We study the radial motion of an object in the gravitational field produced by an isotropic mass density that is singular at the origin. This problem applies to elliptical galaxies and can be used to illustrate motion in a central field appropriate for an intermediate-level mechanics course.

1.
S. T.
Thornton
and
J. B.
Marion
,
Classical Dynamics of Particles and Systems
, 5th ed. (
Thomson-Brooks/Cole
,
Belmont, CA
,
2004
), Chap. 8.
2.
A. P.
French
,
Newtonian Mechanics
(
Norton
,
New York
,
1971
), p.
452
.
3.
L.
Carroll
,
Alice in Wonderland and Through the Looking Glass
(
Oxford U. P.
,
New York
,
2009
).
4.
L. Frank
Baum
,
Dorothy and the Wizard in Oz
(
Harper Collins
,
New York
,
1990
).
5.
The Annotated Alice: The Definitive Edition
, edited by
M.
Gardner
(
W. W. Norton
,
New York
,
1999
).
6.
Galileo
Galilei
,
Dialogue Concerning the Two Chief World Systems–Ptolemaic and Copernican
. Translated by S. Drake (
U. of California Press
,
Los Angeles, CA
,
1953
).
7.
D.
Merritt
, “
Chaos and the shape of elliptical galaxies
,”
Science
271
,
337
340
(
1996
).
8.
H.
Shapley
, “
Galactic and extragalactic studies: XIII. Note on the comparative diameters of spheroidal and spiral galaxies
,”
Proc. Natl. Acad. Sci. U.S.A.
28
(
5
),
186
191
(
1942
).
9.
G.
de Vaucouleurs
, “
Classification and morphology of external galaxies
,”
Handbuch der Physik
(
Springer-Verlag
,
Berlin
,
1959
), Vol.
53
, pp.
275
311
.
10.
L. S.
Sparke
and
J. S.
Gallagher
 III
,
Galaxies in the Universe
, 2nd ed. (
Cambridge U. P.
,
Cambridge
,
2004
).
11.
W.
Dehnen
, “
A family of potential densities pairs for spherical galaxies and bulges
,”
Mon. Not. R. Astron. Soc.
265
,
250
256
(
1993
).
12.
L.
Hernquist
, “
An analytical model for spherical galaxies and bulges
,”
Astrophys. J.
356
,
359
364
(
1990
).
13.
W.
Jaffe
, “
A simple model for the distribution of light in spherical galaxies
,”
Mon. Not. R. Astron. Soc.
202
,
995
999
(
1983
).
14.
MATHCAD, PTC Inc., Needham, MA 02494.
15.
C. W.
Misner
,
K. S.
Thorne
, and
J. A.
Wheeler
,
Gravitation
(
Freeman
,
San Francisco
,
1970
).
16.
M.
Abramowitz
and
I. A.
Stegun
,
Handbook of Mathematical Functions
(
Dover
,
New York
,
1965
), Formula 7.1.23.
17.
T.
de Zeeuw
and
M.
Franx
, “
Structure and dynamics of elliptical galaxies
,”
Annu. Rev. Astron. Astrophys.
29
,
239
274
(
1991
).
18.
M.
Schwarzschild
, “
Triaxial equilibrium-models for elliptical galaxies with slow figure rotation
,”
Astrophys. J.
263
,
599
610
(
1982
).
19.
P. O.
Vandervoort
, “
On Lagrangian methods for the study of small perturbations in elliptical galaxies
,”
Astrophys. J.
341
,
105
112
(
1989
).
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