Parameter study of QPO models - TIFRPrerna Rana Parameter study of QPO models January 21, 2016 8 /...

Parameter study of QPO models

Prerna Rana (JRF)Indian Institute of Astrophysics

Supervisor :Prof. Arun Mangalam

Indian Institute of Astrophysics

January 21, 2016

Prerna Rana Parameter study of QPO models January 21, 2016 1 / 34

Table of contents

Introduction

QPO models in X-ray binaries

Results of parameter study of QPO modelsI E, J orbit classificationI Circular orbitsI e, r space calculationsI Emission region

Model of QPOs in AGNs and microquasarsI Ratios due to photon propagation

Conclusions and Future work


Introduction

Quasi periodic oscillations are broad peaks observed in Fourier power spectrum ofXRBs, microquasars and AGNs.

QPOs - few Hz to kHz in XRBs, kHz in microquasars and µHz to mHz in AGNs.

3:2 ratio of pairs of QPOs.

I XTE J1550-564 - 280 Hz and 180Hz (Belloni 2012)GRO J1655-40 - 450Hz and 300Hz (Belloni 2012)Sco X-1 - 1.1kHz and 0.5Hz (M.Van Der Klis 2005)4U 1636-536 - 1175Hz and 880Hz (Stella Vietri 1998)

Accretion of hot blobs in Schwarzschild geometry and constraining eccentricity oforbits with distance to get required ratio.

In Kerr geometry, constraining spin and distance to get required ratio.


Motivation for our models: kinematic oscillations in theCorona and jet

Figure: Image Courtesy: (Mohan and Mangalam 2015)


Models of QPOs in XRBs

1 Relativistic Precession model

2 Tidal disruption model

3 Non-linear resonance model

4 Warped disc model


1. Relativistic Precession model

Sco X-1 and 4U1608-52 (neutron star LMXRBs) - Separation of QPOs found to decreasewith increasing luminosity.

Higher frequency peak - ν2 = νφ near inner disk boundary.

Lower frequency peak - Periastron Precession frequency of inner edge of accretion disk.(Stella Vietri (1999))

ν1 = νper = νφ − νr (1)

Figure: Change of ∆ν with νφ for different masses 2.2M, 2.0M and 1.8M. Image courtesy: Stella Vietri (1999)


2. Tidal disruption model

Blobs of matter orbiting a Schwarzschild black hole get deformed by tidalinteraction.

Emission from tidally disrupted blob gives rise to observable phenomenon.(Germana et al. (2009))

Observed Fourier power spectrum of XTE J1550-564 was reproduced by thismodel.

Twin peak observed corresponds to

νuppνlow

=νr + νφνφ

(2)


3. Non-linear resonance model

Ratios are non-linear resonances in oscillations of accretion disks. (Abramowicz et al.2004)

Non-linear resonance - For orbital motion of a particle

U(r , l) =1

2

(∂2U

∂r2

)

(r − r)2 +

1

6

(∂3U

∂r3

)

(r − r)3 + ... (3)

Small oscillations around r described by non-linear differential equations.

δθ + ω2θ(1 + hδr)δθ = 0

δθ + ω2θ(1 + hCos(ωr t))δθ = 0

ωr

ωθ=νr

νθ=

2

n(4)

where, n = 1, 2, 3, ... and this is called parametric resonance.

νlow

νupp=νr

νθ=

2

3(5)

Similar parametric resonance between νφ and νr has been suggested.

νlow

νupp=νr

νφ=

2

3(6)


4. Warped disc model

Interaction between small amplitude disk oscillations and relativistic disk gives riseto observable phenomenon.

Figure: Warped disc model. Image courtesy: Kato (2004)

Dispersion relation (Kato (2001))

[(ω −mΩ)2 − κ2][(ω −mΩ)2 − nΩ2⊥] = c2

s k2(ω −mΩ)2 (7)

κ = radial epicyclic frequency, Ω⊥ = frequency of vertical oscillation.k = wavenumber in radial direction, cs = sound speedΩ = angular velocity, m, n = modes of oscillation in radial and vertical direction.


Orbits in Schwarzschild geometry

Figure: Bound orbit in Schwarzschild geometry.


Figure: Scattering orbit in Schwarzschild geometry.


Figure: Innermost stable circular orbit in Schwarzschild geometry.


Figure: Plunge orbit in Schwarzschild geometry.


Circular orbits - Schwarzschild geometry

For circular orbit case, e = 0, equations become,

νφ =

(M

4π2r3

)1/2

(8)

νr = νφ

(1−

6M

r

)1/2

(9)

8 10 12 14 16 18 201

2

3

4

5

6

x (r/M)

Rat

io

Frequency ratio v/s x for circular orbits

Relativistic Precession model

Tidal Disruption model

Non-linear resonance(Keplerian) model

Figure: Frequency ratio for all models for circular orbits in Schwarzschild geometry as function of x .


Kinematic oscillation frequencies - Schwarzschildgeometry

For bound orbits in Schwarzschild metric, eccentricity 0 < e < 1, µ =M

a, l = a(1− e2),

where a is Semi major axis of orbit.

Orbital frequency of a particle,(S. Chandrasekhar, The Mathematical Theory of BlackHoles (1983))

νφ =(M(1− e2))1/2

l3/2[1− e2 − µ(3 + e2)]1/2f −11 (10)

where,

f1 =

∫ χc

0

dχ(1− e2)3/2

[1− e2 − 2µ(3 + e cosχ)]1/2[1− e2 − 2µ(1 + e cosχ)](1 + e cosχ)2(11)

where χc is calculated as

χc = π − 2ψc


where ψc is calculated such that

π(1− 6µ+ 2µe)1/2 =

∫ ψc

π/2

dγ

(1−m sin2 γ)1/2(12)

where, γ = π/2 corresponds to the periastron point of the orbit which is starting point of theintegration and γ = ψc corresponds to the point where particle completes one orbit (φ = 2π).

Radial epicyclic frequency of a particle

νr =(M(1− e2))1/2

l3/2[1− e2 − µ(3 + e2)]1/2f −12 (13)

where,

f2 =

∫ 2π

0

dχ(1− e2)3/2

[1− e2 − 2µ(3 + e cosχ)]1/2[1− e2 − 2µ(1 + e cosχ)](1 + e cosχ)2(14)


1. Relativistic precession model

νupp

νlow=

νφ

νφ − νr=

1

1− (f1/f2)(15)

0.0 0.2 0.4 0.6 0.8 1.0

6

8

10

12

14

16

18

20

e

x(a/

M)

Relativistic Precession model

1.2

1.4

1.6

1.8

Figure: Frequency ratio contours of RP model as function of eccentricity and x in Schwarzschild geometry.



νupp

νlow=νφ + νr

νφ= 1 +

f1

f2(16)

0.0 0.2 0.4 0.6 0.8 1.0

6

8

10

12

14

16

18

20

e

x(a/

M)

Tidal disruption model

1.2

1.3

1.4

1.5

1.6

1.7

Figure: Frequency ratio contours of TD model as function of eccentricity and x in Schwarzschild geometry.


3. Non-linear resonance model - Keplerian

νupp

νlow=νφ

νr=

f2

f1(17)

0.0 0.2 0.4 0.6 0.8 1.0

6

8

10

12

14

16

18

20

e

x(a/

M)

Non-linear Resonance model- Keplerian

1.2

1.4

1.6

1.8

Figure: Frequency ratio contours of NL- Keplerian model as function of eccentricity and x in Schwarzschild geometry.


Kinematic oscillation frequencies - Kerr geometry

In the Kerr geometry for rotating black holes, particles in circular orbits have

Orbital frequency

νφ =1

2π

(GM

rg

)1/2

(x3/2 ± a)−1 (18)

Radial epicyclic frequency

νr = νφ(1− 6x−1 ± 8ax−3/2 − 3a2x−2)1/2 (19)

Vertical epicyclic frequency

νθ = νφ(1∓ 4ax−3/2 + 3a2x−2)1/2 (20)

where, x = r/(GM/c2), rg = (2GM/c2), a is spin of black hole.

Upper sign for Prograde and lower sign for Retrograde orbits.


Emission region - Mohan and Mangalam 2015

A constant spin Ω = dφ/dt = uφ/ut in arotating metric has limits

Ωmin < Ω < Ωmax

which comes from timelike wordline of par-ticles i.e. ~u.~u = −1.

For Kerr case, this leads to the condition

(4a4r + 8a4 + 2a3r7/2 + 8a3r2 + 8a3r5/2 + 8a3r3/2 + a2r5

−a2r4 − 2a2r3 + 2ar11/2 + 4ar9/2 + r7 − 3r6) > 0(21)

And region outside event horizon

r > 1 + (1− a2)1/2 (22)


1. Relativistic precession modelνupp

νlow=

νφ

νφ − νr(23)

3 4 5 6 7 8

0.0

0.2

0.4

0.6

0.8

1.0

x(r/M)

a

Relativistic Precession model (Prograde)

1.2

1.4

1.6

1.8

4 6 8 10 12

0.0

0.2

0.4

0.6

0.8

1.0

x(r/M)

a

Relativistic Precession model (retrograde)

1.2

1.4

1.6

1.8

Figure: Frequency ratio contours of RP model as function of x and spin in Kerr geometry.



νupp

νlow=νφ + νr

νφ(24)

4 6 8 10 12 14 16 18

0.0

0.2

0.4

0.6

0.8

1.0

x(r/M)

a

Tidal disruption model (Prograde)

1.2

1.3

1.4

1.5

1.6

1.7

5 10 15 20

0.0

0.2

0.4

0.6

0.8

1.0

x(r/M)

a

Tidal disruption model (retrograde)

1.2

1.3

1.4

1.5

1.6

1.7

Figure: Frequency ratio contours of TD model as function of x and spin in Kerr geometry.


3. Non-linear resonance model - Keplerian modelνupp

νlow=νφ

νr(25)

5 10 15 20

0.0

0.2

0.4

0.6

0.8

1.0

x(r/M)

a

Non linear resonance model- Keplerian (Prograde)

1.3

1.4

1.5

1.6

1.7

1.8

5 10 15 20 25

0.0

0.2

0.4

0.6

0.8

1.0

x(r/M)

a

Non-linear resonance model - Keplerian (retrograde)

1.3

1.4

1.5

1.6

1.7

1.8

Figure: Frequency ratio contours of NL- Keplerian model as function of x and spin in Kerr geometry.


4. Non-linear resonance model - Epicyclic modelνupp

νlow=νθ

νr(26)

5 10 15 20

0.0

0.2

0.4

0.6

0.8

1.0

x(r/M)

a

Non linear resonance model- Epicyclic (Prograde)

1.3

1.4

1.5

1.6

1.7

1.8

5 10 15 20 25

0.0

0.2

0.4

0.6

0.8

1.0

x(r/M)

a

Non-linear resonance model- Epicyclic (retrograde)

1.3

1.4

1.5

1.6

1.7

1.8

Figure: Frequency ratio contours of NL- Epicyclic model as function of x and spin in Kerr geometry.


Conclusions for orbits in Schwarzschild geometry.

For higher radii r > 18M, range of eccentricity to give required ratio becomes verynarrow near e ∼ 0.7 for all models.

For circular orbits

I Tidal disruption model and Non-linear resonance model give ratiobetween 1 to 2 for wide range of radii.

I Relativistic Precession model give required ratio for radii less than 10M.

For all models eccentricity ranges from 0 to 0.7 to give required ratio.

Emission region: x > (3 + e2)/(1 − e2)


Conclusions for orbits in Kerr geometry.

For high value of spin, smaller radii can emit to give required ratio and vice versafor prograde motion.

For retrograde, required ratio comes from larger radii for high value of spin andvice versa for retrograde motion.

Model Radii range (x) Radii range-prograde retrogradea = 1 to a = 0 a = 0 to a = 1

Relativistic Precession 3 to 7.5 6 to 11

Tidal disruption 3 to 17 6 to 21

Non-linear resonance - K 3 to 20 8 to 25

Non-linear resonance - E 3 to 20 8 to 27


QPO model for AGNs and microquasars

Seyfert 1 galaxy REJ 1034+396 showed 1 hourperiodicity in energy range 0.3-10keV. (Marek2008). GRS 1915+105 microquasar showed 3:2ratio. (Torok et.al 2005)

GR model of jet variability: (Mohan and

Mangalam 2015)I Blob is launched in Zone 1 driven by

radiation pressure, centrifugally driven inZone 2 and finally it reaches final angularmomentum.

I Quasi periodic variability - due to orbitalmotion of blob along helical path as the localangle is close to angle to the observer’s lineof sight.

I In Schwarzschild geometry, they simulatedmotion of such blobs considering GR effects.

I They simulated light curve from such blobsfor a range of mass, inclination angle andlorentz factors.

I They expected QPOs ranging from 1 to 33days periodicities.

(Mohan and Mangalam 2015)


Figure: Image Courtesy: Mohan and Mangalam 2015

Helical flow geometry of lob emitted from source.

tem is time emission, then tobs = (tem − tadv + ∆tLB)(1 + z). (Mohan & Mangalam 2015;Poutanen & Beloborodov 2006)

∆tLB =

(δ

8y2

(1 + y

(1

3−

2

14

)))R

c(27)

where, y = (1− cos Ψ) and δ = 2M/R


Flow trajectories

The instantaneous position of the emitting spot is given by

xs = (x , y , z) = ($(t) cosφ(t), $(t) sinφ(t), z(t)), (28)

where $ is the cylindrical distance to the emitting spot from z(t).

In our variability model, a prescription for $ = $(z) is given by the geometry of themagnetic surface. The velocity components (xs = dxs/dt) are then

xs = (x , y , z) = ($ cosφ(t)−$φ(t), $ sinφ(t) +$φ(t), z). (29)

The angle between the velocity vector of the spot xs and the initial direction vector k0 isgiven by

cos ξ =xs · k0

|xs|=

1

|xs|

(sinα

sinψxs · k +

sin(ψ − α)

sinψxs · n

). (30)


Flow trajectories

The Doppler factor D evaluated in an instantaneous stationary frame at the source isgiven by

D =1

γjet(1− βjet cos ξ). (31)

In the observed frame, the above expression must be modified to account for thegravitational redshift.

The effective redshift factor g is then

g =Eobserved

Eemitted= (1− 2M/R)1/2D =

(1− 2M/R)1/2

γjet(1− βjet cos ξ). (32)

If γjet,i is the bulk Lorentz factor of the flow at the initial launch radius $0, the energyconservation condition along the trajectory can be written as

ut

(1− 2M/R)=

γjet

(1− 2M/R)= ε =

γjet,i

(1− 2M/$0)= γjet,f . (33)

The light curve is given by F(tem) = g2+Γ(tem) where Γ is the slope in the relation

Fν ∝ νΓ between the spectral flux Fν and the emission frequency in the observer frame.

We express tem in terms of tobs to obtain the observed light curve.


Pulse profile from a spot of a rotating disc

Fourier series of a Pulse profile from a spot of a rotating object as a cosine series, includingeffects like Doppler boost, relativistic aberration, gravitational redshift and light bending inSchwarzschild geometry. (Rana & Mangalam in preparation for disc and jet)

F (φ) ≈ C + C1 cos(φ+ ζ1) + C2 cos[2(φ+ ζ2)] + C3 cos[3(φ+

π

6

)](34)

where C ’s are of order unity, For the case of rotating Neutron star (Poutanen & Beloborodov2006)

C = Q + h

(Q2 +

U2

2

), C1 =

U(1 + 2hQ)

cos ζ1

C2 =hU2/2

cos 2ζ2, C3 =

4 + τ

4hTU2

tan ζ1 =T

U

(3 + τ)Q + (4 + τ)h(Q2 + U2/4)

1 + 2hQ, tan 2ζ2 =

T

U

(4 + τ)(1 + 2hQ)− 1

h

IE′ = I(1 + h cosα

′)E′−(τ−1)

U = (1− u) sin i sin θ and Q = u + (1− u) cos i cos θ, T =2πRv

csin i sin θ and u = rs/R

rs is Schwarzschild radius and R is radius of star, i= inclination angle from center of star

θ= angle between rotation axis and spot on the star.


Conclusions and Future work

1 Results suggest that emission region is very near to compact object.

2 Geometric effects to be isolated before calculating radiation driving fluid flows (M& M 2015). This will provide clues from emitting region (close to the horizon andin the ergosphere).

3 (Under development) GR effects: Light bending, Aberration, time delay,gravitational redshift, Doppler & relativistic beaming, frame dragging for QPOs inXRBs and microquasars.

4 CARMA model for timing analysis that jointly characterizes the noise and signal.The resulting comprehensive time series tool will be of utility in characterizingsignals from XRBs observed from ASTROSAT.


THANK YOU


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