Differential Rotation. Oort's Formulas. The flatness of the Milky Way is already suggestive of a general rotation about an axis normal to the galactic plane. Observations of the motions both of stars and of interstellar gas have confirmed this rotation and shown it to be differential. This means that the angular velocity of rotation depends on the distance from the galactic centre (Fig. 17.13). Thus the Milky Way does not rotate like a rigid body. Near the Sun, the rotational velocity decreases with radius.
The observable effects of the galactic rotation were derived by the Dutch astronomer Jan H. Oort. Let us suppose the stars are moving in circular orbits about
a) Velocity w.r.t the Milky Way a) Velocity w.r.t the Milky Way
Fig. 17.13a-d. The effect of differential rotation on the radial velocities and proper motions of stars. (a) Near the Sun the orbital velocities of stars decrease outwards in the Galaxy. (b) The relative velocity with respect to the Sun is obtained by c) Radial velocity
c) Radial velocity
d) Tangential velocity
d) Tangential velocity
subtracting the solar velocity from the velocity vectors in (a). (c) The radial components of the velocities with respect to the Sun. This component vanishes for stars on the same orbit as the Sun. (d) The tangential components of the velocities
Fig. 17.13a-d. The effect of differential rotation on the radial velocities and proper motions of stars. (a) Near the Sun the orbital velocities of stars decrease outwards in the Galaxy. (b) The relative velocity with respect to the Sun is obtained by subtracting the solar velocity from the velocity vectors in (a). (c) The radial components of the velocities with respect to the Sun. This component vanishes for stars on the same orbit as the Sun. (d) The tangential components of the velocities the galactic centre (Fig. 17.14). This approximation is acceptable for population I stars and gas. The star S, seen from the Sun © at galactic longitude l at distance r, has circular velocity V at a distance R from the centre. Similarly for the Sun the galactic radius and velocity are R0 and V0. The relative radial velocity vr of the star with respect to the Sun is the difference between the projections of the circular velocities on the line of sight:
Denoting the angular velocity of the star by w = V/R and that of the Sun by w0 = V0/R0, one obtains the observable radial velocity in the form vr = R0(w — w0) sinl.
where a is the angle between the velocity vector of the star and the line of sight. From Fig. 17.14 the angle CS© = a + 90°. By applying the sine theorem to the triangle CS© one obtains sin (a + 90°) _ R0 sin l R
The tangential component of the relative velocity of the Sun and the star is obtained as follows. From Fig. 17.14, vt = V sin a — V0 cos l = Rw sin a — R0w0 cos l. The triangle ©CP gives
R sin a = R0 cos l — r , and hence vt = R0(w — w0) cos l — wr . (17.9)
Oort noted that in the close neighbourhood of the Sun (r ^ R0), the difference of the angular velocities will be very small. Therefore a good approximation for the exact equations (17.8) and (17.9) is obtained by keeping only the first term of the Taylor series of w — w0 in the neighbourhood of R = R0:
Fig. 17.14. In order to derive Oort's formulas, the velocity vectors of the Sun and the star S are divided into components along the line ©S and normal to it
Vq Rq dV
For the tangential relative velocity, one similarly obtains, since rnr ~ rn0r:
Because 2 cos21 = 1 + cos 2l, this may be written vt ~ Ar cos 2l + Br ,
where A is the same as before and B, the second Oort constant, is
The proper motion ¡x = vt/r is then given by the expression
Equation (17.10) says that the observed radial velocities of stars at the same distance should be a double sine curve as a function of galactic longitude. This has been confirmed by observations (Fig. 17.15a). If the distance to the stars involved is known, the amplitude of the curve determines the value of the Oort constant A.
Independently of distance, the proper motions of the stars form a double sine wave as a function of galactic
Fig. 17.14. In order to derive Oort's formulas, the velocity vectors of the Sun and the star S are divided into components along the line ©S and normal to it vr [km/s]
For R ~ Ro ^ r, the difference R — R0 r cos l. One thus obtains an approximate form vr [km/s]
where A is a characteristic parameter of the solar neighbourhood of the Galaxy, the first Oort constant:
Fig. 17.15a,b. The velocity components due to differential rotation according to Oort's formulas as functions of galactic longitude. (a) Radial velocities for objects at a distance of 1 and 2kpc. (Compare with Fig. 17.13.) Strictly, the longitude at which the radial velocity vanishes depends on the distance. Oort's formulas are valid only in the close vicinity of the Sun. (b) Proper motions
Fig. 17.15a,b. The velocity components due to differential rotation according to Oort's formulas as functions of galactic longitude. (a) Radial velocities for objects at a distance of 1 and 2kpc. (Compare with Fig. 17.13.) Strictly, the longitude at which the radial velocity vanishes depends on the distance. Oort's formulas are valid only in the close vicinity of the Sun. (b) Proper motions v
longitude, as seen in Fig. 17.15b. The amplitude of the curve is A and its mean value, B.
In 1927 on the basis of this kind of analysis, Oort established that the observed motions of the stars indicated a differential rotation of the Milky Way. Taking into account an extensive set of observational data, the International Astronomical Union IAU has confirmed the present recommended values for the Oort constants:
The Oort constants obey some interesting relations. By subtracting (17.13) from (17.11), one obtains
Knowing the values of A and B, one can calculate the angular velocity &>o = 0.0053"/year, which is the angular velocity of the local standard of rest around the galactic centre.
The circular velocity of the Sun and the LSR can be measured in an independent way by using extragalactic objects as a reference. In this way a value of about 220 km s-1 has been obtained for V0. Using (17.15) one can now calculate the distance of the galactic centre R0. The result is about 8.5 kpc, in good agreement with the distance to the centre of the globular cluster system. The direction to the galactic centre obtained from the distribution of radial velocities and proper motions by means of (17.10) and (17.14) also agrees with other measurements.
The orbital period of the Sun in the Galaxy according to these results is about 2.5 x 108 years. Since the Sun's age is nearly 5 x 109 years, it has made about 20 revolutions around the galactic centre. At the end of the previous revolution, the Carboniferous period had ended on Earth and the first mammals would soon appear.
The Distribution of Interstellar Matter. Radio radiation from interstellar gas, in particular that of neutral hydrogen, is not strongly absorbed or scattered by interstellar dust. It can therefore be used to map the structure
Fig. 17.16. Clouds P1, P2, various distances seen in the same direction at of the Milky Way on large scales. Radio signals can be detected even from the opposite edge of the Milky Way.
The position of a radio source, for example an HI cloud, in the Galaxy cannot be directly determined. However, an indirect method exists, based on the differential rotation of the Galaxy.
Figure 17.16 is a schematic view of a situation in which gas clouds on the circles P1, P2,... are observed in the direction l (-90° < l < 90°). The angular velocity increases inwards, and therefore the greatest angular velocity along the line of sight is obtained at the point Pk, where the line of sight is tangent to a circle. This means that the radial velocity of the clouds in a fixed direction grows with distance up to the maximum velocity at cloud Pk :
where Rk = R0 sin l. The distance of cloud Pk from the Sun is r = R0 cos l. When r increases further, vr decreases monotonically. Figure 17.17 shows how the observed radial velocity in a given direction varies with distance r, if the gas moves in circular orbits and the angular velocity decreases outwards.
The neutral hydrogen 21 cm line has been particularly important for mapping the Milky Way. Figure 17.18 gives a schematic view of how the hydrogen spectral line is made up of the radiation of many individual concentrations of neutral hydrogen, clouds or spiral arms. The line component produced by each cloud has a wavelength which depends on the radial velocity of the cloud and a strength depending on its mass and density. The total emission is the sum of these contributions.
By making observations at various galactic longitudes and assuming that the clouds form at least partly continuous spiral arms, the distribution of neutral hydrogen in the galactic plane can be mapped. Figure 15.17 shows a map of the Milky Way obtained from 21 cm line observations of neutral hydrogen. It appears that the neutral hydrogen is concentrated in spiral arms. However, interpretation of the details is difficult because of the uncertainties of the map. In order to obtain the distances to the gas clouds, one has to know the rotation curve, the circular velocity as a function of the galactic radius. This is determined from the same radial velocity observations and involves assumptions concerning the density and rotation of the gas. The interpretation of
Fig. 17.18. Clouds at different distances have different velocities and therefore give rise to emission lines with different Doppler shifts. The observed flux density profile (continuous curve) is the sum of the line profiles of all the individual line profiles (dashed curves). The numbers of the line profiles correspond to the clouds in the upper picture
Fig. 17.18. Clouds at different distances have different velocities and therefore give rise to emission lines with different Doppler shifts. The observed flux density profile (continuous curve) is the sum of the line profiles of all the individual line profiles (dashed curves). The numbers of the line profiles correspond to the clouds in the upper picture the spiral structure obtained from radio observations is also still uncertain. For example, it is difficult to fit the radio spiral structure to the one obtained near the Sun from optical observations of young stars and associations.
The Rotation, Mass Distribution and Total Mass of the Milky Way. In (17.17) the galactic longitude l gives the galactic radius Rk of the clouds with maximum radial velocity. By making observations at different longitudes, one can therefore use (17.17) to determine the angular velocity of the gas for various distances from the galactic centre. (Circular motions must be assumed.) In this way, the rotation curve w = w(R) and the corresponding velocity curve V = V(R) (= wR) are obtained.
Figure 17.19 shows the rotation curve of the Milky Way. Its central part rotates like a rigid body, i. e. the
Fig. 17.19. Rotation curve of the Milky Way based on the motions of hydrogen clouds. Each point represents one cloud. The thick line represents the rotation curve determined by Maarten Schmidt in 1965. If all mass were concentrated within the radius 20kpc, the curve would continue according to Kepler's third law (broken line). The rotation curve determined by Leo Blitz on the basis of more recent observations begins to rise again at 12 kpc
angular velocity is independent of the radius. Outside this region, the velocity first drops and then begins to rise gradually. A maximum velocity is reached at about 8 kpc from the centre. Near the Sun, about 8.5 kpc from the centre, the rotational velocity is about 220 km s-1. According to earlier opinions, the velocity continues to decrease outwards. This would mean that most of the mass is inside the solar radius. This mass could then be determined from Kepler's third law. According to (6.34),
Using the values R0 = 8.5 kpc and V0 = 220 km s one obtains
The escape velocity at radius R is I2GM
such stars would exceed the escape velocity. This has been confirmed by observations.
The preceding considerations have been based on the assumption that near the Sun, the whole mass of the Galaxy can be taken to be concentrated in a central point. If this were true, the rotation curve should be of the Keplerian form, V a R_1/2. That this is not the case can be established from the values of the Oort constants.
The derivative of the Keplerian relation
Using the properties (17.15) and (17.16) of the Oort constants, one finds
This gives an escape velocity near the Sun Ve = 310kms-1. One therefore should not see many stars moving in the direction of galactic rotation, l = 90°, with velocities larger than 90kms-1 with respect to the local standard of rest, since the velocity of for a Keplerian rotation curve. This disagrees with the observed value and thus the assumed Keplerian law does not apply.
The mass distribution in the Milky Way can be studied on the basis of the rotation curve. One looks for
17.4 Structural Components of the Milky Way
a suitable mass distribution, such that the correct rotation curve is reproduced. Recently distant globular clusters have been discovered, showing that the Milky Way is larger than expected. Also, observations of the rotation curve outside the solar circle suggest that the rotational velocity might begin to rise again. These results suggest that the mass of the Galaxy might be as much as ten times larger than had been thought.
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