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SOLUTIONS TO EXERCISE 12
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 Baade 24 11700 22.0 532 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. |