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Galactic Merger Rates of
Pulsar Binaries
Chunglee Kim
Thesis advisor: Dr. Vicky Kalogera
Thesis Defense
April 26, 2006
Pulsar binaries
~20 ms - 200
ms
~1500 known PSRs
(Credit: M. Kramer)
We consider NS-NS, NS-WD, NS-BH. Pulsars in these systems are
- rare (~8 + 30 in the Galactic disk)- typically old, mildly recycled- strong sources of GW
We are interested in ‘inspiral’ signals
GW signals from pulsar binaries
consider
merging binaries
(mrg < Hubble time)
Credit: K. Thorne
Our work (Kim et al. 2003; Kalogera et al. 2004; Kim et al. 2004)
Problems in pulsar binary event rates until recently:
- rate predictions highly uncertain
(by more than two orders of magnitude)
- lack of quantitative understanding of uncertainties
(statistical & systematic)
We introduce an analysis method to give a
statistical significance of the rate estimates.
Small number bias and selection effects for faint pulsars
are implicitly included in our analysis.
Lifetime = current age + remaining time (PSR) (GW emission)
Beaming correction factor = 1/ PSR beaming fraction
Merger rate R
adapted from PSR & binary properties
We calculate the number of sources (Npop) using
SEARCH
Lifetime of a systemNumber of sources
x correction factorR =beaming
Basic strategy
Consider one system at a time (e.g., J0737-3039)
P(Nobs)
PSR population models
(luminosity & spatial distribution)
+PSR survey simulation
obtain Nobs (given Npop)
apply Bayes’ theorem
Input parameters & results are relevant to PSR J0737-3039
PSR population modela PSR population can be defined
Spatial distribution
R=(x2+y2)1/2
Luminosity
distribution
PSR i (R,Z,L)i
fix Ps, pulse width, & Porb
f(R,z) exp
Ro: radial scale length, zo: vertical height
|Z||Z|
ZZoo
RR22
2R2Roo22
-- --
Spatial distribution (Narayan 1991)
Reference model: Ro=4.0 kpc, zo=1.5 kpc
PSR population modelRadio PSR luminosity distribution (Cordes and Chernoff 1997)
power-law:
Lmin: cut-off luminosity (Lmin < L)
Reference model: Lmin=0.3 mJy kpc2, p=2.0
log L
log N
?
Lmin
slope: p
(L; Lmin, p) determines
a fraction of faint PSRs
in a given population
Orbital motion effects are taken into account
PSR B1913+16
credit: M. Kramer
Survey Selection effects
Nobs follows the Poisson distribution, P(Nobs; <Nobs>)
Earth
PSR survey simulation - SEARCH
Calculate Nobs,i varying Npop, i
Same Ps & Pb, but
diff. radio flux densities
S = L/d2L
d
posteria PDF data likelihood x prior PDF
Statistical Analysis
Apply Bayes’ theorem to calculate P(<Nobs>)
P(<Nobs>) P(Nobs; <Nobs>) x P(Nobs)
where P(Nobs; <Nobs>) is obtained from SEARCH.
P(1;
<Nobs>)
P(1; <Nobs>) ; assume P(Nobs)=const.
Nobs = 1
Pi(R)Pi(<Nobs>) chain rule
For an each observed system i,
<Nobs> Npop; and Rlifetime
Npop
For an each observed system i,
Pi(R) = Ci2R exp(-CiR)
where Ci = <Nobs> life
Npop fb i fb: beaming correction factor
Combine individual P(R)’s and calculate P(Rtot)
Individual P(R)
P(Rtot)
NS-NS binaries
LivingstonObservatory
Hanford Observatory
Merging binaries in Galactic disk:PSRs B1913+16, B1534+12, and J0737-3039
Galactic NS-NS merger rate (Myr-1)
P(Rgal )
Detection rate for the initial LIGO (yr-1)
Probability density function of Rgal
NJ0737 ~ 1600 (most abundant)
Lifetime ~ 185 Myr (shortest)
NJ1534 ~ 400
NJ1913 ~ 600
Reference model Detection rate for the initial LIGO (yr-1)
P(Rgal) in a linear scale (reference model)
Detection rate for the initial LIGO (yr-1)
Galactic NS-NS merger rate (Myr-1)
Rpeak
Rpeak (1913+1534+0737) Rpeak (1913+1534)
~ 6-7
Increase rate factor
Detection rate of DNS inspirals for
LIGOdue to the discovery of PSR J0737-3039
Rdet (adv. LIGO) ~ 200 events per yr
Rdet (ini. LIGO) ~ 1 event per 30 yr
The most probable DNS inspiral detection rates for LIGO
Rdet (adv. LIGO) ~ 10 – 500 events per yr
Rdet (ini. LIGO) ~ 1 event per 10 – 400 yr
All models:
Reference model:
Global P(Rgal): calculation
Rpeak is strongly dependent on the PSR luminosity func.
f(R,z) is relatively poorly constrained, but the rate
estimates are NOT sensitive to the assumed
distribution function.
Global probability density function Pglobal(R)
Pglobal(R) =pdp
Lmin
dLmin P(R; Lmin,p) f(Lmin) g(p)
intrinsic functions for Lmin and p following Cordes & Chernoff (1997) – Based on 22 PSRs with spin period < 20 ms
from Tauris & van den Heuvel (2003)
SN rate constraints
Two NS are likely to be formed by SNe type Ib/c. Therefore, SNe (Ib/c) rate can be considered as an upper limit to the NS-NS rate.
SN Ib/c=600-1600 Myr-1 (Cappellaro et al. 1999)
However, the fraction of SN Ib/cactually involved in the formation ofNS-NS systems is uncertain. Based on population syntheses, the fraction could be ~ 5% or less…
Type Ib/c
Type Ib/c
Global P(Rgal) and SN rate constraints Pro
babili
ty D
ensi
ty
Galactic NS-NS merger rate (Myr-1)
SNU5
SNL5
SN Ib/c = 600-1600 Myr-1 (Cappellaro, Evans, & Turatto 1999)
SNL5= SN (lower)x0.05 = 30 Myr-1
SNU5= SN (upper)x0.05 = 80 Myr -1
Suppose, ~5% of Ib/c SNe are
involved in the NS-NS formation.
The empirical SNe rate
Global P(Rgal) and SN rate constraints Pro
babili
ty D
ensi
ty
Galactic NS-NS merger rate (Myr-1)
SNL5 : Conservative upper limit of RNS-NS
SNU5
Implications of new discoveries
(1) PSR J1756-2251 (Faulkner et al. 2005)
(2) PSR J1906+0746 (Lorimer et al. 2006)
Implications of J1756-2251
J1756-2251: The 4-th merging NS-NS known in the Galactic disk (Faulkner et al. 2005)
discovered by the Parkes Multibeam Pulsar Survey with
the acceleration search technique. Detailed simulations for acceleration searches are
needed to calculate P(R) including J1756-2251.
Contribution of J1756-2251 to the Galactic DNS merger rate.
No significant change in the total rate.
Rpeak (3 PSRs + J1756) Rpeak (3 PSRs)
~ 1.04
J1756-2251 ~ another example of 1913-like population
Implications of J1906+0746
J1906+0746: a young pulsar in a relativistic binary in the Galactic disk (Lorimer et al. 2006)
PSR name Ps (ms) Pb (hr) e life (Gyr)
B1913+16 59.03 7.752 0.617 0.365
B1534+12 37.90 10.098 0.274 2.7
J0737-3039A 22.70 2.45 0.088 0.185
J1756-2251 28.46 7.67 0.181 2.0
J1906+0746 144.07 3.98 0.0853 0.82
Characteristic age ~ 112 kyr !
Death time ~ 82 Myr (< tmrg) ~lifetime
J1141-6545 393.90 4.744 0.172 0.105
Implications of J1906+0746
Follow-up (optical/timing) observations are crucial
Rpeak (3 PSRs + J1906) Rpeak (3 PSRs)
~ 2
Assume J1906+0746 is a NS-NS binary:
N1906 ~ 300
t1906 ~ 82 Myr
N0737 ~1600
t0737 ~ 185Myr~ 0.5 x
companion is an NS or WD
total mass ~ 2.61 ± 0.02
M
NS-WD binaries
(1) Merging binaries:
PSRs J0751+1807, J1757-5322, and J1141-6545
(2) Eccentric binaries:
PSRs J1141-6545 and B2303+46
NS-WD binaries as GW sources for LISA
The GW background
due to the large number of sources
limits the detectability of weak sources in fgw < 3 mHz
Calculate the contribution from NS-WD binaries to the GW background for LISA.
In-spiraling NS-WD binaries emit gravitational waves in a frequency range fgw ~ 0.01 – 100 mHz
Consider 3 merging systems
(PSR J0751+1807, J1757-5322, and J1141-6545)
GW signals from NS-WD binaries
The contribution from NS-WD
binaries to the GW background
would be negligible
confusion noise level
due to WD-WD binaries
fmax,0751
fmax,1757
fmax,1141
GW
am
plitu
de (
h rm
s)
1 yr obs
J0751+J1757+J1141J1757+J1141
J1141
chirpmass GW freq. source
number
density
integration time
standard binary scenario predicts - circular orbit
- NS formation first - recycled PSR
“non-zero eccentricity” implies
- WD formed first
- non-recycled PSR
J1141-6545 : e=0.172 (Kaspi et al. 2000, Bailes et al. 2003)
B2303+46 : e=0.658 (Stokes et al. 1985, van Kerkwijk & Kulkarni 1999)
Galactic birthrate of eccentric NS-WD binaries
Empirical estimates
- Kalogera, CK, Ihm, Belczynski 2005 (StarTrack)
Nelemans, Portegies Zwart, & Yungelson 2001 (upper limit)
Tauris & Sennels 2001
Brown et al. 2002
Portegies Zwart & Yungelson 1999
- Davies, Ritter, King 2003
Theoretical predictions on birthrates
Theoretical estimates
Compare theoretical & empirical estimates Empirical estimates -Kalogera, CK, Ihm, Belczynski 2005 (error bar @95% CL) “Lower Limits”
- Kalogera, CK, Ihm, Belczynski 2005 (StarTrack)
Nelemans, Portegies Zwart, & Yungelson 2001 (upper limit)
Tauris & Sennels 2001
Brown et al. 2002
Portegies Zwart & Yungelson 1999
- Davies, Ritter, King 2003
REF :4 Myr-1
Theoretical estimates
No beaming correction
J1141-6545 and B2303+46
~h 2
d
Mchirp fh: GW amplitude
f: GW frequency = 2/Porb
d: distance to the source
Detection distance for advanced LIGO:
NS-NS ~ up to 350 Mpc
NS-BH (10 ) ~ up to 740 Mpc
(almost an order of magnitude increase in Vdet)
M
BH binaries (BH-BH, BH-NS) are
even stronger GW sources than NS binaries.
However, they have not been observed, yet.
NS-BH binaries
Empirical estimates using SEARCH Fix Ps = 50ms,
pulse width = 0.15
Adapt flux degradation factors
from known NS-NS binaries
RNS-BH < 1000 Myr-1
(upper limit @ 95% prob.)
with beaming correction
Pro
babi
lity
dens
ity
0 200 600 1000
Galactic merger rate (Myr-1)
Calculate P(R) given
Nobs=0
Constrain theoretical models
Theoretical predictions on RNS-BH ~ 10-8 – 10-5 yr -1
O’shaughnessy, CK, et al. 2005, ApJ, 633, 1076
accepted range of parameters
parameterspace used in theoretical model (StarTrack)
Calculate Rgal of BH binaries using only those models.
Give strong constraints on
Rdet of BH binaries and
population synthesis models
Establish a set of models (or parameters), which are
consistent with the estimated RNS-NS based on our
empirical method
NS-NS binariesNS-NS binaries
Empirical rate constraints
Galactic merger rate (Myr-1)10-2 0.1 1 10 102 103
10-2 0.1 1 10 102 103
log (Probability Density)
log (Probability Density)
StarTrack results
wide NS-NS
merging NS-NS
Only a few % of modelssatisfy both constraintssimultaneously
MergingB1913+16, B1534+12, J0737-3039
Wide (mrg > Hubble time)J1181-1736, J1518+4904, J1829+2456
Consistent with empirical rates
more than 95% of models
are ruled out;
Still, wide range of
parameters are possible.
Constrained predictions w/ StarTrack lo
g (P
roba
bilit
y D
ensi
ty)
Galactic merger rate in (Myr-1)
10-2 0.1 1 10 102
BH-BH
NS-BH
NS-NS
Dashed lines: unconstrained
Solid lines: constrained (NS-NS)
no recycled PSR-BH
NS-BH binaries: discussions
Pfahl et al. (2005) suggested that the Galactic birthrate of recycled PSR-BH binaries ~ less than 10-7 yr-1
consistent with our work (RMSP-BH < 10-8 yr-1)
If any, presumably, slow/normal PSR-BH binaries
dominate the NS-BH population
Recycled PSR-BH
NS-NS << 10-4 and R NS-NS < 10-4 yr-1
StarTrack results:
NS-BH binaries: discussions
Observational challenges
pulsars in NS-BH binaries are expected to have relatively short observable lifetimes, large accelerations in orbital motions than those of NS-NS binaries.
Large-scale interferometers
Square Kilometer Array (SKA) … radio (EM)
GEO/LIGO/TAMA/VIRGO … GW
death-time (Myr) spin-down age (Gyr)observable lifetime ~ 10% of MSP lifetime
Summary
We study
empirical Rgal of pulsar binaries (NS-NS & NS-WD)
detectability of such systems for GW detectors
constraints on theoretical models and BH rate estimates
Pulsar binaries are one of the most promising targets for GW
detectors, and they are likely to provide some of the first GW
detections.