3. Modeling an LSB galaxy
There are many ways to construct model galaxies. For our purpose we prefer to build a galaxy model after an existing galaxy with well-determined properties. Here we choose LSB galaxy F563-1; this galaxy is a late-type LSB galaxy, representative of the field LSB galaxies found in the survey by Schombert et al. (1992). The optical properties of this galaxy are described in de Blok et al. (1995); measurements of metallicities in H II regions are described in de Blok & van der Hulst (1998a); a neutral hydrogen map and rotation curve are given in de Blok et al. (1996). Parameters such as stellar velocity dispersion, which cannot be measured directly, are set in comparison with values measured locally in the Galaxy. The current star formation rate as deduced from H imaging is taken from van den Hoek et al. (1997). For convenience these data are summarized in Table 1.
Table 1. Parameters for F563-1 (H0 = 75 km s-1 Mpc-1).
The most difficult problem we face in constructing a model is converting the measured luminosity to a stellar disk mass. This is one of the most persistent problems in analyzing the dynamics of galaxies, and, unfortunately, the present observations do not provide a unique answer for this stellar disk mass-to-light ratio . Rather than using the so-called "maximum disk" value , which is an upper limit to the possible values of , we adopt a value based on colors and velocity dispersions of . An extensive motivation for this choice is given by de Blok & McGaugh (1997). The implications of choosing a different value of for the evolution of the stellar disk will be discussed in Sect. 3.2.
3.1. The model
The stellar disk particles are distributed radially according to the (measured) surface density profile. We adopt a vertical distribution of the form (e.g. van der Kruit & Searle 1982), with a constant vertical scale height kpc The disk is truncated at 25 kpc. The luminosity yields a total stellar mass of . The velocity dispersion of the stars is fixed via the relation
where is the stellar surface density. This implies a stability parameter throughout for the disk (dotted line in Fig. 1).
Particle velocities are assigned according to the (gas) rotation curve corrected for asymmetric drift. Dispersions in directions are drawn from Gaussian distributions with dispersions of respectively, using the relations valid for the solar neighborhood
with and the orbital and epicyclic frequencies respectively.
The gas surface density distribution is modeled after the H I distribution. The total gas mass is (this is the total H I mass multiplied by 1.4 to include He); the surface density decreases almost linearly with radius out to 33 kpc. The vertical distribution is assumed to decline exponentially. The scale height of the gas layer can be calculated using
(Dopita & Ryder 1994) where km s-1 is the (adopted) gas velocity dispersion. The gas particles are assigned velocities according to the rotation curve, with isotropic dispersion .
The halo is included in the calculations as a rigid potential. This is justified since the galaxy model evolves in isolation. We will thus also ignore any contraction of the halo under the influence of the disk potential. We do not expect this effect to be important anyway as the mass of the disk (assuming ) is only 4 per cent of the measured halo mass (de Blok & McGaugh 1997). As advantages we do not have to make assumptions about halo particle orbits, and we do not have to spend time in calculating the force of the halo particles on the galaxy. We assign an isothermal density distribution to the halo,
with central volume density and core radius kpc. These are the values derived from a rotation curve decomposition assuming . This will correctly put the maximum rotation velocity at 113 km s-1.
For the simulations we use 40,000 SPH particles and 80,000 star particles initially. This corresponds to an SPH particle mass of and a star particle mass of .
3.2. Implications of for star formation history
We can show that a maximum disk value for is not a plausible option for our modeling exercise, and that in fact the "most likely" value we use in constructing models may still overestimate the true value of , making the stellar disk a really insignificant component of the whole galaxy system.
where is the star formation time scale (e.g. Guiderdoni & Rocca-Volmerange 1987 and Charlot & Bruzual 1991). This ignores any gas locked up in long-lived stars, but this effect will not be important in LSB galaxies due to their large gas-fractions. A small value for means that star formation proceeds rapidly, while a large indicates that star formation proceeds very slowly.
For HSB galaxies the star formation time scale is normally a few Gyr (Kennicutt et al. 1994). For F563-1 the measured current SFR as derived from H imaging is approximately /yr; this yields a star formation time scale of 77 Gyr. Thus star formation in this disk proceeds much slower than in HSB galaxies.
Since we know the stellar and gas mass, and hence the total disk mass of the galaxy, we can use the exponential parameterization and the known SFR to compute the "age" of the disk (i.e. the time elapsed since star formation started):
For the adopted stellar mass of F563-1 Eq. 8 yields an age for the galaxy of 37 Gyr, much larger than the age of the universe.
If the exponentially declining SFR describes the star formation history adequately and the galaxy is not in a state of unusually low star formation activity (note that the blue colors for these LSB galaxies suggest a relatively high current star formation activity), then the old implied age of the disk could mean that the mass of the disk should in reality be even less.
We show this in Fig. 2 where we present a few solutions for Eqs. 7 and 8 for various values of the SFR. If we adopt /yr as the true SFR then if the galaxy is 15 Gyr old. Higher values for would make the galaxy older than the universe. A maximum disk solution (van Albada & Sancisi 1986) for the stellar mass-to-light ratio yields , which leads to an age of 110 Gyr. A current SFR of order /yr is needed to reconcile the maximum disk with the exponentially declining SFR model, which implies that the measurements underestimate the true SFR by an order of magnitude. Thus seems highly unlikely (it would give LSB galaxies SFRs comparable to those of actively star forming late type galaxies), and therefore we reject the maximum disk as an acceptable solution for the stellar disk mass.
The adopted which we use in the galaxy model is only consistent with the age of the universe if the true SFR is of order /yr, otherwise the stellar mass is probably overestimated. However if the stellar disk is indeed less massive than our galaxy model, then any conclusions concerning the disk stability and thickness of the disk will be even stronger.
It might be argued that the exponential star formation history is not the right choice for this type of galaxy, and that other functional forms of star formation history, in combination with the measurements, will yield reasonable ages. This is not true, as any other (reasonable) star formation history will result in even larger ages for the disk. Taking for example another extreme functionality - a constant SFR of 0.05 /yr - yields an age Gyr. The only way to get a reasonable age, given the measured luminosity and assumed , is to assume that the current SFR in F563-1 (and LSB galaxies in general) has been underestimated, which is unlikely. We refer to McGaugh & de Blok (1997) for an extensive discussion on the various functionalities of the star formation history.
We therefore conclude that the disk of F563-1 must have a small stellar mass-to-light ratio.
© European Southern Observatory (ESO) 1999
Online publication: February 23, 1999