Milky Way Galaxy

The Distance and Mass of Sagittarius A^*

Direct, geometric measures of distance in astronomy are limited to a small number of objects, such as bodies within the Solar System, stars within several hundred parsecs, and simple stellar systems, such as resolved binary stars (visibly-separated stars as seen in a telescope). One particularly important object whose distance can be derived geometrically is Sagittarius A^* (Sgr A^*), the massive black hole candidate at the center of the Galaxy.^[1]

The importance of knowing the distance R₀ to Sgr A^* cannot be overstated. This one measurement sets the scale for every other measurement of the Milky Way Galaxy. Other characteristics of our Galaxy, such as the Galaxy's mass and the Sun's orbital velocity around the Galactic center, rely upon the accurate measurement of R₀.

The ability to accurately measure R₀ through geometric means is a recent development, courtesy of advances in infrared astronomy. New instruments in infrared astronomy take advantage of adaptive optics to create high-resolution images of the galactic center. The optics of these instruments remove a star's twinkling, allowing astronomers to photograph the stars at the Galactic center with a resolve of approximately 0.025".^[2] This corresponds to a spatial resolution of less than 200 AU. With these instruments, the orbits of stars around Sgr A^* can be precisely measured. Because Sgr A^* provides virtually all of the gravitational attraction at the very center of the Galaxy, the motion of nearby stars is governed solely by this object, and observation of this motion is a direct measure of the properties of Sgr A^*.

The velocities of the stars near Sgr A^* can be above 1,000 km s^-1. While these are very high velocities, they are not so high that one sees the effects of General relativity in their motions around the black hole. The stars close to Sgr A^* therefore move in Keplerian orbits; the mass of Sgr A^* can therefore be directly measure by measuring the size and period of the orbit of a single nearby star.

The problem with applying this in astronomy is that we do not directly measure the size of the orbits of distant stars; instead, we measure the proper motions of the stars, which is a measurement of angle rather than length. If our information were limited to only the proper motion and orbital period of the stars orbiting Sgr A^*, then we could only derive the ratio of the mass of Sgr A^* to the cube of the distance R₀. We have, however, one more piece of information: a Doppler shift for each star as it moves toward or away from us. This additional piece of information permits us to directly calculate both the mass of Sgr A^* and the distance R₀ from Earth to Sgr A^*.

For instance, if a star were in a purely circular orbit around Sgr A^*, and if Earth were in the plane of that orbit, then the Doppler shift would give us a direct measure of the orbital velocity. The velocity V is directly related to the radius of the orbit r through the orbital period P: V = 2 r/P. Once one knows the radius of the orbit from the velocity and period, we can derive the distance R₀ by measuring the size in arc-seconds of the orbit on the sky. The mass of Sgr A^* can then be derived directly from the velocity and radius through the equation relating centrifugal force to the gravitational force: G M_bh = V²r, where G is the gravitational constant and M_bh is the mass of the black hole. In this way both the distance and the mass of Sgr A^* are found.

In practice, the stars orbiting Sgr A^* move in highly-elliptical orbits that are tilted with respect to the Sun. There remains, however, enough information in the time-dependent velocity profiles and in the shape of the orbits on the sky to derive a values for R₀ and M_bh as well as the tilt of the orbit relative to Earth, the orbit's semimajor axis, and other parameters that describe the orbit.

We can visualize why this is so by thinking about the appearance of a circular orbit that is tilted relative to us. If the obit's rotational axis were tilted by 45, for instance, the the shape of the orbit would appear elliptical to us. But Sgr A^* would sit at the very center of the ellipse, rather than at one focus, and the orbital speeds at the two maximum distances from Sgr A^* would be equal, which does not occur in a true elliptical orbit. These things would tell us that the orbit is a circle and not an ellipse. As these examples show, our knowledge that the orbits of stars around Sgr A^* are Keplerian permits us to derive their characteristics.

The parameters that describe the orbits of half a dozen stars close to Sgr A^* can be completely determined.^[2,3] These stars, which all lie within 0.5" (3700 AU) of Sgr A^*, move in highly elliptical orbits, more like the orbits of comets around the Sun than like orbits of the the planets. The orbital periods of these stars ranging from 15 to 94 years around Sgr A^*. The star with the shortest period, labeled S2 by the astronomers that study it, has already been observed through most of its orbit. It has a semimajor axis of 0.1220.0025 arc seconds (900 AU), an eccentricity of 0.8760.0072 (very elliptical), and a period of 15.240.36 years. This star approaches to within 110 AU of Sgr A^*. From this single star is derived the distance R₀ from Earth to Sgr A^*; the current best value is 7.620.32 kpc. With this distance, the mass of Sgr A^* is found to be 3.610.32 million solar masses.