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\centerline{\bf Astronomy 101U Homework \#4 -- Is the Universe Really Expanding?}
\bigskip
(Original conception by Bruce Balick.  Modified by Doug Ingram with
help from Woody Sullivan.  Modified again and updated by Doug Ingram
It is originally modelled on a lab using the Voyager
software for the Mac.  In that version, the students took the data
themselves using the program to guide them.)

\bigskip
\noindent {\bf I. Introduction}

	Is the tranquil universe that you observe at night really a 
universe at rest?  You have heard (or will hear) that it is not; 
other people and the text claim that it is expanding.  Any rational, 
observant person should be puzzled by this assertion.  After all, 
you have no direct sensory evidence of your own to support such an 
outlandish notion.	

	The goal of this exercise is to make you challenge any claim 
about the overall state of motion of the universe.  Let us pose the 
hypothesis "The universe expands," and ask you, on the basis of data 
you will collect, to be very skeptical of its validity.  To what 
extent can you find the weak points underlying this assertion?  Can 
you synthesize an equally tenable but quite different hypothesis or 
model consistent with the same set of data?  Incidentally, Hubble 
and others first uncovered a pattern of apparent expansion in the 
1920's.  Hubble's "law" of expansion was originally just his 
interpretation of a pattern of data; later work with far more data 
has continued to give us confidence in Hubble's Law.

\medskip
\noindent {\bf II. Before Your Begin}

	There are some concepts that you must master before tackling 
this exercise.  You might need to review some of the relevant 
material.  The items below are important for this assignment and for 
the course as a whole, so be sure you understand them!

\begin{description}
\item
1.  \underline{Luminosity} (L):  A measure of the total power, or amount of 
luminous energy (light) emitted per second by an object.  For 
example, common light bulbs produce 100 watts of power.  Compare 
luminosity to apparent brightness (B), which is a measure of the 
amount of light an observer receives per second per unit area at 
some distance from an object.

\item
2.  \underline{Inverse Square Law}:  The apparent brightness (B) of an object of 
luminosity (L) changes as the inverse square of the distance to 
the object (r):  B is proportional to L/r$^2$.

\item
3.  \underline{Magnitude} (m):  The magnitude of an object is a way of 
describing the apparent brightness of an object.  (It's a lousy 
way, but astronomers can't break the habit.)  A star is five 
magnitudes brighter than another if that star is 100 times 
brighter than the other.  Bright stars are of 1st or 2nd 
magnitude.  At a dark site, the eye can detect stars as faint as 
  6th magnitude.  A small telescope can reach 10th magnitude.  A 
large telescope with a very sensitive detector and ideal observing 
conditions can reach 28th magnitude.  

\item
4.  \underline{Standard Candle}:  Objects of nearly the same luminosity (such as 
auto headlights) are "standard candles."  You can gauge the 
distance of a standard candle by its apparent brightness (i.e., 
magnitude).  That is, headlights that appear faint are inferred to 
be far away.
{\bf For standard candles, the apparent magnitude of an object is a 
measure of its distance.  Be sure that you understand why!}

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\item
5.  \underline{Doppler shift}:  The spectrum of light from a moving object 
appears to undergo a wavelength (or frequency, or photon energy) 
shift.  Receding objects have spectra shifted to longer 
wavelengths (lower photon energies), and approaching objects have 
spectra shifted to shorter wavelengths.  The amount of shift in 
the spectrum as measured by an observer is related to the velocity 
of the emitter.  By convention, the velocity is positive 
(negative) when the emitting object is receding (approaching).  
Velocities of galaxies in your data table have been measured from 
the observed shifts of spectral lines.
\end{description}

\bigskip
\noindent {\bf III. What to Do With the Table}

On the back of the last page of this exercise, you'll find 
a large table of data for 100 galaxies.  This data is 
collected from a general sky survey that had a limiting 
magnitude for object detection of around 16.  Anything fainter 
than 16th magnitude just wasn't bright enough to be picked up, 
and many objects close to 16th magnitude were probably ``lost 
in the noise'' since they are so faint.  

		You'll be extracting velocity, magnitude and positional 
information from the large data table for a random set of 
galaxies that you choose.  Select 20 galaxies at random and 
copy down the data for these galaxies into a table of your own 
(keep this table on a separate page).  Using the small table 
at the end of this write-up, convert the magnitude to a 
distance and add this information as an extra column to your 
table.  If you have a fractional magnitude (e.g. 14.7) then 
interpolate as best you can to get the estimated distance.  
Keep in mind any errors that may be involved in your 
estimation for later.

		Bear in mind that you're looking for some sort of trend of 
galaxy velocity with distance as far out as you can see.  Once 
you're done with your table, plot velocity vs distance for 
your 20 galaxies.  This is a Hubble diagram.  Put velocity on 
the y-axis and extend the y-axis from 0 to 10,000 km/sec.  Put 
distance on the x-axis and extend the x-axis from 0 to 500 
million light years.

\underline{Below your graph, answer the following questions}:

\begin{description}
\item
(1)  Is there a trend in your graph that shows galaxy velocity 
increasing or decreasing with distance?  Be honest!

\item
(2)  Draw your best fit straight line (try to get the line to 
match the slope of the points as best you can) through your 20 
data points and measure the slope.  Recall that slope can be 
found by calculating rise over run, so to get the slope, just 
use the starting and ending point of your line.  Figure out 
what the change in y is for the line and divide that by the 
change in x.  That's the slope, and that is Hubble's constant 
(more on Hubble's constant, H, later).  If you took out the 
three points furthest to the top left, would this change the 
slope by much?  What about the three points furthest to the 
bottom right?
\end{description}

Add another 20 galaxies to your own data table, paying 
careful attention to the Magnitudes.  Try to arrange your data 
taking so that you get roughly an equal number of galaxies in 
each magnitude interval.  In the end, you'll want at least 5 
in each interval, so after you've collected data for 40 
galaxies, if you have fewer than five in an interval, add more 
galaxies to the table until each interval has at least 5.

Now draw a new Hubble diagram with your 40 selected galaxies, 
and then answer the following questions:

\medskip
\begin{description}
\item
(3)  Draw your best fit straight line through the 40 data points 
and measure the slope.  Does this differ significantly from 
the slope you measured in question (5)?  If so, why?  If not, 
why not?

\item
(4)  Again, take out the three furthest points to the top left 
and then to the bottom right.  Will this change the slope much 
in either case?  Explain whether (and why) it is good or bad 
to have the slope of your graph depend so much on these points 
at either end of your distribution.

\item
(5)  In your final graph, we took care to arrange things so that 
you would get an equal number of galaxies in each magnitude 
interval.  It might have crossed your mind that we are 
manipulating the results of this exercise by giving you 
``artificial'' instructions so that you get the results we want 
and prove our standard models correct.  Are we introducing an 
unfair bias into your analysis by giving you these instruc-
tions?  Explain (don't skip this one, it's important!).

\item
(6)  Using your velocity vs. distance graph from (3) above, you 
have drawn your best fit line through the points.  Write this 
slope down as part (a).  Next, draw another line which 
represents to you the lowest possible slope of line that is 
still plausible in light of the data you have (the best way to 
do this is to take out the 3-5 highest points and then treat 
the data set as you would normally).  Write down this slope as 
part (b).  Finally, draw a line which represents that highest 
possible slope for the line to still represent the data and 
write down this slope as part (c).

\item
(7)  What you've just measured is the Hubble constant, H!  It's 
a measure of how long the Universe has been expanding.  Thus, 
as we have seen (or will see) in lecture, H is just a measure 
of how old the Universe is.  Convert your three values for H 
into ages for the Universe (in years).  You will need some 
familiarity with scientific notation as well as the following 
numbers:

		1 million light years = 9.46 x 10$^{18}$ km
		1 billion years = 3.16 x 10$^{16}$ sec

Converting your slope (in units of km/sec over Million 
light years) is a simple process, provided we take it one step 
at a time.  First, substitude the number of kilometers in a 
million light years for the Million light years unit in the 
denominator.  Now, assuming you got a slope of 20 from your 
graph, you should have something like this:

$${\rm H} = {20\ {\rm km/sec}\over 9.46 x 10^{18}\ {\rm km}}.$$

	At this point, the kilometer units cancel out, so you're 
left with some number in units of (1/sec).  Now invert your 
value for H to get an age for the Universe, like this:

$${\rm Age} = {9.46 x 10^{18}\ {\rm sec} \over 20}$$

	Now you want to convert your answer into billions of years, 
so just divide by the number of seconds in a billion years, 
and you've got an age for the Universe in the units we want!

\item
(8)  You will probably notice that your values for the age of 
the Universe have a large potential for error.  Take a look at 
your upper and lower values for the age and compare them to 
what is typically quoted (12-18 billion years) in textbooks.  
List all possible sources of error in this homework assignment 
that you can think of (at least 4, but the more the better) 
that could explain the difference between the accuracy of our 
measurement and the accuracy one normally sees quoted.

\item
(9)  Why do you suppose the 11th-12th magnitude galaxies are 
more numerous than the (fainter) 13th-14th magnitude galaxies 
in the large data table even though galaxies grow more 
numerous as they grow fainter (further away)?

\item
(10)  If brightest galaxies are the easiest ones to find (hence, 
they wouldn't be missed in data collection for our table), why 
are there so few (brighter) 8th-9th magnitude galaxies 
relative to 11th-12th magnitude galaxies in the table?

\item
(11)  Your final graph should show with some confidence that the 
Universe is expanding based on the pattern of motion you were 
able to measure.  What do you suppose the graph would have 
looked like if the Universe were contracting instead of 
expanding?  Draw a graph as part of your answer.

\item
(12)  What if the Universe had no large-scale motion?  Again, 
draw a graph as part of your answer.
\end{description}

\noindent {\bf Appendix: Converting Magnitudes to Distances}

	If you assume that galaxies are standard candles then you can 
convert magnitudes into distances.  If your galaxy has a magnitude 
between two of the numbers on this table, estimate the distance to 
the galaxy by using interpolation (itŐs ok to be off by +/- 5 Mly on 
such large scales in this experiment since there are other, much 
larger errors involved).

\begin{tabular}{c c c c c c c c c}
Magnitude & 8 & 9 & 10 & 11 & 12 & 13 & 14 & 15 \\
Distance in Mly & 21 & 33 & 52 & 82 & 130 & 210 & 330 & 520 \\
\end{tabular}

	We use distances measured in millions of light years (Mly).  
Textbooks occasionally use Mpc (or million parsecs); 1 Mpc = 3.26 
Mly if you ever want to convert units.

\bigskip
\centerline{\bf Data on 100 Galaxies}
\begin{center}
\begin{tabular}{c c c c c c}
\underline{Galaxy name} & \underline{Radial velocity} & \underline{Magnitude} &
\underline{Galaxy name} & \underline{Radial velocity} & \underline{Magnitude} \\
 & (km/sec) & & & (km/sec) & \\
NGC 4047 & 3426 & 13.1 & NGC 5452 & 2272 & 13.3 \\
NGC 3938 & 838 & 10.9 & NGC 5144 & 3187 & 13.2 \\
NGC 5055 & 587 & 9.3 & M 51 & 565 & 9.0 \\
NGC 2880 & 1620	& 12.5 & NGC 5308 & 2132 & 12.2 \\
NGC 3512 & 1352 & 13.0 & NGC 5473 & 2156 & 12.3 \\
NGC 4395 & 307 & 10.7 & UGC 8658 & 2155 & 13.3 \\
NGC 2903 & 467 & 9.5 & M 60 & 1128 & 9.8 \\
UGC 3973 & 6650 & 13.5 & NGC 5256 & 8371 & 14.1 \\
UGC 3691 & 2102 & 13.2 & NGC 5660 & 2483 & 12.3 \\
NGC 1270 & 5048 & 14.0 & NGC 5633 & 2451 & 12.9 \\
NGC 7457 & 790 & 11.7 & NGC 5772 & 5033 & 13.9 \\
NGC 428 & 1266 & 11.9 & NGC 5529 & 2971 & 12.6 \\
NGC 622 & 5483 & 14.1 & NGC 5544 & 3292 & 13.2 \\
NGC 584 & 1868 & 11.3 & NGC 5380 & 3037 & 12.8 \\
NGC 5236 & 337 & 8.2 & NGC 5383 & 2354 & 12.1 \\
NGC 1042 & 1360 & 11.5 & NGC 5297 & 2507 & 12.2 \\
NGC 720 & 1820 & 11.1 & NGC 5377 & 1950 & 12.0 \\
NGC 1199 & 2528 & 12.5 & NGC 4736 & 329 & 8.9 \\
\end{tabular}
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\begin{tabular}{c c c c c c}
\underline{Galaxy name} & \underline{Radial velocity} & \underline{Magnitude} &
\underline{Galaxy name} & \underline{Radial velocity} & \underline{Magnitude} \\
 & (km/sec) & & & (km/sec) & \\
NGC 1179 & 1716 & 11.8 & NGC 5033 & 961 & 10.6 \\
NGC 1350 & 1649 & 11.4 & NGC 4357 & 4332 & 13.5 \\
NGC 7626 & 3638 & 12.2 & NGC 3982 & 1208 & 11.7 \\
NGC 7785 & 4019 & 12.6 & NGC 3780 & 2865 & 11.8 \\
NGC 169 & 4682 & 13.3 & NGC 3945 & 1340 & 11.5 \\
UGC 1551 & 2817 & 13.3 & NGC 3725 & 3303 & 13.6 \\
NGC 7332 & 1451 & 11.9 & UGC 6162 & 2268 & 13.7 \\
NGC 7742 & 1818 & 12.3 & NGC 3415 & 3279 & 12.8 \\
NGC 1449 & 4131 & 14.4 & NGC 5128 & 323 & 8.0 \\
NGC 1395 & 1583 & 11.3 & NGC 2445 & 3936 & 13.6 \\
NGC 3115 & 476 & 10.1 & NGC 3694 & 2250 & 13.5 \\
NGC 3078 & 2231 & 12.1 & NGC 3801 & 3174 & 13.0 \\
NGC 2924 & 4362 & 13.3 & NGC 2403 & 259 & 8.9 \\
NGC 2855 & 1660 & 12.4 & IC 529 & 2417 & 12.7 \\
NGC 5128 & 323 & 8.0 & UGC 5707 & 2816 & 13.7 \\
NGC 3885 & 1706 & 12.8 & NGC 2693 & 5006 & 12.7 \\
NGC 4936 & 3065 & 12.4 & NGC 2776 & 2643 & 12.2 \\
NGC 5328 & 4610 & 12.6 & NGC 253 & 259 & 8.0 \\
NGC 5017 & 2384 & 13.1 & NGC 2782 & 2529 & 12.1 \\
NGC 4786 & 4509 & 12.8 & NGC 2719 & 3185 & 13.7 \\
NGC 4636 & 869 & 10.5 & NGC 2415 & 3779 & 12.8 \\
NGC 5812 & 2028 & 12.2 & NGC 2424 & 3307 & 13.6 \\
NGC 5921 & 1503 & 11.5 & UGC 3730 & 2868 & 14.2 \\
NGC 4486 & 1180 & 9.6 & NGC 5457 & 388 & 8.2 \\
NGC 6758 & 3237 & 12.6 & UGC 3975 & 2639 & 13.9 \\
NGC 7144 & 2062 & 11.6 & IC 2656 & 2743 & 14.0 \\
NGC 5820 & 3443 & 12.9 & NGC 2655 & 1623 & 10.9 \\
NGC 5866 & 874 & 10.9 & NGC 2276 & 2579 & 11.9 \\
NGC 6703 & 2590 & 12.5 & NGC 2268 & 2466 & 12.2 \\
NGC 6824 & 3675 & 12.7 & NGC 6217 & 1586 & 11.9 \\
NGC 6643 & 1736 & 11.8 & NGC 6643 & 1736 & 11.8 \\
NGC 6217 & 1586 & 11.9 & NGC 4472 & 817 & 9.3 \\
\end{tabular}
\end{center}

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