2 Determination of the MOND rotation curve

The procedure followed when determining a MOND rotation curve has been described previously (e.g., BBS). In the context of MOND, the true gravitational acceleration g, is related to the Newtonian acceleration $g_{\rm n}$ as

$$% \mu (g/a_0)g = g_{\rm n}$$ (1)

where a₀ is the acceleration parameter and $\mu(x)$ is some function which is not specified but has the asymptotic behavior

$$% \mu(x) = 1,\;\;\;\; x > 1\;\;\; {\rm and}\;\;\; \mu(x) = x,\;\;\; x < 1$$ (2)

(Milgrom 1983); a convenient function with this asymptotic behavior is

$$% \mu(x) = x(1 + x^2)^{-1/2}.$$ (3)

The circular velocity is given as usual by

$$% v = \sqrt{ r g}.$$ (4)

From Eqs. (1), (2) and (4) it is evident that the rotation curve about a finite bounded mass M in the low acceleration limit is asymptotically flat at a value given by

v⁴ = GMa₀, (5)

which forms the basis of the observed Tully-Fisher (TF) relation. The Newtonian acceleration $g_{\rm n}$ is determined, as usual, by applying the Poisson equation to the mass distribution deduced from the distribution of the observable matter (disc, bulge, and gas). The surface density distribution of the stellar disc is assumed to be traced by the distribution of visible light (i.e., no variation of M/L within a given component of a given galaxy), but then the question arises as to which photometric band is most appropriate. The near-infrared emission (e.g., $K^{\prime }$-band) is considered to be a better tracer of the old dominant stellar population, and less susceptible to position-dependent extinction, but this is not generally available. Below, we use the r-band as a tracer of the form of the mass distribution in the stellar disc, but, with respect to NGC 3198, we also consider more recent $K^{\prime }$-band photometry.

The stellar disc may be assumed to be asymptotically thin or have a finite thickness related to the radial scale length of the disc by an empirical rule (van der Kruit & Searle 1981); this makes little difference in the final result. Applying Eqs. (1), (3), and (4), a least squares fit is then made to the observed rotation curve v(r) where the single free parameter of the fit is the mass-to-light ratio of the disc; in cases where there is an indication of a bulge from the light distribution, M/L of the bulge enters as a second parameter.

For the gaseous component a surface density distribution equal to that of the H  I is taken, multiplied by a factor 1.3 to account for primordial helium. The gas layer is taken to be infinitesimally thin. The contribution of the gas to the total rotation is fixed, but does depend on the distance to the galaxy.

In principle, the parameter $a_{\rm o}$ should be universal and, having determined its magnitude, one is not allowed to adopt this as a free parameter. But as noted above, the derived value of $a_{\rm o}$ does depend upon the assumed distance scale. Sanders & Verheijen (1998) give MOND fits to the rotation curves of 30 spiral galaxies in the UMa cluster which they assume to be at 15.5 Mpc. The preferred value of $a_{\rm o}$ with this adopted distance is equal to the BBS value of $1.2\times 10^{-8}$ cm s^-2. However, based upon the Cepheid-based re-calibrated Tully-Fisher relation (Sakai et al. 2000), Tully & Pierce (2000) argue that the distance to UMa should be taken to be 18.6 Mpc. We have recalibrated the Tully-Fisher law using this same sample of galaxies but with the three test galaxies (NGC 2403, NGC 3198, NGC 7331) left out of the fitting. Within the errors, the slope and intercept of the Tully-Fisher relation are the same as that found by Sakai et al. (2000), and the distance to Ursa Major is only 1% smaller than that found Tully & Pierce (2000). In that case the MOND fits to the UMa galaxies imply that the value of $a_{\rm o}$should be adjusted to $0.9 \times 10^{-8}$ cm s^-2. This is also the preferred value of $a_{\rm o}$ from MOND fits to rotation curves of a sample of nearby dwarf galaxies with distances taken primarily from group membership (Swaters & Sanders 2002, in preparation).