Limited to 10 people
The needed data for some homogeneous sample of objects are 1) redshifts; 2) distances measured independently of the redshift; 3) some assurance that the sample really is homogeneous. In another course I have had students measure redshifts of galaxies. We found this to be experimentally quite challenging with roof-top equipment, but still possible. However, it is in fact the distances that are the most difficult to extract from accessible observations, and historically this is the value that Hubble added, not the redshifts. Hence in the project described below, we adopted the measures of redshift from the research literature and concentrated instead on the measurement of distances using techniques of direct imaging.
The telescope was mounted inside a pre-existing dome on the roof of the Kersten Physics Teaching Center on the campus of the University of Chicago. Light pollution is terrible - about 100 times the brightness of a dark sky. Some of the light is local - there are rooftop greenhouses a block away that are brilliantly lit for at least a few hours during the night. The eastern sky, looking over Lake Michigan, is substantially darker. Altogether the sky brightness is a strong function of many variables.
The telescope mount was secured to a massive metal work surface (as opposed to the default tripod). Despite this, vibrations from people walking near the telescope were obvious, and we had to adopt a "keep still" policy during exposures. In practice, our exposures were an automated series of ten 40-second snapshots, providing the flexibility to discard any images before co-addition that were affected by tracking errors, excessive vibration, clouds, etc. (Also, the short exposures provide the opportunity to measure unsaturated bright stars.)
I found that the visual limit for the 2-inch finder telescope was mag = 7.5. This is at least a couple magnitudes worse than would be possible in a dark sky, and is a serious limit to accurate pointing because so few stars can be seen, especially at the high Galactic latitudes where we were normally working. The field of view of the CCD is 8 x 12 arcminutes, which is a small target at the scale of the finder telescope, so this was a definite problem.
The solution to the problem of properly centering a faint galaxy in the 8-inch telescope was provided by a second optical system consisting of a 135-mm focal-length camera lens attached to a separate but otherwise identical CCD camera and data system. The whole lens + CCD package was piggy-backed on the Celestron telescope using a mounting bracket already there for mounting a 35-mm camera. (This arrangement was intended for another purpose: measuring the thickness of the disk of the Milky Way using star counts, but it turns out to be ideal in this other application.) The field of the lens is 2 x 3 degrees, comparable to the field visible in the finder telescope. In only a few seconds exposure, more than enough stars can be seen to identify the field. The relative orientations of the 135-mm lens field and the 8-inch telescope field were fixed. The procedure was to move the telescope such that, in the wide-field camera, the target position would be at some particular pixel location, guaranteeing that it would then appear centered in the 8-inch field of view. Calibrating the effect of the fine adjustment controls took a bit of practice; with some experience the students could center a new target in about 30 minutes, start-to-finish. In other words, on a clear night we could observe about three objects on the Humason/Hubble diagram.
The ratio of focal lengths of the two optical systems was about a factor of 15. This suggests that galaxies in the wide-field camera at v = 1000 km/sec (Virgo) should look similar to galaxies in the 8-inch telescope at v = 15,000 km/sec (Ursa Major I). While we could have made more of this feature, the filters were different and the comparison would not have been truly accurate.
The Millennium Star Atlas (1997 Sky Publishing Corp.) has a depth, scale, and other attributes that I found very useful for navigating in terms of the wide-field camera. For the 8- inch telescope exposures, I prepared finding charts covering 10 x 15 arcmin from the compressed CD-ROM version of the Palomar Sky Survey (RealSkyView).
To accommodate the piggybacked system as well, appropriate counterweights had to be made and attached. Other custom machining related to mounting the 135-mm lens on the ST-7 camera, and providing for a holder that enabled 2-inch square glass filters to be mounted in front of the 135-mm lens. We used Schott filters BG-38 and RG-650; these are complementary and give comparable count rates for both the sky and for stars of intermediate color. We used the red filter despite the worse image quality because the galaxies appeared with better contrast. As indicated above, these images were used only for pointing the telescope - the actual analysis was done on the 8-inch telescope exposures.
We used a computer lab two floors below the telescope, one Macintosh per student. These machines were networked together, so that the data on the Zip disks could be shared easily, and to enable use of a common printer.
As already mentioned, the "ccdops" program provided by SBIG for the ST-7 and other camera models has some image processing and data analysis capabilities: we used mostly dark subtraction, flat-fielding, and co-addition. ccdops has the capability to write FITS-formatted files, which are readable by the main analysis package we used, MAIA (Macintosh Astronomical Image Analysis, inexpensive share- ware authored by Tim DeBenedictis). MAIA runs best on PowerPC's, of which we had only two, and our work-around was to compress the file size for the sake of the less-capable machines. MAIA writes out text files, which can then be imported into things like Microsoft Excel for any further analysis.
While the complement of software worked for us, there are many options, of course. The SBIG product line of CCD cameras can be controlled by PC's as well, so an all-PC version of the project could be easily implemented.
The largest-redshift cluster successfully observed by the students was Corona Borealis (Abell 2065) at v = 23,000 km/sec, but it was apparent that, given adequate observing conditions, we could also have detected (if not measured accurately) the most distant Humason clusters Bootes and Ursa Major II. In total the students obtained images in about 6 clusters, with more than one pointing in the nearer clusters to get more galaxies per image. The range in distance was about a factor of 25, i.e., quite large.
Their task was to replicate the "Hubble diagram" presented in Humason (1936) using their own assignments of distances and using the redshifts obtained by others, thus re-discovering the linear nature of the redshift-distance relation.
We did not instruct the students what to do in detail, other than consider using the angular size and/or flux of the brightest cluster galaxies (as did Hubble). Measuring a number of galaxies per cluster gives some appreciation of the inherent scatter in this approach.
MAIA contains a Gaussian-fit routine that is intended for stellar photometry. In the event, most of the galaxies in our images were in fact not just ellipticals, but fairly round ellipticals. We found that the Gaussian fit routine worked very well on these galaxies. The output quantities are thus size and peak intensity, from which a flux can be easily extracted. The size measurement is expected to be relatively robust with respect to variations in the atmospheric conditions (sky brightness and transparency), while of course the flux would depend on the transparency. The field of view of the 8-inch telescope was not large enough to include stars with accurately known magnitudes, in general. In principle we could have calibrated by deliberately looking at standard stars each night, but we did not bother with this step. Assuming no change in atmospheric transparency still led to a convincingly linear Hubble diagram. The redundancy of the size and flux information allows the students, in principle, to check whether the surface brightnesses of the galaxies are similar (this is an intrinsic quantity of a galaxy that is directly measurable without knowledge of the distance, so it is a valuable check), but we did not have time in the quarter to get to this level of refinement.
The idea was briefly mentioned earlier: use the wide-field CCD camera - the one behind the 135-mm lens - to measure star counts above the plane of the Milky Way, thereby obtaining a measure of the thickness of the disk. Do this for each of the two filters (BG-38 and RG-650, roughly Johnson V and I), including a high-latitude cluster of stars like Coma Berenices or Praesepe in the field. The cluster color- magnitude diagram establishes a built-in calibration of the distances to each of the field stars, assuming they are all on the main sequence - an assumption that is justified for at least the bluer stars.
It was necessary to measure stars at least as faint as V = 12 because of needing to sample sufficiently distant volumes. At the same time, we needed to cover a fairly large area of sky to get adequate star-count statistics. These requirements oppose each other, since the large pixel size needed for the large area yield so many sky counts under each star image that it becomes the limiting factor in the signal- to-noise ratio. The focal length of 135 mm was intended to be a reasonable compromise.
Then, we obtained images of several edge-on disk galaxies with known redshifts. The Hubble Constant comes from assuming the Milky Way has a thickness that is similar to these other disks.
The students did obtain values for the angular widths of the edge-on disks, and they did produce a color-magnitude diagram for the members of the Coma Berenices star cluster, but they did not get as far as measuring the drop in stellar density with increasing height above the plane of the Milky Way, and relating that to the other galaxies.