Astronomy 162: Professor Barbara Ryden
The Sun, for instance, a fairly typical disk star,
is orbiting with a speed
v = 220 km/sec = 0.000225 parsecs/year.
The radius of the Sun's orbit around the galactic center is
a = 8000 parsecs.
The circumference of the Sun's orbit is then
2 pi a = 50,300 parsecs.
The orbital period of the Sun thus turns out to be
P = 2 pi a / v = (50,300)/(0.000225) = 220,000,000 years.
It takes the sun 220 million years to circle once around the center of our galaxy. During the 4.6 billion years that the Sun has been in existence, it has gone around the center just over 20 times.
Each star in the disk is on a very nearly circular orbit, anchored by all the mass enclosed within its orbit, whether it's luminous or not. Thus, the amount of mass within a star's orbit can be determined from Kepler's Third Law:
M + M* = a3 / P2,
where M = mass inside star's orbit (in solar masses)
M* = mass of the star (in solar masses)
a = radius of the star's orbit (in AU)
P = orbital period of star (in years)
A few clarifying words:
In the above equation, M is the total mass in a sphere of radius a, centered on the galactic center. (The mass outside the sphere doesn't have any net effect on the star's orbit).
Since the mass M includes the mass of the supermassive black hole at the galactic center, M is guaranteed to be much much greater than M*, the mass of a single star.
For the Sun's orbit:
a = 8000 parsecs = 1.65 billion A.U.
P = 220 million years
THEREFORE (get out your calculators if you want to check these numbers), the mass inside the Sun's orbit is M = a3 / P2 = 90 billion Msun.
Ninety billion solar masses is a lot of stuff, but this just represents the mass inside the Sun's orbit. What's the TOTAL mass of our galaxy, out to its very farthest edge? Most luminous matter (stars, gas, and dust) lies within 15,000 parsecs of the galactic center. Therefore, if luminous matter were the only matter present in our galaxy, the orbital speeds of stars and gas clouds would decrease beyond 15,000 parsecs, just as the orbital speeds of planets in the Solar System decrease as you go outward from Mercury to Pluto. BUT (and this is a big but!) orbital speeds of star are constant, or actually slightly rising, as you go more than 15,000 parsecs from the galactic center. Those few lonely stars stars and gas clouds at a distance of 25,000 parsecs are zipping around at 300 kilometers per second. There must be a great deal of dark matter in the outer regions of our galaxy in order to keep these high speed stars from escaping.
The exact extent of the dark halo around our galaxy is poorly known. The high orbital speeds of globular clusters indicate that the dark halo may extend as far as 200,000 parsecs from the center of our galaxy (that's nearly a third of the distance to our neighbor, the Andromeda Galaxy). The total mass of our galaxy, in that case, is 1 Trillion solar masses, of which 90 percent is dark rather than luminous.
What you see is more than what you get!
Neutrinos are elementary particles. They snub other elementary particles such as electrons, neutrons, and protons, very rarely interacting with them. Neutrinos also snub photons, very rarely absorbing, scattering, or emitting them. In other words, since they rarely emit light, neutrinos are dark matter. The main drawback to neutrinos as dark matter is that an individual neutrino has very little mass. The exact mass of a neutrino is so tiny it hasn't yet been measured accurately. However, the upper limits on neutrino mass tell us that it takes at least 4 billion neutrinos to equal the mass of a single proton. Neutrinos partially make up for their low mass by the fact that they are extremely numerous; in total, neutrinos may contribute a few percent of the dark matter.
MACHO is a (rather contrived) acronym for MAssive Compact Halo Object. MACHOs are dim, dense objects with masses comparable to, or somewhat smaller than, the Sun. For example, brown dwarfs, if they exist in the halo, would qualify as MACHOs, as would old cold white dwarfs, and isolated black holes. MACHOs can be detected because they act as gravitational lenses, briefly amplifying light from distant stars as they pass in front of them. Obsessive-compulsive astronomers have carefully monitored the apparent brightness of millions of stars in the Magellanic Clouds, waiting for MACHOs to pass in front of them. The result of their watching and waiting is an estimate of the number of MACHOs in the halo. The verdict: about half the dark matter in the halo is made of MACHOs.
WIMP is an acronym for Weakly Interacting Massive Particle. (The name MACHO was, in fact, first proposed as a humorous riposte to the name WIMP.) WIMPs are elementary particles proposed by the theory of particle physics. They have not, however, been seen yet in laboratory experiments, so their existence should still be regarded as hypothetical. WIMPs resemble neutrinos, in that they rarely interact with other particles, including photons. Their main difference from neutrinos is that, as their name implies, they are massive. One WIMP is equal in mass to as much as 10,000 protons (or 40 trillion neutrinos). WIMPs are thought to contribute about half the dark matter. (If the non-MACHO half of the dark matter doesn't consist of WIMPs, then it must be made of something even stranger!)
Thus, the question ``WIMP or MACHO?'' probably has the answer, ``Some of each''.
Prof. Barbara Ryden (firstname.lastname@example.org)
Updated: 2003 Feb 19
Copyright © 2003, Barbara Ryden