1. Orbits in stellar systems
ASTC22 - Lecture 10
Orbits in stellar systems
The Kepler problem
Orbits in spherical and axisymmetric systems
Epicycles: wheels upon wheels
Orbits in more complicated potentials

2. The Kepler problem
a.k.a.
2-Body problem
this is a partly self-study part following exactly ASTB21
The most important things to know is the use of energy and
angular momentum principles.
The interesting but least important part, which I keep here for fun, is the derivation of the elliptical orbit from the Laplace
vector. This is not a required knowledge for you in ASTC22.

3. The problem of 2 bodies (2-Body problem) was first studied
in the context of planetary system, starting from planetary
and cometary orbits. However, it also describes the orbit of
an isolated pair of compact galaxies, for example.

5. Equation of relative motion

6. our textbook
(p. 116) calls it Lz,
and the radius R.

10. One vector is so special that it had to be discovered and
re-discovered a couple of times:
Laplace vector = Laplace-Runge-Lenz vector
Pierre-Simon, Marquis de Laplace
( )

13. ..so
Apocenter Pericenter

15. 2

16. centrif.force
grav.force
(Make sure you
know how to derive
Keplerian speed !)

18. The full set of orbital elements (constants describing a Keplerian orbit)
includes the two omegas and the inclination angle I, describing orbit’s
orientation (shown below), two parameters describing its size and shape:
semi-major axis a and eccentricity e, and finally the time of perihelion
passage t0
Alternatively, the orbit could be described by 3 initial components of
position vector, and 3 velocity components (there are 6 variables
describing the position and velocity).

19. Motion in axisymmetric potentials. The epicyclic
approximation [epi (Gr.) = upon, kyklos (Gr.) ~cakra (Snskr.) =wheel]

20. (phi)
(R)
(Energy int.)

21. Minimum of
effective potential
for the particular
choice of E and Lz

22. Numerical solution for the
motion in the (R,z) plane
in the potential similar to
the one shown in
the previous slide.

23. Up to ~here
no approx.’s

24. Epicycles...
(z)
(approx.
z-equation)

25. (dR/dt=0)
(epic.
x-eq.)

27. Epicyclic
oscillation

29. Three important cases:1 2
sqrt(2)
in flat
rot. Curve
vc=const

31. Asymmetric drift in y is caused by the imbalance of the number
of stars on
orbits with the
guiding center
inside and
outside the
position of sun.
(The `insiders’
dominate! They
lag behind the sun
in their y-motion)

32. 2-D and 3-D orbits in nonaxisymmetric (triaxial) potentials