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Advanced Celestial Mechanics. Questions. Question 1 Explain in main outline how statistical distributions of binary energy and eccentricity are derived in the three-body problem. Escape cone. Density of escape states. Question 2. Calculate the potential above an infinite plane. . . .

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advanced celestial mechanics questions

Advanced Celestial Mechanics. Questions

Question 1

Explain in main outline how statistical distributions of binary energy and eccentricity are derived in the three-body problem.

question 2
Question 2

Calculate the potential above an infinite plane.

  • .
question 3
Question 3
  • Write the acceleration between bodies 1 and 2 in the three-body problem using only the relative coordinates i.e. in the Lagrangian formulation. Use the symmetric term W.
question 4
Question 4
  • Show that in the two-body problem the motion takes place in a plane. Derive the constant e-vector, and derive its relation to the k-vector. Draw an illustration of these two vectors in relation to the orbit.
question 5
Question 5
  • Define true anomaly and eccentic anomaly in the two-body problem. Derive the transformation formula between these two anomalies. Define also the mean anomaly M.
  • .
question 6
Question 6
  • Define the scattering angle in the hyperbolic two-body problem, and derive its value using the eccentricity. Derive the expression of the impact parameter b as a function of the scattering angle.
question 7
Question 7
  • Derive the potential at point P, arising from a source at point Q, a distance r’ from the origin. Define Legendre polynomials and write the first three polynomials.
question 8
Question 8
  • Show that the shortest distance between two points is a straight line using the Euler-Lagrange equation.
question 9
Question 9
  • Write the Lagrangian for the planar two-body problem in polar coordinates, and write the Lagrangian equations of motion. Solve the equations to obtain Kepler’s second law.
question 10
Question 10
  • If the potential does not depend on generalized velocities, show that the Hamiltonian equals the total energy. Use Euler’s theorem with n=2.
question 11
Question 11
  • Write the Hamiltonian in the planar two-body problem in polar coordinates. Show that the is a constant.
  • .
question 12
Question 12

The canonical coordinates in the two-body problem are

Use the generating function

To derive Delaunay’s elements.

question 13
Question 13
  • Show that the Hamiltonian in the three-body problem is
  • Write the Hamiltonian equations of motion for the three-body problem
  • .
question 14
Question 14

Show that in the hierarchical three-body problem

Make use of the canonical coordinates and the Hamiltonian