## Info

5 10 15

Galactic Radius (kpc)

Rotation curve for the Milky Way.The curve inside the Sun's orbit is well determined from 21 cm studies. Outside the Sun's orbit, we use HII regions in molecular clouds.We use spectroscopic parallax on the exciting stars to give the distance to the HII regions, and then CO observations to measure the Doppler shift. [Daniel Clemens (Boston University) Clemens, D., Astrophys. J., 295,422, 1985]

5 10 15

Galactic Radius (kpc)

Rotation curve for the Milky Way.The curve inside the Sun's orbit is well determined from 21 cm studies. Outside the Sun's orbit, we use HII regions in molecular clouds.We use spectroscopic parallax on the exciting stars to give the distance to the HII regions, and then CO observations to measure the Doppler shift. [Daniel Clemens (Boston University) Clemens, D., Astrophys. J., 295,422, 1985]

When we eliminate these ends, a reasonable rotation curve has been derived for R in the range 3 to 8 kpc. This is shown as part of the curve in Fig. 16.9.

Using HI to determine the rotation curve outside the Sun's orbit is more difficult. There is no maximum Doppler shift along any line of sight. It is therefore necessary to measure independently v and d, the distance from the Sun to the material being studied. From d and €, we can deduce R (see Problem 16.6). Until molecular observations, there was no reliable rotation curve for R > R0. The best astronomers could do was to derive the mass distribution for R < R0, and then make some assumptions about how the mass distribution would continue for R > R0. From the assumed mass distribution, a rotation curve could be derived. It was assumed that there was relatively little mass outside the Sun's orbit, so the rotation curve was characterized by a falloff in v(R) that was close to that predicted by Kepler's third law.

However, recent observations of molecular clouds have provided a direct method of measuring the rotation curve outside the Sun's orbit. Molecular clouds associated with HII regions were studied. The radial velocities were determined from observations of the carbon monoxide (CO) emission from the clouds. The distance to the stars exciting the HII regions were determined by spectroscopic parallax. This gives a reliable rotation curve at least out to about 20 kpc. The combined rotation curve (using HI inside the solar circle and CO outside, is shown in Fig. 16.9. There is no falloff in v(R) out to 20 kpc, and there may even be a slight rise. This means that there is much more mass outside the Sun's orbit than previously thought!

We can see from equation (16.3) that if v(R) is constant from 8 to 16 kpc, then M(16 kpc) will be twice M(8 kpc). This means that there is as much mass between 8 and 16 kpc as there is out to 8 kpc. However, the luminosity of our galaxy is falling very fast as R increases. Since the luminous part of the matter is mostly in the disk, it would seem that this extra mass cannot be part of the disk. Current thinking places the extra mass in the halo of the galaxy. We still have little idea of what form this matter takes. It has been suggested that it can be in the form of faint red stars, but recent results make this seen unlikely. This is our first encounter with dark matter, matter whose gravitational effects are felt, but which is not very luminous. (Astronomers used to call this "missing" matter, but it is not missing. We can tell that it is there by its gravitational effects. We just can't see it.) We will see that there is strong evidence for dark matter in other galaxies, and we will discuss it farther in Chapter 17.

Example 16.2 Galactic mass distribution

For what (spherically symmetric) mass distribution is v(R) constant?

solution

From equation (16.3), we know how M(R) is related to v(R), and from equation (16.1) we know how p(r) is related to M(R).

For a spherical coordinate system, the volume element is 4^r2dr, so equation (16.1) becomes

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