4. ROTATION AND MASS:

We infer the masses of astronomical bodies by measuring orbital motions and using Newton's Laws. We can infer the mass of the Earth by measuring the Moon's orbit. Likewise, we can infer the mass of Jupiter by measuring the orbits of any of its moons; the mass of the Sun from the Earth's (or any other planet's) orbit; and the masses of stars in binary systems from their orbital motions about each other. By the exact same principle we can infer the mass of the Milky Way from measurements of the orbital motion of the Sun (or any other star) about its center.

The Milky Way rotates, and that rotation is nothing more than the orbits of the stars around its center. But not all stars orbit with the same period. The Sun makes one full orbit around the Milky Way in approximately 250 million years. But the Milky Way does not rotate like a rigid body (such as a phonograph record). Stars closer to the center take less than 250 million years to make one full orbit, and stars more distant from the center take more than 250 million years. This swirling motion, in which the inner parts rotate faster than the outer parts, is called differential rotation, as illustrated below.

This differential rotation is also illustrated nicely by this Java simulator of rotation velocities. Because of this differential rotation, when we look to the right of the center of the Milky Way, we would see that the stars (in orbits closer to the center of the Milky Way) are pulling away from us; but when we look to the left, we would see the stars moving toward us. (If the Milky Way rotated like a rigid body, however, the distance between any two stars would never change, and we would not be able to measure its rotation rate.)

We can measure these motions from the Doppler Shifts of spectral lines in the stars, and this is how Jan Oort first measured the rotation rate of the Milky Way. But we can measure the rotation of the Milky Way much better by observing the Doppler shifts of the 21-cm radio spectral line of interstellar hydrogen gas (which orbits the center of the Milky Way just as the stars do). That is so because the 21-cm line is not obscured by dust in the Milky Way, so we can see 21-cm emission from the interstellar hydrogen gas at distances where the stars are invisible.

The physical principle by which we can infer the mass of the Milky Way follows from Newton's laws of motion. The idea is this: for a star to remain in a circular orbit, the centrifugal force (the outward force you feel when going around a sharp curve at high velocity) must be in exact balance with the attractive force of gravity. If an object of mass m is moving in a circle of radius R with velocity V, the centrifugal force is given by

F_Cent = mV²/R.

But the gravitational attraction between a central mass, M, and an orbiting mass, m, separated by a distance, R, is given by

F_Grav = GMm/R²,

Where G is Newton's constant of universal gravitation. By equating F_Cent = F_Grav we derive the equation

M = V²R/G, (1)

which enables us to infer the mass of the central object from the velocity and radius of the orbiting body. In the solar system, the central object is the Sun. We can infer the Sun's mass from the orbital velocity (30 km/s) and radius (1 astronomical unit) of the Earth. We could also infer the mass of the Sun from Jupiter's orbit, for example. Obviously, since Jupiter is further from the Sun (11.2 astronomical units) than the Earth, its orbital velocity must be less [by a factor (11.2)^-1/2 = 0.3] so that the mass on the left hand side of equation (1) will still equal the Sun's mass. Like the Milky Way, the solar system is in differential rotation: the inner planets orbit faster than the outer planets. In exactly the same way, we can use equation (1) to infer the mass of the Milky Way from the velocity of the Sun's orbit about the center of the Milky Way. But there is one important difference: in the solar system, almost all the mass is concentrated in the Sun; but in the Milky Way, the mass is distributed in a bulge, disk, and halo, as illustrated below.

In this case, equation (1) tells us the net mass contained within a sphere of radius equal to the orbital radius of the star. (That is true because the force of gravitational attraction is only sensitive to the mass inside the orbit.) Since the mass of the Milky Way is spread out, the mass inferred from equation (1) will increase with increasing radius. This is illustrated in the two graphs below.

The solid curve on the left is called the rotation curve of the Milky Way. We can use the data of this curve in equation (1) to calculate the net mass of the Milky Way as a function of distance from the galactic center. The result is plotted in the curve on the right. This exercise gives a remarkable result: the mass of the Milky Way is several times greater than the sum of the masses of the visible stars and interstellar gas! We don't know what this matter is. We only know it is there because we can see the effects of its gravity on the orbits of the stars and gas. Since we can't see it, we call it dark matter.

Before reading on, you should experiment with the Java simulator of rotation velocities of galaxies to see how the rotation curve of a galaxy depends on its mass distribution.

DARK MATTER: MACHOS AND WIMPS

This is our first exposure to the concept of dark matter in the universe, a theme that will recur throughout the remainder of this course. As you will see, dark matter is everywhere. In fact, more than 80% of all the matter in the universe is dark matter. What is this dark matter? That is one of the outstanding scientific questions of the day. What we do know is that the dark matter is terribly important: it not only holds the Milky Way together, it controls the evolution and fate of the entire universe. This is a good time to read an overview: Dark Matter in the Universe by Vera Rubin.

One idea is that the dark matter of the Milky Way might reside in the form of planets, brown dwarf stars, or even black holes. Not knowing what they are, astronomers call them MACHOS ("MAssive Compact Halo ObjectS). Even if such an object is invisible, there's a chance that we may detect one indirectly if it happens to pass nearly directly in front of a star. In that case, the gravity due to the invisible object can focus the light of the star, making its image brighten for several days. This focusing is called gravitational microlensing. We already discussed the principle of gravitational lensing in Lesson 7. But in this case, the lensing object is not necessarily a black hole, but more likely an invisible star or planet.

It is very unlikely that a macho will pass close enough to the line of sight to a star to make the star's image brighten appreciably. Therefore, to have a reasonable chance of catching such an event, astronomers must build a special telescope camera and computer software that are capable of continually monitoring the brightness of several millions of stars. This has been a great technical challenge, but astronomers have succeeded in finding hundreds of microlensing events. An example is shown below.

Astronomers are confident that these events are due to gravitational lensing for two reasons: (1) the symmetric curve of brightening and dimming that you see in the upper right panel has the exact shape that was expected from the theory of gravitational lensing; and (2) there was no color change during the brightening event. The absence of color change rules out the possibility that the events are due to a sudden flare of the star. In fact, some stars do flare (for example, binary stars sometimes flare due to a mass transfer event), but the star's color always changes during a flare.

What are these machos? They are not normal stars. If they were, they would be bright enough to see. For a while, astronomers thought that they might be very faint red dwarf stars, but recent observations by the Hubble Space Telescope rule that out (see Search for Dark Matter). Another possibility is that they might be white dwarf stars; but it is very difficult to understand how there could be enough white dwarf stars in the halo of the Milky Way to cause as many events as have been seen. Right now, we don't know what they are.

Can the machos account for the dark matter inferred from the rotation of the Milky Way? We're not sure. According to best estimates today, if the objects responsible for the observed microlensing events are invisible massive objects in the halo of the Milky Way, their net mass is less than the mass of the dark matter by about a factor of 0.5. But these estimates still have substantial uncertainties. Astronomers are now building more powerful telescope systems that should be capable of detecting many thousands of microlensing events, and they hope that such observations will provide the crucial clues to answer these questions.

Another popular idea is that the dark matter of the Milky Way might be diffuse matter (some kind of gas). But astronomers have extremely sensitive techniques for detecting ordinary matter (made of the known elements) in gaseous form by looking for its emission or absorption at radio, infrared, optical, ultraviolet, or X-ray wavelengths, and they have searched hard to find such gas in the halo of the galaxy. Indeed, they have observed gas of ordinary matter in the galactic halo, but its density is far less than required to account for the dark matter that we know must be there. Therefore, if the dark matter is in gaseous form, it can't be ordinary matter. Perhaps it is some kind of nearly invisible matter such as neutrinos or some other kind of subatomic particle, as yet unknown. The generic term for this speculative new form of matter is WIMPS -- Weakly Interacting Massive ParticleS. Physicists have built very sensitive experiments to try to detect WIMPS, but they haven't seen anything yet.

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Last modified March 9, 2002
Copyright by Richard McCray