5th Lec

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5th Lec

• orbits

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Stellar Orbits

• Once we have solved for the gravitational
  potential (Poissons 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

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Motions in spherical potential
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In static spherical potentials star moves in a
plane (r,q)

• central force field
• angular momentum
• equations of motion are
• radial acceleration
• tangential acceleration

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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
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Energy Conservation (WHY?)
?eff
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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.

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6th Lec

• Phase Space

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• 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

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• 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

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• 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.

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Examples homogeneous sphere

• potential of the form
• using xr 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.

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homogeneous sphere cont.
B
A
A

• orbit is ellipse
• define t0 with xA, y0
• One complete radial oscillation A to -A
• azimuth angle only increased by

t0
B
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Radial orbit in homogeneous sphere

• equation for a harmonic oscillator
• angular frequency 2p/P

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Altenative equations in spherical potential

• Let

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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

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Tutorial Question 3 Show in Isochrone potential

• radial period depends on E, not L
• Argue , but for
• this occurs for large r, almost Kepler

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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