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Astron. Astrophys. 352, 129-137 (1999)

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2. Background

In an extensive study of stellar kinematics of spiral galaxies, Bottema (1993) presented measurements of the stellar velocity dispersions in the disks of twelve spiral galaxies. This first reasonably sized sample represented a fair range of morphological types and luminosities, although it was not a complete sample in a statistical sense. In each galaxy he determined a fiducial value for the velocity dispersion, namely at one photometric (B -band) scale length. He then found that this fiducial velocity dispersion correlated well with the absolute disk luminosity as well as with the maximum rotation velocity of the galaxy. His sample contained both highly inclined galaxies (where the velocities in the plane are in the line of sight) and close to face-on systems (where one measures the vertical velocity dispersion); when he forced the relations for the two classes of galaxies to coincide he found that a similar ratio between radial and vertical velocity dispersion as applicable for the solar neighbourhood was needed.

Bottema's empirical relation for velocity dispersion versus rotation velocity is

[EQUATION]

whereas for velocity dispersion versus disk luminosity it reads (in the form of absolute magnitude)

[EQUATION]

These relations can -for any galaxy for which the photometry is available or for which the rotation curve is known- be used to estimate the radial stellar velocity dispersion of the old disk stars at one photometric scale length from the center. Doing this for the de Grijs sample of edge-on galaxies and estimating the vertical velocity dispersion from the vertical scale height, one can in principle determine the axis ratio of the velocity ellipsoid for this entire sample. It would appear that this is a rather uncertain procedure, since one will have to assume a mass-to-light ratio ([FORMULA]) in order to calculate the vertical velocity dispersion from the photometric parameters. We will show, however, that [FORMULA] does not enter explicitely in the formula for the ratio of velocity dispersions.

We will list our assumptions:

[FORMULA] The surface density of the disk has an exponential form as a function of galactocentric distance:

[EQUATION]

[FORMULA] The vertical distribution of density can be approximated by that of the isothermal sheet (van der Kruit & Searle 1981), but we will use instead the subsequently suggested modification (van der Kruit 1988)

[EQUATION]

A detailed investigation of the sample (de Grijs et al. 1997) shows indeed that the vertical light profiles are much closer to exponential than to the isothermal solution, although the mass density distribution most likely is less peaked than that of the light, since young populations with low velocity dispersions add significantly to the luminosity but little to the mass. Then the vertical velocity dispersion [FORMULA] can be calculated from

[EQUATION]

The usual parameter [FORMULA] used in the notation for the isothermal disk (and in de Grijs 1998) is [FORMULA]. It is important to note that this formula assumes that the old stellar disk is self-gravitating. Although this can be made acceptable for galaxies like our own at positions of a few radial scale lengths from the center (see van der Kruit & Searle 1981), it is improbable in late-type galaxies, which have significant amounts of gas in the disks, and we will need to allow for this.

[FORMULA] The mass-to-light ratio [FORMULA] is constant as a function of radius. Support for this comes from the observation by van der Kruit & Freeman (1986) and Bottema (1993) that the vertical velocity dispersion in face-on spiral galaxies falls off with a scale length about twice that of the surface brightness (but note that Gerssen et al. 1997, could not confirm this for NGC488), combined with the observed constant thickness of disks with galactocentric radius.

[FORMULA] We are not making any assumptions on the functional form of the dependence of the radial velocity dispersion or the axis ratio of the velocity ellipsoid. The observed radial stellar velocity dispersions in Bottema's sample are consistent with a drop-off [FORMULA], in which case this axis ratio would be constant with galactocentric distance. However, over the range considered the data can be fitted also with a radial dependence for the radial velocity dispersion in which the parameter Q for local stability against axisymmetric modes (Toomre 1964) is constant with radius ([FORMULA]; see van der Kruit & Freeman 1986). The definition of Toomre's (1964) parameter Q for local stability against axisymmetric modes is

[EQUATION]

Disks are stabilised at small scales through the Jeans criterion by random motions (up to the radius of the Jeans mass) and for larger scales by differential rotation. Toomre's condition states that the minimum scale for stability by differential rotation should be no larger than the Jeans radius.

[FORMULA] We assume that spiral galaxies have flat rotation curves with an amplitude [FORMULA] over all but their very central extent. This assumption implies that we may write the epicyclic frequency [FORMULA] as

[EQUATION]

where A and B are the Oort constants.

First we will look into the background of the Bottema relations (1) and (2) (see also van der Kruit 1990; Bottema 1993, 1997).

Evaluating Toomre's Q at [FORMULA] and using the expression for the epicyclic frequency above, we find

[EQUATION]

Using [FORMULA] and the total disk luminosity from [FORMULA] we get

[EQUATION]

Neither in the sample of galaxies that Bottema used to define his relations, nor in our sample of edge-on systems do we have galaxies with unusually low surface brightness. It seems therefore justified to assume that for the galaxies considered we have a reasonably constant central surface brightness (Freeman 1970; van der Kruit 1987)

[EQUATION]

where [FORMULA] stands for B -magnitudes arcsec-2. So, if [FORMULA], Q and [FORMULA] are constant between galaxies, we see that the fiducial velocity dispersion depends only on the disk luminosity and the rotation velocity.

Bottema's relation (1) can then be reconciled with Eq. (9), if we have

[EQUATION]

This is approximately the Tully-Fisher relation (Tully & Fisher 1977); not precisely, since we use the disk luminosity and not that of the galaxy as a whole (however, for late-type galaxies this would be a minor difference) 1.

So, we see that Bottema's relation (1) follows directly from Toomre's stability criterion in exponential disks with flat rotation curves as long as Eq. (11) holds. The proportionality constant in Eq. (11) can be fixed using the parameters for the Milky Way Galaxy and for NGC 891 as given in van der Kruit (1990). These two galaxies have [FORMULA] and [FORMULA] 2. This gives

[EQUATION]

and using also Eq. (10) we get

[EQUATION]

From this we find with Bottema's relation (1), that [FORMULA]. In a somewhat different, but comparable manner, Bottema (1993) has also concluded that this product is of order 5.

We now turn to the vertical velocity dispersion. Evaluating the equation for hydrostatic equilibrium (5) at galactocentric distance [FORMULA] we find

[EQUATION]

and can thus calculate the vertical velocity dispersion from

[EQUATION]

Finally we examine the ratio of the two velocity dispersions. If we eliminate [FORMULA] between Eqs. (8) and (14) we obtain

[EQUATION]

and with Eq. (1)

[EQUATION]

Note that due to the elimination of the surface density also the mass-to-light ratio has dropped out of this equation and the result is independent of any assumption on [FORMULA]. Eq. (17) translates the ratio of the two length scales to that of the corresponding velocity dispersions and the underlying physics can be summarized as follows. In the vertical direction the length scale and the velocity dispersion relate through dynamical equilibrium. In the radial direction the velocity dispersion is related to the epicyclic frequency through the local stability condition, which is proportional to the rotation velocity. The "Tully-Fisher relation" then relates this to the integrated magnitude and hence to the size and length scale of the disk.

One should be careful in the use of Eq. (17), since in practice photometric scale lengths are wavelength dependent and its derivation -and therefore the numerical constant- is valid only at one exponential surface density scale length. The purpose of presenting it here is only to show that, if the two velocity dispersions are derived in a consistent manner from Eqs. (8) and (14) -or alternatively Eqs. (9) and (15)-, the assumption used for [FORMULA] drops out in the resulting ratio.

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© European Southern Observatory (ESO) 1999

Online publication: November 23, 1999
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