1. 5th Lec
orbits

2. Stellar Orbits
Once we have solved for the gravitational potential (Poisson’s eq.) of a system we want to know: How do stars move in gravitational potentials?
Neglect stellar encounters
use smoothed potential due to system or galaxy as a whole

3. Motions in spherical potential

4. In static spherical potentials: star moves in a plane (r,q)
central force field
angular momentum
equations of motion are
radial acceleration:
tangential acceleration:

5. Orbits in Spherical Potentials
The motion of a star in a centrally directed field of force is greatly simplified by the familiar law of conservation (WHY?) of angular momentum.
Keplers 3rd law
pericentre
apocentre

6. Energy Conservation (WHY?)
eff

7. Radial Oscillation in an Effective potential
Argue: The total velocity of the star is slowest at apocentre due to the conservation of energy
Argue: The azimuthal velocity is slowest at apocentre due to conservation of angular momentum.

8. 6th Lec
Phase Space

9. at the PERICENTRE and APOCENTRE
There are two roots for
One of them is the pericentre and the other is the apocentre.
The RADIAL PERIOD Tr is the time required for the star to travel from apocentre to pericentre and back.
To determine Tr we use:

10. The two possible signs arise because the star moves alternately in and out.
In travelling from apocentre to pericentre and back, the azimuthal angle  increases by an amount:

11. The AZIMUTHAL PERIOD is
In general will not be a rational number. Hence the orbit will not be closed.
A typical orbit resembles a rosette and eventually passes through every point in the annulus between the circle of radius rp and ra.
Orbits will only be closed if is an integer.

12. Examples: homogeneous sphere
potential of the form
using x=r cosq and y = r sinq
equations of motion are then:
spherical harmonic oscillator
Periods in x and y are the same so every orbit is closed ellipses centred on the centre of attraction.

13. homogeneous sphere cont.B A A
orbit is ellipse
define t=0 with x=A, y=0
One complete radial oscillation: A to -A
azimuth angle only increased by š
t=0 B

14. Radial orbit in homogeneous sphere
equation for a harmonic oscillator
angular frequency 2p/P

15. Altenative equations in spherical potential
Let

16. Kepler potential Equation of motion becomes: with and thus
solution: u linear function of cos(theta):
with and thus
Galaxies are more centrally condensed than a uniform sphere, and more extended than a point mass, so

17. Tutorial Question 3: Show in Isochrone potential
radial period depends on E, not L
Argue , but for
this occurs for large r, almost Kepler

18. Helpful Math/Approximations (To be shown at AS4021 exam)
Convenient Units
Gravitational Constant
Laplacian operator in various coordinates
Phase Space Density f(x,v) relation with the mass in a small position cube and velocity cube