\magnification=\magstephalf
\def\pp{\noindent\parshape 2 0truecm 15truecm 2truecm 13truecm}
\def\apjref#1;#2;#3;#4 {\par\pp#1, {\it #2}, {\bf #3}, #4. \par}
\def\header#1{\goodbreak\centerline{#1}}
\def\heading{\header}
\def\center{\centerline}
\def\np{\vfill\eject}
\hoffset=0.0 true cm
\voffset=0.0 true cm
\vsize=23.5 true cm
\hsize=17.0  true cm
\parskip=5pt
\parindent=20pt
\raggedbottom
\baselineskip=11pt

\newdimen\llsize \llsize=\hsize \advance\llsize by-2.22223pt
\def\lline{\hbox to \llsize}
\def\makeheadline{\vbox to  0pt{\vskip-31pt
  \line{\vbox to8.5pt{}\the\headline}\vss}
  \nointerlineskip}
\headline={\ifnum\pageno>1
	\vbox{
	\lline{\tenrm Astr. 218 Student Guide \hfil }
%	\lline{\tenrm\modtest\hfil}
}
	\else\hfil\fi}

\hrule
\bigskip

\noindent
{\sl This is a course guide for the Berkeley almost-equivalent of our
Astro 154 (only Galactic, and it takes a full semester, 4 hours/week).
The emphasis and the coverage are thus somewhat different, but I thought
that it is a very good summary of what is important, and what is not so
much, and which parts of M&B should be taken with some caution.  I hope
this is useful for your studies. --- G.}

\bigskip
\hrule
\bigskip

\centerline{\bf A Student Guide to Astronomy 218}
\bigskip

  (Written by I.~R.~King, to try to acquaint students with what they do
and don't need to study, and perhaps even how much.  This is the 1989
update.) 

     Astronomy 218 covers a broad area that is a general introduction to
stellar systems.  Its three general foci are basic stellar data, the
distribution and motions of stars, and stellar dynamics. 

     The Mihalas and Binney text is an excellent and comprehensive one,
but there are places where it goes into more detail than is needed in
this course, and other places where I feel it needs to be filled out
somewhat. This document is an attempt to tell students what is more
important and what is less important.  It is organized lecture by
lecture, since in a sense the amount of time spent in class is a first
indicator of importance. I hope that the summary below will be generally
useful to students in organizing their study of the subject.  (Note: the 
numbering of lectures corresponds to an intended schedule and may differ 
a little from actual unit lectures.)

  If a section is referred to in brackets, you are not expected to have 
studied it as carefully as the others.

     Remember also that I have made additions, corrections, and updates
in class at numerous points.  These you will have to recover from your
notes, because they will not be mentioned here unless they are really
major. 

\bigskip\centerline{1. Introduction; history; overall picture (1--1,2)}

The introductory 4 paragraphs are quite good.

     Although you are not likely to be examined on it, a historical view
of modern astronomy is useful for understanding it, and may eventually
be of use in teaching it.  The one place where it may really matter is
in understanding some of the obscure terminology, especially where the
latter has evolved with time.  The most notorious example of this is the
terminology of stellar populations. 

     Sec.~1--2 may serve as a good vehicle at the end of the semester for
asking, ``What is it all about, and what have I learned about it?'' 
(This is for the first two parts of the course only, of course.) 

\bigskip\centerline{2. Coords; p.m.'s; px, L--K corr; r.v. (2--1,2,3,4 + L--K notes)}

     Lectures 2 -- 6 introduce the basic stellar data. 

     You ought to be able to define all the various coordinate systems,
but remember that you will probably be able to use somebody's program
for actual conversions. 

     Proper motions:  Know what they are, how they are measured, what
accuracies are realistic, and what general sorts of data are available.
(For data you will have to rely mainly on Sec.~2--8, which will not be
otherwise commented on in this summary.)  Be able to distinguish between
relative and absolute proper motions; and know what fundamental systems
are, what their problems are, and why they are important.  Remember the 
updates on the Lick survey and Hipparcos.

     Parallax:  The basic idea is Astro-10-level.  Again, know the
practical problems, the limitations, typical accuracies, and where to
find the data.  You should understand the Lutz--Kelker correction 
(separate handout --- not in M\&B) and its importance. 

     (Note the common astrometric problems in proper motions and
parallaxes.  Be aware of the effects of seeing and how it limits 
measurements.) 

     Take a good look at the list of stars nearer than 5 parsecs, and 
note what types predominate.

     Radial velocities:  Again, measurement, practicalities,
limitations, and data.  Be sure also that you understand the general
workings of the digital cross-correlation method. 


\bigskip\centerline{3. Spectra; mags \& colors; absorption (2--5,6,7,8; 3--11)}

     Spectra as a purely heuristic study.  You should be able to
classify spectra on sight to a whole or even half a Henry Draper class 
(i.e., F0 vs.~F5).  Try to recognize at least some luminosity criteria. 

     You should have manipulation of magnitudes at the tip of your
tongue (e.g., 5 mag $\leftrightarrow 100\times$, 2 mag $\leftrightarrow
6.3\times$, etc.). It would be nice also if you were familiar with
approximate limiting magnitudes for various observational techniques
(e.g., 20 or 21 for the coarse-grain Sky Atlas, 24 for a CCD on a 
4-meter, etc.).

     Understand what magnitude systems are and how they are related by
color equations.  Be familiar with $UBV$, and $uvby$ with its $c_1$ and 
$m_1$ indices.  (Note that their applications are introduced later, in
3--11 and elsewhere.)  Note how $B-V$ colors are often used instead of
spectral types. 

     Be thoroughly familiar with the relation between absolute
magnitude, apparent magnitude, and distance.  You should know instantly
the distances that correspond to distance moduli of 0, 5, 10, \dots, 30
mag. 

     Note the brief Sec.~2.7, on absolute energy distributions. 

     Sec.~3--11, on absorption, is a long one.  You should understand why
absorption in magnitudes is proportional to column density, and where
the cosecant law comes from.  Color excesses are important:~the
reddening line in Fig.~3--26, what determines its slope, and what its use
is.  Know how the ratio of absorption to reddening is found, and why it
is so important. 

\bigskip\centerline{4. Masses and radii: binaries (3--1,2,3[,4])}

     The chapter actually begins with a brief section on stellar
distances, a terribly important topic that threads through all of
stellar astronomy.  Note the equivalence between determining absolute
magnitudes and determining distances (again, $M = m + 5 - 5 \log r$). 
The part on moving clusters refers basically to the distance of the
Hyades. As a method it really belongs with the later discussion of
statistical parallaxes. 

     The remainder of this lecture deals with masses and radii --- and
thus necessarily with binaries. 

     Remember here that our objective is to study galactic structure;
regard the stars as the actors on the stage:~what are their personal
characteristics? 

     Note the separation here:~the previous chapter dealt with the
quantities that are directly observed; here we take up quantities that
are derivative, even though they are physically more fundamental. 

     Stellar masses are really fundamental and really scarce.  They come
from careful analysis of binaries.  Know the 3 kinds of binaries and how
they are observed and exploited (and how they evade exploitation).   
Note the importance of good parallaxes in visual-binary masses.  Note 
also the increasing importance of speckle interferometry.  And note how 
the different kinds of binaries complement each other in giving mass 
information.

     You should know the stellar form of Kepler's 3rd law but need not
bother with other mathematical details of binaries. 

     You must be acquainted with the mass-luminosity relation. 
Recognize it as empirical, but shun the blind empiricism of your text
and understand it as a causal relationship for main-sequence stars;
other masses follow from those of the main-sequence progenitors (allowing
for mass-loss where necessary --- which is usually).  Know the 
approximate power law.

     The radii (another scarce commodity) are astrophysically important,
and astrophysicist Mihalas makes much of them {\it per se}; but their
importance for us is in the knotty problem of transforming between the
observational and theoretical HR diagrams.  Otherwise, skip the
astrophysics for now; it will come back soon enough to haunt or
fascinate you in other courses.  Note how the constant-$R$ lines are 
straight and parallel in the theoreticians' HR diagram.

  You need not remember the mathematical discussion of eclipsing 
binaries, but be familiar with the logical relation between the geometry 
and the light curve (Fig.~3--3).

     In addition, be familiar with the stellar-census aspect of binaries
--- how common they are. 

     The material in Sec.~3--4 you should know thoroughly in a
qualitative way, but you need not know anything quantitative except 
Eq.~3--25 and some rough picture of relative abundances.  Remember again
that one of the central importances of this material is to establish the
vital connection between the observational and theoretical HR diagrams. 

\bigskip\centerline{5. HR diagram, populations (3--5,6,7)}

     The HR diagram is of course fundamental to nearly everything in
stellar and galactic astronomy.  You should be able to draw a pretty
good main sequence of $M_V$ against either spectral type or $B-V$ and
also have a reasonable mental picture of Fig.~3--8.  Likewise the class-V
sequence in Fig.~3--7.  Know the general run of temperatures for class V
in Table 3--5, and appreciate how the bolometric corrections go, but
don't dwell on all the tables. 

  Distinguish clearly between the observer's coordinates used in 
Sec.~3--5 and the theoretician's coordinates used in Sec.~3--7.  (Note 
that they are confused in Fig.~3--5, which is a curious fossil.)

     Why is there a main sequence?  What does it signify physically?

     Have a good picture of Fig.~3--6 in your head; it wouldn't hurt to
be familiar with the names of most of these clusters.  (Have you ever
looked in the Hagen Atlas?) 

     Note the quantitative relation (not given explicitly in M\&B)
between the absolute magnitude of the turnoff and the age. 

     In Sec.~3--6, know what blanketing is (and why), how the blanketing
vector works, and what $\delta(U-B)$ and [Fe/H] are; but skip the rest
of the math. Globular-cluster color-magnitude diagrams and their
morphology are important, as are abundance differences.  (Note the
various observed quantities that are used to estimate [Fe/H].  Note the 
importance --- and rarity --- of real abundance determinations.) 

     Be familiar also with the $XYZ$ notation, and know its relation to 
[Fe/H] and to $n$(He)/$n$(H).

  You should know all the sequences that are labeled in Fig.~3--13, and 
approximately where they lie in $M_V$ and $B-V$.

     The second parameter is important, even though no one has 
identified it.

     Sec.~3--7 is much more detailed than we need.  You ought to know all
of stellar evolution on the level of being able to handle all the
questions of Astr.~10 students competently, and it is highly desirable
to have a general physical understanding of what happens.  (Remember how
good Shu's book is for this.)  But you certainly don't need the details
in M\&B for this course, especially the quantitative ones. 


\bigskip\centerline{6. Variable stars (3--8,9,10)}

     Sec.~3--8 deals with stellar pulsation in a quantitative way, where
I was pretty much descriptive.  Your keynotes should be the use of these
stars as distance criteria and as population indicators.  Remember that
I also gave a general survey of some other types of variable stars,
which should be in your notes. 

     In Sec.~3--9, be aware of the general problems of age-dating, both
in practice and in principle, but skim the details lightly.  Variation
of chemical composition with time is pretty hazy now, but you'll hear a
lot more of it in the future.  Note the spatial gradients.  Almost the
same sort of remarks apply to Sec.~3--10. 

     Remember that a lot of the M\&B discussion of RR Lyrae types is 
messed up, and rely on your notes for that.

     Note the c-m differences in Fig.~3--14.  Remember $\Delta V$, the 
blueness of the HB, and the second parameter.

\bigskip\centerline{7. Stellar statistics: star-count eq.; Malmquist corr. (4--1,2)}

     Much of the early part of Chap.~4 is archaic and even misleading.
The only worth-while equation in the first part is the star-count eq.
4-19 (which I prefer, for clarity, to derive without including $a(r)$
and $S$); you should be able to derive it.  From this (or else directly,
as in M\&B) you should be able to derive the $10^{0.6m}$ law for uniform
densities. Note the corresponding behavior of contributions to the total
light, and the resulting Olbers' paradox.  Note that the star-count
equation is a convolution and is thus hopeless to solve except with a
very narrow luminosity function.  Similarly, on pp.~234--239 you should
know what an $m,\log \pi$ table is, but ignore the solution method that
is based on it. 

     Also buried in the discussion of solutions is the Malmquist
correction (Eq.~4--47, then top of p.~241 to Eq.~4--54), which you should
be able to derive and interpret. 

     Regarding absorption, note Seeliger's relation as a clever formal
trick, and note the powerful effect of absorption in Table 4--3.  Know
what a Wolf diagram does. 


\bigskip\centerline{8. Luminosity function (4--2)}

     Initial motivation: what is the luminosity function, and why does 
one want to know it?  Note that in its practical determination the 
definition of number/volume can almost never be applied directly; 
statistical selection effects rule everything.

     The discussion of the luminosity function is also somewhat archaic;
it may well be that the methods explained on pp.~217--222 will never be
seriously used again.  Future efforts will probably use approaches like
that of Wielen (W2, which you should read thoroughly, in the reading
folder) and Schmidt (explained in class as a $V/V_m$ method), and also
the photometric interpretation of colors and magnitudes of faint stars
(also in your notes).  Luyten's ``reduced proper motion'' will probably
continue to be a useful quantity. As for results, you should be familiar
with the general characteristics of Table 4--5; note the discussion on
pp.~223--226.  Don't take the remainder of pp.~226--229 too seriously,
except to notice how important white dwarfs are. 

     Again, be acquainted with van de Kamp's 5-pc list, and know where 
to find Gliese's 20-pc list.

\bigskip\centerline{9. Init'l LF, IMF; space distr \& gradients (4--2,3,4)}

     Read carefully the material on pp.~229--232 about the initial LF and
the IMF.  Recognize the importance of the logical break where MS life
equals the age of the MW. Note the Salpeter function, and understand why
it is sometimes quoted with a 1.35.  Read the Miller-Scalo article
(reading folder), and appreciate what they are up to at each stage of
their odyssey. 

     Note the brief but important discussion of Pop II. 

     Appreciate the methods of Sec.~4--3; note the qualitative behavior
of the numbers in Table 4--16, but take them with a large grain of salt
quantitatively.  Note the discussion of composition gradients. 

\bigskip\centerline{10. Models of the MW; pops overall (4--(3,4),5)}

     Secs.~4--4,5 are rather long for their content.  See how the density
distribution in the halo is traced by means of globular clusters and RR
Lyraes, and note that the distance to the Galactic center must be
determined first. Note the metallicity differences among globulars. 

  Note that we will return to overall models of the Milky Way after 
finishing the M\&B book.

     Regarding pp.~266--268, remember that this is being superseded by
current work that is not all in.  Similarly Baade's window, where there 
is much work (including the not-yet published Terndrup and Rich theses)
that postdates the M\&B text. 

     The first part of 4--5 is a recap of the historical development of
population ideas, which I elaborated on in the first lecture.  Note the
population groups summarized in Table 4--18, which really go back to a
discussion by Oort in 1958; it is much less clear today that they are
basic distinctions. 


\bigskip\centerline{11. Galaxy types: classes \& properties (5--1)}

     The Hubble-Sandage classification is the basic one, but note how it
has evolved with time.  You should certainly look through the {\it
Hubble Atlas} and the classification section of the {\it Revised Shapley
Ames}, and you should be familiar with the classification even if you do
not know it in exquisite detail.  Know how de Vaucouleurs' system
differs.  Morgan's system is never used, but note his introduction of
cD's and N's.  Note that none of the systems mention the existence of
the dwarf spheroidals but that their existence is important.  Van den
Bergh's anemic classification is rarely used, but his luminosity classes
are important and are in the RSA.  Be acquainted with the zoo of queer
objects on pp.~300--302. 


\bigskip\centerline{12. Density profiles; pops \& content (5--2[,3,4],5)}

     For galaxy surface brightnesses, de-emphasize photography; the
future belongs to the electronic devices.  Know de Vaucouleurs' law and
Hubble's law, and know what King-model profiles look like.  (Watch out
for graphs whose coordinate scales differ.)  Be aware of the effect of
seeing, but merely skim the details of the discussion of its effect on
measured core radii.  Note the exponential law for discs, and the idea
of disc-to-bulge ratio.  Regarding elliptical isophotes, be aware of the
projection problem.  Isophotal twists and triaxial configurations are
important too, but you don't need to know the detailed mathematics of
any of this.  For the relation between projected and true shapes of 
ellipticals, note Eqs.~5--21, and note the existence of the integral 
equations 5--27,28, but don't emphasize any of that math, nor the 
quantitative results in Fig.~5--29, which are quite uncertain because of 
selection effects.  Note the bit on photometry of spiral arms. 

     Skip 5--3 and 5--4; they are part of Astr.~228.

     You should know the basic idea of the selection effects in 5--5 but 
can gloss over the details.


\bigskip\centerline{13. Solar motion \& mean vels (6--1,2,3,4,5)}

     At the start, be sure that you appreciate what a distribution
function really means, and what a velocity distribution is.  Understand 
why we emphasize the first 2 moments of the distribution.  Get all the 
mis-chosen terminology under control.  For standards of rest, and solar 
motion terminology, please adopt my definitions, and not the mixed-up 
ones in M\&B.

     Skim 6--2 quickly; it is Astr.~10 stuff.

     Understand why rv's and pm's are treated separately in solar-motion 
solutions.  Remember that the discussion of least squares on p.~392 is 
replaced by a handout.  Note that solar motion from pm's is really 
superseded by the later discussion of statistical parallaxes (6--6).

     Remember that all the numerical tables in Chaps.~6 and 7 are much 
too detailed; of the differences between types, only the gross ones are
real.

     Regarding basic solar motion, rely on the method on pp.~399--400, 
not on the definition on p.~398.

     The discussion of observational results in 6--5 is important, but 
we will really take it up as part of the discussion of rotation 
constants in 8--2 (for which we will rely on the Kerr and Lynden-Bell 
reprint in your reading folder).


\bigskip\centerline{14. Stat px; vel ellipsoid (6--6,7--1)}

     For this subject it's a good idea to have good Astr.-10-level 
explanations in mind before digging into the details of the diagrams and 
the mathematics.  Gloss over the detailed trigonometry on p.~411; also 
supplement (or replace) Figs.~6--9,10,11 by the ones that I gave.  Know the 
essential difference, and the uses, of tau and upsilon components.

     Note the similarity (and the differences) of the cluster parallax 
method that is used for the Hyades (pp.~76--79).

     Recognize how important statistical parallaxes are in determining 
absolute magnitudes of many types of stars.

     Sec.~7--1 introduces the velocity ellipsoid.  Review the terminology,
and the idea of principal axes and directions, but skim over the 
mathematical expressions on pp.~420--421.  Again, emphasize the 
summarizing text on pp.~422,424, rather than the details in Table 7--1.  
Remember the vertex deviation; it's important, even if we don't 
understand it.

\bigskip\centerline{15. Moving grps; old-disc vel distr; sph'd v distr; vel disp 
(7--2,3)}

     Remember that I gave a more detailed discussion of moving groups
(particularly Eggen's work) than M\&B do.     

     Know the general dependence of velocity dispersion on age, and the 
general explanations that have been suggested for it, but not with all 
the detail that M\&B introduce.  Note that we will be taking up the 
theoretical explanation of the speedup of older stars after we have done 
the theory of stellar encounters.  The details of pp.~438--441 can also be 
skipped, as long as you appreciate the idea that different populations 
(in chemical abundance, that is) have different motions.

     In Sec.~7--2, skip the Oort discussion.  Then note the useful 
characteristics of the Bottlinger diagram, but remember its defect of
ignoring the $w$ component.  Put most of your attention on the Eggen,
Lynden-Bell, and Sandage work.  Look carefully at Figs.~7--11 and 7--12,
think about selection effects, and read ``Dynamical Implications''
carefully.  Their paper has sloppy aspects, but it took up an important
problem. 

     For Sec.~7--3, rely on your lecture notes, which should be more 
detailed than the material in the book.

\bigskip\centerline{16. Basic gal rot; rot law of MW (8--1,2,3)}

  Sec.~8--1 starts with a qualitative Astr.-10-level part that you can
skip.  The radial and tangential-velocity formulas are fundamental, and
you should be able to derive them.  So is the $R_{min} \leftrightarrow
V_{max}$ idea, even though the book doesn't get back to using it till
p.~492.  You might do well to jump to that point and study pp.~490--497. 
Be aware of the north-south discrepancy for the inner Galaxy.  Remember
that I gave a more thorough discussion of the combined optical (H II
regions) and radio (CO) determination of the outer rotation curve. 

     You should be able to derive the Oort formulas and should know them 
both in the theta and in the omega form.  Note also the $AR_o$ formula, and 
the similar formula derived by Kerr and Lynden-Bell.


\bigskip\centerline{17. Discussion of rot consts; other gals (8--(2),4)}

     For the rotation constants, read the M\&B discussion for the 
explanations of the general methods and reasoning, but rely completely 
on the Kerr \& Lynden-Bell paper for the results.  (Watch out for the 
fact that M\&B, and much of the literature, still use the old IAU 
values.)

     Know the elementary physics of the 21-cm line, but let pp.~485--489 
wait for Astr.~216.

     For other spirals, be aware of the long-slit and Fabry-Perot 
techniques, and look carefully at the results in Fig.~8--18.  Don't 
bother with the projection geometry; just be aware that the practical 
problem exists.  Give a light reading to the part on 21-cm rotation 
curves.  Be aware of warps, and their observations and their theoretical 
problems.  (Note the MW warp, on p.~536.)  Note the jiggles in the 
equal-velocity curves of Fig.~8--24 where they cross spiral arms.

     Note the relative lack of rotation in giant ellipticals, and the 
empirical $\sigma^4$ law (as well as the $V^4$ law for spirals).


\bigskip\centerline{18. Obs gas distr \& sp str in MW (9--1)}

  You should know in a general way how distances are found for the
hydrogen along each line of sight.  Take the maps of spiral structure in
the Milky Way with a large grain of salt, but don't underestimate the
importance of the problem. Likewise, be aware of its difficulty:~look at
Figs.~9--3, 9--6, and 8--11. 

     Note the general radial distribution, in Fig.~9--7.


\bigskip\centerline{19. HI in other gals; GMC's \& CO; gal ctr (9--2,3,4)}

     For other galaxies, the data on 21-cm distribution are in process 
of being supplanted by much better VLA data.  Look only at the nice M81 
map in Fig.~9--10 (and also 8--24).

     Note the existence and nature of the Magellanic Stream.

     Giant molecular clouds have become very important in studies of star
formation and of structure of galaxies.  Note the nature of the CO 
observations, the role of $^{13}$CO, and the fact that CO is a tracer rather 
than a major actor.

     The final subject, the Galactic center, is another one where new 
knowledge is rapidly accumulating.  For this discussion, try to identify 
in Fig.~9--20 the features that are referred to in the text.  Keep the 
various size-scales clear in your mind, from the 3-kpc arm down to the 
central parsec of the nucleus.  Look at the handout montage of diagrams. 
A good exercise would be to locate the source of each of them in the 
Townes Symposium volume; by then you will probably know your way around. 
(Note also the Genzel \& Townes article in the 1987 {\it Annual 
Reviews.})

\bigskip
\hfill ********** \hfill ********** \hfill ********** \hfill

     At this point the basic nature of this outline changes, because of 
the switch from a textbook to handout notes.  This outline will continue
to use lecture numbers as major headings, but each lecture-note handout 
will appear as a subhead.  (Note that the numbering is only approximate; 
actual dates will deviate a little.)

\bigskip\centerline{20. Basic nature of st dyn; collisionless Boltz eq}

\noindent ``Introduction,'' ``Relation to Other Branches of Physics,''
``The Idealizations of Stellar Dynamics''

     This is material that was merely intended to orient you; there is
nothing specific to learn. 

\noindent ``The Collisionless Boltzmann Equation''

     You should know how to do this derivation; but note that it is 
something that you probably have had before in hydrodynamics, because 
the cBe is nothing but an equation of continuity in the 6-dim phase 
space.  You should remember the cartesian vector-mnemonic form of the 
cBe.

\bigskip\centerline{21. Sol'n of cBe, Jeans' thm; integrals \& ergodicity}

\noindent ``Solution of the Collisionless Boltzmann Equation''

     The general method of solving this kind of p.d.e.~had to be dragged
in, but you are not responsible for knowing it, just as mathematics.  
What you should be sure of is the specific application here:~the form of 
the subsidiary eqs.~and their relation to the eqs.~of motion --- and thus
the relation of the cBe to Liouville's theorem.

\noindent ``Jeans' Theorem and the Concept of Isolating Integrals''

     This is deep stuff and important.  You should be able to state
Jeans' theorem and explain what each part of the statement means. 
Remember that it refers to isolating (!) integrals, and understand why
there is such a restriction (and what an isolating integral is).


\bigskip\centerline{22. cBe in gal coords; 3rd integral}

\noindent ``The Collisionless Boltzmann Equation in Galactic Coords.''

     You should be familiar with how this form is derived, but it would 
be tedious to ask you to reproduce the whole thing.  Be able to derive 
the 2 integrals from the subsidiary eqs., and to state the 
$\langle{\P^2}\rangle=\langle{Z^2}\rangle$ paradox.  Be able to show where the
Oort--Lindblad third integral comes from and how it deals with the paradox. 

  The galactic third integral is another big, deep, and complicated
topic.  You should be able to define, describe and explain surfaces of
section, box and tube orbits, non-classical integrals, regular and
irregular orbits, and the general results of the H\'enon--Heiles paper. 

     The references are, as almost everywhere, for bibliographic 
purposes, and are not meant as a reading list.

\bigskip\centerline{23. The Oort substitution}

\noindent ``Local Galactic Dynamics''
     Many important results are derived here.  Know the assumptions and 
the general course of the solution, but you will never be asked to 
reproduce more than a small part of it at once.  Be sure that you 
appreciate the astronomical content of the equations near the bottom of
p.~22, and also Eqs.~(2.24) and (2.28).  Remember the difference between
constants like $h^2$ and constants like $c_1$. 

\bigskip\centerline{24. Oort subst. (cont'd); quadr 3rd integral}

     Again, know where the asymmetry equation comes from, but
concentrate on its astronomical content, and its applications. 

     The final equation really leads into the following notes.

\noindent ``The Quadratic Third Integral''

     This is a topic of which you should be moderately familiar, without
necessarily mastering all its details.  Know its relation to the Oort
substitution and to the Eddington potential.  (Note that this topic is 
likely to become more important in the near future, as applications of 
St\"ackel potentials are developed.)

\bigskip\centerline{25. Moment eqs (\& hydrodynamic analogy)}
\noindent ``Moment equations''

     This is another of the important methods of stellar dynamics.  Know
the motivation, and the rules that make it possible to integrate by
inspection. Note the parallels with (and differences from)
hydrodynamics.

      Be familiar with the content of Eqs.~(3.12) and (3.13) (without
feeling any need to memorize them).

\bigskip\centerline{26. Asymmetry eq; pi-Z terms}

     Be able to derive this form of the asymmetry equation by deriving 
and manipulating the corresponding moment equation.  Remember the
previous discussion of its applications. 

\noindent ``Higher Moment Equations''

     An insertion for the sake of illustration, and completeness.  (But
note that you could reproduce it by pure stamina, if you were told what
to multiply by before integrating.) 

\noindent ``Pseudomoments''

     Again, know of its existence, and general nature, but you needn't 
remember the details.

\noindent ``The $\langle{\P Z}\rangle$ Correction, as Derived from the Quadratic
Third Integral'' 

     You should read through these and understand them, but need not be 
able to reproduce them.


\bigskip\centerline{27. Differential motions, epicycle orbits; 28. Associations,
Blaauw's equations}
\noindent (These two are commented on together)

\noindent ``Differential Motions in a Rotating Galaxy'', ``Equations of 
Differential Motion''

     This is introductory material for epicycles and for tidal fields in
clusters.  The unperturbed equations you should have seen in a physics
course, but here they are.  Note the definition of the differential
quantities and the way in which the equations for them are derived.
Recognize the physical significance of each of the terms in Eqs.~(5.9). 

\noindent ``Lindblad's Epicycle orbits''

     Another basic approach to galactic dynamics; you should be able to 
reproduce any part of the solution that you are asked for.  Some 
familiar results appear again.  New: (1) $\kappa$ as an important
natural oscillation frequency of the Galaxy, (2) Blaauw's equations for
the diffusion of an association.  Note how this leads us into moving
groups again. 

\bigskip\centerline{29. Z-motions, Oort limit}

\noindent ``Motions Perpendicular to the Galactic Plane''

     Remember that the basic idea is to measure the gravitation of the 
Galactic plane by seeing how far up the velocities carry the stars.

     The section on forces is important, the harmonic-motion part less 
so.  Going to the full Poisson's equation also makes little difference,
and the $\langle{\P Z}\rangle$ terms (discussed later) likewise.  The basic idea
is reiterated in practical terms in the paragraph that begins at the end
of p.~56. Oort's approach is important {\it per se}, and also because it
is at the basis of most other treatments that have been done.  Get its
flavor by reading his BAN 494 paper (and skimming Hill's paper), and
then read from ``The trouble'' (p.~60) to the end. 

\noindent ``The Gravitational Field of the Milky Way''

     This is mainly an addition to your lecture notes.  You ought to be 
able to write a coherent explanation, for an uninformed graduate 
student, of what these general ideas are, but the mathematics is only 
for reference.  What does matter more is the density-and-force models of 
the Milky Way (for which lecture notes are not yet available).

\bigskip\centerline{30. Jeans length, Toomre criterion}

\noindent ``The Jeans Length and the Toomre Criterion''

     If you haven't seen the Jeans length before, you had better learn 
it now, even though it will appear in different terms in star formation 
and galaxy formation.

     Watch out for the confusing terminology: an unstable medium makes 
condensations that are then stable.

     Try to concentrate on the intuitive physics --- which disruptive 
(stabilizing) processes fight against gravitational condensation 
(instability), and what determines in each case who wins?  Look at how 
it's treated mathematically, but don't expect to have to reproduce the 
complicated rotational argument.  Recognize how the Toomre criterion 
probably forces discs to have a high enough velocity dispersion.

\bigskip\centerline{31. Density-wave theory; SSPSF}

\noindent ``The Dynamics of Spiral Arms''

     You basically appreciate the density-wave theory if you can lucidly
explicate the diagram in Fig.~1.  The text starting at the bottom of
p.~74 says some important things, however.  Also note the scraps of
other dynamics in ``Other Theories \dots''  Stochastic star formation is
important to know about, in a general way.  The jury is still out ---
probably without a lot of the evidence. 

     ``Formulas'' is for your reference notes.

\bigskip\centerline{32. Stellar encounters: basic}

     ``The Circumstances of an Encounter'' gives some basic formulas which
all the following discussions will use.  The frequency of encounters is 
also fundamental.

     ``Collisions'' is a digression, but you ought to be familiar with
its basic conclusion about cross-sections, and have some intuitive
feeling of why they go that way. 

  For time of relaxation you should be thoroughly comfortable with the
intuitive arguments, but you can skip the mathematical details of the
calculation; remember, it's only an order of magnitude.  But understand
the argument about setting the size of $D_0$. 

     ``Equipartition rates'' is only for your reference notes.

\bigskip\centerline{33. Fokker--Planck eq}

\noindent ``The Fokker--Planck Equation''

     Meaty stuff.  Understand the conceptual basis: velocity diffusion, 
the Taylor-series expansion of the integral equation, and the role of 
the central limit theorem (whose meaning you should know and appreciate 
forever).  Watch out for the rates versus the $\Delta t$ average values.

\bigskip\centerline{34. Encounter integrals; dyn friction}

\noindent ``The Gravitational Encounter Integrals''

     (Read the RMJ paper, but rely on these notes, because the 
gravitational case doesn't appear anywhere else in print.)

     The derivation of Eq.~(9.45) is full of clever tricks. Admire them
(and understand them, of course), but don't memorize them. 

     Note the stages of specialization and simplification on pp.~94--95; 
their end result is the Spitzer form that we will use further.

     ``The F--P Eq as a Continuity Eq'' is a minor addition to your
reference notes. 

\noindent ``Dynamical Friction''

     This is included not for the sake of having a separate derivation, 
but because dynamical friction is a subject important enough to stand by
itself.  Appreciate the quantities and circumstances that it depends on.

\noindent ``Close Encounters \dots the K--F Eq''

     This is an important topic; it has had relatively little 
application but will surely take an important place in stellar dynamics.
Concentrate on its concepts, not on the mathematical details.

\noindent ``Binaries and Encounters''

     This probably has too much emphasis on the breakup of soft binaries
and not enough on the role of hard binaries in clusters, which is
becoming an increasingly important topic.  Be sure that you appreciate
the difference and the different behavior of hard and soft binaries. 

\bigskip\centerline{35. Virial thm \& applic; proj dens}

\noindent ``The Virial Theorem''

     This is the basic derivation; note that it is the tensor virial 
theorem.  Don't feel responsible for the tricky details of the 
derivation, but look carefully at the result and the components of the 
kinetic and potential tensors.  The scalar form, $2T+U=0$, you should
know at all times. 

\noindent ``The Virial Theorem in Spherical Systems''

     This is an important application to star clusters and spherical galaxies. 
Note the $\langle{v^2}\rangle$ and $\langle{v_e^2}\rangle$ formulas, and the
relaxation time. Don't worry about the strip-density formulas. 

  ``The Flattening of a Rotating Star Cluster'' is inserted as an 
example of the use of the tensor virial theorem.  (Note also the use of 
the potential theory of ellipsoids.)  You should follow and understand 
what is done, but you need not be able to reproduce all the details.

  ``Relations between Space Density and Projected Density'' is inserted
only for completeness.  Note its cautionary paragraphs, however. 

\bigskip\centerline{36. Escape rate; F--P sol'n}

\noindent ``The Basic Escape Rate in Star Clusters''

     Here we begin the serious dynamics of star clusters.  Think 
carefully about Ambar\-tsumyan's simple reasoning, which has in it the 
essential physics.  But the real stuff is the Fokker--Planck treatment.  
Look carefully at the separation of variables, with the time-decay 
equation and the eigenvalue equation for the steady-state velocity 
distribution.  Study also the series approximation that leads to the 
lowered-Gaussian approximation.  You will not be held responsible for 
its derivation, but you must know its form.

\bigskip\centerline{37. Models of clusters}

\noindent ``Self-consistent Models of Star Clusters''

     These are the ``King models.''  You should know their physical
basis, the general cluster-building algorithm, the general outline of
the calculation of these models, and the characteristics of the family
that they give. 

\noindent ``Cluster Models with Anisotropic Velocities''

     You should know the general results, particularly the 1+(r/ra)**2 
formula, but don't worry about detailed integrations.  The table of 
formulas is given only for reference.


\bigskip\centerline{38. Tidal Forces}

  Be sure to recognize (and understand) the distinction between the 
deforming effect of a steady tidal force and the energy-adding effect of 
an impulsive tidal force.

\noindent ``Restricted 3-body problem''

     This is not strictly a problem of stellar dynamics, but it is 
something that  every astronomy student should be acquainted with ---
and it is a good introduction to tidal forces in clusters. 

     You should be thoroughly familiar with the idea of a Jacobi 
integral and how it limits the motion of a star.  Note also the 
existence and locations of the Lagrangian points.

\noindent ``Application to a Cluster''

     If given the equations of motion, you ought to be able to derive 
the Jacobi integral and find the location of the tidal limit.  
Similarly, the minimum density for stability.

\bigskip\centerline 39. Tidal shocks \& other effects

\noindent ``Tidal shocks''

     This derivation is too complicated to be expected to be remembered 
in detail, but you ought to be able to give a general outline of it, at 
least as far as the basic physics is concerned.  Be able also to explain
why a dense cluster is immune.

\noindent ``Tidal-Shock Dissolution''

  This is the first example of using the virial theorem to follow the 
changes in radius of the cluster.  You need not be familiar with these 
particular mathematical details.  Note, however, the brief mention of 
testing the impulsive approximation; also the competition between 
tidal-shock evolution and star-escape evolution.

\noindent ``Tidal Shocks in a Globular Cluster''

     Understand the general idea and the process, but don't memorize 
the details.

\noindent ``Dynamical Evolution in Clusters''; ``Stellar Mass Loss''

  The reasoning leading up to Eq.~(13.43) is important.  Note the 
stellar mass loss as a particular case, without dwelling on the details.

\bigskip\centerline{40. Simulations; core collapse}

\noindent ``Simulations''

     This is pretty straightforward text, and you ought to feel 
responsible for the general material.

\noindent ``Core collapse''

     Just one simple derivation, then a lot of talk.  You should be able
to reproduce the general arguments.

\bigskip\centerline{41. Dynamics of elliptical galaxies}

     As I said, these notes repeat a lot of what was in Chap. 5 of M\&B, 
particularly in ``Radial Profiles'' and ``Velocity Dispersions and 
$M/L$ Values.''  Note also that Fig.~1 is the replacement for Fig.~5--14 
of M\&B.

\noindent ``Violent Relaxation''

  Be sure that you understand the physics of this process.  Note 
carefully the lack of equipartition, and its reason.

\noindent ``Flattened Dynamical Models''

       The Prendergast--Tomer method and the Wilson models are to be 
understood in a general way, but of course you are not responsible for 
them in detail.

\noindent ``Triaxiality''

     The section on triaxial dynamics by Schwarzschild's method is 
important.

\bigskip\centerline{42. St\"ackel potentials; perfect ellipsoid}

  I can't be very explicit about the lectures given by Tom Statler.  You
should have a general acquaintance with St\"ackel potentials and the
classes of orbits in them. 

\bigskip\centerline{43. Dyn of clusters of gals}

\noindent ``Clusters of Galaxies''

  Be familiar with these generalizations and numbers.

\noindent ``Dynamical Modeling of the Coma Cluster''

  This is one of the important lines of evidence about the invisible 
mass in the universe.

\noindent ``Mergers and Cannibalism,'' etc.

  Be familiar with the general ideas and arguments.

\noindent ``Formation Time of a Cluster of Galaxies''

     This is a small point but is too little appreciated.  Don't worry 
about the details of the derivation.

\end
