In Fig. 5 we present plots of the total mass-to-luminosity ratios Mo / LB and Mf / LB, also listed in Table 1, versus Hubble type, colour and blue luminosity. LB has been calculated from the corrected magnitudes BTo given by De Vaucouleurs et al. (1976), and the distance; mB = 5.43. The magnitude refers to the light integrated to but it is derived using mean radial luminosity profiles and may therefore not refer to the same radius as does Mo or Mf. But the scatter thus introduced is likely to be small. We conclude from Fig. 5 that M / LB is not a function of type, colour or luminosity.
Figure 5. Mass-to-luminosity ratios as function of Hubble type (upper panels), colour (middle panels) and luminosity (lower panels). Mo is the mass within the radius of the last measured point on the rotation curve, determined with Nordsieck's method, Mf is the mass of the model fit. LB (units 109 L) has been derived from the total corrected blue-luminosity BTo given by De Vaucouleurs et al. (1976). The thin dashed line in panel c refers to a prediction from Sargent and Tinsley, 1974 (see text).
Brosche and Reinhardt (1977) have argued that M / LB is a function of type. They have calculated masses of galaxies from integral HI profiles: M = kG-1 Vmax2 Rmax, with k a constant of order unity, and Vmax the maximum rotational velocity, which occurs at radius Rmax. Such a formula implicitly assumes that the rotation curve declines past Rmax, contrary to the observations. Moreover, they use a correlation of Rmax / Ropt with type, Ropt being the Holmberg radius, noted by Brosche (1973) and Huchtmeier 17975). Thus for early type spiral galaxies they assume a rotation curve with a small value of Rmax, which thus declines in the outer parts, and for late type spirals a rotation curve which reaches its maximum in the outer parts. It is therefore not surprising that they find M / LB increasing with type. In the calculation of masses from integral HI profiles it seems better to use a constant Rmax / Ropt since for early type galaxies the measured Rmax is more related to the central component in the mass distribution, and not to the main disk.
Nordsieck (1973) reported that M / LB is correlated with colour. His sample of galaxies has been selected mainly from the list of galaxies for which Burbidge, Burbidge and Prendergast measured rotation velocities from optical spectroscopy of HII-regions. This sample of galaxies is biased towards luminous Sbc galaxies, and the radius of the last point on the rotation curve is about 5 to 10 kpc, hence the M / LB's are smaller. Omission of the only blue galaxy in his sample (NGC 4490, which forms an interacting pair with NGC 4485) more or less destroys the correlation.
We have compared our results for the dependence of M / LB on colour with predictions made by Sargent and Tinsley (1974). They have calculated simple galactic evolution models, using the Salpeter mass function with an upper mass limit mU and a lower mass limit mL; the luminosity of a star in these models varies with the fourth power of its mass m, and the lifetime of a star is proportional to m-3.08. This gives a relationship between M / L and mL. The colour B - V depends on the coefficient of the mass function. The dashed line in Fig. 5c gives the predicted relation for mL = 0.08 M. Since most of the observed values lie above this line this appears to imply that the contribution of low mass stars to the mass function must be higher than given by a Salpeter function with a cut-off of mL = 0.08 M. Perhaps mL has to be lowered, or the mass function has a coefficient different than 2.35. It could even be that the entire model approach is simply wrong. Nevertheless, the fair amount of scatter in the diagrams in Fig. 3 suggests that there are different mass functions in different galaxies.
For the mean M / LB we find < log Mo / LB > = 0.77 ± 0.32 , and < log Mf / LB > = 1.04 ± 0.33. In Fig. 6 we present the variation of the mass-to-light ratio with radius for those galaxies for which we could find photometric data. The photometric data are not very homogeneous: high quality photometry exists only for a few galaxies. We did not correct for galactic absorption except for IC 342: most galaxies are at high latitude, and the correction for internal absorption is difficult to estimate. Nevertheless, there is a clear trend that M / L, increases with radius, although large differences among galaxies are found. The recent, preliminary, data on M / LJ by Bosma and Van der Kruit (1978) shown in Fig. 6b confirm the trend already present in the data on M / LB shown in Fig. 6a. Note that large M / L ratios occur in the outer parts, where the optical light is below the sky-level and the mass models are uncertain. We estimate that there is a factor of 3 uncertainty in the M / L ratios at about 20 to 30 kpc.
Figure 6. Radial variation of the mass-to-luminosity ratio in various galaxies. The luminosity data are not corrected for internal or galactic absorption, except IC 342 (AB = 2.2m). The photometric data in the upper panel have been taken from the literature: M31 (De Vaucouleurs, 1958); M33 (De Vaucouleurs, 1959); M81 (Schweizer, 1976 - the dashed line refers to a Toomre disk model, the dotted line to an exponential model for the mass distribution cf. Visser, 1978); M101 (Okamura et al., 1976); NGC 2403 (Okamura et al., 1977); NGC 4151 (Simkin, 1975 and priv. comm.); IC 342 (Ables, 1971). Data in the lower panel are preliminary results from Bosma and Van der Kruit (1978).
Having found evidence for an increase in M / L in the outer parts we might ask the question: what is this dark material and is it distributed in a halo or not. The nature of the dark material is unclear but red M-dwarfs could produce such high M / L ratios. The increase in M / L not only occurs in galaxies which are warped, but also in galaxies with oval distortions (NGC 4151, NGC 4258). If all the dark material was distributed in a halo ten times as massive as the disk we would expect the disk to be stable against bar instabilities; the increase in M / L in galaxies with oval distortions is therefore puzzling in this picture. The alternative extreme - a massive disk - presents more difficulties for explaining the existence of warps in relatively isolated galaxies. Note that although warps and ovals occur frequently they do not occur in all spiral galaxies; it is possible that more than one explanation is required to account for invisible mass, warps and ovals at the same time.
Our results on < Mo / LB > and < Mf / LB > can also be compared with the results from binary galaxy statistics. Turner (1976) finds from his sample of binaries a value of < M / L > of 65. Corrected to Ho = 75 km s-1 Mpc-1 this value becomes 97.5. Recent work by Van Albada and Freeman (1978) indicates that this high value must be lowered to < M / LB > 13.5 for pure spiral pairs, in good agreement with our value for < Mf / LB >. Turner and Ostriker's (1977) recent value from the same sample, but luminosity weighted instead of number weighted, is 150. The mean luminosity in Turner's sample is about LB = 2 × 1010 L, from Fig. 2a we find then a value for Vm of 180 km s-1. Since < M / L > is independent of LB in our sample, we expect for a galaxy of this, luminosity that its rotation curve is flat to 0.3 Mpc if we have to reach a < M / LB> of 100 at that radius, which means that Turner's value can only be reached with a rotation curve that is still using beyond 20 to 30 kpc. If we have to reach a < M / LB > of 13.5 at 0.3 Mpc, then the rotation curve may decline according to a Toomre n = 1 disk after about 1.5 - 2 times the optical radius as given by De Vaucouleurs et al. (1976). Note that in both cases we must have an increase in the mass-to-luminosity ratio in the outer parts, and that massive haloes are not excluded. From Van Albada and Freeman's value of M / L the halo mass may still be 1/2 to 2 times the disk mass. On the other hand one of the attractive features of a super massive halo is that the high M / L ratios in clusters of galaxies can be explained by having most of the (unseen) mass in galaxies. High sensitivity 21-cm line observations, and deep surface photometry of a few regular galaxies, like NGC 2403 or NGC 3198, out to a radius of 5 Ropt may distinguish between the possibility that M / LB is "normal" (10 to 20) or extremely high (~ 100).