Most galaxies in the Local Group are dwarf systems, fainter by factors of 100 to more than 10,000 than giant galaxies. These insignificant objects nonetheless provide a strong constraint on the nature of dark matter.
While most of the stars in the Universe reside in giant galaxies like the Milky Way, numerically the most common galaxies are dwarf systems hundreds to many thousands of times fainter. We can only detect such dim galaxies if they are relatively nearby; the local group, for example, contains about two dozen low luminosity galaxies (Hodge 1995).
Luminosity and gas content are key properties of dwarf galaxies. Dwarf ellipticals (dE) are about 100 times fainter than giant galaxies; these gas poor systems typically rotate slowly. Their luminosity profiles are often better fit by exponential laws than by the de Vaucouleurs profiles which fit giant ellipticals. At the other extreme of a simplified low-luminosity Hubble sequence are dwarf spiral (Sm) and irregular (Irr) galaxies with similar luminosities; these systems are gas rich and rotate rapidly. The smallest systems, dwarf spheroidals (dS), are another factor of 100 fainter; they are barely visible against the stellar background of the Milky Way. Below is a summary of these properties:
M < -19: E -- S0 -- Sa -- Sb -- Sc Sd Sm M < -14: dE -- dS0 Irr . . . M < -9: dS ....................... Irr gas poor --------------> gas rich
Three of the four low-luminosity Sd & Sm spiral galaxies studied by Carignan & Freeman (1985) have rotation curves which rise gently out to the last point measured. While a pure-disk mass model fits the declining rotation curve of NGC 7793, massive halos with central densities of about 0.003 M_sun/pc^3 are apparently required in NGC 247, NGC 300, & NGC 3109. Within their Holmberg radii, all three of these galaxies have halo-to-luminous mass ratios of order unity, as does the fainter spiral UGC 2259. In this respect these galaxies are similar to giant Sc galaxies.
Further studies indicate that while some dwarf spirals appear to be scaled-down versions of giant spirals, others are completely dominated by their dark halos. Deeper 21-cm observations of NGC 7793 imply that this galaxy has a massive halo after all, and yield central halo densities an order of magnitude higher than those reported above (Carignan & Puche 1990). Still more impressive is the faint dwarf galaxy DDO 154, where 21-cm observations reveal a very regular gas disk extending beyond 5 Holmberg radii (Carignan & Beaulieu 1989). In this galaxy the stars and neutral hydrogen together amount to only about 10% of the mass required to explain the rotation curve; the other 90% of some 4*10^9 M_sun total mass is dark. The central halo density is about 0.015 M_sun/pc^3.
As already noted for giant galaxies, there is little direct information on the shape of the dark matter distribution. The halo densities quoted above are generally derived from models based on isothermal sphere. An alternate interpretation is that the dark mass is associated with the neutral hydrogen (Freeman 1993). For example, scaling up the HI mass of DDO 154 by a factor of about 7 yields a good fit to the rotation curve without invoking an additional halo (Carignan & Beaulieu 1989).
Dwarf spheroidal galaxies are one to two orders of magnitude fainter than the spiral systems just discussed. The mere presence of such galaxies in the vicinity of the Milky Way is evidence for dark matter; visible stars don't provide enough mass to hold these diffuse objects together against the tides of our galaxy (Faber & Lin 1983). It's impossible to measure rotation curves or integrated velocity dispersions for such faint galaxies; instead, line-of-sight velocities must be measured for individual stars (Aaronson 1983).
The core radius r_c and central 1-D velocity dispersion sigma_0 together provide a dynamical estimate of the central mass density rho_0. In a constant-density sphere the gravitational potential well is parabolic:
2pi 2 (1) Phi(r) = Phi_0 + --- G rho_0 r , 3where Phi_0 is the potential at r = 0 (see Lecture 7). The core radius is roughly where Phi(r_c) - Phi_0 = sigma_0^2; solving for the central density yields
3 sigma_0^2 (2) rho_0 = ------------ . 2 pi G r_c^2A more accurate derivation, based on the isothermal sphere, yields
9 sigma_0^2 (3) rho_0 = ------------ 4 pi G r_c^2(BT87, Eq. 4-124b). Thus from the core radius and central velocity dispersion one can directly infer the total central mass density. Comparing this with the observed central luminosity density gives an estimate for the central mass-to-light ratio, (M/L).
Results for a number of dwarf spheroidal galaxies are listed in the review articles by Kormendy (1987) and Pryor (1992). V-band mass to light ratios range from 5.7 (Fornax) to 94 (Draco). The central mass densities of these systems range from 0.073 M_sun/pc^3 (Fornax) to 1.3 M_sun/pc^3 (Draco). The faintest dwarf spheroidals such as Draco (M_V = -8.9) seem to be completely dominated by dark matter; the visible stars are basically a population test of test objects moving in a potential well generated almost entirely by dark matter.
The halos of dwarf galaxies provide an important constraint on non-baryonic forms of dark matter. Cowsik & McClelland (1973) suggested that neutrinos with nonzero rest mass could provide the dark matter in clusters of galaxies. The comoving density of light neutrinos has been constant since the Universe had a temperature of roughly 1 MeV; in physical coordinates the present mean neutrino density is a few hundred per cubic centimeter. This background of low-energy neutrinos is undetectable with present technology. But if these neutrinos have rest masses m_nu of a few tens of eV then they dominate the mass density of the Universe; estimates of the density parameter Omega thus provide upper bounds on neutrino mass. Assuming for simplicity that all neutrino species have the same mass, this bound is
-3 2 (4) m_nu < 100 eV (T/2.7 K) (H_0/100 km/s/Mpc) Omega/g ,where T is the the microwave background temperature, H_0 is Hubble's constant, and g = g_nu_e + g_nu_mu + g_nu_tau + ... where g_nu is the number of spin states of neutrino species nu; as far as we know, only left-handed neutrinos exist (except in Deep Space Nine; see Krauss 1995). Present values for H_0 imply that m_nu < 50 eV/g in a critical-density world-model, or m_nu < 10 eV/g in an Omega = 0.2 low-density model.
Tremaine & Gunn (1979) pointed out that halos of individual objects provided a complementary limit, which on the face of it rules out stable neutral leptons with masses less than about 1 MeV. In the early Universe the neutrino momentum distribution is specified by Fermi-Dirac statistics; the number-density of neutrinos of species nu with momenta in the range p to p+dp is
3 g_nu dp (5) n_nu(p) d p = ---- ------------- , h^3 exp(p/kT) + 1where T is the temperature, k is Boltzmann's constant, and h is Planck's constant. The maximum of this function at p = 0 is just one-half the limit implied by the Uncertainty Principle. Changing variables from momentum to velocity, the corresponding maximum phase-space mass density is
-- 1 \ 4 (6) f_max = --- ) g_nu m_nu . h^3 / -- nu
The Collisionless Boltzmann Equation preserves phase-space mass density not only in already formed galaxies but even during the collapse of dark halos, so Eq. (6) is a firm upper bound to the maximum phase space density of dark matter in galactic halos. For an isothermal sphere halo model, the latter density is
rho_0 (7) f_max = ---------------------- . (2 pi sigma_0^2)^(3/2)Requiring that Eq. (7) bs less that the upper bound given by Eq. (6) thus implies a limit on the neutrino mass:
-3/4 1/4 -1/4 (8) m_nu > 106 eV (sigma_0/100 km/s) (rho_0/1 M_sun/pc^3) g ,where once again all neutrino species are assumed to have the same mass. This limit is strongest for those systems with low velocity dispersion and high central density (Tremaine & Gunn 1979). Now if the dark halos of dwarf spheroidals like Draco have the same velocity dispersions as the luminous components then the neutrino mass must be m_nu > 500 eV, contradicting the cosmological limit derived above (e.g. Lin & Faber 1983). This would appear to rule out light leptons, including neutrinos, as the dark matter in dwarf spheroidal galaxies.
Because neutrinos are such attractive and elegant dark matter candidates, people have looked hard for loopholes in this argument. Tremaine & Gunn modeled the neutrino halo with an isothermal sphere, and assumed that their velocities are comparable to the stellar velocities. If the neutrinos have a larger velocity dispersion, the limit implied by Eq. (8) is lowered. But to make halos with central densities of 1 M_sun/pc^3 out of neutrinos with cosmologically reasonable masses, the neutrino velocity dispersion would have to be at least 100 km/s; dwarf galaxies like Draco would have halos with core radii of 10 kpc and masses comparable to the mass of the entire Milky Way (Gerhard & Spergel 1992). Such high masses are quite implausible!
Halo models with anisotropic velocity distributions may also modify Tremaine & Gunn's limit. In particular, hollow-halo models with tangentially anisotropic velocities permit halos of the necessary mass to be constructed without exceeding the cosmological limit on maximum phase-space density (Ralston & Smith 1991, Madsen 1991). But such anisotropic halo models often turn out to be dynamically unstable (Barnes 1993). It thus appears that Tremaine & Gunn's limit, though somewhat model dependent, precludes the possibility that the halos of dwarf galaxies are composed of massive neutrinos.
Last modified: April 18, 1997