It is important to realise that stellar disks are often remarkably flat. This can be studied in edge-on systems by determining the centroid of the light distribution in the direction perpendicular to the major axis at various galactocentric distances (e.g. Sanchez-Saavedra et al. 1990, Florido et al. 1991, de Grijs 1997, chapter 5). Apart from some minor warps in the outer parts of the stellar disks, in the inner parts the systematic deviatons are very small.
We may also look at the flatness of the layers of the ISM within them, such as dustlanes. In fig. 5 I collected some images of edge-on disk galaxies. At the top are two `super-thin' galaxies; the disks are straight lines to within a few percent. The same holds for the dustlanes in NGC 4565 (allow for the curvature due to the imperfect edge-on nature) and NGC 891. In the third row the peculiar structure of NGC 5866 has no measuable deviation from a straight line, while for the Sombrero Nebula the outline of the distlane fits very accurately to an ellipse. In the bottom row, NGC 7814 (right) is straight again to wthin a few percent, but NGC 5866 is an example of a galaxy with a large warp in the dust layer.
Figure 5. Selected images of edge-on disks and dust lanes from various public Web-galleries. Top: `Superthin' galaxies IC 5249 (from the Sloan Digital Sky Survey, van der Kruit et al. 2001) and UGC 7321 (http://www.cosmo.nyu.edu/hogg/rc3/UGC_7321_irg_hard.jpg); second row: NGC 4565 (www.cfht.hawaii.edu/HawaiianStarlight/AIOM/English/2004/Images/Nov-Image2003-CFHT-Coelum.jpg) and NGC 891 (www.cfht.hawaii.edu/HawaiianStarlight/Posters/NGC891-CFHT-Cuillandre-Coelum-1999.jpg); third row: NGC 5866 (heritage.stsci.edu/ 2006/24/big.html) and M104 (heritage.stsci.edu/2003/28/big.html); bottom row: ESO 510-G013 (heritage.stsci.edu/2001/23/big.html) and NGC 7814 (www.cfht.hawaii.edu/HawaiianStarlight/English/Poster50x70-NGC7814.html).
The HI kinematics provide probably the strongest indications for flatness. In three almost completely face-on spirals (NGC 3938, 628 and 1058), van der Kruit & Shostak (1982, 1984) and Shostak & van der Kruit (1984) found that the residual velocity field after subtraction of that of the rotation field has an r.m.s. value of only 3 - 4 km/s (or a few pc per Myr) without any systematic pattern. A vertical oscillation with a similar period as that for stars in the Solar Neighborhood (107 years) or even of that of rotation around the Galactic Center (108 years) would correspond to a vertical amplitude ten to a hundred pc. The absence of such residual patterns shows that the HI layers and the stellar disks must be extraordinarily flat, except maybe in their outer regions or when they have recently been in interaction.
Recently, Matthews & Uson (2008a, 2008b) have found evidence for a pattern of corrugation in the disk of the edge-on galaxy IC 2233 with an amplitude up to 250 pc, especially in HI and young stars. IC 2233 is a rather small galaxy (radius 7 or 8 kpc and rotation velocity about 100 km/s) unlike the ones discussed in the previous paragraph.
I have indicated above that the flattening of the stellar disk hz / h is smallest for systems of late Hubble type, small rotation velocity and faint (face-on) surface brightness. It is of interest then to look more closely at systems at this extreme end of the range of flattening: `superthin' galaxies. A prime example is the galaxy UGC 7321, studied extensively by Lynn Matthews and collaborators (Banerjee et al. 2010 and references therein). The picture that appears is that this is a very low surface brightness galaxy (the face-on B-band central surface brightness is ~ 23.4 mag arcsec-2) and a scalelength of about 2 kpc, but a projected vertical scaleheight of only 150 pc. It appears to have vertical structure since there is a color gradient (bluer near the central plane) and appears to consist of two components. Its HI is warped in the outer parts, starting at the edge of the light distribution.
Another good example of a superthin galaxy is IC 5249 (Byun 1998, Abe et al. 1999, van der Kruit et al. 2001). This also is a faint surface brightness galaxy with presumably a small fraction of the mass in the luminous disk. However, the disk scaleheight is not small (0.65 kpc). It has a very long radial scalelength (17 kpc); the faint surface brightness then causes only the parts close to the plane to be easily visible against the background sky, while the long radial scalelength assures this to happen over a large range of R. Therefore it appears thin on the sky. The flattening hz / h is 0.09 (versus 0.07 for UGC 7321). The stellar velocity dispersions are similar to those in the Solar Neighborhood; disk heating must have proceeded at a pace comparable to that in the Galaxy.
The flattest galaxies on the sky have indeed very small values of hz / h. However, these two examples show that superthin galaxies share at least the properties of late type, faint face-on surface brightness and small amounts of luminous disk mass compared to that in the dark halo.
Truncations in stellar disks were first found in edge-on galaxies, where the remarkable feature was noted that the radial extent did not grow with deeper photographic exposures (van der Kruit 1979). Prime examples of this phenomenon of truncations are the large edge-on galaxies NGC 4565 and NGC 5907 (see fig. 6). The truncations appear very sharp, although of course not infinitely so. Rather sharp outer profiles are actually obtained after deprojecting near-IR observations of edge-on galaxies (e.g. Florido et al. 2006).
Figure 6. NGC 4565 and NGC 5907 at various light levels. These have been produced from images of the Sloan Digital Sky Survey, which were clipped at three different levels (top to bottom) and turned into two-bit images adn subsequnetly smoothed (see van der Kruit 2007 for an explanation of the details). Note that the disks grow significantly along the minor axes but not in radial extent.
Figure 7. Correlations of Rmax / h with scalelength h and face-on central surface brightness µofo for a sample of edge-on galaxies. The cross-hatched regions show the prediction from a collapse model as in van der Kruit (1987) and Dalcanton et al. (1997); the dotted and dashed lines show predictions from the star formation threshold model of Schaye (2004) for three different values of the disk mass (from Kregel & van der Kruit 2004, see there for details)
Due to line-of-sight integration, truncations will be more difficult to detect in face-on galaxies. The expected surface brightness at 4 scalelengths is about 26 B-mag arcsec-2 or close to sky. In face-on galaxies like NGC 628 (Shostak & van der Kruit 1984, van der Kruit 1988) an isophote map shows that the outer contours have a much smaller spacing than the inner ones. The usual analysis uses an inclination and major axis determined from kinematics (if available, otherwise this is estimated from the average shape of isophotes) and then determines an azimuthally averaged radial surface brightness profile. But this will smooth out any truncation if its radius is not exactly constant with azimuthal angle. The effects are nicely illustrated in the study of NGC 5923 (Pohlen et al. 2002, their fig. 9), which has isophotes in polar coordinates. The irregular outline shows that some smoothing out will occur contrary to observations in edge-on systems.
Pohlen & Trujillo (2006) studied a sample of moderately inclined systems through ellipse-fitting of isophotes in SDSS data. They distinguish three types of profiles: Type I: no break; Type II: downbending break; Type III: upbending break. Pohlen et al. (2007) have reported that the same types profiles occur among edge-on systems; however, of their 11 systems there were only one for each of the types I and III.
Various correlations have been reviewed in van der Kruit (2009). In general, the edge-on and face-on samples agree in the distribution of Rmax / h; however the fits in moderately inclined systems result in small values of the scalelength compared to the edge-on sample. S. Peters, R. S. de Jong and I have re-analysed the Pohlen et al. data using two approaches: (1) mimick an edge-on view by collapsing the data onto the major axis and (2) calculate a radial profile using equivalent profiles. The luminosity profiles from ellipse fitting (Pohlen et al.) and that using equivalent profiles agree well, in spite of the difference that the first assumes a position for the center and the method with equivalent profiles does not. Often the `major-axis-collapse' method shows in Types I and II truncations when seen `edge-on'. So, there are truncations in the stellar disks but less symmetric than one might expect. Finally, Type III galaxies do not show 'edge-on truncations', but invariably evidence for interaction or other disturbances of the outer parts. A prime example of a Type III profile is NGC 3310, which is a well-known case of a disturbed, probably merging galaxy (van der Kruit 1976, Kregel & Sancisi 2001).
There is a good correlation between Rmax and the rotation velocity (van der Kruit 2008). On average a galaxy like our own would have an Rmax of 15 - 25 kpc (and a scalelength of 4 - 5 kpc). Now look at NGC 300, which has no truncation even at 10 scalelengths (Bland-Hawthorn et al. 2005), so that Rmax > 14.4 kpc. In spite of that it is not outside the distribution observed in edge-on systems between Rmax and Vrot (NGC 300 has ~ 105 km/s and this would give an Rmax of 8 - 15 kpc and an h of 2 - 4 kpc ). So it has a unusually small h for its Vrot; not an unusual Rmax for its rotation! At least some of the Type I galaxies could have disks with normal truncation radii, but large Rmax / h and small h so that the truncations occur at much lower surface brightness.
I note, but cannot discuss in detail, that truncations in stellar disks and warps of HI layers are often associated, and refer to my discussion in van der Kruit (2007).
Acknowledgements Ken Freeman is an expert in many area's of astronomy, but he is in particular known for his research in that of disks of spiral galaxies, and I feel fortunate to have been able to work with him on projects related to that. Many congratulations, Ken, and thanks for all the years of friendship and stimulating collaboration.