Rotation curve of galaxy Keplerian method

[Shailesh Pincha]

There are 2 unknowns in the formula. The time period of rotation and the mass enclosed by orbit is Star. So how could we calculate the expected time period of rotation of stars in a galaxy and thus velocity of stars.

 

[mfb]

We can measure the velocities, and we can measure the amount of visible matter. They don't fit together.

 

[D H]

The title of this thread, "Rotation curve of galaxy Keplerian method," hints at a big misunderstanding. Stars in a spiral galaxy do not have Keplerian orbits. The concept of a Keplerian orbit implicitly assumes a spherical mass distribution. Normal matter in a spiral galaxy does not have anything close to a spherical mass distribution. Instead, there's a central bulge that contains a tiny fraction of the galaxy's mass. Most of the normal matter is in a fairly thin disk. The gravitational potential of that central bulge + disk is not anything close to the conditions for Keplerian orbits.

What can be done is to assume that the concentration of stars hints at the total mass of the galaxy. (A good portion of normal matter is in interstellar gas clouds rather than stars, but presumably the concentration of stars is an indicator of the concentration of those gas clouds.) From this, one can calculate how fast a star in a roughly circular orbit should be going. It's more complex than simple Keplerian orbits, but it is doable.

This is where the problem arises. No matter how much fudging one does regarding the amount of normal matter in those interstellar gas clouds, the numbers don't add up. Stars + gas clouds + dust don't explain the observations. There are only two explanations. One is that our concept of gravitation is fundamentally incorrect at the galactic levels and larger. The other is that our concept of gravitation is correct, but that some other form of mass exists that we can't see in galaxies and that isn't distributed the way stars are (hence the term "dark matter").

There are very few indicators that we don't know how gravity works at galactic scales. There are a large number of indicators that the second explanation is the correct one.

 

[Shailesh Pincha]

The title of this thread, "Rotation curve of galaxy Keplerian method," hints at a big misunderstanding. Stars in a spiral galaxy do not have Keplerian orbits. The concept of a Keplerian orbit implicitly assumes a spherical mass distribution. Normal matter in a spiral galaxy does not have anything close to a spherical mass distribution. Instead, there's a central bulge that contains a tiny fraction of the galaxy's mass. Most of the normal matter is in a fairly thin disk. The gravitational potential of that central bulge + disk is not anything close to the conditions for Keplerian orbits.

What can be done is to assume that the concentration of stars hints at the total mass of the galaxy. (A good portion of normal matter is in interstellar gas clouds rather than stars, but presumably the concentration of stars is an indicator of the concentration of those gas clouds.) From this, one can calculate how fast a star in a roughly circular orbit should be going. It's more complex than simple Keplerian orbits, but it is doable.

This is where the problem arises. No matter how much fudging one does regarding the amount of normal matter in those interstellar gas clouds, the numbers don't add up. Stars + gas clouds + dust don't explain the observations. There are only two explanations. One is that our concept of gravitation is fundamentally incorrect at the galactic levels and larger. The other is that our concept of gravitation is correct, but that some other form of mass exists that we can't see in galaxies and that isn't distributed the way stars are (hence the term "dark matter").

There are very few indicators that we don't know how gravity works at galactic scales. There are a large number of indicators that the second explanation is the correct one.

That is precisely my question. We hypothesise the existence of dark matter or some new gravitational theory based on the deviation of rotation curve from what we expect from Kepler's III Law. But how do we formulate the Kepler's law to be applicable in that condition?

 

[D H]

That is precisely my question. We hypothesise the existence of dark matter or some new gravitational theory based on the deviation of rotation curve from what we expect from Kepler's III Law. But how do we formulate the Kepler's law to be applicable in that condition?

That is not what is done. Kepler's laws derive from Newtonian gravitation assuming a very, very large central mass. Kepler's laws don't quite work even in the solar system. Jupiter's mass is about 1/1000^th of that of the Sun. This means that deviations from Keplerian orbits are easily observable even in the solar system because scientists do much better than three place accuracy nowadays.

However, it is use possible to Newtonian gravity to predict what a star's orbit about the galaxy would look like. To do that, one needs a model of the mass distribution in the galaxy. An obvious model is to use the stars as a proxy for the mass distribution. And then there's a problem. No matter how one fudges the numbers, this approach just doesn't work. That means one of two things: Either Newton's law of gravitation is fundamentally incorrect at galactic scales, or that stars are not a good model of how mass is distributed in the galaxy. Assuming the first is correct leads to alternative formulations of gravitation. Assuming the second is correct (that, ignoring relativistic effects, Newtonian gravitation is a good model of how gravitation works, even at galactic scales) leads to some other form of matter that doesn't behave the way "ordinary" matter does.

There are problems with both. The Bullet Cluster is a big problem for alternative gravity models. That physicists have not seen anything like "dark matter" is a big problem for assuming we do know how gravitation works. However, most people assume that we do know how gravitation works. The Bullet Cluster (and others) shoot big holes into those alternatives to Newtonian gravity.