This preview shows pages 1–3. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
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 OWERL 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; 1storder 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 ....
View
Full
Document
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.
 Spring '07
 SFGull

Click to edit the document details