SOLUTIONS TO EXERCISE 12

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In each case, use the equation v = Hr  (or H = v/r) to compute the
Hubble Constant for each galaxy.  Thus,

for NGC 4471,  H = (850 km/s) / (1.8 Mpc) = 472 km/s/Mpc
for NGC 3193,  H =  1300      /  2.2        591 km/s/Mpc
NGC 6824        3440         4.2        819
NGC 7619        3800         7.3        520
NGC  384        4500         7.0        643
NGC 2563        4800         9.0        533
NGC 1277        5200        11.0        473
NGC 4853        7600        13.8        551
Leo 1          19700        32.0        616

If we try to graph this and try to get a bell curve, we just don't have
enough points to have a clear interpretation.  We can get a mean value
for H by adding up the individual measurements and divide by 10 to get

H = 5750/10 = 575 km/s/Mpc.  Or, we can notice that the measurement
for NGC 6824 is far off from the rest;
thus we can choose to exclude it, to get
H = (5750-819)/(10-1) = 548 km/s/Mpc.  Or we can try a median, and find
that the halfway point of the sequence is
between 533 and 551 and thus estimate that
H = (533+551)/2 = 542 km/s/Mpc.  Or we can make a graph by plotting the
velocity on the vertical axis and distance on
the horizontal axis for each galaxy, drawing
a straight line that best fits the data, and
then computing the slope of that line to get
H = 570 km/s/Mpc or so.  In any case, it looks like H is in that sort of
range.  So what's the error?  For a widespread data
set like this, it's acceptable to look at the range
within which two-thirds of the data points lie as the
"full width at half max" to get a "best value" and "margin
of error."  This gives, as a "final" answer (although any
answer close to this is acceptable),

H = 570 +/- 70 km/s/Mpc (margin of error is 70/570 or 12%).

If, by the way, you do a formal calculation of the standard deviation, you
get happily close to the same result.  That calculation is, however, not
required for this class.

The potential sources of systematic error are many, and include a broken
telescope or spectrograph, a mistake in the Cepheid period-luminosity relation,
and selection of galaxies that all had similar peculiar velocities.  The
telescope being too small, however, is a source of random error, not systematic,
since that leads to low signal-to-noise of each measurement.  Also, having only
a small number of galaxies also is a source of random error, because then the
"bell curve" is not well filled-out, thus leading to a less precise estimate
of the full (and half) width at half maximum.

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