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The Disk Rotation of the Milky Way Galaxy 103 THE DISK ROTATION OF THE MILKY WAY GALAXY 103 The Disk Rotation of the Milky Way Galaxy Vincent Kong Physics George Rainey Physics The rotation of the disk of the Milky Way Galaxy is analyzed. It rotates neither as a solid disk, nor in accordance with individual keplerian orbits. Rather the disk executes a form of differential rotation which suggests that a considerable fraction of its mass resides in the outer portions of the Galaxy. The Milky Way Galaxy is a large disk-shaped system of several hundred billion stars with associated interstellar gas and dust, surrounded by a diffuse spherical halo having a relatively low density of stars. The diameter of the galactic disk is at least 50 kiloparsecs (kpc; a parsec [pc] is the distance at which the radius of the Earth’s orbit about the Sun, 1 astronomical unit [A.U.], subtends an angle of 1 second of arc; 1 parsec = 206,265 A.U.) The Sun is located in the galactic disk at a point about 8.5 kiloparsecs from the galactic center, and is moving in a nearly circular orbit at about 220 km/s. Studies of the radial velocities and proper motions of other stars in the galactic disk indicate that they too are moving in nearly circular orbits in the disk (while stars in the galactic halo tend to have highly eccentric orbits). Thus the entire galactic disk seems to be rotating, and it is of interest to examine the nature of this rotation. In this paper we shall consider the observational consequences of galactic rotation, and then present a galactic rotation curve which is a composite of results obtained by various investigators over the past few years. Kinematics of Galactic Rotation In order to model the situation we assume that objects in the galactic disk are moving in perfectly circular orbits about an axis through the galactic center perpendicular to the galactic plane. Let Θ be the circular orbital speed at radial distance R in the galactic plane. Then the ω Θ ω Θ angular speed is = /R, and for the Sun, denote 0 = 0/R0 . Now if the Galaxy were to rotate as a solid disk, we would have ω = constant, and thus Θ would increase directly with R. On the other hand, for purely keplerian orbits, if we equate the centripetal acceleration to the gravitational acceleration due to a point mass at the galactic center, we find Θ ~ R-1/2 , and ω ~ R-3/2. The galactic disk does in fact exhibit differential rotation, but it is not keplerian due to the appreciably extended mass distribution. For the Galaxy, it is desirable to obtain Θ as a function of R, Θ (R), called the rotation curve. Consider a star at distance r from the Sun and at galactic longitude l as shown in Figure 1. (Galactic longitude is measured in the plane of the Galaxy eastward from the galactic center.) Using the auxiliary angle α we find that the observed radial and tangential velocities with respect to the Sun are, respectively, Θ α Θ ω α ω vr = cos - 0 sin l = R cos - R0 0 sin l (1) Θ α Θ ω α ω vt = sin - 0 cos l = R sin - R0 0 cos l . (2) But from Figure 1 we have α R cos = R0 sin l α R sin = R0 cos l - r hence ω ω vr = ( - 0) R0 sin l (3) ω ω ω vt = ( - 0) R0 cos l - r . (4) 104 KONG, RAINEY Fall 1999 Sun Θ0 l r S R0 R R α min α Θ C Θ Figure 1. The Sun is moving at speed 0 in a circular orbit of radius R0 about the galactic center C. Star S at distance r from the sun is moving at speed Θ in a circular orbit of radius R about the galactic center. Equations (3) and (4) are general, assuming only concentric circular motion in the galactic plane, and are called Oort’s (general) formulae, after the Dutch astronomer Jan Oort (1900- 1992), an early investigator into the problem of galactic rotation. Unfortunately the parameters Θ ω R0 and 0 (and therefore 0) are not precisely known, and there are considerable uncertainties in stellar distances, in general. A further complication arises from the fact that interstellar extinction of light due to dust particles in the galactic disk limits our ability to observe to large distances at optical wavelengths. Oort has derived limiting approximations to equations (3) and (4) which are valid ω only locally. The idea is to obtain a Taylor expansion of the function (R) about R0 , viz. ω ω ω = 0 + (d /dR)0 (R - R0) + .... (5) and to neglect the higher order terms, since (R - R ) will be small for nearby stars. Indeed ≈ 0 for r << R0 we have R0 - R r cos l , and equation (5) becomes approximately ω ω ≈ - 0 -(dw/dR)0 r cos l . (6) THE DISK ROTATION OF THE MILKY WAY GALAXY 105 Substituting equation (6) into equation (3), we obtain ≈ ω vr - R0 (d /dR)0 r sin l cos l = A r sin 2l (7) ω where the Oort constant A is given by A = -(R0/2) (d /dR)0 . Similarly substituting equation (6) into equation (4), we find ≈ ω 2 ω vt - R0 (d /dR)0 r cos l - 0 r where we have neglected terms in r2 . Then using 2 cos2 l = 1 + cos 2l , this becomes ≈ vt A r cos 2l + B r (8) ω ω Θ where the Oort constant B is defined as B = A - 0 . Note that since d /dR = d( /R)/dR = (1/ R)dΘ/dR - Θ/R2 , we can write the Oort constants as Θ Θ A = (1/2)[ 0/R0 - (d /dR)0] (9) Θ Θ B = (-1/2)[ 0/R0 + (d /dR)0] (10) Θ Θ with A - B = 0/R0 and A + B = -(d /dR)0 . The values accepted by the International Astronomical Union (IAU) in 1985 for these local rotation constants are: Θ ± 0 = 220 15 km/s ± R0 = 8.5 0.5 kpc A = 14.4 ± 1.2 km/s/kpc B = -12.0 ± 2.8 km/s/kpc . However, Reid (1993) has made a more recent case for a value of R0 = 8.0 + 0.5 kpc. The local Oort formulae, equations (7) and (8), are double sinusoids. This is, of course, just what one would qualitatively expect as a result of differential galactic rotation. For example, there would be no radial velocities for stars in the directions l = 0° or l = 180°, or nearby stars in direction l = 90° or l = 270°. Nearby stars in directions l = 45° or l = 225° would have positive radial velocities, while nearby stars in directions l = 135° or l = 315° would have negative radial velocities. Tangential velocities (proper motions) would be positive (i.e. in the direction of increasing galactic longitude) for stars in directions l = 0° or l = 180°, but superimposed on this is a negative effect due to the clockwise rotation of the Galaxy as a whole relative to an inertial frame formed by external galaxies; thus the formula for tangential velocities involves two terms, the non-sinusoidal term being negative. For a sample of stars at a common distance r the amplitude of these sinusoidal variations as a function of galactic longitude can be used to estimate the Oort constant A, and the mean offset from vt = 0 can be used to estimate the Oort constant B. The results are somewhat uncertain due to possible systematic errors in proper motions or in the distance scale. 106 KONG, RAINEY Fall 1999 The Galactic Rotation Curve What really needs to be determined, however, is not just the values of the local rotation constants, but the rotation curve Θ(R) for the Galaxy. Because of the difficulty in penetrating the interstellar medium with optical observations, we must rely primarily on radio observation, especially the 21-cm line due to a hyperfine transition in neutral hydrogen. A typical line profile might consist of several peaks due to different clouds with different radial velocities along the same line of sight. For observations in quadrants 0° < l < 90° and 270° < l < 360° there will be α some maximum (absolute) value for the radial velocity corresponding to = 0 and R = Rmin = R0 sin l . Then from equation (1) we have Θ Θ vr,max = (Rmin) - 0 sin l . (11) Assuming that R and Θ are known, Θ(R) can be determined from equation (11) for R < 0 0 ° R0. The method does not work well within about 20 of the galactic center because of markedly non-circular motions there. It also does not work well near l = 90° and l = 270°, since maximum radial velocities are rather poorly defined there. For R > R0 there will be no maximum radial velocity. Here one must rely on objects in the galactic plane of known distance (such as Cepheid variable stars), since r, R0, l, and the law of cosines enable R to be determined, hence Θ(R) from equations (1) or (3). Among the more important early investigations of galactic disk rotation are those of 275 250 Θ(km/s) 225 200 175 150 0246810121416 R (kpc) Figure 2. The rotation curve for the disk of the Milky Way Galaxy. The uncertainty in the rotation speed is of the order of 10 - 20 km/s for R < R0 = 8.5 kpc. THE DISK ROTATION OF THE MILKY WAY GALAXY 107 Schmidt (1965) for R > 5 kpc and Simonson and Mader (1973) for R < 5 kpc.
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