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5 th Lec

Explore how stars move in gravitational potentials and understand the simplified motion in spherical potentials. Study the conservation laws of energy and angular momentum and analyze the radial oscillations in effective potentials. Learn about the phase space at pericentre and apocentre and the periodicity of stellar orbits.

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5 th Lec

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  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 PERIODTris 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: • 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

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