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Physics 681: Solar Physics and Instrumentation – Lecture 19 Carsten Denker NJIT Physics Department Center for Solar–Terrestrial Research Oblateness A rotating, non-rigid body must have an oblateness Consider an inviscid star with rigid rotation Equilibrium in a frame of rest v 0,0, r sin vv P Constant pressure P at the surface 1 2 r sin const. 2 Continuous transition of Φ to the outer gravitational potential ext November 8, 2005 2 Gm r 1 J 2 P2 r r Center for Solar-Terrestrial Research Approximation of the oblate surface ignoring differential rotation while including a quadrupole moment arising from a more rapidly rotating core as compared to the surface rotation r r 1 cP2 and r / r requator rpole / r 3c / 2 with sin 2 2 1 P2 / 3 1 r 3 2 Gm 1 P2 r Gm J 2 P2 const. 1 cP2 r 2 Gm r 1 r 3 J 2 with g 2 r 2 g 2 r Synodic angular velocity (Carrington): Ωsyn = 2.67 10-6 Hz Sidereal angular velocity: Ωsid = 2.87 10-6 Hz Oblateness: 1.04 10-5 or 14 km or 0.02 arcsec The oblateness is difficult to measure! Perihelion of Mercury & General Theory of Relativity Oblateness seems to be related to only the surface rotation November 8, 2005 Center for Solar-Terrestrial Research Oblateness of Jupiter Equatorial radius: Re = 71,370 km Polar radius: Rp = 66,750 km Oblateness: (Re Rp) / Re = 0.0648 First order correction term in gravitational potential: U / m 2 4 GM Re Re ( ) 1 J 2 P2 cos J 4 P4 cos r r r P0 (cos ) 1 1 P2 (cos ) 3cos 2 1 2 1 P4 (cos ) 34 cos 4 30 cos 2 3 8 November 8, 2005 Legendre Polynomials Center for Solar-Terrestrial Research Gravitational Moments J2: oblateness and moment of inertia J4: mass distribution in outer regions, equatorial bulge, and planets thermal structure November 8, 2005 Center for Solar-Terrestrial Research Rotational History Determine the evolution of a rotating star from its initial angular velocity (T Tauri stars ≈ 15 km/s) The initial angular momentum of the Sun was much larger than that of the whole present solar system Magnetic breaking! The rotation rate of main-sequence stars similar to the Sun decreases with age (stellar activity Ca II H & K emission) Stars earlier than F5 have no deep outer convection zone no magnetic field generation no breaking rapid rotators (O to F stars rotate up to 100 times faster than the Sun) Pre-main-sequence solar models are fully convective Turbulent friction leads to a uniform angular velocity Total angular momentum: J0 = 5 1042 to 5 1043 km m2/s (more than 260 times today’s value) November 8, 2005 Center for Solar-Terrestrial Research Skumanich (1972) November 8, 2005 Center for Solar-Terrestrial Research Torques Magnetic breaking Material escaping the rotating solar surface carries some angular momentum with it The magnetic lines of force act as a lever arm that forces the escaping material to rotate rigidly with the solar surface far out to the Alfvén radius rA Beyond rA the magnetic field is too weak to enforce rigid rotation Total loss of angular momentum dJ dm rA2 dt dt Parameterization of torque by a power law dJ K dt November 8, 2005 Center for Solar-Terrestrial Research Angular momentum transport in the interior Uniform rotation is maintained as long as the Sun was fully convective Internal torque required for angular momentum transport is provided by turbulent friction slowing the Sun as a whole in the early phase of the evolution Development of a radiative core only the outer convective shell rotates uniformly and loses angular momentum Core contracts and rotates more rapidly Splitting of p-mode frequencies? Oblateness? Stability? No evidence for a fast rotating core! Instabilities in the presence of strong shear motion! Internal magnetic torque Bp r / t 1010 T 1/ 2 Magentic torque also acts as a restoring force of torsional oscillations November 8, 2005 Center for Solar-Terrestrial Research Evolution of Solar Rotation Pinsonneault et al. (1989) November 8, 2005 Center for Solar-Terrestrial Research