The methods used in stellar dynamics are adopted from theoretical physics. The student is expected to have a fundamental understanding of statistical physics, (vector) calculus, electrodynamics, classical mechanics and should be able to program a computer

Teacher: Simon Portegies Zwart

Assistant: Moein Mosleh


Time & Place:

Each friday from 13:45 to 15:30 in HL427



Example Exam

- The example exam can be found here

- The REAL exam can be found here

Lecture information

Lecture 1


- Gravitational Force

- The 3 laws of Newton

-Energy conservation


- At which mass does gravity become important. (How many protons are required before they become self gravitating.)


A family of models for Spherical Stellar Systems

Lecture 2


- Potential Theory

- Newtons first and second theorm


- Given p(r) = R / (r^2(r+R^2)) where R = constant

  • Whhat is normalisation p(r)
  • What is M(r)
  • What is phi(r)
  • What can say about circular speed at:
    • r << R (deep in the potential well)
    • r >> R (far away from the center)


Standardised Units and Time Scales (Heggie et. al.)

Lecture 3


- What to do with Potentials

- Keplers problem (Orbits)

- Equation of motion (in circular coordinates)

- Angular Momentum


- Given F = -(GM / r2) , rdot = phidot(dr/dphi) , (1/r) = C cos (phi - phi0) + GM/L2

  • Demonstrate that: sin (phi - phi0) = (Lrdot) / eGM
  • Give an expression for V∞ (t -> ∞ ) = V∞(e, L)
  • Give an expression for lcord = a(1-e2)
    • Is it true?
    • What is special about it?


Ejection of Hypervelocity Stars By The (Binary) Block Hole In The Galactic Center (section 1)

Lecture 4


- Motion of the center of mass

- Deflection Angle

- Encounter time

- Coulomb logarithm



Discovery of an Unbound Hypervelocity Star in the Milky Way Halo



Read the following paper:

Mass Segregation, Relaxtion And the Coulomb Logarithm in N-body Systems

- Now measure the Coulomb logarithm in a computer simulation.

Make use of a N-body code, examples can be found, for example, on

Input files can be found here: (Note that these files are in the format as descripbed on )


Lecture 5


- Dynamical friction


Gravitational focusing and shielding of meteoroid streams


Lecture 6


- Distribution functions

- Continuity function

- Collisionless Boltzman Equation


Self-consistent determinations of the total amount of matter near the sun


Reading material for lecture 6 and 7

(Read in the order as written down here)

B&T first edition:

  • 8.2.1
  • 1.E.a
  • 4.1
  • 4.2
  • 2.2.3
  • 4.2.1b
  • 4.2.1d

B&T second edition:

  • 7.2.2
  • 7.2.3
  • 7.2.4
  • F.1.1
  • F.1.2
  • 4.1
  • 4.1.2
  • 2.3.3


Lecture 7


- Collisionless Boltzman Equation

- Jeans equation

- Distribution functions


Assignments: Error propegation in the Oort limit

- Rewrite the Jeans equations for a Cylindrical symmetric system

- Use the Poisson equations for a flattend stellar system to eliminate the potentioal term in the Jeans equations

- Calculate the Oort limit by introducing an error ' y' in the observed distance to a tracer populations

- How does the energy propagate in the calculation of the Oort limit?



The kinematics and dynamics of the galactic globular cluster system


Lecture 8


- Sphericla Stellar Systems

- Plummer

- Distribution functions


Assignments: None

Paper: None


Lecture 9




Stability of Lagrange points and core collapse

In the original 1987 edition of BT, chapt. 2 (p. 140), the authors
proved that the Lagrange points L4 and L5 of the rotating logarithmic
potential are always stable. Accoeding to Pfenniger (1990, A&A 230,
55) this proof was incorrect according to was modified in the new
edition of BT.

1. What was wrong with the original proof?

2. Settle any doubts we still might have by numerically solving an
initial-value problem of Newton's equation of motion in the
corotating frame of reference for the appropriately chosen
potential and rotation speed to demonstrate the instability around
L4 and L5. Use one of the N-body codes from to settle
the issue. Explain how you select initial conditions and
demonstrate what orbits are stable, and when they become unstable.
Start with the stability around L1, L2 and L3, and then construct
initial conditions for L4 and L5.

3. Integrate a 1000 particle Plummer sphere using one of the Nbabel
N-body codes until core collapse. Does the code crash?

4. Reproducing fig.8.1 (BT87) using this simulation.
Explain what happens.


On Core Collaps


Lecture 10


- Binaries


The evolution of a primordial binary population in a globular cluster

Monte Carlo Simulations of Globular Cluster Evolution. III. Primordial Binary Interactions


Added on 10 -december.

Page 140 of the old book. Used for assignment of lecture 9. Click here for the page