THE FIVE COLLEGE ASTRONOMY DEPARTMENT

"The Unseen Universe"

George Greenstein

Amherst College

Amherst, MA 01002

CATALOG DESCRIPTION

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.

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SYLLABUS

CALENDAR: the semester is about 12 weeks long. The

indicated timing adds up to 10 weeks.

(I) SYSTEMATICS OF ORBITS (2 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 --

(II) THEORY OF ORBITS (1 WEEK)

which operates not so much in the discovery-mode as lectures

combined with laboratories, since we don't have much time.

(III) BINARY STARS (1 WEEK)

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.)

(IV) THEORY OF BINARY ORBITS (1 WEEK)

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.

(V) DOPPLER EFFECT (2 WEEKS)

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.

(VI) PROJECTS (2 WEEKS)

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.

(VII) WRAPUP (1 WEEK)

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.