*King:* One of the tightest correlations you showed was
between *R*_{max} and *h*_{R}. What does
*R*_{max} mean, and
is there any obvious reason for such a tight correlation?

*van der Kruit:*
*R*_{max} is the truncation radius.
The tightness of the correlation results from the fact that over
scalelengths in the range from 1 to 15 kpc or so, the truncation radius
occurs at 3 - 4 scalelengths. I have commented on this in the paper
above.

*Byun:* In the 1970 Freeman paper, disk galaxies were
divided into two groups, type I and type II. Type II galaxies had
exponential disks which *do not* continue into the centers. What are
our current understandings of these type II galaxies?

*van der Kruit:* Although there still are some systems
having profiles that were called type II, it appears that these no
longer make up a substantial fraction of the observed profiles. The
prime example, M83, is barred and the bar may be the cause of this
behavior, but there are also galaxies that are unbarred and display the
type II characteristic. But again it no longer is an important class in
modern surface photometry profiles.

*Fall:* My impression is that realistic angular momentum
distributions *M*(*h*) produced by hierarchical clustering do
not have the sharp
features that would lead to edges in the disks. My preferred explanation
for the edges at
*R*_{edge}
4*r*_{disk} is in terms of
local instabilities in the disks (Goldreich & Lynden-Bell and
Toomre-type instabilities
^{(2)}).
George Efstathiou and I showed these would give rise to
*R*_{edge}
4*r*_{disk}.

*van der Kruit:* I am aware of the
Fall & Efstathiou (1980)
explanation. Actually, in my third paper with Leonard Searle on edge-on
galaxies
(A&A, 110, 61, 1981)
we refer to this and find that it predicts truncations at
0.8 ± 0.2 times the observed ones for our sample.

*van der Kruit:* After the session Mike Fall suggested
that the conclusion that the maximum disk hypothesis does not apply to
most galaxies using the disk flattening may be affected by underestimating
the scaleheight due to the presence of stellar generations in the disks
with a range in vertical velocity dispersions and scaleheights. The
younger of
these are brighter and their smaller dispersions lead to a lower scaleheight
in the photometry than in the mass distribution. It is well-known that the
observed diffusion of stellar random motions gives - according to
observations in the solar neighborhood - a velocity dispersion
proportional to sqrt[age] and Mike points out that the
stellar generations therefore have a scaleheight proportional
to age. I was urged by some participants to reply to this.

According to eq. (20) a systematic underestimate of the scaleheight
*h*_{z} results in an *overestimate* in the ratio
*V*_{max}^{disk} / *V*_{max} and
allowing for it then takes the disks even
further from maximum disk. However, the effect must also
occur for the observed velocity disperions in face-on disks,
so Bottema would systematically have
underestimated the dispersions resulting in an underestimate of the
coefficient in eq. (8) and (20). I have returned to some old notes to
myself on this and updated these. In view of the potential importance of the
effect I document this here.

The luminosity of a generation of
stars as a function of age can be estimated in three ways. First look at
the luminosity of the main sequence (MS) turn-off stars. The stellar MS
luminosity is roughly proportional to *M*^{3} and MS
lifetime to
*M*^{-2}. So luminosities
*L*_{MS} are roughly proportional to
^{-3/2}. Secondly, we
may look at the giants only. I estimate from evolutionary tracks that at
the tip of the giant branch the luminosity
*L*_{Giant} is crudely
proportional to the mass on the MS (at least for stars between one and a
few solar masses; MS lifetimes of 1 to 10 Gyr), so that
*L*_{Giant}
^{-1/2}. Finally, single
burst models of photometric evolution can be
used. A nice example is illustrated in Binney & Merrifield (Galactic
Structure; Princeton Univ. Press, 1998) in Fig. 5.19 on page 318. The
(*M*/*L*)_{B} is proportional to age, so that the
luminosity of a single burst
*L*_{SB}
^{-1}. This is -as
expected- nicely in between *L*_{MS} and
*L*_{Giant}.

To estimate the effect it is most practical to look at the integrated
velocity dispersion in a face-on disk. The weighted velocity dispersion
can then be estimated by integrating over the relevant ages
in

To estimate the error this should then be compared to the case of equal weighing of the generations

We had
^{2}
and may take for late type
disks as a reasonable approximation a constant star formation rate
SFR().

In order to check the effect on the scaleheights the best approximation is
to use *L*_{SB}. The results depend on the range in
we take to
perform the integration. Since fits for
*h*_{z} are made above the
dust lanes we need to ignore the youngest generations. As examples I
take the integration from 2 and 3 to 10 Gyr ; this means ignoring
generations in which the scaleheights are
less than 0.2 and 0.3 times that of the oldest generations. The values for
sqrt[<>^{2}]
are then underestimated by
respectively 9 and 6% and the scaleheights by 17 and 11%.

For the effect on the velocity dispersions we have to take
*L*_{Giant}, since thes are measured by comparing to a
late-GIII or
early-KIII template star. The galaxy spectra refer in practice for this
mostly to interarm regions, so it
is fair to take the lower limit in the integration now at 1 or 2 Gyr.
Then
sqrt[<>^{2}]
is underestimated by
respectively 8 and 4%. Even if the lower integration limit is set to zero
(assuming red supergiants fit the luminosity class III template star),
the error is only 18%.

So, the effects are small and furthermore in an application of eq. (20)
we need to take the square root of the
scaleheight, while the two effects work in opposite directions.

^{2} The references to the papers by
Goldreich & Lynden-Bell (1965)
and Toomre (1964)
appear in Table 1 [PCvdK].
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