Using Newton's 3rd Law, we can calculate the total galactic mass within the orbit of an object if we know the rotational velocity of the object and it's distance from the galactic centre. Apparently this is done routinely and the results indicate there must be a huge halo of dark matter extending well beyond the halo of visible galactic matter.

 

My question is:

 

The "distance" in Newton's 3rd Law is the semi-major axis. When we establish the distance of an object from the galactic centre, how do we know that distance is also the semi-major axis? Could not the object in question have an orbit that brings it much closer to the galactic centre than it is "now". In that case the semi-major axis would be smaller and the resulting calculations would be quite different.

 

Enlightenment would be most welcome. Thanks!

Surely you mean Kepler and not Newton.

This is a very good question, and I am not sure how they know. They can in principle measure the red-shift/blue-shift of an object, but this will only give you the velocity along the line between the object and the observer. Since the rotation is so slow, they obviously can't wait for it to move around a bit to see where it goes! There is probably an assumption of continuity somewhere, so another star in a similar orbit close by should have a similar velocity. Since the direction is slightly different, this should give a limit on the velocity of the original star in the direction perperdicular to line of sight.

 

This is just a guess though...

Surely you mean Kepler and not Newton.

I mean Kepler's 3rd Law as revised by Newton.

Yes, Kepler's 3rd Law is just a relation between the period and the semi-major axis length. You need Newton to incorporate masses.

Since the rotation is so slow, they obviously can't wait for it to move around a bit to see where it goes!

Yes, exactly. I thought they'd need at least 2 points to plot the orbit. How do they get the 2nd point when everything is moving so slow?

you can find the motion with redshifts over a few years i think.

If an object's current distance and its semi-major axis are significantly different, then the object has high eccentricity. I'm under the impression that objects in the galaxy's disk are in very circular orbits, while the objects in the galaxy's halo have high eccentricities as well as high inclinations.

I'm under the impression that objects in the galaxy's disk are in very circular orbits, while the objects in the galaxy's halo have high eccentricities as well as high inclinations.

 

There's the rub. The proposed existence of Dark Matter is due to the unexpected high orbital velocities of objects far from the Galactic Centre (in the halo). But if those orbits are highly eccentric, does that not mean that the calculations have assumed an incorrect semi-major axis? And that not quite so much Dark Matter is required?

 

I must be missing something...