In article ,
jacobnavia writes:
Suppose some star S at 60 thousand light years from the center of the
galaxy. A normal star whose mass can be accurately determined.

Its speed can be measured, and its mass is known.

You don't need the mass. At these scales, stars are effectively
massless "test particles," and only accelerations are relevant.

Then, we subtract gravity effects

The "gravity effects" depend on the mass distribution, which is
unknown. (This is what jt was getting at.) In other words, if you
knew the accelerations and assumed a gravity law, you could derive
the mass distribution of the Milky Way. Or if you knew the mass
distribution from other data, you could derive the gravity law. But
you can't derive both. What you could do is look for the _simplest_
set of assumptions that would explain all the accelerations. That
would be a huge advance, of course.

As jt also mentioned, the observables (in principle!) are the three
spatial coordinates and the three components of velocity. With GAIA
and ground-based radial velocity surveys, those will soon be measured
for millions of stars! What you want, though, are the
_accelerations_, which can't be directly measured. However, the
statistical distribution of velocities should at least put
constraints on _either_ the mass distribution _or_ the gravity law,
and the task again will be to look for the simplest assumptions that
explain the data.

In article ,
jacobnavia writes:
Mmm the galaxy has a plane of rotation. Th center of the galaxy, that
star and we are rotating around the same central object, the galaxy, in
a plane.

Any three points define a plane, but neither the Sun nor an arbitrary
star is in general moving in that plane. To put it another way, the
Milky Way disk has a finite height, and the halo is roughly
spherical, and neither stellar component has motions perpendicular to
the vector towards the Galactic center.

[Moderator's note: A pedantic note to avoid posts pointing this out:
any three points define a plane if they are not colinear. -P.H.]

"We don't know that is moving in a circular orbit"... wow, I always
thought that they are doing so,

I'm not sure why you thought that. The orbits are roughly
elliptical, generally with modest eccentricities, but the ellipses
are not closed as for solar system planetary orbits. That's because
the mass distribution of the Galaxy is not spherically
symmetric.

and that the "arms" we see are density
waves in the disc of stars circling the center.

Indeed so. These density waves perturb the stellar orbits. The bar
is also an important perturber.

You might look up "Local Standard of Rest" and "Solar Motion".

The stars must be doing "some" kind of circle around the center since
the form of the galaxy (a rotating plane of stars ) indicates so.

The flatness of the disk indicates that the orbital inclinations are
small (not zero, though) but says nothing about eccentricities. In
fact, you could even have large inclinations if the eccentricities
were also large, but that's not what's observed.

I even thought that the sun was rotating about 1 rotation per 250
million years, so it is around 20 galactic years old. I thought
that the orbit was a circle. Is that not correct?

The time scale is about right (I get 220 Myr), but "circle" is an
approximation even rougher than "ellipse."

Dark matter scenarios suppose some form of invisible matter outside

Not only "outside." The morphology of the putative dark matter halo
is unknown, but simulations say it ought to be concentrated toward
the Milky Way center, roughly but not exactly spherically symmetric,
and more extended than the stellar distribution.

A second star we could use of course, is the nearest one, the sun. The
sun's orbit could tell us about the force effects here. Here we have
more data and less problems than with the star's far away. We know very
precisely the distance to the center, and the mass between us and the
center, so the effects of gravity could be calculated much more easily.

The mass is derived from the solar motion; we don't know it
independently. Unless I'm missing something. One standard reference
on the subject is at
http://iopscience.iop.org/article/10...ta#apj490685s4
but the authors don't derive masses from the rotation curve.

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