Pseudo Potential in Kerr Geometry and itsApplications

Soumen Mondal 1,2

1. Korea Astronomy and Space Science Institute, Daejeon, Korea.

2. R. K. Mission Residential College, Narendrapur, Kolkata, India.

. – p.1/33

INTRODUCTION

In astrophysical problems it is often advantageous to use

pseudo-potentials instead of full general relativity. An well

known example is so-called ‘Paczyn’ski-Wiita potential’ in

which the gravitational potential of the star of mass M

namely,[

−GMr

]

is replaced by

ΦPW = −GM

r − 2GMc2

We propose an effective potential Φeff (gravitational pot.

+ centrifugal pot.) which adequately describes the space-

time around a rotating black-hole.

We call this as ‘Pseudo-Kerr potential’ .

. – p.2/33

The Pseudo-Kerr Potential: PK-Potential.

The form of our potential (in the units of[

GMBH

c2

]

) is,

Φeff = 1 − 1r − r0

+2a

(

l + ωr2)

r3 sin θ+

α2(

l + ωr2)2

2r2 sin2 θ

where, a is Kerr Parameter,

r0 = 0.04 + 0.97 a + 0.085 a2,

α2 = r2 − 2r + a2

r2 + a2 +2a2

r

, ω = 2ar3 + a2r + 2a2

where, l = −uφ is the conserved specific angular momentum for

the particle whereas l = −uΦ/ut is the conserved specific angular

momentum for fluid. Symbol uµ being the covariant components

of the four velocity.. – p.3/33

(3): Major Applications of Pseudo-Kerr Potential

1. Viscous Effect:Fluid Dynamics in the presence of viscosity to study shock wa ves, 2009 MNRAS.in Collaboration with Dr. Prasad Basu (Indian Institute of S cience, India) and Prof. S.K. Chakrabarti.

2. Smooth Particle Hydrodynamics Simulations:S.P.H simulations in 1-Dimension and in 2-Dimensions.in Collaboration with Prof. Diego Molteni (Dipartimento di Fisica e Tecnologie Relative, Palermo,Italy) and Prof. S.K. Chakrabarti.

3. Gravitational Wave:Emission of gravitational wave from a binary black hole in th e presence and in the absence ofaccretion disk, 2008 MNRAS.in Collaboration with Dr. Prasad Basu (Indian Institute of S cience, India) and Prof. S.K. Chakrabarti.

4. Radiation Hydrodynamics:Analysis of the spectrum for different Kerr parameter in the presence and in the absence of viscosity.in Collaboration with Dr. Samir Mandal (Ben-Gurion Univers ity of the Negev Beer-Sheva, Israel) andProf. S.K. Chakrabarti.

. – p.4/33

The Plan of the talk is the following

(1): Particle Dynamics in Pseudo-Kerr Potential,

(2): Fluid dymamics: Accretion around Black holes,

(3): Shocks in Accretion Flows: SPH results, QPOs

(4): Conclusions.

The results obtained from our potential match with the general

relativistic results for the values of Kerr parameter (a) ranging

from −1 to 0.8.

. – p.5/33

Effective Potential. .... Reference: MNRAS-06.

The curves are drawn for (from top to bottom) l = 3.2, (ms)3.4872(pk)3.4641(gtr), 3.6, 3.8,(mb)4.0201(pk)4.0(gtr), 4.2 respectively for the Kerr parameter value a = 0.0.

. – p.6/33

Effective Potential. .... Reference: MNRAS-06.

The curves are drawn for (from top to bottom) l = 2.7, (ms)2.8978(pk)2.9029(gtr), 3.1, 3.3,(mb)3.4237(pk)3.4142(gtr), 3.5 respectively.

. – p.7/33

Geometrical Points. .... Reference: IJP-05.

Fig. 2(a-b): (a) The locations of the marginally bound ( rmb) and marginally stable(rms) orbits and (b) the angular momenta lmb and lms at these orbits. Resultsgenerally agree for −1 < a < 0.8.

. – p.8/33

Marginally Stable Values of Effective Potential

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1a

0.7

0.75

0.8

0.85

0.9

0.95

1

Φms

Marginally Stable Values of Effective Potential or

Maximum Energy Dissipation with Kerr parameters ( a).

Results generally agree for −1 < a < 0.8.

. – p.9/33

Particle Trajectory: .... Reference: MNRAS-06.

-8 -4 0 4 8x

-8

-4

0

4

8

y rrr s- +

-12 -8 -4 0 4 8 12x

-12

-8

-4

0

4

8

12

y rrr- + s

0 4 8 12 16 20 24x

-12

-8

-4

0

4

8

12

z

r

rr

s

+

-

The particle trajectories using pseudo-Kerr potential. Th e parameters are a = 0.8, l = −4.5.. – p.10/33

Gravitational Wave Emission from binary black holes

0

20

40

60

80

100

120

140

160

0 100 200 300 400 500 600

radi

us(in

geo

met

ric u

nit)

time(year)

a=0.5 a=0.4 a=0.3

Figure shows infall time decreses with increase of Kerr parameter.

Ref: Basu, P.; Mondal, S.; Chakrabarti, S. K.,2008, MNRAS, 388.

. – p.11/33

Height of the Accretion Disk

The height of the disk in the vertical equilibrium (NT73) is given by

h(r) =(

p

ρ0

)12 r

32

Γ

[

(r2+a2)2−∆a2

(r2+a2)2+2∆a2

]12

where, Γ =[

1 − A2

∆r4 (Ω − ω)2]

−

12

and A = r4 + r2a2 + 2ra2

In our case the vertical height of the disk is given by

h(r) =(

p

ρ0

)12

r12

(

∂Φeff

∂R

)

−

12|R=r and R2 = r2 + z2

. – p.12/33

Height of the Accretion Disk

The height of the disk in the vertical equilibrium (NT73) is given by

h(r) =(

p

ρ0

)12 r

32

Γ

[

(r2+a2)2−∆a2

(r2+a2)2+2∆a2

]12

where, Γ =[

1 − A2

∆r4 (Ω − ω)2]

−

12

and A = r4 + r2a2 + 2ra2

In our case the vertical height of the disk is given by

h(r) =(

p

ρ0

)12

r12

(

∂Φeff

∂R

)

−

12|R=r and R2 = r2 + z2

. – p.12/33

Accretion around Black holes

(2): Fluid Dynamics

• Basic Equations: (Sonic point analysis, nature of the sonic

points and different solutions)

• Parameter Space : ( for different Kerr parameter. )

• Topology of the Flow : ( accretion and wind topology. )

• Shock Waves : ( [a] the region of the parameter’s space

responsible for the formation of shock waves and [b] the

variation of the location of shock waves with Kerr parameter

and viscosity of the flow.)

. – p.13/33

Shock conditions in the flow

The Rankine-Hugoniot conditions across the shock are given by

1. E+ = E−

(Energy)

2. M+ = M−

(Mass flux)

3. P+ + ρ+v2+ = P

−+ ρ

−v2−

(Total Pressure).

. – p.14/33

Importance of the shock in accretion flow

• The hot puffed up post-shock matter intercepts and inverse

-Comptonizes a significant portion of the soft photons from

the disk, to produce high energy power law tail.

• Excess thermal gradient force in the post-shock drives a part

of the incoming matter as outflow/jet along axis.

• Origin of QPO can be explain by shocks.

• It can produce non-thermal electrons through shock

acceleration, which is essential to explain the non-thermal

power-law spectrum in high energy.

Reference:

M. L. McConnell et al., 2002, ApJ, 572, 984

Ling & Wheaton, 2003, ApJ, 584, 399

Chakrabarti & Mandal, 2006, ApJ, 642, L49. – p.15/33

Variation of Energy with the Sonic points

E =(

1 + 12n+1

) Φ′

eff(r,l)

[ 2(2n+1)

H′

H ]+ Φeff (r, l).

The curves are drawn for (from top to bottom) l = 2.3, 2.488, 2.7,

2.9, 3.1, 3.3 ,3.5,3.7 respectively.

. – p.16/33

Variation of Sonic points with Energy and Ang. mom.

0.92

0.94

0.96

0.98

1

1.02

1.04

1.06

1.08

1.1

0.4 0.6 0.8 1 1.2 1.4

en

erg

y E

log10(x)0.5 1 1.5 2

Log(x)

1

2

3

4

5

sp. an

gula

r m

om

entu

m

variation of energy with distances

l=3.1l=3.0

l=2.3 a=0.5

variation of sp. ang. mom. with distances

1.008

1.0051.004kepla=0.5

. – p.17/33

Variation of Sonic points with Energy and Ang. mom.

0.92

0.94

0.96

0.98

1

1.02

1.04

1.06

1.08

1.1

0.4 0.6 0.8 1 1.2 1.4

en

erg

y E

log10(x)0.5 1 1.5 2

Log(x)

1

2

3

4

5

sp. an

gula

r m

om

entu

m

variation of energy with distances

l=3.1l=3.0

l=2.3 a=0.5

variation of sp. ang. mom. with distances

1.008

1.0051.004kepla=0.5

0.5 1 1.5 2 2.5Log10(X )

0.97

0.98

0.99

1

1.01

1.02

1.03

1.04

I M O

εε=1.005

l=2.9a=0.5

απ=0.0

c

. – p.17/33

Flow Topologies at various regions of the Parameters Space

Region of the Parameters Space available Standing and Oscil lating Shocks.

. – p.18/33

Different Regions of the Parameters Space

2 2.2 2.4 2.6 2.8 3 3.2 3.4 3.6 3.8 4 4.2 4.4Angular Momentum( )

1

1.01

1.02

1.03

1.04

1.05

1.06

1.07

1.08E

nerg

y( )

0.0

0.50

0.75

0.90

0.95

0.990.998

0.999

-0.999-0.50

l

ε

Region of the Parameters Space available Shocks in Accretio n flow.

. – p.19/33

Parameters space contains shocks in viscous accretion flow.

2.5 3 3.5 41

1.01

1.02

1.030.1

ε 0.05

0.0

a=0.0

lin

i

2 2.5 3 3.51

1.01

1.02

1.03

1.04

1.05

ε

l

i

in

0.0

0.05

0.1 a=0.5

The different regions of the parameters space shrink as theviscosity parameter αΠ(= 0.0, 0.05, 0.1) increases and moved upto the high energy regions. The parameter spaces also shiftstowards the region of higher energy and lower angular momentumwith the increase of Kerr parameter while their overall-shaperemain almost unchanged.

. – p.20/33

Shocks in Accretion and Wind flows.

The solution topology (arrowed curve) includes a standing shock.

. – p.21/33

Shocks in Accretion and Wind flows.

The solution topology (arrowed curve) includes a standing shock.

2 2.05 2.1 2.15 2.2Angular Momentum

0.75

0.8

0.85

0.9

0.95

1

1.05

Ene

rgy

5 10 15 20Shock location

1.01

1.02

1.03

NSA SA SWNSW

a=0.99

SA

SW

mb

ms

I

O

N O

I

O

OI

M

M

M

M

1

1

. – p.21/33

Chapter 6. 100π π π π

Fig. 2(a): Variation of the global solution topologies with the vis osity parameter ofthe a retion ow are shown for a moderate value of kerr parameter (a=0.5). The otherrequired parameters are also mentioned above.

0 0.5 1 1.5 20

1

2

3

M

0.5 1 0.5 1 0.5 1log(x) log(x) log(x) log(x)

α π=α π= 0.03 α π=0.040.0 =α π 0.075

l in= 3.245 x in= 3.1,0.5a= ,

Fig. 2(b): Variation of the topologies with the vis osity parameter for same valuesof kerr parameter but dierent values of initial angular momentum and inner soni point(mentioned above).. – p.22/33

General Questions

Q1: How does the shock location vary with

A. the flow parameter ?

B. the spin of the black hole ?

C. the viscosity of the flow ?

. – p.23/33

Analysis of Shock Locations

The variation of shock location with energy of the flow ( αΠ = 0).

(a) Accretion shock ( Solid Curve ),

(b) Winds shock ( Dashed Curve ).

Shocks form at a very low angular momentum for a=0.5.

. – p.24/33

Variation of Shock Locations with Kerr parameter

1 1.002 1.004 1.006 1.008 1.01 1.012Energy

0

20

40

60

80

Sho

ck L

ocat

ion

a=0.55

a=0.50

a=0.45

Shock in accretion

Shock in wind

ang.mom.=2.95

. – p.25/33

Variation of Shock Locations with Kerr parameter

0 10 20 30 40 50 60 70 80 90 100 110 120Shock location (X )

1

1.005

1.01

1.015

1.02

1.025

a=0.0a=-0.5a=-0.8

a=0.5

a=0.65

a=0.75

a=0.8

εs

s

l = 90% l ms

Shock locations shift rapidly towards BHfor higher values of Kerr parameter ( a).

. – p.26/33

Extreme shock locations Viscosity and Spin.

6 8 10 12 14 16 18Shock location (X )

2

2.25

2.5

2.75

3

3.25

3.5

An

gu

lar

mo

men

tum

(

)

(0, 0)

(0.25, 0)

(0.50, 0)

(0.75, 0)(0, 0.1)

l in

s

(0.5, 0.1)

(0.5, 0.05)

(0, 0.

05)

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1Kerr parameter(a)

0

2

4

6

8

10

12

14

16

18

20

Extr

em

e s

hock l

ocati

ons

r

ms

mb

r

Extreme shock forms close to the horizon for rapidly rotating BHbut move outwards with the increase of viscosity.

. – p.27/33

General Questions

Q2: Is there any correlation between QPO

frequency and spin of the black hole ?

If yes, then

Q3: Is it possible to estimate the spin of the

black hole from its QPO frequency ?

. – p.28/33

SPH simulation produces Shocks at different time steps

-1

0

1

2

3

4

5

6

7

8

9

0 5 10 15 20 25 30 35 40 45 50 55

-30

-20

-10

0

10

20

30

0 10 20 30 40 50

M

X

Y

X

a=0.0 a=0.0

Smooth Particle Hydrodynamics Simulation shows Standing S hocks in the accretion flow

. – p.29/33

SPH simulations consistant with analytic solution.

0

0.5

1

1.5

2

2.5

3

3.5

0 5 10 15 20 25 30 35 40

--->

Ma

ch n

o. M

---> r/2

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

3000 3500 4000 4500 5000 5500 6000 6500

--->

En

erg

y

---> Time

Ene

rgy

Time

Shocks at different time steps.

. – p.30/33

SPH simulations produces Oscillation of Post-shock matter

0

0.5

1

1.5

2

2.5

0 5 10 15 20 25 30 35

0.3

0.4

0.5

0.6

0.7

0.8

22 23 24 25 26 27

Radius

Mac

h N

o.

. – p.31/33

Summary:Important results and conclusions :

• Accretions/winds occur at lower values of angular momentum

and higher energy as spin of the black hole increases.

• Flow variables such as, shock locations, sonic points etc.

shift rapidly towards the black hole for highly spining Kerr

black hole.

• Extreme shocks occure in case of inviscid flow. Shocks

moves outwars with the increase of viscosity of the flow .

• These extreme shocks may important to introduce oscillations

in the disk flow very close to the centre and therefore, may

excite QPOs mechanism observed in many Galactic Micro-

quasars.

. – p.32/33

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THANK YOU. . – p.33/33