Motions in the Milky Way are commonly expressed with respect to the
Since the MW is approximately rotationally symmetric, we adopt a cylindrical coordinate system; let z be distance above the plane of the MW, R be the distance from the galactic center in the plane of the MW, and phi be the azimuthal coordinate, measured in the direction of the MW's rotation.
The LSR moves about the galactic center at the local value of the circular velocity. Velocities with respect to the LSR are represented by vertical, radial, and azimuthal components
dz dR dphi (1) w = -- , u = -- , v = R ---- - V(R_0) , dt dt dtrespectively, where V(R) is the rotation curve of the MW, and R_0 is the R component of the Sun's position.
The determination of the LSR is a two-step process. Basically, we assume that on average stars in the Sun's vicinity are moving in the phi direction but have no net motion in the R or z directions. It might also seem natural to assume that on average the stars are also moving in the phi direction with the local circular velocity, but this is not true because of the asymmetric drift (see below); the greater the random velocities of individual stars, the more the net motion lags behind the local circular velocity. Thus the steps are
2 (2) v_a = sigma_R / (110 km/sec)(MB81, Figure 6-7).
Results for different ensembles of stars may be combined to obtain the motion of the Sun with respect to the LSR:
(3) (u, v, w) = (-9, 12, 7) km/s(MB81, Chapter 6-4).
To relate the LSR to the FSR, we need the local circular velocity, V_0 = V(R_0). In 1985, IAU Commission 33 recommended
(4) V_0 = 220 km/s , R_0 = 8.5 kpc(Kerr & Lynden-Bell 1986). Note that MB81 use an older convention.
The rotation of the galaxy gives rise to an organized pattern of stellar motions in the vicinity of the Sun. In this section, these motions are described under the assumption that random velocities are zero.
A physical understanding of the kinematic consequences of galactic rotation may be gained by considering separately the local effects of solid-body and differential rotation (MB81, Chapter 8-1).
If the MW rotated as a solid body, with angular velocity omega independent of R, then distances between stars would not change, and all radial velocities would be zero. However, stars would still show proper motions (with respect to an external frame of reference); the transverse velocity of a star at a distance r from the Sun would be
(5) v_t = - omega r .
Of course, the MW does not rotate as a solid body; the orbital period is an increasing function of R in the vicinity of the Sun. Stars at radii R < R_0 therefore catch up with and pass us, while we catch up with and pass stars at radii R > R_0. This results in non-zero radial velocities,
(6) v_r ~ r sin(2l) ,where l is the galactic longitude of the star under observation.
Consider a star at galactic radius R moving at the circular velocity appropriate to that radius, V(R) = R omega(R). With respect to the LSR at the Sun's galactic radius R_0, the radial and transverse components of the star's motion are
(7) v_r = (omega - omega_0) R_0 sin(l) , (8) v_t = (omega - omega_0) R_0 cos(l) - omega r ,where omega_0 = omega(R_0).
In the local limit, where r/R_0 is a small parameter, the above expressions become
(9) v_r = A r sin(2l) , (10) v_t = (A cos(2l) + B) r ,where A and B are the Oort constants, given by
1 V dV (11) A = - (- - --) , 2 R dR 0 1 V dV (12) B = - - (- + --) , 2 R dR 0where the subscript 0 indicates that the expressions in parenthesis are evaluated at the solar radius, R_0. In brief, A is a measure of the shear of the MW's rotation, while B is a measure of the vorticity.
Observations of local stellar motions allow a direct estimate of the Oort constants. In practice these quantities are subject to a number of constraints; the recommendation of the IAU commission is
(13) A = 14 km/sec/kpc , B = -12 km/sec/kpc .
Like the Sun, other stars have random velocities with respect to the LSR. Complementing the discussion above, this section will discuss random motions of stars in our immediate neighborhood, while neglecting the larger-scale effects of rotation.
Consider an ensemble of stars with orbits passing through the vicinity of the Sun. Since the Milky Way is ~50 rotation periods old, it is reasonable to assume that stars are well-mixed; that is, slight differences in orbital period will have had enough time to spread out initially-correlated groups of stars, in effect assigning stellar orbits randomly-chosen phases. By symmetry, the velocity ellipsoid at z = 0 should have principal axes aligned with the R, phi, and z directions (this statement is intuitive when random velocities are analyzed using epicyclic theory).
To a first approximation, histograms of random stellar velocities do not differ much from gaussians, so a convenient approximation to the velocity distribution with respect to the LSR of a well-mixed stellar ensemble is
-Q(u,v-v_a,w) (14) f(u,v,w) = f_0 e ,where v_a is the asymmetric drift velocity of the ensemble, the function
(15) Q(u,v',w) = [u,v',w] . T . [u,v',w] ,and the symmetric tensor
1 [sigma_R^-2 0 0 ] (16) T = - [ 0 sigma_phi^-2 0 ] . 2 [ 0 0 sigma_z^-2]describes the shape of the velocity ellipsoid. Note that T is proportional to the random part of the kinetic energy tensor of the stellar ensemble; it is diagonalized as a consequence of the mixing assumption made above. For this T, the quantity Q(u,v-v_a,w) is
1 u^2 (v-v_a)^2 w^2 (17) Q = - (--------- + ----------- + ---------) . 2 sigma_R^2 sigma_phi^2 sigma_z^2
In practice the velocity ellipsoid for a given ensemble of stars (e.g. all stars of a given stellar type) is never quite diagonalized because of imperfect mixing. The most significant term to be added to Eq. 17 is proportional to u(v-v_a), indicating that the velocity ellipsoid lies in the plane of the disk, but is not precisely oriented toward the galactic center. The angle between the long axis of the velocity ellipsoid and the R direction is called the longitude of vertex, l_v.
Results for a wide range of stellar types are listed in Table 7-1 of MB81. For dwarf stars, the various components of the velocity dispersion become greater progressing from early to late spectral types. This is an age effect: late spectral classes include more old stars, and the random velocities of stars increase with time (presumably due to gravitational scattering, although a satisfactory theory is still lacking). The angle l_v becomes smaller progressing from early to late types since older stars are more completely mixed. For giant stars, kinematic parameters reflect those of the dwarf stars they evolved from. The largest random velocities belong to evolved objects such as white dwarfs which include the largest fractions of very old (e.g. 10^10 year-old) stars.
Empirically, the net lag of a given ensemble of stars with respect to the LSR is approximated by Eq. 2 above. This relationship is a consequence of the collisionless Boltzmann equation. In cylindrical coordinates, the CBE for a steady-state axisymmetric system is
df df v_phi^2 dPhi df (18) 0 = v_R -- + v_z -- + (------- - ----) ---- dR dz R dR dv_R v_R v_phi df dPhi df - --------- ------ - ---- ---- R dv_phi dz dv_z(e.g. BT87, Chapter 4.1a).
To derive the relationship for the asymmetric drift we take the radial velocity moment of Eq. 18; multiplying by v_R and integrating over all velocities, the result is a Jeans equation:
R d _______ d _______ (19) 0 = -- -- (nu v_R v_R) + R -- (v_R v_z) nu dR dz _______ ___________ dPhi + v_R v_R - v_phi v_phi + R ---- . dRThe azimuthal motion can be divided into net streaming and random components:
___________ _____ 2 2 (20) v_phi v_phi = v_phi + sigma_phi 2 2 = (V_0 - v_a) + sigma_phi ,where the second equality follows from the definition of the asymmetric drift velocity v_a. For the radial motion,
_______ 2 (21) v_R v_R = sigma_R ,since there is no net streaming motion in the radial direction. Combining Eqs. 19, 20, & 21, using the identity V_0^2 = R dPhi/dR, and assuming that v_a/V_0 is small, we obtain
sigma_R^2 sigma_phi^2 d ln (22) v_a = --------- [----------- - 1 - ------ (nu sigma_R^2) 2 V_0 sigma_R^2 d ln R R d _______ - --------- -- (v_R v_z)] . sigma_R^2 dzThis equation relates the asymmetric drift velocity to the radial component of the velocity dispersion. If we compare ensembles of stars which have the same radial distribution and velocity ellipsoids of similar shapes, the expression inside square brackets is constant and we recover Eq. 2.
Several of the terms in Eq. 22 may be further simplified. By multiplying the CBE by v_R v_phi and integrating over all velocities it is possible to show that
sigma_phi^2 - B (23) ----------- = ----- , sigma_R^2 A - Bwhere A and B are the Oort constants. The dispersions listed in table 7-1 of MB81 show that for almost all stellar types, sigma_phi^2/sigma_R^2 = ~0.5 as expected from Eqs. 13 & 23.
The most problematic part of Eq. 22 is the last term within the square brackets. This term represents the tilt of the velocity ellipsoid at points above (and below) the galactic midplane, z = 0. We do not presently know very much about the orientation of the velocity ellipsoid away from the midplane. If the ellipsoid remains parallel for all z values then this term is identically zero, while if the ellipsoid tilts to always point at the galactic center then
R d _______ sigma_z^2 (24) --------- -- (v_R v_z) = 1 - --------- . sigma_R^2 dz sigma_R^2Numerical experiments indicate that the most likely behavior is somewhere between these two extremes.
Due date: 3/16/95
15. Suppose that the principal axes of the velocity ellipsoid near the Sun are always parallel to the unit vectors of a spherical coordinate system. Show that for |z|/R small,
_______ z 2 2 v_R v_z = - (sigma_R - sigma_z) . RSee BT87, Prob. 4-5 for a hint.
Last modified: March 1, 1995