The most striking feature of the rotation curves in Fig. 1 is that V(R) declines very slowly, if at all. Consequently the curves in Fig. 3, giving the mass inside radius R, do not converge to a finite value. Before we attempt to interpret these results in chapter 7 we first discuss the uncertainties in our results. We first comment on the dependence of our results on the treatment of the various types of non-circular motions:
a Spiral arm motions: we corrected the effect in M81 using
Visser's (1978)
data; no correction has been applied for the other
galaxies. We estimate that this leads to a possible error in
M(R)
of at most 20%.
b Oval distortions: we did not correct for these. On the basis of the simple model calculations described in chapter 4 the error is estimated to be about 20% at most.
c Warps: 50% of the galaxies in our sample are thought to be
warped in the outer parts. Most of them have amplitudes
(
z(Ro) / Ro)
of about 10 - 20%. We do not know whether a systematic error in
the rotation curve is made because of our assumption that the
circular velocity in a ring probes the mass distribution, even for
the case that the ring is not in the plane of the galaxy. It can
be argued that part of the radial velocities in the outer parts
consist of motions perpendicular to the plane. (z-motions). As an
example for a warped galaxy consider NGC 2841: Suppose the true
mass distribution is that of an exponential disk with the same
scalelength as that of the light. Then the turn-over point of the
rotation curve will be at a radius of about 5', and at
Ro = 14'
the difference between the observed radial velocity and the
expected radial velocity from the exponential disk will be
80 km s-1. If this difference is entirely due to z-motions
seen at an inclination of 68° the true amplitude will be about 215
km s-1
and this is uncomfortably high. Similar arguments can be made for
other warped galaxies. Also in the case of a disk with a less
rapid decrease of the rotation curve in the outer parts we always
find positive z-motions. It is very unlikely that the z-motions
should always be directed away from the main plane. The presence
of non-circular motions in the gaseous disk is more difficult to
assess, but they are unlikely to exceed 10 to 20 km/s. We estimate
that the error in
M(R)
may be about 50%, in the outer parts. This
is based on a comparison of two curves of
M(R)
derived for NGC 2841; one derived for
Ro = 14', and one for
Ro = 7' (at this
radius the warp starts). For R = 3.5' the curves agree within 15%,
near R = 7' they differ by 40%, and if we extra polate to
R = 14', the difference is 125% (i.e.
(1 -
2 /
(1/2)(1 +
2) is
1.25). Note that this is an extreme example.
d Large scale asymmetries. Regions with large scale asymmetries have been excluded, therefore we underestimate the mass of the galaxy in the outer, distorted parts. This error might be large, but the mass in the inner parts should be reasonably well determined.
e Small scale asymmetries. We believe that these are unimportant in this analysis.
In the cases where we could not determine the rotation curve
in the inner parts properly a large error in the curve of
M(R)
can be made. In chapter 4.3 we discuss this
uncertainty for NGC 5033. Omission of the optical data reduces
M(R =
0) with a factor
10. Therefore it is very important to determine the rotation
curve in the inner parts if conclusions on the distribution of
mass are to be made. The curve of M(R), however, is hardly
affected by this problem. It might be argued that the thin disk
approximation, implicit in Nordsieck's method, breaks down in the
inner parts of early type spirals. Comparison with the spheroid +
disk method shows, however, that even in the inner parts
Nordsieck's method gives an adequate result. Since we adopted an
eccentricity of 0.866 (b/a = 0.5) for the inner spheroid,
a nearly spherical nuclear bulge should be reasonably well
approximated. A correction for a bulge is difficult to make anyway: in NGC 5055
the bulge is very small, yet the increase in
M(R)
to the centre is as large as in M31 which has a large bulge. Nordsieck's
solution of this problem, which involves the measurement of the
extent of the nuclear bulge on plates, fails in view of the
"growth" of the bulge with exposure time (see NGC 7331,
chapter 4.7).