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

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.

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.

Equation of relative motion

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

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 (1749-1827)

..so Apocenter Pericenter

2

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

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).

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

(phi) (R) (Energy int.)

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

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

Up to ~here no approx.’s

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

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

Epicyclic oscillation

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

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)

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