Gravitational and electromagnetic emission by magnetized coalescing binary systems

0.2 MB·English

by  S. Capozziello

Gravitational and electromagnetic emission by
magnetized coalescing binary systems
S. Capozziello1,2, M. De Laurentis1,2, •
I. De Martino3, M. Formisano4, D. Vernieri1
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Abstract We discuss the possibility to obtainanelec- 1 Introduction

] tromagnetic emission accompanying the gravitational
E
waves emitted in the coalescence of a compact binary According to General Relativity, compact binary sys-
H
system. Motivatedbytheexistenceofblackholeconfig- tems (composed of neutron stars (NS), white dwarfs
h. urations with open magnetic field lines along the rota- or black holes (BH)) are strong emitters of gravita-
p tion axis, we consider a magnetic dipole in the system, tional waves (GWs), the ripples of space-time due to
- theevolutionofwhichleadsto(i)electromagneticradi- the presence of accelerated masses (coalescence). As a
o
r ation, and (ii) a contribution to the gravitational radi- consequenceofthegradualinspiralling,thesystemloses
st ation, the luminosity ofboth being evaluated. Starting energy,linearandangularmomentum(Hulse & Taylor
a from the observations on magnetars, we impose upper (1975); Nice (2003); Taylor (1989); Stairs (2004);
[
limits for both the electromagnetic emission and the Weisberg & Taylor (2003)). The coalescence of a

1 contributionofthemagneticdipoletothegravitational compact binary system can be split into three dis-
v wave emission. Adopting this model for the evolution
tinct, but not sharply delimited phases: the inspi-
6
of neutron star binaries leading to short gamma ray
0 ral, the plunge/mergerand the ring-down. For the
bursts, we compare the correction originated by the
3 inspiral phase an accurate analytical description via
5 electromagnetic field to the gravitational waves emis- the so-calledPost-Newtonian(PN) expansionhasbeen
. sion, finding that they are comparable for particular
1 developed (for a review see (Blanchet (2006)). Spin
0 values of the magnetic field and of the orbital radius and mass quadrupolar contributions to this dynam-
1 of the binary system. Finally we calculate the electro-
ics are also well-known (Barker & O’Connell (1975)),
1 magnetic and gravitationalwave energy outputs which
: withinclusionofleadingorderspin-orbit(Ryan(1996);
v resultcomparableforsomevaluesofmagneticfieldand
Majar & Vasth(2008);Cornish & Shapiro Key(2010);
i radius.
X Kidder et al. (1993, 1995); O’Connell (2004); Rieth
r Keywords Coalescing binary systems, gravitational (1997); Will (2005); Zeng & Will (2007)), spin-spin
a
waves, electromagnetic emission; (Wang & Will(2007);Arun et al.(2009);Klein & Jetzer
(2010); Apostolatos et al. (1995, 1996); Majar (2009))
and mass quadrupole mass monopole (Poisson (1998);
S.Capozziello,M.DeLaurentis,
Racine (2008)) couplings.
I.DeMartino,M.Formisano,D.Vernieri
Similarlytotheinspiral,thering-downphaseadmits
1Dipartimento di Scienze Fisiche, Universit di Napoli “Federico
a perturbative analytical model which describes the
II”
dampedoscillationsofthesinglecompactobjectresult-
2INFNSez. diNapoli,Compl. Univ. diMonteS.Angelo,Edifi-
ing from the binary coalescence (Berti et al. (2009)).
cioG,ViaCinthia,I-80126, Napoli,Italy.
3Fisica Teorica, University of Salamanca, 37008 Salamanca, The plunge/mergerphaseis howevernon-perturbative,
Spain having no analytical description for generic systems,
4Dip. diFisica,Universit`adiRoma”LaSapienza”,PiazzaleAldo therefore it is studied through numerical simulations.
Moro5,I-00185Roma,Italy Templates on binary inspiral (Buonanno & Damour
(1999,2000);Buonanno et al.(2003);Buonanno & Damour
(2006);Cutler et al.(1993);Damour et al.(1998,2000);
Damour(2001);Damour et al.(2002,2003);Finn & Chernoff
2
(1993); Flanagan & Hughes (1993); Pan et al. (2004)) the individual magnetic momenta of a binary neutron
and robust search algorithms were developed for GWs star system, we will introduce a global measure of the
sourcesobservablewithground-based(LIGO Collaborationelectromagnetic properties of the system, given by a
(2009); Sengupta (2009); di Serafino et al. (2010); single magnetic dipole.
di Serafino & Riccio (2010)) and space-based interfer- Gravitational radiation transports away energy, lin-
ometers (Arun et al. (2009); Lang & Hughes (2009)). ear and orbital angular momentum from the system.
In addition to GWs emission, these systems emit Whilethetotalangularmomentum(definedasthesum
electromagnetic radiation(Capozziello et al. (2011); of the orbital angularmomentum and possible individ-
Capozziello et al. (2010)). This emission is due to dy- ual spins of the NS) exhibits instantaneous changes
namics of system and to behavior of plasma around during orbital evolution, its secular evolution over
the system. There are many different systems that timescales larger than the orbital period is a pure
emit gravitational and electromagnetic waves, and in decrease in magnitude, with the direction kept fixed
many case the electromagnetic emission is very ener- (Apostolatos et al. (1994)).
getic (AGNs, QSOs, GRBs). Having in mind (i) the configuration of open field
Pulsars with surface magnetic fields in the range lines aligned with the spin of the final compact object
108÷12Gandmagnetarswithsurfacemagneticfieldsup resulting from the coalescence,and (ii) the property of
to1014÷15Ghavebeenreported(Duncan & Thompson the gravitational radiation to conserve the direction of
(1992);Sinha & Mukhopadhyay(2010);Kouveliotouet al. the totalangularmomentumontimescales longerthan
(1998); Thompson & Duncan (1995); Usov (1992); the orbital period, we chose the magnetic field aligned
Vasisht & Gotthelf (1997); Woods et al. (1999)), as with the total angular momentum of the system.
summarized in (Sinha & Mukhopadhyay (2010)). An In Sec. 2, we present an overview on GWs emission
upper limit of 1018 G was also advanced from the re- fromcoalescingbinarysystems. InSec. 3,weintroduce
quirementtoavoidexoticstructureintheneutronstars anddiscussthemagneticfieldandwealsocalculatethe
(Sinha & Mukhopadhyay (2010)). An interaction of correction induced in the GW luminosity and ampli-
the magnetic dipoles modifies the gravitational radia- tude. For comparison, in Sec. 4, we present the lumi-
nosity of the accompanying electromagnetic radiation.
tion leaving the system, and hence the back-reaction
These are compared and the conclusions presented in
to the orbit is slightly different. Limits on magnetar
Sec. 5.
magnetic fields render these corrections to the second
or even higher PN order. Neutron star binary coa-
lescence, leading to a final black hole with a torus is
2 Gravitational wave emission from coalescing
a configuration believed to be the progenitor of short
binaries
GRBs. In the black hole-torus central engine part of
the torus matter can be convertedinto radiation, lead-
Let us summarize some basic concepts s of the gravi-
ing to the observed bursts (Belczynski et al. (2008)).
tationalradiationgeneratedby the evolutionofa com-
Fully general-relativistic simulations of binary neutron
pact binary system. A wide discussion about the GWs
starcoalescenceleadingtothisconfigurationhavebeen
emission can be found in literature (Maggiore (2007);
presented in (Rezzolla et al. (2010)), with a detailed
Misner et al. (1973); Weinberg (1972)). The starting
and accurate study of the torus. It was found that un-
point for any discussion of GWs is the approximation
equal mass neutron star binaries led to a larger torus
to General Relativity (GR), when the space-time met-
and an empirical formula in terms of the mass ratio
ric can be treated as deviating only slightly from a flat
and total mass was advanced for the torus mass. Nev-
metric (Maggiore (2007); Shapiro & Teukolsky (1983);
ertheless, the effects of the magnetic fields were not in-
Misner et al.(1973)). WestartfromlinearizedEinstein
cludedinthetreatmentofRef. (Rezzolla et al.(2010)).
equation
Many efforts are done in order to study the electro-
magnetic emission from these systems, in particular 16πG
(cid:3)h =− T , (1)
from a computational point of view. It was showed µν c4 µν
how the dynamics of the binary systems induce the where 2=η ∂µ∂ν is the d’Alembertian, expressed in
µν
electromagnetic emission and its link with GWs emis-
terms of the trace reversed metric perturbation
sion (Palenzuela et al. (2009, 2010); Anderson et al.
(2008)). hµν =hµν − 1ηµνh , (2)
Inthispaperweinvestigateasimplemodelforthein- 2
spiral,whichalsotakesintoaccountthemagneticfield.
where h =g −η and h=η hµν (such that h=
µν µν µν µν
Rather then monitoring the complicated evolution of
−h). Due to the imposed Lorentz (harmonic) gauge
3
µν
∂ h = 0, only 6 of the 10 components of h are ial components
ν µν
independent. BysettingT =0weobtainthevacuum
µν
1
wave equation: M = µr2(1+cos2ωt),
11
2
(cid:3)hµν =0. (3) M22 = 1µr2(1−cos2ωt),
2
GWs thus propagate at the speed of light. M = 1µr2 sin2ωt. (10)
12
We evaluate the leading order contribution to the 2
metricperturbationsundertheassumptionthatthein-
From which we obtain
ternal motions of the source are slow compared to the
speed of light. We also assume that the source’s self- h (t)= 4G µr2ω2 (1+cos2θ) cos(2ωt) , (11)
gravityisnegligible. Letusfirstintroducethemomenta + c4d 2
of the mass density
4G
M = c12 d3xT00(t,x), (4) h×(t)= c4dµr2ω2 cosθ sin(2ωt). (12)
Z
1 Here the time is shifted in order to eliminate the de-
Mi = d3xT00(t,x)xi, (5)
c2 pendence on φ (Buonanno (2007)). From Eq. (8) the
Z total radiated power emerges as,
1
Mij = d3xT00(t,x)xixj , (6)
c2 Z 32 Gµ2r4ω6
P = . (13)
and impose the conservation law ∂ Tµν = 0, valid in 5 c5
µ
linearized gravity. For sources having strong gravita-
InordertocompensateforthelossofenergybyGW
tional field, as NSs or BHs, the mass density depends
emission,theradialseparationrbetweenthetwobodies
alsoonthe binding energy,howeverforweakfields and
mustdecrease. Theorbitalfrequencyandconsequently
smallvelocitiesT00/c2 reducestotherest-massdensity
the GW frequency also changes in time, and can be
ρ. Denoting the field h which satisfies the transverse
µν derived using from the balance equation
and traceless gauge conditions (Weinberg (1972)), as
transverse-tracelesstensorhTijT,itisconvenienttocom- dEorbit =−P . (14)
putefromitexplicitlythetwoindipendentpolarization dt
states h+ and h× (Buonanno (2007)).
In terms of the traceless quadrupole tensor
3 Gravitomagnetic corrections on gravitational
1 wave emission
Q =M − δ M, (7)
ij ij ij
3
Inthissectionweintroducethemagneticfieldwiththe
(M denoting the trace of M ), the leading order ex-
ij desiredstructure,namelyalignedwiththetotalangular
pression of the total radiated power is
momentum J of the system.
G ... ...
P = hQ Q i. (8)
5c5 ij ij 3.1 Gravitomagnetic-electromagneticanalogy
Henceforth, we willapply this formalismfora compact
The gravitomagnetic potential characterizing the com-
binarysystemwithmassesm andm ,totalmassM =
1 2 pact binary is A =(0,A), with
µ
m + m and reduced mass µ = m m /(m + m ).
1 2 1 2 1 2
With the plane of motion chosen as the x-y plane and J×r
A=α , (15)
inthe circularmotionapproximationthe reducedmass r3
particle has the coordinates r=(x,y,z) given by
where α = G/c Tartaglia & Ruggiero (2002), and r is
x(t)=r cosωt, y(t)=r sinωt, z(t)=0, (9) theseparationbetweenthebodiesofthebinarysystem.
Due to the fact that the motion of the binary system
r being the relative distance between the two bodies. is entirely confined into the orbital plane, we evaluate
Withρ=µδ(r)wefindMij =µxixj withthe nontriv- all the physical quantities of interest in the plane it-
self. This procedure is analogue to that used in GWs
emission calculations, where one reduces the problem
4
only to the planar relative motion. The gravitomag- whilefortheelectromagneticcontributioninthesystem
netic field is found as inwhichthemagneticfieldisalongthez-axis[suchthat
J=(0,0,J)] we find
2α
B=∇×A= J . (16)
r3 T00 = −µ2α3ǫ , (23)
EM EM
We have used the assumption, that neutron stars ro- J 2 3
ǫ = 2c2− r2ω2 . (24)
tate slowly, therefore the proper spin contributions to EM µr3c 2
(cid:18) (cid:19) (cid:20) (cid:21)
J arenegligible,alsothatthe relativistic(PNand2PN
corrections)arealignedtothe Newtonianorbitalangu- Despite apparences, TE00M does not depend on masses,
larmomentum, thus J·r=0(for amotionconfinedto but it was written this way for an easy PN order eval-
the x-y plane the only nonvanishing component of J is uation: O(J/µrc)2 = O(v/c)2 = O GM/c2r . We
J ≡J). Similarly, a gravitoelectric field emerges as also remark, that O(rω/c)2 = O(v/c)2, therefore the
z (cid:0) (cid:1)
respective term of the expression (24) could be safely
α
E= v×J . (17) dropped.
r3
Inserting
and they combine to a gravito-electromagnetic tensor
T00 =T00 +T00 =ρc2−µ2α3ǫ (25)
F =∂ A −∂ A . PF EM EM
µν µ ν ν µ
The remarkable property of this gravitomagnetic
into(6),thecorrectionstothegravitationalwavesemis-
field B is its alignement to J, a property we seek
sion can be derived. In a binary system, the pres-
for the magnetic field in our model. Therefore we
ence of an additional dipole-electromagnetic field has
willintroduceanelectromagnetic fieldanalogoustothe
to be taken into account. The correction can be de-
gravitoelectric and gravitomagnetic fields, by changing
rived by considering also the electromagnetic stress-
α=G/c intoα=µ /4π,whereµ is the vacuummag-
0 0 energytensorbesidetheperfectfluidone. stressenergy
netic permeability. In what follows, by α we mean this
tensor. Thanks to the gravito-magnetic ormalism dis-
expressionandwe refer to A,B,E,F as true electro-
µν cussedabove,wecanconsiderthe electromagneticcon-
magnetic quantities.
tribution under the same standard of the gravitational
one: thismeansto takeintoaccountthe00-component
3.2 Gravitational wave energy loss and polarizations of the total stress-energy tensor given by the gravita-
tional part (the mass density) and the electromagnetic
We characterize the source by the total stress-energy contribution. In order to get the momenta (obtained
tensor by integrating the total stress-energytensor in the vol-
ume where the binary system is contained) we have to
Tµν =Tµν +Tµν . (18)
PF EM compute an integral consisting of a gravitational and
an electromagnetic part:
composed by a perfect fluid part
Tµν = ρ+ p uµuν −pgµν (19) Mij = d3xρ(x)xixj+
PF c2 Z
(cid:16) (cid:17)
Gravitational part
and an electromagnetic part
2
| {z J } 1 3
−µ2α3 d3x r2ω2−2c2 x x
TEµνM = µ1 FµαFνβηαβ − 14ηµνFσλFσλ . (20) (cid:18)µc2(cid:19) Z r6 (cid:18)2 (cid:19) i j
0 (cid:18) (cid:19) Electromagneticpart
In order to calculate the momenta (4)-(6) we need the | {z (26})
00-components of these tensors. The 00-component of
where µ is the reduced mass of the binary system. We
the perfect fluid contribution in a comoving frame de-
obtain
fined as
1+cos(2ωt) sin(2ωt) 0
uα = u0,0,0,0 , u0u0 =c2 (21) Mij = M sin(2ωt) 1−cos(2ωt) 0 , (27)
2  
0 0 0
(cid:0) (cid:1)
simply becomes
 
where M is its trace, given by
T00 =TPF =ρc2, (22)
PF 00
4πµα3 J 2 3
M=µr2 1− 2c2− r2ω2 . (28)
3r3 µc2 2
(cid:18) (cid:19) (cid:20) (cid:21)!
5
The nontrivial quadrupole tensor components are 4 Electromagnetic emission by the magnetized
binary
1
Q = M[1+3cos(2ωt)] ,
11
6
Now we calculate the electromagnetic energy emission
1
Q = M[1−3cos(2ωt)] , bybinarysystemsinordertocompareittoGWsenergy
22
6
emission. We consider the electromagnetictensor com-
1
Q33 = − M , ponents in orderto calculate the time energyemission:
3
1
Q12 = 2Msin(2ωt) . (29) αJ 2
T01 =−2αx˙(t) , (33)
E cr3
Inserting them into Eq. (14) we obtain the loss of en- (cid:18) (cid:19)
ergy
2
αJ
T02 =−2αy˙(t) , (34)
dE dE dE E cr3
= + = (cid:18) (cid:19)
dt dt dt
(cid:18) (cid:19)GWT (cid:18) (cid:19)GW (cid:18) (cid:19)GWB and finally
32Gµ2r4ω6 8πr3
= − 5c5 1− 15c2µα4ǫEM , TE03 =0. (35)
(cid:18) (cid:19)
(30) Calculating the Poynting vector P~, whose components
are Pi = cT0i, it is possible to obtain the electromag-
dE E
where stands to indicate the total energy netic energy emitted in the unit-time as
dt
(cid:18) (cid:19)GWT
output of gravitationalwaves emission due to the pure
dE dE
gravitationalterm , and the electromagnetic EM =− P~ ·dS~,
dt dt
(cid:18) (cid:19)GW ZΣµ
dE
correction . The polarizations result to be
where Σ is the surface of the reduced mass object,
dt µ
(cid:18) (cid:19)GWB which gives
Gµr2ω2 4πµr3α4
h+ = c4d 1− 3c ǫEM × dEEM =−2αcrω αJ 2Σ . (36)
(cid:18) (cid:19) dt cr3 µ
×[3+cos(2θ)]cos(2ωt), (31) (cid:18) (cid:19)
From eq. (16), it is possible to get the dipole magnetic
momentum J:
Gµr2ω2 4πµr3α4
h× =− c4d 1− 3c ǫEM cosθsin(2ωt). J = Br3. (37)
(cid:18) (cid:19)
2α
(32)
Nowweareableto comparethe differentcontributions
Calculatingh+ andh× foraNSbinarysystemusing to energy emission: the standard GWs term, the elec-
tromagnetic correction to it and then the pure electro-
r =109÷1012m, d∼1019m, m1 =m2 =1,4M⊙, magnetic one. The results are shown in Tab. 1.
afterafewcalculationsweobtainthatthecorrective
electromagnetictermis initially negligible with respect
5 Conclusions
to the gravitational one, but it becomes not negligible
whenweincreasetheorbitalradius,aswellasthemag-
GWssciencehasenteredaneweraandtherecentyears
netic field, as shown in Fig 1. Similar results can be
have been characterized by several major advances.
obtained also for an equal-mass BH or White Dwarfs
For what concerns the most promising GW sources for
binary system for which the corrective terms are neg-
ground-based and space-based detectors, notably, bi-
ligible. However, it is possible to obtain for a binary
nary systems composed of NS and BHs, our under-
systemwitha verystrongdipole magneticmomentum,
standing of the two-body problem and the GW gen-
anelectromagneticcontributiontoGWsemissionwhich
eration problem has improved significantly. The best-
is comparable to the standard gravitationalone.
developed analytic approximation in General Relativ-
ityis undoubtablethe post-Newtonianmethod. Inthis
6
Table1 Thecolumnsofthetable,fromlefttoright,indicatetheradialseparationbetweenthetwobodies(r),theorbital
dE
frequency (ω),the electromagnetic energy emitted in the unit-time EM , the loss of energy in gravitational emission
(cid:18) dt (cid:19)
dE dE
GW and its electromagnetic correction GWB . The ranges are for magnetic fields B=(1012÷1018) G.
(cid:18) dt (cid:19) (cid:18) dt (cid:19)
r (m) ω (Hz) − dE (erg/s) − dE (erg/s) − dE (erg/s)
dt EM dt GW dt GWB
109 3.04×10−4 1.14×1021÷1.14×1033 6.72×1029 6.88×1012÷6.88×1024
(cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1)
1010 9.61×10−6 3.60×1020÷3.60×1032 6.72×1024 6.88×109÷6.88×1021
1011 3.04×10−7 1.14×1020÷1.14×1032 6.72×1019 6.88×106÷6.88×1018
1012 9.61×10−9 3.60×1019÷3.60×1031 6.72×1014 6.88×103÷1.64×1015
paper, we have calculated the electromagnetic correc-
tionstotheGWsemittedbyacoalescingbinarysystem.
Wehaveconsideredanelectromagneticdipole-typefield
and calculated the electromagnetic contribution to the
stress-energy tensor. We have obtained a correction to
the standard gravitational energy-loss which becomes
null setting the magnetic field to zero. In general, the
electromagnetic correction term is negligible with re-
specttothegravitationalone. Thisresultholdsalsofor
black hole and white dwarfs binary systems. However,
there could be coalescing binary systems with a large
dipole magnetic momentum and a large orbital radius
where the electromagnetic correction to GWs emission
is relevant. In particular, the electromagnetic energy
emitted by these binary systems is comparable to the
gravitational wave output for given values of the mag-
netic field and of the orbital radius. Revealing such a
phenomenon by observations could be an electromag-
netic signature for the gravitationalwaves emission.
7
Fig.1 Profilesoftheenergylossintimewhenweincreasetheorbitalradius(r)andthetotalemittedgravitationalenergy
with respect to the electromagnetic correction to GWs emission: (a) the magnetic field is B =1012G; (b) B =1015G; (c)
B=1018G.
(a) (b)
(c)
8
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