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Astrophysics 405 Homework Assignment 4
Tully­Fisher Galaxies in the Expanding Universe
Due 1999 April 15
This assignment is a ``dry lab'' simulation of a galaxy observation project. It provides real
data from the research literature, upon which you will perform measurements and analysis.
In place of telescopes and detectors, your tools will be a ruler, pencil, and calculator.
Images, spectra, and apparent magnitudes mB for a collection of nearby galaxies have been
provided. The images are scanned R­band photographs extracted from the Digital Sky Sur­
vey database; larger versions are available at www.ras.ucalgary.ca/ # gibson/asph405.
The spectra are neutral hydrogen 21cm emission line profiles obtained with the Green Bank
300­ft. and 140­ft. radio telescopes by Brent Tully and Rick Fisher for their original research.
Some spectra contain foreground Milky Way Hi emission or absorption near 0 km s -1 , which
should be ignored.
1. For each galaxy, carry out the following steps. Report your results in a table for the
entire set, supplementing with explanatory notes on calculations where appropriate.
Also summarize and explain the amount of error you feel is associated with each result.
(a) Measure the mean radial velocity of the galaxy.
(b) Measure the apparent minor/major axial ratio #/#; this is the complement of the
ellipticity used in the Hubble classification scheme (Equation [23.1] in the text).
(c) The apparent magnitude mB needs to be corrected for dust extinction to accu­
rately represent the luminosity of the galaxy. This extinction is the sum of two
components: A g from foreground Milky Way dust, and A i from dust internal to
the target galaxy. Estimate each component:
i. # 0.4 B mag at the Galactic poles; for this exercise, assume an infinite,
plane­parallel Galactic ``atmosphere'' to find A g as a function of Galactic
latitude b. This approximation has two main flaws: (1) the distribution of
dust in the ISM is lumpy, not smooth, and (2) the Milky Way's disk has
an outer edge. The latter is not a concern for this galaxy sample, while the
former has been glossed over for simplicity.
ii. A plane­parallel model won't work for A i , since many of the galaxies in this
set are viewed close to edge­on. Instead, use the relation
A i = 0.40 · (#/# - 1)
and the axial ratio found above. This corrects the galaxy to a face­on ori­
entation but leaves the ``one airmass'' internal extinction in place, since it is
incorporated in the standard Tully­Fisher empirical calibrations.
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(d) Assuming the galaxy has a circular disk, calculate the angle # by which the plane
of the disk is tilted with respect to the plane of the sky. Use Erik Holmberg's
formula
cos 2 # =
(#/#) 2
- r 2
0
1 - r 2
0
,
where r 0 = 0.20 is the assumed axial ratio for a system seen completely edge­on.
(e) Measure the Hi line full width at half maximum (FWHM). From this, calculate
the maximum rotational velocity V max of gas in the galaxy, correcting for the
projection e#ects (think carefully about how your angles are defined!).
(f) Calculate the absolute magnitude MB via the Tully­Fisher relation, using Equa­
tions [23.4] in the text. Where possible, choose the equation whose Hubble type
best matches that of the galaxy; assume Sa if the object appears earlier than Sa,
and Sc if later than Sc. For those galaxies you are unable to classify, assume type
Sc.
(g) Compute the T­F distance for each galaxy.
2. Make a plot of radial velocity vs. distance for the entire dataset. Do the data show
a linear relation? If deviations from linearity are present, are they within the scatter
you might expect from the uncertainties you estimated, or do they show evidence for
acceleration or deceleration? (Note that the Universe has been ``adjusted'' for this
exercise to make possible significant cosmological measurements which normally would
not be readily obtainable for such a local sample of galaxies.)
3. Calculate the Hubble constant H 0 based on these data. If there is evidence for a non­
unity acceleration/deceleration term q 0 , calculate that as well, using text Equation
[27.100], neglecting terms higher than 2nd­order in redshift. The ambitious can perform
a full # 2 ­minimization to find the best values of q 0 and H 0 , but all that is required here
is a fit which looks reasonable to the eye.
4. Assume # = 0 to
find# 0 from q 0 . What is the geometry of the Universe according to
your measurements? What does this predict for the ultimate fate of of the cosmos?
5. Use this result to estimate the age of the Universe t 0 from text Equations [27.35­37],
selecting the expression appropriate for the geometry you have found. Show how this
expression derives from the Friedmann Equation ([27.8]), i.e., carry out the parts of
Problems [27.4­7] and [27.10] in the text which apply to this geometry.
6. Are your results for H 0 , q 0
,# 0 , and t 0 consistent with currently held estimates? Would
you reasonably expect to be able to measure all of the quantities you have here with a
similar dataset for the ``real'' Universe? Why or why not?
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Galaxy Data
Use the following data in your calculations. There are 32 galaxies in this sample. Images
and Hi 21cm line spectra are given on subsequent pages. Some of the 21cm spectra are for
galaxies not included in the sample, and can be ignored.
Name Coord (l, b) mB
NGC 2366 146.42 # +28.54 # 11.49
NGC 3556 148.32 # +56.26 # 10.45
NGC 3782 154.44 # +65.96 # 13.77
NGC 3877 150.72 # +65.96 # 10.52
NGC 3893 148.15 # +65.23 # 9.67
NGC 3917 143.65 # +62.79 # 11.53
NGC 3953 142.21 # +62.59 # 9.74
NGC 3972 138.85 # +60.06 # 12.66
NGC 3992 140.09 # +61.92 # 8.68
NGC 4010 146.68 # +67.36 # 12.35
NGC 4013 151.86 # +70.09 # 11.33
NGC 4085 140.59 # +65.17 # 11.08
NGC 4088 140.33 # +65.01 # 10.52
NGC 4100 141.11 # +65.92 # 10.90
NGC 4157 138.47 # +65.41 # 11.03
NGC 4178 271.86 # +71.36 # 9.68
NGC 4183 145.39 # +71.73 # 13.44
NGC 4192 265.44 # +74.96 # 10.95
NGC 4206 270.20 # +73.55 # 12.60
NGC 4217 139.90 # +68.85 # 10.63
NGC 4236 127.41 # +47.36 # 10.05
NGC 4498 277.92 # +78.75 # 12.80
NGC 4501 282.33 # +76.51 # 9.71
NGC 4532 291.02 # +68.94 # 14.52
NGC 4535 290.07 # +70.64 # 10.19
NGC 4651 293.07 # +79.12 # 7.71
NGC 4654 295.43 # +75.89 # 9.79
NGC 4758 304.53 # +78.72 # 14.06
NGC 5204 113.50 # +58.01 # 11.02
NGC 5585 101.00 # +56.47 # 10.26
IC 769 269.74 # +72.44 # 12.00
IC 2574 140.20 # +43.60 # 8.75
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