Handout_2 - Classical Dynamics and Fluids P 25 O RBITS C...

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Unformatted text preview: Classical Dynamics and Fluids P 25 O RBITS C ENTRAL F ORCE F IELD F O P r v Particle moving in central force field. Potential U ( r ) yields radial force F =- U =- d U d r e r . Motion remains in the plane defined by position vector r and velocity v . No couple from central force angular momentum is conserved: mr 2 = J = constant (Kepler II) Total energy is conserved: E = U ( r ) + 1 2 m ( r 2 + r 2 2 ) = 1 2 m r 2 + U ( r ) + J 2 2 mr 2 The effective potential U eff ( r ) has a contribution from the angular velocity. U eff ( r ) = U ( r ) + J 2 2 mr 2 The effective potential has a centrifugal repulsive term 1 r 2 Classical Dynamics and Fluids P 26 N ON- CIRCULAR O RBITS IN P OWER-L AW F ORCE We can gain a lot of insight into orbits by studying the force law F =- Ar n with A positive, so force is attractive. The effective potential is then U eff ( r ) = Ar n +1 n + 1 + J 2 2 mr 2 the only exception being n =- 1 ( U eff then contains a log r term). The centrifugal potential is repulsive and r- 2 . A plot of U eff ( r ) shows which values of the index n lead to bound or unbound orbits, and which lead to stable or unstable orbits. For n - 1 (including the log r potential), the potential increases as r and the orbits are bound and stable. For n <- 3 the attraction at r overcomes the centrifugal repulsion and the orbits are not stable (this is the case for the central region of black holes in GR). For- 3 < n < 1 the potential goes to zero at r = and the orbits can either be bound or unbound. The potential is qualitatively different for different values of n : n >- 1 : Orbit at r stable. All orbits bound.- 3 < n <- 1 : Orbit at r stable. Unbound orbits for E > . n <- 3 : Orbit at r unstable. Will go to r = 0 or r = Classical Dynamics and Fluids P 27 N EARLY C IRCULAR O RBITS IN C ENTRAL P OWER- LAW F ORCE U Let F =- Ar n , n = index, with common cases n = +1 (2D SHM) and n =- 2 (gravity, electrostatics). U eff = Ar n +1 n + 1 + J 2 2 mr 2 Nearly circular orbits are oscillations/perturbations about r . Taylor expansion of U eff gives U eff = U + ( r- r ) d U eff d r r + 1 2 ( r- r ) 2 d 2 U eff d r 2 r + d U eff d r is zero at r giving d U eff d r = Ar n- J 2 mr 3 = 0 at r The second derivative of U eff is d 2 U eff d r 2 = nAr n- 1 + 3 J 2 mr 4 = ( n + 3) J 2 mr 4 at r . Using the energy method d d t ( 1 2 m r 2 + U eff ) = 0 we get the SHM equation m r + ( n + 3) J 2 mr 4 ( r- r ) | {z } = 0; 1st-order Taylor expansion of d U eff / d r i.e. SHM about r with angular frequency p = n + 3 J mr 2 Classical Dynamics and Fluids P 28 N EARLY C IRCULAR O RBITS IN P OWER- LAW F ORCE II How does p of the perturbation compare with c of the circular orbit at r ? c = = J/mr 2 . Therefore p = n + 3 c ....
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This note was uploaded on 05/09/2009 for the course DAMTP NST 1B Phy taught by Professor Sfgull during the Spring '07 term at Cambridge.

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Handout_2 - Classical Dynamics and Fluids P 25 O RBITS C...

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