"The Unseen Universe"

George Greenstein
Amherst College
Amherst, MA 01002


In recent years astronomers have come to realize that
the view of the universe that we get through telescopes is not
telling the whole story. Rather, in addition to all the
astronomical objects that we can observe, the universe
contains an enormous number of unseen things: objects which
we have never directly detected and, in some cases, which we
never will. Some of these objects are black holes, some are
planets orbiting nearby stars, and the nature of the rest -- the
mysterious "dark matter" -- is entirely unknown.

In this course, working with real and simulated data,
students will retrace the path whereby we have come to this
remarkable conclusion. Much of the course takes an inquiry-
based approach to learning, in which students forge their own
understanding through seminar discussions and their own
efforts. This is a first course in Astronomy; and while much of
the work will involve computers, no previous programming
experience is required.

Two class meetings per week plus computer laboratories.
First Semester. Professor Greenstein and Astronomy Education
Fellow Lovell.


CALENDAR: the semester is about 12 weeks long. The
indicated timing adds up to 10 weeks.


First class meeting: a lecture in which the instructor
talks about the philosophy behind the course, demonstrates the
software for the first exercise, and prompts a class discussion
of what can be determined from this data. This software
exhibits moons orbiting all those planets of the Solar System
which have more than one satellite -- including the Earth,
where we'll use a few of its artificial satellites -- together
with the Sun and its satellites (the planets). Only objects in
circular orbits are animated.

For each system, the software animates the satellites
orbiting over a period of time judiciously chosen such that
they all complete at least half an orbit. There is a clock
indicating time: each satellite leaves a trail, so one can see
the orbit.

The user clicks on a satellite. The program responds with
the distance to the system, and

-- the time since observations began (t=0)
-- the initial and current positions of the moon relative
to the planet

Each team of students works with a different system, and is
asked to measure for each satellite:

-- its period of orbit
-- its distance from its primary

and to graph these [using MATLAB]. They are encouraged to
make these plots both on a linear scale and a log-log scale.

Comparing notes, the students realize that the graphs are
all parallel, but with differing coefficients. It is a delicate
matter to help them figure out the power in the power law, for
they are not necessarily all that comfortable with logs.

Our method is to have them use Matlab to make lots and
lots of plots of various functions, searching by trial and error
for the one with the same graph as their observed data. They
simply try a guess for the P(R) function -- say, P = R ^ 4 --
normalize their proposed function to agree with their first
data point, and see if it fits the other data points. If it doesn't
they try something else -- P = R ^ 3 say -- and keep going till
they meet with success.

Once this is figured out, the students brainstorm about
what it could be which determines the coefficient in front of
the power law. We collect as many hypotheses as possible:
many can be shown by the students to be untenable, but many

The net result is that the students have discovered for
themselves a general regularity concerning orbits. This
motivates the next portion of the course --


which operates not so much in the discovery-mode as lectures
combined with laboratories, since we don't have much time.


This segment begins with observations of binaries in
which the orbits of both members can be seen. The critical
point here is that both stars move, as opposed to the previous
situation in which the satellites moved but the primary did
not. Only main sequence stars are illustrated. In fact, all the
exhibited systems are real, so that students are working with
real data -- they are binaries whose orbital elements have
been determined by Hipparchos, although in the software they
are seen face-on rather than tilted (we do not want the
students to spend time worrying about angles of inclination at
this stage).

We have lots of systems -- enough so that the class as a
whole can come up with a mass-luminosity relation. The
software here is closely analogous to that developed for the
satellites of planets. For each system, the software animates
the stars orbiting over a period of time judiciously chosen
such that their component stars complete more than half an
orbit. There is a clock indicating time: each star leaves a trail,
so one can see the orbit. The distance to the system is given.

The user clicks on a star. The program responds with

-- the star's luminosity
-- the time since observations began (t=0)
-- the initial and current angular positions of the star
-- the star's spectrum

Each team of students works with a different system,
and is asked to measure

-- the period of orbit
-- the linear separation between stars.

(Students might get the idea to put together all their results,
so as to assemble a plot of period versus separation for
binaries. They will see that there is no simple relationship
between them -- because period also depends on M, and this
has been left out of the analysis. Perhaps this will come as a
surprise to them and perhaps not.)


which, since we have only one week, is done in lecture mode.
The critical issue here is to teach the concept of CENTER OF
MASS. Once this is done, it is easy to re-do the previous work
to get the formulae for the velocity of each star in a binary

With this theory, students then return to the binary
system they had previously analyzed and determine the masses
of each star. Combining their results, the class as a whole
comes up with a mass-luminosity relation for main sequence


We now turn to the spectra shown in the binary star
exercise, and to the theory of the Doppler effect -- again, via
lecture format. The presentation of spectroscopy is minimal:
the theory of the Doppler effect is presented in detail, but
spectral lines are presented simply as a given. (Nothing in
what follows in the course requires students to understand the
origin of the lines.)

Once this theory is in hand, there are two projects:

(A) Students return to their previous binaries, which are
now shown edge-on rather than face-on. Consulting their
spectra, students now measure directly the stars' velocities.
They then compare these results with those obtained by simply
tracking their orbits over one period, to make sure the two

(B) Next we present images of two circumstellar disks:
one seen edge-on and the other inclined (these are actual
astronomical images of real disks). At selected data points
along the major axis spectra are available, from which
students determine Doppler velocities. From the image,
students determine the angle of inclination, and from this plus
the Doppler velocities the orbit velocity and therefore the
primary's mass.


The class splits into thirds, so that each group has many
members. Each group has a system to work on. These systems

(A) a single star wiggling back and forth
(B) a rotating ring at the center of M87
(C) a spiral galaxy

The (A) system turns out to have a planet around it, the (B)
system a giant black hole at its center, and the (C) system dark
matter. Each group analyzes its system and, at the conclusion
of the semester, reports on its results in a "conference."

(A) A Single Star Wiggling To And Fro

The data here consist of

-- Doppler velocities of a nearby star as a function of
-- the star's luminosity and distance

Students are able to leap right in, since this is so reminiscent
to the work the class did on binary stars. Unfortunately, they
very quickly come grinding to a halt, since they only have
limited data: what they easily did before, now is impossible.
The main work they face is figuring out how to get results
from limited data. They ultimately determine the unseen
companion's mass and distance from the primary, subject to an
ambiguity involving the angle of inclination of the orbit.

(B) Rotating Ring at the Center of M87

The data here is taken from HST observations of M87. It
consists of

-- an image showing the overall galaxy with its jet
-- an image of the nuclear regions, showing a blob with a
major and minor axis. The major axis is inclined
perpendicularly to the jet
-- spectra of the blob's center, left- and right-hand sides
-- the blob's total luminosity
-- the galaxy's distance

Analyzing the spectral data, students find the Doppler
velocities corresponding to the nucleus, and the left- and
right-hand edges of the blob. Because the blob is so irregular,
and because we do not know its intrinsic thickness (it may not
be paper-thin), it is difficult to measure its angle of
inclination to the line of sight. Thus there is some ambiguity in
measuring the actual rotational velocity of the blob. Thus,
what the measurements tell us is a lower limit to this
rotational velocity.

At any rate, students can now

-- determine from the orbital velocities a lower limit to
the mass of whatever lies in the blob's center
-- determine from the galaxy's distance and the central
region's angular diameter its linear diameter
-- show that whatever constitutes this central mass
cannot be composed of solar-type stars (the measured
luminosity is far too low)

They are then on their own.

(C) Spiral Galaxy

The data here consists of

-- an image of a (real) spiral galaxy
-- its total luminosity
-- five points on its major axis for which we have a
spectrum. (The spectrum is "contaminated" by the galaxy's
systematic velocity.)

From this, students find

-- the galaxy's inclination angle
-- the Doppler velocity at each data point
-- the orbital velocity at each data point
-- the galaxy's recessional velocity

Plotting the orbital velocities, students find

-- they do not show the expected fall-off with increasing
R: indeed, the rotation curve is flat
-- they imply a mass which depends on R
-- they imply a mass greater than that derived from the
galaxy's luminosity, if the stars are assumed to have one solar

From here on, students explore whatever they want to.


After the conference is over, the course ends with the
instructors giving a series of lectures in which these final
projects are put in perspective: at this point, but not before,
readings are distributed on the search for extra-solar planets,
for black holes, and dark matter.