Milky Way Galaxy

The Local Motion of the Galaxy

The Sun and its neighboring stars, lying in the Galactic disk, constitute a stellar fluid orbiting the Galactic center. As with molecules in a gas, the stars in this fluid have random thermal motions that they acquire over billions of years of numerous weak encounters among themselves. The fluid itself moves in swirls, shears, and divergences with a velocity that is an average of the velocities of its stellar atoms. How this fluid moves as it flows in a disk around the Galactic center tells us something about how the gravitational field and the mass generating the field are distributed within the Galaxy.

The very form of the galactic disks in other galaxies suggests that the Milky Way's disk is rotating around the Galactic center. Within the stellar fluid the rotation of the disk appears as a vorticity, a property that describes the rotations within a fluid. Analogy to other astrophysical disks, such as accretion disks around neutron stars and the planets orbiting the Sun, suggests that the Milky Way's disk is differentially rotating, with the inner regions completing a rotating in less time than the outer regions. This form of rotation gives a shear to the stellar fluid. The shear is greatest if the stars obey Kepler's laws, like the planets orbiting the Sun. A planet's orbital period increases proportionally with the orbital semimajor axis to the 3/2 power. If the stars in the Galactic disk were orbiting the Galactic center in circular Keplerian orbits, the rotation velocity of the disk would fall as the inverses square root of the distance from the Galactic center. But unlike the gravitational field in the Solar System, which is almost entirely generated by the Sun, the gravitational field in Galactic disk depends on the distribution of mass within the Galaxy. At the extreme, the gravitational field generated by this extended mass distribution could cause every star to have the same orbital period, and the Galactic disk would rotate like a solid body, with the rotation velocity of the disk increased proportionally with radius. The actual rotation of the disk is somewhere between differential Keplerian rotation and solid body rotation.

The shear and vorticity within the Galactic disk do not give large relative velocities to the stars we see. Our sight within the Galactic disk is limited to a distance of about 1 kpc from the Sun, compared to a distance of 7.6 kpc to the Galactic center. This means that the stars we see are influenced by a Galactic gravitational field that varies by an order of 15% along the direction to the Galactic center. The stellar velocities change direction by a comparable amount over 1 kpc along the direction of rotation. The distances, and therefore the gravitational gradient and the curvature of the orbital paths, are smaller for the subset of stars that have their motions on the sky measured to a reasonable precision.

The small motions of the stellar fluid relative to the Sun are described by the vorticity, the shear in two directions within the plane of the disk, and the divergence of the flow along the Galactic plane. This description suppresses the motion of the fluid perpendicular to the Galactic disk. The properties of the fluid are parameterized by four constants that are collectively called the Oort constants.

If the motion of the stellar fluid in the Galactic disk were axisymmetric around the axis of the disk, the fluid would only have vorticity from the rotation of the disk and a shear across the radius vector from the disk's differential rotation?the simple expectations we set out at the beginning of our discussion. But two features of our Galaxy should break this axisymmetry: our Galaxy is a barred-spiral galaxy, so that a large bar of stars rotate like a solid body at the very center of the Galaxy, and the Sun lies between two spiral arms, offset more towards one arm than the other. The gravitational fields generated by the central bar and by the spiral arms should influence the motion of the stars around the Sun. We expect to see a more complex motion within the local stellar fluid than given by axisymmetric motion.

The vorticity and shear appear in the longitudinal proper motion?the rate of change in longitude?of the stars, and the flow divergence and shear appear in the doppler shift of the stars. The proper motion is much easier to measure for a large collection of stars than the redshift, so in practice astronomers have better measurements of the vorticity and shear of the stellar fluid than the flow divergence. The effect of vorticity and shear appear as trigonometric variations of the average proper motion with Galactic longitude. The vorticity adds a component to the proper motion that is constant with Galactic longitude. It represents the constant rotation of the sky as we orbit the Galactic center. An analogy in our Solar System is the daily 1 rotation of the sky relative to the Sun. The shear adds terms that are the sine and cosine of twice the Galactic longitude. This expresses the absence of a contribution to the proper motion along planes of shear, and a contribution to the proper motion that is of the same sign in either direction perpendicular to the planes of shear. If the shear is purely from the differential rotation of the Galactic disk, then the stars in the direction of the Galactic center move ahead of the Sun and the stars away from the Galactic center lag behind the Sun; in terms of a change in Galactic longitude, the shear changes the longitude of both sets of star in the same direction without changing the longitude of the stars at the same radius in the disk as the Sun.

The complication to this picture is the Sun's own motion. It adds a component to the proper motion of a star that depends on the star's distance that varies as the sine and cosine of the longitude. For instance, if the Sun were moving faster that the surrounding stars, the stars perpendicular to that motion would fall behind the Sun, so that the stars on either side of the sun move in opposite directions in longitude.

In principle the differences in sinusoidal behavior makes disentangling the Solar motion, the vorticity, and the shear from the observed average proper motion of a group of stars a simple matter: simply break the variations in proper motion with latitude into a constant plus sine and cosine terms, which is done through a Fourier transform, and read the coefficients for each trigonometric function. In practice, the life of an observer is not so simple. Because distance plays a role in how the Sun's motion affects a star's proper motion, variations in how far one samples the surrounding stellar fluid can cause the Sun's motion to add additional trigonometric terms to a star's proper motion that mimic the effects of vorticity and shear. This pollution by the Sun's motion can be deduced and corrected at the cost of greater uncertainty.

The most recent measurements of local stellar motion are based on the enormous catalogs of stellar proper motion and parallax measured by the Hipparcos satellite.^[1] Using a sample of 1 million stars, the most recent results suggest that the azimuthal velocity around the Galactic center is nearly constant, varying only as 1 km s^?1 kpc^?1, which is only a half percent variation in the rotation velocity over 1 kpc. This constancy of rotation velocity with distance from the Galactic center is a common property of spiral galaxies. The orbital velocity of 250 km s^-1 implied by the local flow is somewhat larger than the accepted value of 210 km s^?1 derived from the proper motion and distance measures of Sgr A*. This difference in velocity may be a consequence of a shear in the radial velocity of the stellar fluid. After correcting for the mimicry by the Sun's motion of the vorticity and shear, one team of astronomers finds a large radial shear. The inference is that the stars in the Galactic disk do not flow axisymmetrically around the Galactic center. The disk stars appears to be gravitationally influenced by the bar in the Galactic center and the spiral arms flanking the Sun.