## 2. The luminosity law for spiral arms## 2.1. Equations and basic assumptionsIn order to determine the features of density waves in the Lin and Shu (1966) approximation, the surface density in a thin disk is represented as the sum of an unperturbed surface density , and a small perturbation , which represents the spiral pattern, When this expression is introduced into the motion equations, one finds that they admit solutions for whose real part has the form: where is the surface density amplitude, and is the shape function. At any time It is then a spiral pattern, with The condition that density waves are self-sustained (that is, the surface-density response is locally consistent with that required to support the spiral gravitational field) allows one to calculate two important relations: the dispersion relation and the equation for the density amplitude . In the standard case of a thin disk with a Schwarzschild velocity distribution, these relations are, respectively (Lin and Shu 1966, Toomre 1969, Shu 1970):
In order to obtain a relatively simple analytical expression for the contribution of spiral arms to the brightness profile of galaxies, we have to adopt some approximations simplifying the theoretical problem. The set of approximations assumed here is the following:
where is the slope of the straight rotation curve, and is the extrapolated velocity at . It is well known that, except for small
where is the constant pitch angle. Although the observed pitch angle often depends slightly on
Together with these approximations, we will also admit throughout this paper that the mass-to-luminosity ratio is a constant for each galaxy and that the hypotheses taken by the density wave theory are valid. ## 2.2. Contribution of the spiral armsAccording to Eq. (13), the epicyclic and intrinsic frequencies (Eqs. 10 and 9) are given by On the other hand, considering the values tabulated by Lin, Yuan and Shu (1969) for the reduction factor , and taking into account that we are here interested in obtaining relatively simple analytical expressions, we can approximate the function by Substituting Eq. (19) into the dispersion relation (Eq. 7), we find is the Toomre wavenumber scale. Expression (20) is formally identical to the dispersion relation of a gaseous disk (Lin and Shu 1966). Hence, to the level of approximation that we consider here, the stellar and gaseous disks are dynamically similar. This explains why we have not tried of decomposing Eq. (7) into these two components. Using Eqs. (11) and (21), and defining the Toomre stability parameter as the dispersion relation can be written as which is an algebraic equation of second order in and, hence, its solution has two branches The solution () in Eq. (24) corresponds to the Long Wave Mode (LWM), while the solution (-) corresponds to the Short Wave Mode (SWM). In order to find the functional form of the density amplitude appearing in Eq. (4), we must solve Eq. (8). To that end, we consider again Eq. (19) and we use . The solution of Eq. (8) is then which, according to Eq. (24), is positive in the LWM and negative in the SWM. Using Eqs. (23) and (26) to eliminate and in Eq. (25), we obtain For patterns with and, assuming a constant ratio for the galaxy disk, the surface brightness profile is where and are arbitrary constants, while is defined bellow. The function appearing in Eq. (29) contains the dependence of on the rotation curve and on the unperturbed disk surface density (see Eqs. 21 and 26). For an exponential unperturbed disk (Eq. 1) with an outer differential rotation given by Eq. (13), this function reduces to and being the blue mass-to-luminosity ratio and the central surface brightness of the unperturbed disk. The range of respectively. Consequently, we will consider that
is zero for any value of © European Southern Observatory (ESO) 1997 Online publication: July 3, 1998 |