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Astron. Astrophys. 318, 741-746 (1997)

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2. The luminosity law for spiral arms

2.1. Equations and basic assumptions

In 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 [FORMULA], and a small perturbation [FORMULA], which represents the spiral pattern,

[EQUATION]

When this expression is introduced into the motion equations, one finds that they admit solutions for [FORMULA] whose real part has the form:

[EQUATION]

where [FORMULA] is the surface density amplitude, and [FORMULA] is the shape function.

At any time t, Eq. (4) represents a density distribution with a geometric shape given by

[EQUATION]

It is then a spiral pattern, with m arms, which rotates with an angular velocity [FORMULA] and whose radial wavenumber [FORMULA] is given by

[EQUATION]

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 [FORMULA]. 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):

[EQUATION]

where

[EQUATION]

G being the gravitational constant, [FORMULA] the intrinsic frequency, [FORMULA] the radial velocity dispersion of stars, [FORMULA] the epicyclic frequency, [FORMULA] the angular velocity, [FORMULA] the stability parameter, and [FORMULA] the reduction factor. This last function, [FORMULA], given by Eq. (12) and tabulated by Lin, Yuan and Shu (1969), describes how much the response to a spiral perturbation is reduced with respect to that found for a cold disk ([FORMULA]).

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:

1. In the regions dominated by the disk, the rotation curve can be approximated by a straight line. In other words, the angular velocity of the disk rotation is given by

[EQUATION]

where [FORMULA] is the slope of the straight rotation curve, and [FORMULA] is the extrapolated velocity at [FORMULA].

It is well known that, except for small r values, Eq. (13) is roughly valid for many spiral galaxies. Since the most inner regions are dominated by the bulge, deviations from Eq. (13) in the limit of small r are not important in the total profile.

2. The shape of the spiral arms corresponds to a logarithmic spiral. That is, the wavenumber and the shape function are given by:

[EQUATION]

where [FORMULA] is the constant pitch angle.

Although the observed pitch angle often depends slightly on r and, hence, the shape of the spiral arms deviates something from a logarithmic spiral, this approximation is reasonably good for most spiral galaxies (Danver 1942, Kennicut 1981).

3. Since Eq. (1) gives very good fits for the disks of S0 galaxies, we will also admit here that the unperturbed disk has a surface density profile which verifies the exponential law (1). This hypothesis will allow us to obtain a contribution of spiral arms which is coherent with the profile commonly considered for the disk component.

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 arms

According to Eq. (13), the epicyclic and intrinsic frequencies (Eqs. 10 and 9) are given by

[EQUATION]

where

[EQUATION]

On the other hand, considering the values tabulated by Lin, Yuan and Shu (1969) for the reduction factor [FORMULA], and taking into account that we are here interested in obtaining relatively simple analytical expressions, we can approximate the [FORMULA] function by

[EQUATION]

Substituting Eq. (19) into the dispersion relation (Eq. 7), we find

[EQUATION]

where

[EQUATION]

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

[EQUATION]

the dispersion relation can be written as

[EQUATION]

which is an algebraic equation of second order in [FORMULA] and, hence, its solution has two branches

[EQUATION]

The solution ([FORMULA]) 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 [FORMULA] appearing in Eq. (4), we must solve Eq. (8). To that end, we consider again Eq. (19) and we use [FORMULA]. The solution of Eq. (8) is then

[EQUATION]

Now, we define

[EQUATION]

which, according to Eq. (24), is positive in the LWM and negative in the SWM.

Using Eqs. (23) and (26) to eliminate [FORMULA] and [FORMULA] in Eq. (25), we obtain

[EQUATION]

For patterns with m logarithmic spiral arms, the wavenumber k and the shape function [FORMULA] are given by Eqs. (14) and (15). Substituting these equations into Eq. (4), we obtain the surface density profile (with fix t and [FORMULA]) of the spiral perturbations:

[EQUATION]

and, assuming a constant [FORMULA] ratio for the galaxy disk, the surface brightness profile is

[EQUATION]

where [FORMULA] and [FORMULA] are arbitrary constants, while [FORMULA] is defined bellow.

The function [FORMULA] appearing in Eq. (29) contains the dependence of [FORMULA] 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

[EQUATION]

where

[EQUATION]

and constants [FORMULA] are defined by

[EQUATION]

[FORMULA] and [FORMULA] being the blue mass-to-luminosity ratio and the central surface brightness of the unperturbed disk.

The range of r values in which Eq. (29) must be taken into account is restricted by the condition [FORMULA] and by the requirement that the wavenumber k is real. From Eqs. (31) and (24), we see that these conditions imply

[EQUATION]

respectively. Consequently, we will consider that [FORMULA] is zero for any value of r not satisfying some of conditions (33), while [FORMULA] is given by Eq. (29) if both criteria are simultaneously satisfied.

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© European Southern Observatory (ESO) 1997

Online publication: July 3, 1998
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