Disk Dynamics Julian Krolik Johns Hopkins University

Central Objects’ Radius of Influence r A ~ GM/v ∞ 2 ~ 4M 7 (v ∞ /100 km/s) -2 pc Bondi accretion rate ~ r A 2 ρ ∞ v ∞ Same dimensional analysis as for r A but with c substituted for v ∞ gives R g = GM/c 2 = 1.5 (M/Msun) km

Angular Momentum Limit Inflow requires angular momentum transport Central question of accretion dynamics: What moves angular momentum from one fluid element to another? j K ep /r 1 = 2 ! r s t op / j 2

At Angular Momentum Limit, Matter Settles into a Ring, then Spreads into a Disk Photons efficiently carry away energy, but not angular momentum

Conservation Laws in a Geometrically Thin, Axisymmetric Disk: Mass ! _ M = ¡ 2 ¼rv r t + r ( rv r § ) = 0

Another Conservation Law: t ¡ r 2 ­§ ¢ + r ¡ r 3 ­ v r § ¢ = 1 2 r w h ere G = t orque = r 2 Z dÁ Z d z T r Á ; an d T r Á = s t ress ! ¡ Z d z T r Á = _ M­ 2 ¼ · 1 ¡ j ¤ r 2 ­ ¸

Significance of Inner Boundary Condition net angular momentum accretion rate; if there is a stress inner edge, it determines both this and the net energy accretion rate. That is, the inner boundary condition determines the potential radiative efficiency _ M j ¤ =

A Third Conservation t µ ¡ 1 2 r 2 ­ 2 § ¶ + r µ ¡ 1 2 r 3 ­ 2 v r § ¶ = 1 r ( G­ ) But---in steady-state, 1 r ( ¡ ¼r 3 ­ 2 v r § ¡ G­ ) = _ M r · ( r 2 ­ 2 ) µ 3 2 ¡ j ¤ r 2 ­ ¶¸ < 0 If---matter is cold, and energy is conserved locally,

Orbital energy must be lost, first by dissipation into heat, and then by radiation if the disk is to stay thin S = 3 4 ¼ GM _ M r 3 µ 1 ¡ j ¤ r 2 ­ ¶

The Thermal Spectrum T e ® / S 1 = 4 / ³ M _ M = r 3 ´ 1 = 4 ¡ 1 ¡ j ¤ = r 2 ­ ¢ 1 = 4 W i t h L E ¾ T =( 4 ¼cr 2 ) = GM ¹ e = r 2, T e ® / ( _ M = _ M E ) 1 = 4 M ¡ 1 = 4 r ¡ 3 = 4 ¡ 1 ¡ j ¤ = r 2 ­ ¢ 1 = 4 k T ¤ » 1 ( M = M ¯ ) ¡ 1 = 4 k e V

What is the Torque? Shakura & Sunyaev (1973) dimensional analysis: stress ~ momentum flux ~ pressure: T = αp

What really is the torque? What is the specific angular momentum flux? Torque often imagined to be molecular viscosity, but this cannot be true Magnetic stress much more likely: Specific angular momentum flux determined by dynamics at inner boundary

Origin of magnetic stress: fieldline stretching by orbital shear Effective tension of magnetic fieldlines means stretching requires a force

Origin of Magnetic Field Amplification: the Magneto-Rotational Instability ¡ » ­ 8 > < > : < 0 ¸ < 2 ¼v A = ­ 1 ¸ » 2 ¼v A = ­ 2 ¼v A = ¸ ­ ¸ > 2 ¼v A = ­

General Relativistic Version Expressions for the disk’s mean torque and dissipation in terms of the accretion rate are basically the same, but for O(1) correction factors if---the integrated stress and the integrated heating rate are evaluated in the fluid frame But close to the black hole, orbital dynamics change qualitatively

Inner Disk Dynamics: the Effects of General Relativity There is an innermost stable circular orbit (often called the “ISCO”)

Inner Disk Dynamics: the Effects of Black Hole Spin Describe spin by a/M (-1 < a/M < 1) Even with j=0, there is rotation: (expression here valid in the equatorial plane) Inside r=2, total orbital energy can be <0! r g ­ ZAMO = c = 2 ( r = r g )( a = M ) ( r = r g ) 4 + ( a = M ) 2 ( r = r g ) ( a = M ) 2 r = r g

Inner Disk Dynamics: the Event Horizon At r=r h, matter and photons can only go inward r h = r g = 1 + p 1 ¡ ( a = M ) 2

Consequences of Relativistic Inner Disk Dynamics Surface density must decline from near the ISCO inward If stress stopped at ISCO, the binding energy there would be the total energy liberated by accretion In practice, continuity of magnetic forces may increase total liberated energy, diminish angular momentum brought to the black hole