George Greenstein

Astrophysicist / Educator / Writer

Sidney Dillon Professor of Astronomy Emeritus at Amherst College

Homepage -- Contact George

A course for non-science majors.


"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.

***************************************** SYLLABUS

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 cannot.

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 system.

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 stars.


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 agree.

(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 are

(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 time -- 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 luminosity.

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.