Does Kepler's Third Law Account for Dark Matter in Galaxy Rotation Curves?

In summary, the conversation discusses the factors that affect the speed of planets in orbit, including distance, mass, and gravitational force. Despite Jupiter and Mercury having very different masses, they would travel at the same speed if they were at the same distance from the Sun due to the offsetting forces of centrifugal and centripetal force. Kepler's third law states that the square of the period of revolution is proportional to the cube of the mean distance between the planet and the sun, but this is only an approximation and can be more accurately calculated using Newton's formula. The concept of dark matter is also mentioned, which explains the faster rotation curves of galaxies due to the presence of additional mass.
  • #1
FeynmanMH42
69
0
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.
 
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  • #2
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.
 
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  • #3
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
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
FeynmanMH42 said:
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.
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.

But why?, you may ask..:smile:

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
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 each other, 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:
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.

I do not agree with this because the forces are not in opposing directions but rather normal to each other, which will make the object with the larger mass move faster than the less massive object, as confirmed by Newton.
 
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  • #7
xAbsoluteZerox said:
p^2 = a^3 is only for bodies orbiting our sun where p must be in years and a must be in AU [
Why? What happens if I use the second and the metre?
 
  • #8
neutrino said:
Why? What happens if I use the second and the metre?

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 don't work in kepler's formula, you come up with nonsensical results.
 
  • #9
xAbsoluteZerox said:
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 don't work in kepler's formula, you come up with nonsensical results.
Oh, it's the equality sign! Sorry, I had [tex]\propto[/tex] in mind.
 
  • #10
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
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
Labguy said:
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.

xAbsoluteZerox said:
I do not agree with this because the forces are not in opposing directions but rather normal to each other, which will make the object with the larger mass move faster than the less massive object, as confirmed by Newton.
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...:confused:

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:
However typically the central body is so much more massive that the orbiting body's mass may be ignored. Newton also proved that in the case of an elliptical orbit, the semimajor axis could be substituted for the radius.
 
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  • #13
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
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
FeynmanMH42 said:
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?
These may not be technical enough, but try http://arxiv.org/PS_cache/astro-ph/pdf/0401/0401088.pdf" [Broken].
 
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  • #16
Labguy said:
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?

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
xAbsoluteZerox said:
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.
In looking at the formulae for two-body systems, it appears that you are right and Labguy blew it, so I was likely wrong...:frown: But, that's Ok, I think I was wrong once before a long time ago on simple, stupid stuff...:biggrin: 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??..:confused:
 
  • #18
xAbsoluteZerox said:
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.

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

[tex]P^2=\frac{4\pi^2 a^3}{G(M_p+M_s)}[/tex]

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).
 
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  • #19
SpaceTiger said:
I find this argument strange, given that you already wrote out Kepler's Third Law. The period of an orbiting body is given by

[tex]P^2=\frac{4\pi^2 a^3}{G(M_p+M_s)}[/tex]

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).

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
xAbsoluteZerox said:
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.

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:

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

...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.
 
  • #21
SpaceTiger said:
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.

I completely agree, however I was asked the difference between Kepler's and Newton's laws earlier and the difference is that Keplers is an approximation. For all practical purposes, as long as a much more massive object is present, much less massive objects's mass can be neglected. I am simply explaining that in theory, they do not orbit at the exact same speed, something which in practice we don't need to worry about.
 
  • #22
SpaceTiger said:
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.
Thanks ST; at least I was mostly right in my earlier posts.

Especially when I posted in # 12:
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:
However typically the central body is so much more massive that the orbiting body's mass may be ignored. Newton also proved that in the case of an elliptical orbit, the semimajor axis could be substituted for the radius.
When the Sun contains more than 99.8% of the total mass of the Solar System, is seems that two orbiting (asteroids, for instance) would have a whole lot more pertubations acting on them from other asteroids, Jupiter, et. al. than any possible effect of the difference in mass between any asteroid and the Sun.(?)
 

1. What are Kepler's Laws and how were they discovered?

Kepler's Laws are three scientific laws that describe the motion of planets around the sun. They were discovered by the astronomer Johannes Kepler in the early 17th century through his detailed observations of the planets' orbits.

2. How do Kepler's Laws relate to dark matter?

Kepler's Laws were originally formulated to describe the motion of visible planets in our solar system. However, they are also applicable to objects orbiting around massive bodies, such as stars and galaxies. Dark matter, which is a type of invisible matter that makes up a large portion of the universe, also follows Kepler's Laws when orbiting around massive bodies.

3. What role does dark matter play in understanding the universe?

Dark matter plays a crucial role in understanding the structure and evolution of the universe. It is believed to be responsible for the formation of large-scale structures, such as galaxies and galaxy clusters, and plays a major role in the dynamics of these structures.

4. How is dark matter detected and measured?

Dark matter cannot be directly detected because it does not interact with light. However, its presence can be inferred through its gravitational effects on visible matter. Scientists use a variety of techniques, such as gravitational lensing, to indirectly measure the distribution of dark matter in the universe.

5. What are the current theories and research surrounding dark matter?

There are many different theories and ongoing research surrounding dark matter, including attempts to directly detect it using specialized detectors, studying its effects on the cosmic microwave background, and testing alternative theories of gravity. Scientists are also trying to better understand the properties and behavior of dark matter through simulations and observations of its interactions with visible matter.

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