spin-down of neutron stars: competing effects of...
TRANSCRIPT
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?
IntroductionSpindown of neutron stars
Competing effectsResults
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 ).
P. Jaikumar Gravitational Waves or Magnetic Braking?
IntroductionSpindown of neutron stars
Competing effectsResults
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 ).
Even isolated neutron stars can radiate gravitational waves -("mountain", r-mode instability )
P. Jaikumar Gravitational Waves or Magnetic Braking?
IntroductionSpindown of neutron stars
Competing effectsResults
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 ).
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 )
P. Jaikumar Gravitational Waves or Magnetic Braking?
IntroductionSpindown of neutron stars
Competing effectsResults
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?
IntroductionSpindown of neutron stars
Competing effectsResults
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?
IntroductionSpindown of neutron stars
Competing effectsResults
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?
IntroductionSpindown of neutron stars
Competing effectsResults
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?
IntroductionSpindown of neutron stars
Competing effectsResults
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?
IntroductionSpindown of neutron stars
Competing effectsResults
Conclusions
What are gravitational waves?Detecting gravitational waves
What are gravitational waves?
Gravitational waves are produced whenever matter isaccelerated in a non-spherical fashion
P. Jaikumar Gravitational Waves or Magnetic Braking?
IntroductionSpindown of neutron stars
Competing effectsResults
Conclusions
What are gravitational waves?Detecting gravitational waves
What are gravitational waves?
Gravitational waves are produced whenever matter isaccelerated in a non-spherical fashion
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)
P. Jaikumar Gravitational Waves or Magnetic Braking?
IntroductionSpindown of neutron stars
Competing effectsResults
Conclusions
What are gravitational waves?Detecting gravitational waves
What are gravitational waves?
Gravitational waves are produced whenever matter isaccelerated in a non-spherical fashion
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
P. Jaikumar Gravitational Waves or Magnetic Braking?
IntroductionSpindown of neutron stars
Competing effectsResults
Conclusions
What are gravitational waves?Detecting gravitational waves
What are gravitational waves?
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)
P. Jaikumar Gravitational Waves or Magnetic Braking?
IntroductionSpindown of neutron stars
Competing effectsResults
Conclusions
What are gravitational waves?Detecting gravitational waves
What are gravitational waves?
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
P. Jaikumar Gravitational Waves or Magnetic Braking?
IntroductionSpindown of neutron stars
Competing effectsResults
Conclusions
What are gravitational waves?Detecting gravitational waves
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 .
P. Jaikumar Gravitational Waves or Magnetic Braking?
IntroductionSpindown of neutron stars
Competing effectsResults
Conclusions
What are gravitational waves?Detecting gravitational waves
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)
P. Jaikumar Gravitational Waves or Magnetic Braking?
IntroductionSpindown of neutron stars
Competing effectsResults
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)
P. Jaikumar Gravitational Waves or Magnetic Braking?
IntroductionSpindown of neutron stars
Competing effectsResults
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)
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)
P. Jaikumar Gravitational Waves or Magnetic Braking?
IntroductionSpindown of neutron stars
Competing effectsResults
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)
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)
P. Jaikumar Gravitational Waves or Magnetic Braking?
IntroductionSpindown of neutron stars
Competing effectsResults
Conclusions
Neutron stars as GW sourcesMagnetic brakingr-modes and Gravitational wave braking
Isolated NS (spindown due to EM radiation)
P-Pdot diagram
P. Jaikumar Gravitational Waves or Magnetic Braking?
IntroductionSpindown of neutron stars
Competing effectsResults
Conclusions
Neutron stars as GW sourcesMagnetic brakingr-modes and Gravitational wave braking
Isolated NS (spindown due to EM radiation)
P-Pdot diagram
dΩdt
= −2B2R6Ω3
3Ic3sin2χ (4)
P. Jaikumar Gravitational Waves or Magnetic Braking?
IntroductionSpindown of neutron stars
Competing effectsResults
Conclusions
Neutron stars as GW sourcesMagnetic brakingr-modes and Gravitational wave braking
r-mode instability
r-mode perturbations are (quasi)-toroidal fluid oscillations
P. Jaikumar Gravitational Waves or Magnetic Braking?
IntroductionSpindown of neutron stars
Competing effectsResults
Conclusions
Neutron stars as GW sourcesMagnetic brakingr-modes and Gravitational wave braking
r-mode instability
r-mode perturbations are (quasi)-toroidal fluid oscillations
r-mode energy grows with GW emission
P. Jaikumar Gravitational Waves or Magnetic Braking?
IntroductionSpindown of neutron stars
Competing effectsResults
Conclusions
Neutron stars as GW sourcesMagnetic brakingr-modes and Gravitational wave braking
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?
IntroductionSpindown of neutron stars
Competing effectsResults
Conclusions
Can r-modes play a role or is it all magnetic?r-mode growth/saturationGravitational strain
Coupled Equations:
Ω
Ω= −Fmag
P. Jaikumar Gravitational Waves or Magnetic Braking?
IntroductionSpindown of neutron stars
Competing effectsResults
Conclusions
Can r-modes play a role or is it all magnetic?r-mode growth/saturationGravitational strain
Coupled Equations:
Ω
Ω= −Fmag−2α2Kc [KjFg + (1 − Kj)[Fv + Fm]]
P. Jaikumar Gravitational Waves or Magnetic Braking?
IntroductionSpindown of neutron stars
Competing effectsResults
Conclusions
Can r-modes play a role or is it all magnetic?r-mode growth/saturationGravitational strain
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))
P. Jaikumar Gravitational Waves or Magnetic Braking?
IntroductionSpindown of neutron stars
Competing effectsResults
Conclusions
Can r-modes play a role or is it all magnetic?r-mode growth/saturationGravitational strain
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′
P. Jaikumar Gravitational Waves or Magnetic Braking?
IntroductionSpindown of neutron stars
Competing effectsResults
Conclusions
Can r-modes play a role or is it all magnetic?r-mode growth/saturationGravitational strain
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)
P. Jaikumar Gravitational Waves or Magnetic Braking?
IntroductionSpindown of neutron stars
Competing effectsResults
Conclusions
Can r-modes play a role or is it all magnetic?r-mode growth/saturationGravitational strain
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
P. Jaikumar Gravitational Waves or Magnetic Braking?
IntroductionSpindown of neutron stars
Competing effectsResults
Conclusions
Can r-modes play a role or is it all magnetic?r-mode growth/saturationGravitational strain
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
P. Jaikumar Gravitational Waves or Magnetic Braking?
IntroductionSpindown of neutron stars
Competing effectsResults
Conclusions
Can r-modes play a role or is it all magnetic?r-mode growth/saturationGravitational strain
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
P. Jaikumar Gravitational Waves or Magnetic Braking?
IntroductionSpindown of neutron stars
Competing effectsResults
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
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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.
P. Jaikumar Gravitational Waves or Magnetic Braking?
IntroductionSpindown of neutron stars
Competing effectsResults
Conclusions
Spindown evolutionGravitational strain evolutionGravitational wave signalChoice of r-mode saturation value and EoS
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
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
grav. wave strain h x 1024
time (s)
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14
15
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
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.
P. Jaikumar Gravitational Waves or Magnetic Braking?
IntroductionSpindown of neutron stars
Competing effectsResults
Conclusions
Spindown evolutionGravitational strain evolutionGravitational wave signalChoice of r-mode saturation value and EoS
Sensitivity to signal
0.1
1
10
100
1000
10000
102 103 104
h~f0.5 (1/Hz0.5) x 1024
Frequency (Hz)
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h~f0.5 (1/Hz0.5) x 1024
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h~f0.5 (1/Hz0.5) x 1024
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h~f0.5 (1/Hz0.5) x 1024
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h~f0.5 (1/Hz0.5) x 1024
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h~f0.5 (1/Hz0.5) x 1024
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h~f0.5 (1/Hz0.5) x 1024
Frequency (Hz)
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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.
P. Jaikumar Gravitational Waves or Magnetic Braking?
IntroductionSpindown of neutron stars
Competing effectsResults
Conclusions
Spindown evolutionGravitational strain evolutionGravitational wave signalChoice of r-mode saturation value and EoS
small αsat is more plausible
10-710-610-510-410-310-210-1100101102
100 102 104 106 108 1010 1012
grav. wave strain h x 1024
time (s)
12131415
10-710-610-510-410-310-210-1100101102
100 102 104 106 108 1010 1012
grav. wave strain h x 1024
time (s)
12131415
10-710-610-510-410-310-210-1100101102
100 102 104 106 108 1010 1012
grav. wave strain h x 1024
time (s)
12131415
10-710-610-510-410-310-210-1100101102
100 102 104 106 108 1010 1012
grav. wave strain h x 1024
time (s)
12131415
Smaller αsat implies r-mode effect appears later in evolution.
This is because of delayed onset of magnetic damping.
P. Jaikumar Gravitational Waves or Magnetic Braking?
IntroductionSpindown of neutron stars
Competing effectsResults
Conclusions
Spindown evolutionGravitational strain evolutionGravitational wave signalChoice of r-mode saturation value and EoS
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
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14
1510-4
10-3
10-2
10-1
100
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102 103 104
h~ x 1024
frequency (Hz)
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1510-4
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h~ x 1024
frequency (Hz)
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1510-4
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102 103 104
h~ x 1024
frequency (Hz)
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1510-4
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10-2
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102 103 104
h~ x 1024
frequency (Hz)
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102 103 104
h~ x 1024
frequency (Hz)
12 13
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10-4
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10-2
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h~ x 1024
frequency (Hz)
12 13
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14
10-4
10-3
10-2
10-1
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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).
P. Jaikumar Gravitational Waves or Magnetic Braking?
IntroductionSpindown of neutron stars
Competing effectsResults
Conclusions
Conclusions
We wanted to see if r-modes can compete with magneticdamping in NS spindown.
P. Jaikumar Gravitational Waves or Magnetic Braking?
IntroductionSpindown of neutron stars
Competing effectsResults
Conclusions
Conclusions
We wanted to see if r-modes can compete with magneticdamping in NS spindown.
For small αsat and B ≤ 1012G, the answer is YES!
P. Jaikumar Gravitational Waves or Magnetic Braking?
IntroductionSpindown of neutron stars
Competing effectsResults
Conclusions
Conclusions
We wanted to see if r-modes can compete with magneticdamping in NS spindown.
For small αsat and B ≤ 1012G, the answer is YES!
Magnetic damping of r-modes (back-reaction) suppresses/delaysr-mode growth.
P. Jaikumar Gravitational Waves or Magnetic Braking?
IntroductionSpindown of neutron stars
Competing effectsResults
Conclusions
Conclusions
We wanted to see if r-modes can compete with magneticdamping in NS spindown.
For small αsat and B ≤ 1012G, the answer is YES!
Magnetic damping of r-modes (back-reaction) suppresses/delaysr-mode growth.
r-mode saturates for B ≤ 1015G, unless EoS is very soft.
P. Jaikumar Gravitational Waves or Magnetic Braking?
IntroductionSpindown of neutron stars
Competing effectsResults
Conclusions
Conclusions
We wanted to see if r-modes can compete with magneticdamping in NS spindown.
For small αsat and B ≤ 1012G, the answer is YES!
Magnetic damping of r-modes (back-reaction) suppresses/delaysr-mode growth.
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)
P. Jaikumar Gravitational Waves or Magnetic Braking?
IntroductionSpindown of neutron stars
Competing effectsResults
Conclusions
Work in Progress
Comparing to observed braking indices
P. Jaikumar Gravitational Waves or Magnetic Braking?
IntroductionSpindown of neutron stars
Competing effectsResults
Conclusions
Work in Progress
Comparing to observed braking indices
Limits on GW energy radiated away
P. Jaikumar Gravitational Waves or Magnetic Braking?