Analytical Model of Disk Evaporation and State Transitions in Accreting Black Holes
Hyerin Cho, Ramesh Narayan
Abstract
Abstract State transitions in black hole X-ray binaries are likely caused by gas evaporation from a thin accretion disk into a hot corona. We present a height-integrated version of this process, which is suitable for analytical and numerical studies. With radius r scaled to Schwarzschild units and coronal mass accretion rate <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msub> <mml:mrow> <mml:mover accent="true"> <mml:mrow> <mml:mi>m</mml:mi> </mml:mrow> <mml:mrow> <mml:mo>̇</mml:mo> </mml:mrow> </mml:mover> </mml:mrow> <mml:mrow> <mml:mi>c</mml:mi> </mml:mrow> </mml:msub> </mml:math> to Eddington units, the results of the model are independent of black hole mass. State transitions should thus be similar in X-ray binaries and an active galactic nucleus. The corona solution consists of two power-law segments separated at a break radius r b ∼ 10 3 ( α /0.3) −2 , where α is the viscosity parameter. Gas evaporates from the disk to the corona for r > r b , and condenses back for r < r b . At r b , <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msub> <mml:mrow> <mml:mover accent="true"> <mml:mrow> <mml:mi>m</mml:mi> </mml:mrow> <mml:mrow> <mml:mo>̇</mml:mo> </mml:mrow> </mml:mover> </mml:mrow> <mml:mrow> <mml:mi>c</mml:mi> </mml:mrow> </mml:msub> </mml:math> reaches its maximum, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msub> <mml:mrow> <mml:mover accent="true"> <mml:mrow> <mml:mi>m</mml:mi> </mml:mrow> <mml:mrow> <mml:mo>̇</mml:mo> </mml:mrow> </mml:mover> </mml:mrow> <mml:mrow> <mml:mi>c</mml:mi> <mml:mo>,</mml:mo> <mml:mi>max</mml:mi> </mml:mrow> </mml:msub> <mml:mo>≈</mml:mo> <mml:mn>0.02</mml:mn> <mml:mspace width="0.25em"/> <mml:msup> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>α</mml:mi> <mml:mrow> <mml:mo stretchy="true">/</mml:mo> </mml:mrow> <mml:mn>0.3</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:mrow> <mml:mn>3</mml:mn> </mml:mrow> </mml:msup> </mml:math> . If at r ≫ r b the thin disk accretes with <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msub> <mml:mrow> <mml:mover accent="true"> <mml:mrow> <mml:mi>m</mml:mi> </mml:mrow> <mml:mrow> <mml:mo>̇</mml:mo> </mml:mrow> </mml:mover> </mml:mrow> <mml:mrow> <mml:mi>d</mml:mi> </mml:mrow> </mml:msub> <mml:mo><</mml:mo> <mml:msub> <mml:mrow> <mml:mover accent="true"> <mml:mrow> <mml:mi>m</mml:mi> </mml:mrow> <mml:mrow> <mml:mo>̇</mml:mo> </mml:mrow> </mml:mover> </mml:mrow> <mml:mrow> <mml:mi>c</mml:mi> <mml:mo>,</mml:mo> <mml:mi>max</mml:mi> </mml:mrow> </mml:msub> </mml:math> , then the disk evaporates fully before reaching r b , giving the hard state. Otherwise, the disk survives at all radii, giving the thermal state. While the basic model considers only bremsstrahlung cooling and viscous heating, we also discuss a more realistic model that includes Compton cooling and direct coronal heating by energy transport from the disk. Solutions are again independent of black hole mass, and r b remains unchanged. This model predicts strong coronal winds for r > r b , and a T ∼ 5 × 10 8 K Compton-cooled corona for r < r b . Two-temperature effects are ignored, but may be important at small radii.