spin-down of neutron stars: competing effects of...

IntroductionSpindown of neutron stars

Competing effectsResults

Conclusions

Spin-down of Neutron Stars:Competing Effects of Gravitational Waves and

Magnetic Braking

Prashanth Jaikumar1

Collaborators (arXiv:1107.1000):

Jan Staff2

Vincent (Paktoo) Chan1

Rachid Ouyed3

1California State U., Long Beach2Louisiana State U.

3 U. of Calgary

4th Mini-workshop on neutron stars and neutrinos, March 26-27, 2012

P. Jaikumar Gravitational Waves or Magnetic Braking?

http://shovkovy.faculty.asu.edu/astro2012/



Conclusions

Motivation

[Image: NASA]

Gravitational Waves (Advanced LIGO) should open up a newwindow to see the sky (eg., Radio - 1930s, COBE - 1990s)

(i) neutron star-neutron star binaries (40 or more @ 〈d〉 ∼200 Mpc )(ii) black hole-neutron star binaries (10 or more ).




Conclusions

Motivation

[Image: NASA]



Even isolated neutron stars can radiate gravitational waves -("mountain", r-mode instability )




Conclusions

Motivation

[Image: NASA]



Even isolated neutron stars can radiate gravitational waves -("mountain", r-mode instability )

The more sources we model (template ), better the prospects ofdetecting signal (matched filtering )




Conclusions

Outline1 Introduction

What are gravitational waves?Detecting gravitational waves

2 Spindown of neutron starsNeutron stars as GW sourcesMagnetic brakingr-modes and Gravitational wave braking

3 Competing effectsCan r-modes play a role or is it all magnetic?r-mode growth/saturationGravitational strain

4 ResultsSpindown evolutionGravitational strain evolutionGravitational wave signalChoice of r-mode saturation value and EoS

5 ConclusionsP. Jaikumar Gravitational Waves or Magnetic Braking?



Conclusions


What are gravitational waves?

Gravitational waves are produced whenever matter isaccelerated in a non-spherical fashion




Conclusions




Spacetime is rigid (c4/8πG as "stiffness constant") → metric isgenerally very close to Minkowski: gαβ = ηαβ + hαβ where|hαβ | ≪ 1 is a small perturbation (weak field limit)




Conclusions




Spacetime is rigid (c4/8πG as "stiffness constant") → metric isgenerally very close to Minkowski: gαβ = ηαβ + hαβ where|hαβ | ≪ 1 is a small perturbation (weak field limit)

Gravitational wave equation hαβ = 0

hTT gaugeαβ =

0 0 0 00 h+ 0 00 0 −h+ 00 0 0 0

+

0 0 0 00 0 h× 00 h× 0 00 0 0 0




Conclusions



The effect of gravitational waves of two different polarizationsfor various phases, when incident on a ring of particles can bevisualized as:

Image: Ju et al. (2000)




Conclusions



The effect of gravitational waves of two different polarizationsfor various phases, when incident on a ring of particles can bevisualized as:

Image: Ju et al. (2000)

Incident wave alters spacing between two masses (strainh = ∆L/L) - this "path length" change can be measured by laserinterferometry




Conclusions


Advanced LIGO

(Left):Louisiana, (Right) Washington St.

Change in length due to passing wave results in light atphotodetectorExtracting gravitational wave signals from noisy data requiresaccurate theoretical waveforms .




Conclusions


Unprecedented Capability

X10 more sensitive than LIGO

Test masses (34" diameter x 20" cm thick mirrors) are stabilizedto 10−14cm

World’s largest ultra-high vacuum chamber (∼ 10−9 Torr)High-power lasers and state-of-the-art optics (surfaceimperfections ∼ λ/1000)




Conclusions

Neutron stars as GW sourcesMagnetic brakingr-modes and Gravitational wave braking

Binary neutron stars as GW sources

Inspiral of neutron star binaries

dEdt

= −325

G4

c5

(mM)2(m + M)

a5, (1)




Conclusions




dEdt

= −325

G4

c5

(mM)2(m + M)

a5, (1)

Orbital period decreases as a consequence

Porb(t) =

(

P8/30 − 8

3kt

)3/8

, k ∝(

Gc3

Mchirp

)5/3

∼ 10−6 /sec (2)

Chirp signal (increasing amplitude AND frequency)


http://demonstrations.wolfram.com/NewtonianGravitationalWaveChirpSignalFromMergerOfACompactBin/



Conclusions




dEdt

= −325

G4

c5

(mM)2(m + M)

a5, (1)

Orbital period decreases as a consequence

Porb(t) =

(

P8/30 − 8

3kt

)3/8

, k ∝(

Gc3

Mchirp

)5/3

∼ 10−6 /sec (2)

Chirp signal (increasing amplitude AND frequency)

For circularized orbit,

PGW(t) = Porb(t)/2 (3)


http://demonstrations.wolfram.com/NewtonianGravitationalWaveChirpSignalFromMergerOfACompactBin/



Conclusions


Isolated NS (spindown due to EM radiation)

P-Pdot diagram




Conclusions


Isolated NS (spindown due to EM radiation)

P-Pdot diagram

dΩdt

= −2B2R6Ω3

3Ic3sin2χ (4)




Conclusions


r-mode instability

r-mode perturbations are (quasi)-toroidal fluid oscillations




Conclusions


r-mode instability

r-mode perturbations are (quasi)-toroidal fluid oscillations

r-mode energy grows with GW emission




Conclusions


Gravitational braking

Above the critical frequency Ωc, rotational energy is dissipated bygravitational wave emission.

1τ(Ωc)

=

[

1τζ

+1τη

+1

τGW

]

(Ωc) = 0 (5)

(Jaikumar, Rupak & Steiner, Physical Review D78, 123007 (2008)).P. Jaikumar Gravitational Waves or Magnetic Braking?



Conclusions

Can r-modes play a role or is it all magnetic?r-mode growth/saturationGravitational strain

Coupled Equations:

Ω

Ω= −Fmag




Conclusions


Coupled Equations:

Ω

Ω= −Fmag−2α2Kc [KjFg + (1 − Kj)[Fv + Fm]]




Conclusions


Coupled Equations:

Ω

Ω= −Fmag−2α2Kc [KjFg + (1 − Kj)[Fv + Fm]] and (6)

α

α= [Fg − [Fv + Fm]]−

Ω

2Ω,

Fg(t) = Fg(Ω(t), α(t)) , Fv(t) = Fv(Ω(t), α(t),T(t))




Conclusions


Coupled Equations:

Ω

Ω= −Fmag−2α2Kc [KjFg + (1 − Kj)[Fv + Fm]] and (6)

α

α= [Fg − [Fv + Fm]]−

Ω

2Ω,

Fg(t) = Fg(Ω(t), α(t)) , Fv(t) = Fv(Ω(t), α(t),T(t))

Fm(t) ∝ R3α2(t)Ω(t)B2(t) (7)

B2(t) = B2(0)∫ t

0α2(t′)Ω(t′)dt′




Conclusions


Growth phase of the r-mode

Conditions:

1. Fv (viscous damping) is small since mode amplitude is small (andassume rapid cooling)

2. Differential rotation is small

α

α= (Fg − Fm)−

Ω

2Ω. (8)




Conclusions


Saturation phase of the r-mode

Conditions:

1. α = 0 mode amplitude is large (∼ O(0.01)−O(1))

Ω

Ω=

2α2satKcFg − Fmag

1 − α2satKc(1 − Kj)

. (9)

αsat is the saturation amplitude of the r-mode.

2. Differential rotation is large




Conclusions


Gravitational strain amplitude

Strain amplitude h(t):

h(t) =

√

380π

ω2(t)S22

D. (10)

r-mode angular frequency ω(t) = 4Ω(t)/3, D = source distance

S22 = current multipole:

S22 =√

232π15

GMc5

αΩR3J . (11)

J is EoS-dependent quantity ∼ 0.1




Conclusions


Gravitational waveform

h(f ) is the Fourier Transform of h(t) (steepest descent)

|h(f )| ≈√

|h(t)|2|f |

. (12)

Knowing the time evolution of Ω determines the gravitationalwaveform




Conclusions

Spindown evolutionGravitational strain evolutionGravitational wave signalChoice of r-mode saturation value and EoS

Period vs time

0

0.0005

0.001

0.0015

0.002

0.0025

0.003

0.0035

100 102 104 106 108 1010 1012

Period (s)

time (s)

12131415

0

0.0005

0.001

0.0015

0.002

0.0025

0.003

0.0035

100 102 104 106 108 1010 1012

Period (s)

time (s)

12131415

0

0.0005

0.001

0.0015

0.002

0.0025

0.003

0.0035

100 102 104 106 108 1010 1012

Period (s)

time (s)

12131415

0

0.0005

0.001

0.0015

0.002

0.0025

0.003

0.0035

100 102 104 106 108 1010 1012

Period (s)

time (s)

12131415

Period vs time (APR EoS), constant temperature T = 109 K.

For B ≥ 1013G, magnetic braking starts to dominate the r-mode.




Conclusions


Strain amplitude vs time

10-710-610-510-410-310-210-1100101102

100 102 104 106 108 1010 1012

grav. wave strain h x 1024

time (s)

1213

14

15

10-710-610-510-410-310-210-1100101102

100 102 104 106 108 1010 1012


time (s)

1213

14

15

10-710-610-510-410-310-210-1100101102

100 102 104 106 108 1010 1012


time (s)

1213

14

15

10-710-610-510-410-310-210-1100101102

100 102 104 106 108 1010 1012


time (s)

1213

14

15

Gravitational wave strain as a function of time (APR EoS), constanttemperature T = 109 K.

Larger B fields have lower peak strain because r-mode neversaturates.




Conclusions


Sensitivity to signal

0.1

1

10

100

1000

10000

102 103 104

h~f0.5 (1/Hz0.5) x 1024

Frequency (Hz)

0.1

1

10

100

1000

10000

102 103 104

h~f0.5 (1/Hz0.5) x 1024

Frequency (Hz)

0.1

1

10

100

1000

10000

102 103 104

h~f0.5 (1/Hz0.5) x 1024

Frequency (Hz)

0.1

1

10

100

1000

10000

102 103 104

h~f0.5 (1/Hz0.5) x 1024

Frequency (Hz)

0.1

1

10

100

1000

10000

102 103 104

h~f0.5 (1/Hz0.5) x 1024

Frequency (Hz)

0.1

1

10

100

1000

10000

102 103 104

h~f0.5 (1/Hz0.5) x 1024

Frequency (Hz)

0.1

1

10

100

1000

10000

102 103 104

h~f0.5 (1/Hz0.5) x 1024

Frequency (Hz)

0.1

1

10

100

1000

10000

102 103 104

h~f0.5 (1/Hz0.5) x 1024

Frequency (Hz)

0.1

1

10

100

1000

10000

102 103 104

h~f0.5 (1/Hz0.5) x 1024

Frequency (Hz)

0.1

1

10

100

1000

10000

102 103 104

h~f0.5 (1/Hz0.5) x 1024

Frequency (Hz)

GW signal (frequency space) for three different EoS: APR(red), BBB2(cyan) and EoS A(blue)

Left panel (B=1013 G), right panel (B=1014 G), αsat=0.01.




Conclusions


small αsat is more plausible

10-710-610-510-410-310-210-1100101102

100 102 104 106 108 1010 1012


time (s)

12131415

10-710-610-510-410-310-210-1100101102

100 102 104 106 108 1010 1012


time (s)

12131415

10-710-610-510-410-310-210-1100101102

100 102 104 106 108 1010 1012


time (s)

12131415

10-710-610-510-410-310-210-1100101102

100 102 104 106 108 1010 1012


time (s)

12131415

Smaller αsat implies r-mode effect appears later in evolution.

This is because of delayed onset of magnetic damping.




Conclusions


EoS effects

Gravitational wave strain as a function of frequency.

10-4

10-3

10-2

10-1

100

101

102

103

102 103 104

h~ x 1024

frequency (Hz)

12

13

14

1510-4

10-3

10-2

10-1

100

101

102

103

102 103 104

h~ x 1024

frequency (Hz)

12

13

14

1510-4

10-3

10-2

10-1

100

101

102

103

102 103 104

h~ x 1024

frequency (Hz)

12

13

14

1510-4

10-3

10-2

10-1

100

101

102

103

102 103 104

h~ x 1024

frequency (Hz)

12

13

14

1510-4

10-3

10-2

10-1

100

101

102

103

102 103 104

h~ x 1024

frequency (Hz)

12 13

15

14

10-4

10-3

10-2

10-1

100

101

102

103

102 103 104

h~ x 1024

frequency (Hz)

12 13

15

14

10-4

10-3

10-2

10-1

100

101

102

103

102 103 104

h~ x 1024

frequency (Hz)

12 13

15

14

10-4

10-3

10-2

10-1

100

101

102

103

102 103 104

h~ x 1024

frequency (Hz)

12 13

15

14

Stiffer EoS (Left: APR) displays larger r-mode effect than soft (Right:EoS A).

Gravitational braking is larger for less compact stars (w/ samebaryonic mass).




Conclusions

Conclusions

We wanted to see if r-modes can compete with magneticdamping in NS spindown.




Conclusions

Conclusions


For small αsat and B ≤ 1012G, the answer is YES!




Conclusions

Conclusions



Magnetic damping of r-modes (back-reaction) suppresses/delaysr-mode growth.




Conclusions

Conclusions




r-mode saturates for B ≤ 1015G, unless EoS is very soft.




Conclusions

Conclusions




r-mode saturates for B ≤ 1015G, unless EoS is very soft.

For ideal scenario, GW emission can last years (above thresholdin 2nd, 3rd gen. detectors)




Conclusions

Work in Progress

Comparing to observed braking indices




Conclusions

Work in Progress


Limits on GW energy radiated away




Conclusions

Work in Progress


Limits on GW energy radiated away

GW signal once spindown achieved deconfinement density(Paktoo Chan’s talk)


spin-down of neutron stars: competing effects of...

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