George Greenstein

Astrophysicist / Educator / Writer

Sidney Dillon Professor of Astronomy Emeritus at Amherst College

Homepage -- Contact George

A first course for science majors.


The Five College Astronomy Department

Instructors: George Greenstein and Amy Lovell
Amherst College, Amherst, MA

This course explores the currently unsolved mystery of dark matter in the universe using an inquiry-based approach to learning. Working with actual and simulated astronomical data, students will explore this issue both individually and in seminar discussions. The course will culminate in a "conference" in which teams present the results of their work.

SYLLABUS (Spring 1999)

CALENDAR: the semester is about 12 weeks long. The indicated timing adds up to 11 weeks.

TEXTBOOK: Frank Shu, "The Physical Universe: An Introduction to Astronomy"



"Other Suns, Other Planets?" by David Black, in Sky and Telescope August 1996 pg 20+
"New Worlds" by Geoffrey Marcy and R. Paul Butler, in Sky and Telescope March 1998 pg 30+

First class meeting: a lecture in which the instructors talk about the philosophy behind the course, demonstrate the software for the first exercise, and prompt a class discussion of what can be determined from this data. This software exhibits planets orbiting nearby stars (lots of such planetary systems -- enough for each team of students to have its own to work on). The inclinations of these systems are randomly distributed, taking care that a few systems are face-on, and a few edge-on.

For each star, the software exhibits

(i) the star's name, distance, and luminosity

(ii) an animation showing the planets orbiting over a period of "observations." The inner planets are seen to move, the outer do not. There is a clock indicating time: each planet leaves a trail, so one can see the orbit.

(iii) the user clicks on a planet, and specifies a date. The program responds with

  • the time since observations began (t=0)
  • the X, Y coordinates of planet and star
  • the spectra of the planet, showing Doppler shifts [contaminated by the V sin (i) effect]

These will be used in this segment of the course. Later sections will come back and look at --

(iv) each planet's IR brightness (from which information on the planet's temperature can be gained). Spectra of the planets show spectral features and the overall Planck function.

Using (i) - (iii), students are asked to do five things:

(A) They determine the angle of inclination of the system (ONE DAY)

(B) They determine the planets' orbital periods and explore the period distance relation. They are expected to come up with the power-law relation between period and distance. This work should take TWO DAYS

Comparing notes, the class finds that each team has come up with the same power law, but with different coefficients out front. Students are led to notice the correlation between this coefficient and the star's luminosity. They are guided to remember what they already know of Kepler's laws, orbit theory, etc.

gravity: Shu pgs 33-38
Kepler's laws: Shu pgs 463-466

(C) They are thus able to determine the stars' masses, and relate L to these derived masses. They may also be able to come up with a mass function for the stars. This work should take ONLY ONE DAY.

This work relates to future portions of the course in two ways:

-- their mass-luminosity relation will be used when they compare the dynamically determined galactic masses with those determined from the galaxies' optical luminosities

-- the mass function, which rises as we go to lower masses, suggests low-mass stars as possible candidates for the dark matter. At any rate, once they do begin to explore this possibility, their mass function will be essential.

(D) They recall what they learned of the Doppler Effect, and directly measure planetary velocities from their spectra. This work should take ONE WEEK.


Doppler effect: Shu pgs 53-60
spectral lines: Shu pgs 40-47

This technique, of course, will be used later on when they measure rotation curves of galaxies. These measurements can also be verified by comparison with velocities derived from their previous determination of the planets' orbital periods and distances from the star.

Students are asked to plot velocity versus distance from the star. These graphs

-- should also include theoretical predictions, derived from their previous determination of the star's mass

-- will be essential later on, when they do the same for galactic rotation curves: they will be surprised to see that the curve for the planetary system drops with r, but the curve for galaxies is flat.

Students also realize that such Doppler measurements are best taken when the planets are at the outermost points in their orbits, which will be useful later on when they are determining galactic rotation curves.

These measurements can also be made on the more distant planets, which do not move appreciably during the relatively short time span of the observations. In these cases, students have to deal with the V sin (i) effect, since they cannot wait to take a measurement when each planet is at the most opportune point in its orbit.

Later in the course, when teams are exploring various hypothesis of the nature of the dark matter, they will return to section (iv) of the software and will be able to do two more things:

(E) They determine each planet's temperature from its IR brightness. This requires them to recall what they know about sigma * T4, which will be used later on when they evaluate the hypothesis that the dark matter is composed of various warm things such as dust, planets, etc. This work should take ONLY ONE DAY


Blackbody radiation: Shu pgs 77-80



Galaxies in general: Shu Chapter 12

Software exhibits spiral galaxies (lots of them-- enough for each team of students to have their own to work on). The software gives for each galaxy

-- its name, distance and luminosity

-- upon clicking a cursor at any point on the galaxy, the program shows

  • the angular distance from the galactic center
  • a velocity

[These are real galaxies and the data is real, so that here too the inclinations are randomly distributed.]

Using this data, students do three things:

(A) From the luminosity, students measure the optically determined mass of their galaxy. This work should take LESS THAN ONE DAY.

-- The easy way to do this is to simply assume all stars have the same mass, one solar mass, say and trivially get the mass from L.

-- The more sophisticated way is to use their derived mass function and mass-luminosity relation.

(B) From the velocity data, students extract the systemic velocity and the rotation curve. This work should take ONE DAY.

[Because the students are working here with real data, the Hubble expansion is included in the systemic velocities.]

Plotting their rotation curves, students immediately realize that they are flat, rather than falling off as did their analogous curves for planets around stars. This is the first surprise.

(C) Students also determine the galaxies' masses from their rotation curves. (We ignore the fact that the theory whereby one goes from rotation curve to mass distribution is only valid for spherical distributions.) From this work the students find that their derived masses

-- depend on where in the galaxy you take your data -- the mass keeps increasing as you take data farther out. This is the second surprise.

-- are greater than the masses determined in (A): here is the third surprise. We have reached the dark matter problem at last!

This work should take ONE DAY.


We are now nearly half-way through the semester, and we have reached a major watershed in the course. Here is a good point for a discussion, summarizing what has happened and placing it in context. This should take ONE DAY


"Vera Rubin: An Unconventional Career" by Sally Stephens, Mercury Jan/Feb 1992 pgs 38+


In seminar discussion, the instructors ask "what is the dark matter?" Many suggestions come up, and the students' first propose far-out things like axions, cosmic strings and the like. The response is simply to say that Yes, these are viable candidates -- but so are plenty of more pedestrian possibilities, and they should work on what they can work on. Another point is that the quantity of the dark matter within galaxies is sufficiently modest that there is no compelling argument against the hypothesis that it is all baryonic. This work should take ONE WEEK.

Students then assemble into teams, each charged with working out the details of a possible dark matter candidate as suggested in the seminar discussion. Each team meets regularly together, and with the instructor or the TA: these replace the regular class meetings. These projects should take THREE WEEKS.

Data and suggestions for simple calculations have been assembled for the following dark matter candidates:

(A) COULD THE DARK MATTER CONSIST OF DIFFUSE INTERSTELLAR GAS? If so, the gas will very likely be hydrogen, since hydrogen is by far the most abundant constituent of the universe. The 21-cm hyperfine transition of hydrogen provides a useful probe of this hypothesis. Details have been worked out of a simple calculation of this line's intensity, and 21-cm data for each galaxy is presented by the software.


21-cm line: Shu pgs 227-231
interstellar medium: Shu chapter 11 in general


A calculation can be made of the expected infrared emission from such matter, and comparisons can then be made with IR observations of the galaxies as exhibited by the software.

  • Chapters 1 and 2 of "Infrared :the New Astronomy" by David Allen (John Wiley, 1975.)
  • "Infrared and Submillimeter Astronomy From Space" by Stephen Strom, Sky and Telescope April 1983 pgs 312+
  • "Charting the Infrared Sky" by Michael Skrutskie, Sky and Telescope August 1997 pgs 46+

Simple calculations can predict the maximum distance starlight could propagate through such a medium (the scattering cross section is roughly the geometric size of the scatterers) and a comparison with observations can be made.

If one is thinking of meteors or planets, there is another test: how often should the inferred objects collide with the Earth, or disrupt orbits in the solar system?


(a) The students have assembled their own mass function: they show from it that, while faint stars are more common than bright ones, there are not enough of them to be the dark matter. Of course this conclusion only holds for the faintest stars we are presently capable of detecting. Students ask how far beyond this limit the same trend must persist for ultra-faint stars to constitute the dark matter. They then ask whether such a great number of low-luminosity stars can be ruled out in any other way (for example by star-star collisions, or galactic obscuration effects).

(b) Gravitational lensing of background stars is presently the best method we have for direct detection of a uniform population of brown dwarfs. Details are available of a simple calculation of this effect, and observations of such lensing events have been assembled.

  • GR: Shu pgs 364-367
  • Chapter 4 of Clifford Will, "Was Einstein Right?" Basic Books 1986.
  • "Searching for Dark Matter" by Mario Mateo in Sky and Telescope January 1994 pgs 20+
  • "Bonuses of the Microlensing Business" by Mario Mateo in Sky and Telescope September 1997 pgs 38+

(D) COULD THE DARK MATTER CONSIST OF A UNIFORM DISTRIBUTION OF BLACK HOLES? Again, there are two probes of this hypothesis.

(a) The precursors of black holes are high-mass stars: there are today very few such objects. On the other hand, their lifetime is very short. Is it possible that there have been so many generations of such stars that their black-hole remnants could be the dark matter? If not, how many more such high-mass stars must have existed in the past? What must have been the luminosity (both due to their steady shining, and due to the supernova explosions whereby they collapsed) of galaxies in the past?

  • Stellar evolution: Shu chapter 8
  • Stellar evolution: "The Lives of Stars" by Icko Iben and Alexander Tutukov in Sky and Telescope December 1997 pgs 36+

(b) Gravitational lensing of background stars is also the best method for direct detection of such a uniform population of black holes. The materials assembled to test hypothesis (C) are also be useful here.


Teams present their results in a conference. This should take ONE WEEK


The instructor delivers a lecture -- this is the only lecture of the entire course, other than the first day's orientation. In it the semester's work is placed in context. This should take ONE DAY