter would go around the sun at slightly different speeds if placed at the same distance.

It's important in science to understand which effects can be neglected in which regimes. In the example of the golf ball and shuttle, you'll find that other physical effects will act to separate the golf ball and spacecraft long before the difference in center-of-mass of the idealized two-body systems.

I think it is a good rule of thumb to remember that two bodies will orbit at the same rate if placed at the same distance from a much more massive object.