As we have discussed above, we can only measure uncontaminated stellar disk kinematics by looking away from the central bulge region. Unfortunately .
since the luminosity of a disk drops exponentially with radius , these outer parts of disks are quite faint. We must therefore either make very long observations to obtain the requisite signal in the outer parts of galaxies, or try to decompose the complex kinematics at smaller radii into the separate disk and bulge contributions.
Figure 1 The mean rotation velocity for both gas (open circles) and stars (filled squares) in the disk galaxy NGC 488. The gaseous rotation curve is adapted from Peterson (1980a), and the stellar data are taken from Gerssen, Kuijken & Merrifield (1997). The dotted line shows the circular speed for this galaxy predicted purely from the stellar kinematics.
since the luminosity of a disk drops exponentially with radius , these outer parts of disks are quite faint. We must therefore either make very long observations to obtain the requisite signal in the outer parts of galaxies, or try to decompose the complex kinematics at smaller radii into the separate disk and bulge contributions.
Rotational motion Kinematic studies of the disks of external galaxies reveal that, in all but face-on systems 
at all radii: the stars in other disk galaxies, like those in the disk of the Milky Way (§10.3), follow approximately circular orbits with little random motion. We can therefore apply methods developed for calculating rotation curves from gas disks (§8.2.4) to the values of 
obtained from stellar spectra. Generally, the rotation curves derived from stellar observations have similar properties to those obtained from gas: the stellar rotation increases rapidly with radius near the center of the galaxy, and then flattens out to an approximately constant value. The similarity between the rotation curves from these two different populations confirms that the disk stars follow orbits similar to the circular motions of the gaseous component. Note, though, that the outer parts of disk galaxies are optically very faint, so it is not possible to measure stellar kinematics to the large radii that the HI data allow. Stellar rotation curves do not, therefore, provide any evidence for dark halos.
Figure 1 shows the rotation curves of both the gaseous and stellar components of the disk of the Sb galaxy NGC 488. The stellar kinematics are limited in extent by the faintness of the disk at large radii, and contain-
Figure 2 Stellar PV diagram for the major axis of the Sa galaxy NGC 3593 (as marked on the accompanying image). The kinematics were derived using the UGD algorithm. [DSS image from the Palomar/National Geographic Society Sky Survey, reproduced by permission; the kinematic data were obtained by M. Merrifield and K. Kuijken]
ination by the bulge component for R < 20 arcsec. The gaseous kinematics contain a gap at radii around 20 arcsec since there are no measured emission lines in this region. Where the data do overlap, it is clear that there is a small systematic offset between the two rotation curves. This offset is the asymmetric drift that we have already encountered in the solar neighborhood . It arises because stars do not follow purely circular orbits, but have some random component to their motions . As in the Milky Way, the measured asymmetric drift can be used to characterize the size of this random motion relative to the ordered streaming (see below).
... comparisons between the gaseous and stellar rotation curves of SO galaxies have turned up the unexpected result that, in a significant minority of cases, these species rotate in opposite directions. These small counter-rotating gas disks in SO galaxies could well arise from the late accretion and tidal disruption of small satellite galaxies. Rather harder to explain is the case of NGC 3626, in which as much as 
of hydrogen (atomic and ionized) counter-rotates relative to the stars across a wide radial range (Ciri, Bettoni & Galletta 1995).
The strangest counter-rotating disk phenomenon of them all, however, is exemplified by NGC 4550. Absorption features in spectra of this apparentlynormal edge-on SO galaxy were found to have a double structure (Rubin, Graham & Kenney 1992), and analysis of the galaxy's LOSVDs showed that these features arise because 50% of the stars in the disk of this galaxy follow approximately circular orbits in one direction, while the other 50% follow similar orbits in the opposite direction (Rix et al. 1992).4 A handful of similar systems have now been identified. For example, Bertola et al. (1996) found clear signs of a small counter-rotating stellar disk in the kinematics of
the Sa galaxy NGC 3593. As Figure 11.18 illustrates, this counter-rotation shows up clearly in the galaxy’s stellar PV diagram. Since there are relatively few galaxies for which spectra have been obtained of the quality necessary to produce PV diagrams, the fraction of galaxies that contain counter-rotating disks remains uncertain. However, a study by Kuijken, Fisher & Merrifield (1996) indicates that less than ~ 10% of S0 galaxies contain even modest counter-rotating components, so these objects are probably quite rare.
Random motions As discussed above, we can infer the presence of non-circular stellar motions by observing asymmetric drift. In principle, we can obtain a direct measure of the random component of the stellar motions by extracting values of 
from spectra. If we define the usual cylindrical polar coordinate system, 
aligned with the axis of symmetry of the disk, then spectra of face-on galaxies provide a measure of the 
-component of the stars’ velocity dispersion, 
, whereas observations of edge-on systems allow us to measure a combination of the 
- and 
-components, 
and 
. Of course, all that we can observe directly is the average velocity dispersion integrated along the line of sight. Further, the usual techniques for measuring kinematics do not allow for the differences that we would expect to find in stars of different ages (c.f. the Milky Way, §10.3.2), but rather they produce some weighted average kinematics depending on the choice of template star. Some care must therefore be taken when converting the observed kinematics into localized measurements of the dynamics of disks.
In practice, even line-of-sight averaged quantities are difficult to observe. Face-on disks are intrinsically very faint, and the random motions in the -direction tend to be small. It is therefore difficult to obtain spectra at sufficiently high signal-to-noise ratios for to be measurable. For edge-on systems, we have seen that the broadening of the LOSVD is dominated by the different projections of the rotational velocity that one sees along any given line of sight. In order to measure the relatively small contribution to the total broadening of the LOSVDs due to the random stellar motions in the - and -directions, we must be able to characterize accurately the total width and shape of the velocity distributions. Once again, data obtained using high throughput spectrographs on large telescopes are beginning to overcome these difficulties. Observations of a dozen or so galaxy disks with a range of orientations have allowed us to characterize the variations in the components of velocity dispersion with radius – for a summary, see Bottema (1993).
Observations of face-on galaxies show that declines exponentially with radius. This relation has a simple physical interpretation: from the equations of stellar dynamics, it is straightforward to show that, for a thin disk,
where 
is the surface density of the disk, and 
is a characteristic measure of the scale-height of the disk perpendicular to the plane .
If does not vary with radius (as observations of edge-on galaxies suggest ), then 
. We also know that the surface brightness of galaxies follows the exponential law, 
(§4.4). Therefore, if mass follows light so that 
we would expect 
to decline exponentially with radius with an e-folding length of 
. Comparisons between kinematic and photometric measurements reveal that does, indeed, decline with a scale-length of approximately twice that of the surface brightness (Bottema 1993). Note, however, that the photometric scale-lengths used in this analysis were generally based on optical measurements,... the infrared scale-lengths of disks tend to be systematically smaller. Since the infrared photometry is more likely to reflect the underlying mass distribution, the above simple analysis is not secure.
Evaluation of the other two components of the velocity dispersion is more complicated since both contribute to 
in observations of inclined galaxies, and there is the additional contribution to the broadening of the LOSVD from the projection of the mean streaming. Modeling of the observed values of from edge-on galaxies has revealed that, like , 
and 
decline approximately exponentially with radius with an e-folding length twice that of the disk surface brightness (Bottema 1993). We might therefore tentatively conclude that the ratio between the different components of random stellar motions remains approximately constant, and hence that the shape of the velocity ellipsoid in disk galaxies does not vary dramatically with radius.
Note that the inference that we have drawn as to the constant shape of the velocity ellipsoid is rather indirect. We have measured the -component of the random motions from observations of face-on galaxies, and the other two components from observations of other systems with edge-on orientations. In fact, it is possible to estimate all three components in a single a disk-dominated galaxy that has intermediate inclination 
. When we observe such a galaxy at some point R from its center along its major axis the observed dispersion is given by
The annulus of radius R within the disk cuts the apparent minor axis at a distance R cos i from the center. Here we will find a line-of-sight velocity dispersion
Thus, by measuring the variations in the line-of-sight dispersions along the principal axes, we can measure two linear combinations of 
. To close this system of equations, we can turn to the theory of stellar dynamics, which shows how dispersion components are related to one another. For example, if the epicycle approximation is valid, then Figure 3 The line-of-sight stellar velocity dispersion observed along the major and minor axes of NGC 488. The dotted lines shows a simple stellar kinematic model fitted to these data. [After Gerssen, Kuijken & Merrifield (1997)]
the radial and tangential dispersions are simply related to the circular speed, 
:
... The circular speed itself can be obtained from the rotational velocities of the stars, corrected for asymmetric drift . It is thus possible to solve for 
(R) and
just from the observed mean velocity along the major axis, 
), combined with the line-of-sight dispersions
The first application of this technique has been made by Gerssen, Kuijken & Merrifield (1997). Figure 3 shows the results of fitting a simple model that meets the above requirements to the observed major- and minor-axis dispersions of NGC 488 [as well as ) – see Figure 1]. A check on this model fit is provided by the predictions made for the circular speed, 
which can be compared to the observed gaseous rotation velocities – see Figure1. The results of this analysis indicate that the components of the stellar velocity ellipsoid in NGC 488 are in the approximate ratio 
Thus, the ordering of these components is the same as for the giant stars in the solar neighborhood of the Milky Way (Table 10.3), the only other place in which this ratio has been measured. More detailed comparisons between the velocity ellipsoids of disk galaxies must await observations of larger samples of these systems, and more sophisticated modeling of higher-quality data.
Figure 3 The line-of-sight stellar velocity dispersion observed along the major and minor axes of NGC 488. The dotted lines shows a simple stellar kinematic model fitted to these data. [After Gerssen, Kuijken & Merrifield (1997)]
