1. Dependence of Multi-Transonic Accretion on Black Hole Spin
Paramita Barai
Graduate Student, Physics & Astronomy
Georgia State University
Atlanta, GA, USA
06/29/2005
12/24/2018
P. Barai, GSU

2. Contents Introduction & Motivation Our system formulations Results
What are we doing?
Our system formulations
Results
Conclusions
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3. Introduction Mach number Accretion flow Transition
Subsonic: M < 1
Supersonic: M > 1
Transonic: crosses M = 1
Transition
Smooth -- Sonic point
Discontinuous -- Shock
Multi-transonic flow  3 sonic points
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4. Motivation
BH inner boundary condition : Supersonic flow at Event Horizon (EH)
Far away from EH – subsonic
Hence BH accretion is essentially transonic
Except cases where already supersonic initially
Multi-transonic flow  Shock Formation
Transonic Flow is relevant in various astrophysical situations:
Black Hole (BH) or Neutron Star (NS) accretion
Collapse / Explosion of stars
Solar / Stellar winds
Formation of protostars & galaxies
Interaction of supersonic galactic (or extragalactic) jets with ambient medium
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5. Draw fig in board
BH spin & angular momentum of accreting matter – directions
Show prograde & retrograde accretion flow explicitly (or, co & counter rotating BH)
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6. Highlights
Study variation of dynamic and thermodynamic properties of accretion flow very close to event horizon & their dependence on spin of BH
Found : Higher shock formation possibility in retrograde accretion flow (i.e. BH spin is opposite to rotation of accreting matter) as compared to prograde flow
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7. Can be done at any length scale Notations Gravitational Radius
What do we do?
Formulate & solve conservation equations governing general relativistic, multi-transonic, advective accretion flow in Kerr metric
Calculate flow behavior very close ( 0.01 rg) to event horizon as function of a
Can be done at any length scale
Notations
Gravitational Radius
Black Hole Spin / Kerr Parameter = a
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8. Describing Our System No self-gravity, no magnetic field
Units: G = c = MBH = 1
Boyer-Lindquist coordinates (– + + +)
Observer frame corotating with accreting fluid
 = Specific angular momentum of flow -- aligned with a
Stationary & axisymmetric flow
Euler & Continuity eqns :
Polytropic equation of state
K ~ specific entropy density
 = adiabatic index, n = polytropic index
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9. Our System … Specific proper flow enthalpy, h
Polytropic sound speed, as
Frame dragging neglected
Weak viscosity limit
Very large radial velocity close to BH  Timescale (viscous >> infall)
Effect of Viscosity     Flow behavior as function of  provides information on viscous transonic flow
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10. Metric & Others
Kerr metric in equatorial plane of BH (Novikov & Thorne 1973)
Angular velocity, 
Normalization:
4th Component of velocity:
Just glance thru the eqns (this slide & the 2 next) while giving the talk finally – no need to say in detail ..
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11. Accretion Disk Vertically integrated model (Matsumoto et al. 1984)
Flow in hydrostatic equilibrium in transverse direction
Equations of motion apply to equatorial plane of BH
Thermodynamic quantities vertically averaged over disk height
Calculate quantities on equatorial plane
1-dimensional quantities, vertically averaged
Disk height, H(r) from Abramowicz et al. 1997
Show fig in black-board: showing the disk heights –H/2 to +H/2,
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12. Conserved Quantities Specific energy (including rest mass)
Mass accretion rate
Entropy accretion rate
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13. Quantities at Sonic Point
Radial fluid velocity
Sound speed
Quadratic eqn for (du/dr)c
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14. Sonic Point Quantities
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16. Results For some [P4] -- get 3 sonic points on solving eqn
rout > rmid > rin
rout, rin : X-type sonic points
rmid : O-type sonic pt (unphysical – no steady transonic soln passes thru it)
Multitransonic Accretion: (rin) > (rout)
Multitransonic Wind : (rin) < (rout)
General astrophysical accretion  Flow thru rout
Flow thru rin possible only in case of a shock
If supersonic flow thru rout is perturbed to produce entropy = [(rin) – (rout)], it joins subsonic flow thru rin forming a standing shock
Shock details from GR Rankine-Hugoniot conditions
Don’t show this slide to audience .. Just remember what is written in the text here .. Show previous fig & say this text in words, showing the fig
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18. E- Multi-Transonic Space For Different 
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20. Variation of E- Multi-Transonic Space for Change in Kerr Parameter, a
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22. Angular Momentum We found:
At higher values of angular momentum multi-transonicity is more common for retrograde flow
Weakly rotating flows, found in several physical situations:
Detached binary systems fed by accretion from OB stellar winds
Semidetached low-mass non-magnetic binaries
Supermassive BHs fed by accretion from slowly rotating central stellar clusters
Turbulence in standard Keplerian accretion disk
Draw Fig in Black-Board : BH spin (a) in 1 direction, & accretion disk angular momentum is another or same direction depending on if retro or prograde accretion flow
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23. Important Conclusion: Differentiating Accretion Properties of Co & Counter Rotation of Central Black Hole
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27. Differentiating Pro & Retro-Grade Flows
Prograde Accretion Flow
Multi-transonic regions at lower value of 
Retrograde Accretion Flow
Multi-transonicity much more common
Covers higher value of angular momentum ()
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28. Terminal Behavior of Accretion Variables
Terminal value of accretion variable, A
A = A (at r = re + )
Integrate flow from rc down to r  get A
Studied variation of A with a  BH spin dependence of accretion variables very close to event horizon
Can be done for any r  dependence of flow behavior on BH spin at any radial distance from singularity
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31. Discussion & Summary
Study dependence of multi-transonic accretion properties on BH spin at any radial distance / accretion length scale
Very close to event horizon
Possible shock location
Found non-trivial difference in the accretion behavior for co & counter rotating BH
Prograde flow (co-rotating BH) – low 
Retrograde flow (counter-rotating BH) – high  & greater shock formation possibility
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32. Astrophysical Implications
Preliminary step towards understanding how BH spin affects astrophysical accretion and related phenomenon
Shock waves in BH accretion disks must form through multi-transonic flows
Study of the post shock flow helpful in explaining:
Spectral properties of BH candidates
Cosmic (galactic & extragalactic) jets powered by accretion (formation & dynamics)
Origin of Quasi Periodic Oscillations in galactic sources
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33. Thank You All
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34. Observational Consequences
Debate: Determining BH spin from observations
Most popular approach: study of skew shaped fluorescent iron lines
Our approach presents the potential to deal with this problem
We predict behavior of flow properties
For hot, low- prograde accretion flow & high- retrograde flow
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