## Determination of the rotation curve

maximum Doppler shift. We then assign that Doppler shift to material at the subcentral point (the point of closest approach to the galactic center) for that particular longitude. We can see from Fig. 16.6 that the distance of the subcentral point to the galactic center, Rmin, is

From equation (16.11), we see that if vmax is the maximum radial velocity along a given line of sight, then the angular speed ^(Rmin) for that line of sight is given by n(R0 sin Í) = (Vmax/R0 sin I) + ^0

The rotation curve for material within the Sun's orbit can be determined reasonably well from 21 cm line observations. In determining the rotation curve, we make two important assumptions: (1) The orbits are circular. This means that we need to determine v(R) at only one point for each value of R. (2) There is some atomic hydrogen all along any given line of sight. It is especially important that there be some hydrogen at the subcentral point of each line of sight.

The method takes advantage of the fact that, for lines of sight through the part of the galaxy interior to the Sun's orbit, there is a maximum Doppler shift. It is easy to inspect the 21 cm spectrum at each longitude and determine the

By studying lines of sight with longitudes ranging from 0° to 90°, the corresponding value of Rmin will range from zero to R0. This means that we measure Q(R) once for each value of R from zero to R0. However, we have already said that, if the material is moving in circular orbits, one measurement per orbit is sufficient to determine the rotation curve.

There are some limitations to this technique. We have already seen that the distribution of interstellar gas is irregular. If there happens to be no atomic hydrogen at the subcentral point for some line of sight, we will see a vmax which is less than the value that we would see if there were material at the subcentral point. There are also problems arising from non-circular orbits. The effect of both of these problems can be reduced by repeating the procedure for the fourth quadrant. Because of the inclination of the galactic plane relative to the celestial equator, the fourth quadrant studies must be performed in the southern hemisphere, and have been done in Australia. For a number of years it seemed that there were disagreements between the first and fourth quadrant rotation curves, but we now think we understand them in terms of non-circular motions. There is further evidence for such motions in the large radial velocities observed for t close to 0° and 180°.

Another problem is that we cannot really cover the full range from zero to R0. For t close to zero, sin t is close to zero, and the Doppler shift is very small. Random motions of clouds are much larger than the radial velocity due to galactic rotation. Similarly, for t near 90°, Q is close to Q0, providing a small radial velocity due to galactic rotation.

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