Earth and sun

The masses of the sun and earth may be determined from their mutual motions about their center of mass (barycenter). In general for elliptical orbits, the analysis is relatively involved. But if the orbits are circular and if one of the two objects is much more massive than the other, M > m, it is quite straightforward to find the mass of the heavier object. In this limiting case, the mass M will be essentially at rest while mass m orbits it.

Write the radial component of Newton's second law,

Substitute for Fr the radial component of the gravitational force on mass m, and substitute for ar the inward acceleration, — œ2r, of a body (mass m) in circular motion at radius r with angular velocity œ (rad/s),

Solve for M, a>2r3 4^2r3 4^2a3 M = —— = —^r ^ 2 (kg; mass of central (9.7)

The third term is obtained from the relation between the angular velocity m and the period P of the orbit, m = 2^/P. If the orbit is elliptical, the semimajor axis a takes the place of r (not proven here).

This relation (7) yields, for a circular orbit and M > m, the mass of the more massive object M in terms of the radius of the orbit and the angular velocity of the less massive object, or in terms of the radius and orbital period. In this limiting case (M > m), the result does not depend on the mass m! Consider the earth-sun system. These assumptions are quite valid because the sun is about 300 000 times more massive than the earth, and the earth's elliptical orbit has a distance of closest approach to the sun (perihelion) only 3% less than the distance at the farthest point (aphelion). Thus (7) yields the solar mass if its distance is known (see below).

Similarly, the mass of the earth can be obtained from the moon's motion. In this case, our assumptions are somewhat less valid but still adequate. The ratio of the masses is 81, and the closest and farthest approaches of the moon, perigee and apogee respectively, differ by 12% (eccentricity 0.0549).

Moon

The mass of the moon is obtained from measurements of the monthly oscillation of the earth as it orbits the barycenter of the earth-moon system every lunar month. The oscillation is revealed by observations of the motions of the planets against the background stars, especially Mars. (This is due to the phenomenon of parallax.) The oscillation indicates that the barycenter is 4700 km from the earth center; compare to the 6400 km radius of the earth. The known distance to the moon leads, by subtraction, to the barycenter-moon distance. This turns out to be 81.3 times greater than the 4700-km earth-barycenter distance. Accordingly, from the definition of the barycenter, the moon is known to be 81.3 times less massive than the earth.

258 9 Properties and distances of celestial objects Spiral galaxies and the Galaxy

The mass of the (MW) Galaxy and the masses of other spiral galaxies may be estimated in a manner similar to that used for the earth and sun. Stars in a spiral galaxy rotate in a more or less organized manner about the center at some radius with a corresponding period of rotation. Individual stars can not be resolved except for very bright stars in close galaxies, and the rotational motion of the Galaxy is much too small to be observed directly.

Nevertheless, the orbital speed v of the aggregate of stars in some small region of the galaxy can be obtained from Doppler shifts of the spectral lines from the constituent stars if the inclination (tilt) of the galaxy relative to the line of sight is taken into account. The radius r of the orbit is obtained from the angular displacement of the region from the galaxy center, if the distance from the earth to the galaxy is independently known. The speed and the radius yield the orbital period P . In the crude approximation that most of the mass is at the center of the galaxy, the expression (7) yields the mass.

In fact, a substantial portion of the mass of a galaxy is distributed throughout the galaxy (e.g., the stars themselves). Thus the rotation of the stars observed gives the mass of the galaxy within the orbit of the test star observed. The formula (7) is technically valid only if the galaxy is spherical and if the force is strictly an r-2 force. These conditions are not met in the typical case because of the disk-like shape of many galaxies and because of the continuous distribution of matter.

Nevertheless, given all the above, one can obtain an estimate of the mass within the orbital radius with (7) if one assumes a perfectly circular orbit that lies in the plane of the Galaxy. Take the speed of our star, the sun, around the center of the Galaxy, Vq = 220 km/s, as determined by spectroscopic Doppler-shift observations of distant galaxies. The distance of the sun from the galactic center is —25 000 LY = 2.4x 1020 m. The result, from (7), for the galactic mass is —1 x 1011 Mq for the matter in the disk within the solar radius. Stellar and gas motions at greater radii indicate that all the mass out to about 50 kLY radius is —3.5 x 1011 Mq. Out to 100 kLY, it approaches 1012Mq . Keep in mind that these mass values were obtained solely from the motions of stars in their orbits.

Comparison of these masses to the luminosity of the Galaxy, —1.4 x 1010Lq, tells us that much of the matter in the Galaxy has less luminosity per kg than the sun. Indeed, the many stars that are less massive than the sun are less luminous per kilogram than the sun.

The amount of emitted light in our and other galaxies is far too little to explain all the gravitation indicated by stellar and gas motions. There are just not enough stars, even including dead remnants of normal stars (white dwarfs, neutron stars, or black holes), to explain the orbital speeds. The unknown non-emitting material giving rise to this excess gravity is called dark matter, as noted previously in Section 4.2.

The masses of galaxies without organized orbital motion can not be obtained in this manner because individual stars can not be resolved and tracked. In these elliptical galaxies, the stars orbit the barycenter more or less in random planes and directions. The virial method described below can be used to estimate roughly their masses; the widths of spectral lines (velocity dispersion) arising from an aggregate of stars in the galaxy provide information about the galaxy mass.

The masses of the most massive galaxies can range up to >1013 solar masses; the smallest "galaxies" have ~108 M©. Dwarf galaxies can range down to less than 106M©. Smaller groupings of stars are called "clusters" (of stars); they are usually associated with galaxies. They are not to be confused with "clusters of galaxies" discussed below.

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