On dynamics of relativistic shock waves with losses in gamma-ray burst sources

0.08 MB·English

by  E. V. Derishev

5 ONDYNAMICSOFRELATIVISTIC SHOCKWAVES
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0 WITHLOSSESINGAMMA-RAYBURSTSOURCES
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E.V.Derishev,Vl.V.Kocharovsky,K.A.Martiyanov
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InstituteofAppliedPhysics
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46Ulyanovst.,603950NizhnyNovgorod,Russia
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[email protected]

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v
2 Abstract Generalizationoftheself-similarsolutionforultrarelativisticshockwaves(Bland-
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ford&McKee, 1976) isobtainedinpresenceoflosseslocalizedontheshock
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front ordistributedinthedownstreammedium. Itisshownthattherearetwo
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qualitativelydifferentregimesofshockdeceleration,correspondingtosmalland
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largelosses. Wepresent thetemperature, pressureanddensitydistributionsin
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0 thedownstreamfluidaswellasLorentzfactorasafunctionofdistancefromthe
/ shockfront.
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p
- Keywords: relativisticshockwaves,gamma-raybursts
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Introduction
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Theprogenitorsofgamma-raybursts(GRBs)arebelievedtoproducehighly
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v relativistic shocks at the interface between the ejected material and ambient
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X medium (see, e.g., Meszaros, 2002; Piran, 2004 for review). Non-thermal
r spectraandshortdurationofGRBsplaceafirmlowerlimittothebulkLorentz
a
factorofradiatingplasma,whichmustexceedafewhundredtoavoidthecom-
pactnessproblem (e.g.,Baring&Harding,1995).
Consider arelativistic spherical blastwaveexpanding intoauniform ambi-
ent medium withthe Lorentz factor Γ 300. The average energy per baryon
∼ 2
in the fluid comoving frame behind the shock front is of the order of Γm c
p
(Taub, 1948), where m is proton mass, and the plasma in the downstream
p
presumably forms a non-thermal particle distribution extending up to very
highenergies. Undertheseconditions, medium downstream issubject tovari-
ous loss processes. The non-thermal electrons produce synchrotron radiation,
which accounts for GRB afterglow emission, and (at least partially) for the
prompt emission. Apart from the synchrotron radiation of charged particles
there is another mechanism of energy and momentum losses connected with
inelastic interactions ofenergetic protons withphotons. Thesereactions cause
2
proton-neutronconversionasaresultofchargedpioncreation. Itshouldbeno-
ticed that for typical interstellar density the Coulomb collisions are inefficient
and the charged particles instead interact collectively through the magnetic
field. Thisallowstodescribe plasma motion using hydrodynamical approach,
thoughitcanbreakforasmallfractionofthemostenergetic particles.
When a proton turns into a neutron or another neutral particle is born, it
does not interact with the magnetic field and hence the energy spent for its
creation is lost from the hydrodynamical point of view. The synchrotron and
inverse Compton emission, as well as energetic photons, neutrinos and neu-
tronsproduced viaphotopionic reactions, escapefromdownstream givingrise
to non-zero divergence of the energy-momentum tensor. The creation of en-
ergetic neutrons is also afirststep inthe production ofhighest-energy cosmic
raysthrough theconverter mechanism (Derishevetal.,2003).
Ejection from theGRBprogenitor ofamassM0 withinitial Lorentz factor
Γ0 results in two shocks propagating asunder from the contact discontinuity.
The forward shock moves into the external gas and has a much greater com-
pression ratio at its front than the other, reverse shock, which passes through
theejectedmatter. Astheshockedexternalgashasatemperaturemuchhigher
thanthatinthevicinityofthereverseshock,weneglectthelossesintheejecta.
Wediscuss two models. In the first one we assume the energy losses to be
localized close to the shock front, whereas the matter downstream the shock
is considered lossless. In another model the shock front is treated as non-
dissipative andthelossesaredistributed allovertheshocked gas.
Following the recipe of Blandford and McKee (1976) we generalize their
well-knownself-similarsolutionsforrelativisticblastwavesforthecase,where
the energy and momentum of the relativistic fluid is carried away by various
speciesofneutralparticles.
Self-similar solutions
Westartfromtheenergy-momentumcontinuityequations,whereinthecase
ofdistributed lossesanon-zero r.h.s. isincluded:
∂T00 1 ∂(r2T0r) ∂T0r 1 ∂(r2Trr) 2p
c∂t + r2 ∂r = −ϕ0T00, c∂t + r2 ∂r − r = −ϕ1T0r,
T00 = wγ2 p, T0r = wγ2β, Trr = wγ2β2+p,
−
whereT istheenergy-momentum tensor,γ theLorentzfactor,w=e+ptheen-
thalpydensity,pthepressure,etheenergydensity. Allquantitiesaremeasured
inthefluidcomovingframe. Inthefollowinganalysisweusetheultrarelativis-
ticapproximation oftheseequations, obtainedbyexpandingvelocityuptothe
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third contributing order in γ and the equation of state up to the first order.
−
Becauseofthelackofspace,hereweconsideronlyequallossesfortheenergy
andmomentum(ϕ0=ϕ1=ϕ).
RELATIVISTICSHOCKWAVESWITHLOSSESINGRBSOURCES 3
1.2
1
r 1 a =10 a =10
cto 0.8
a
z f0.8 a =5
orent0.6 ssure0.6
red L0.4 e =0.8e =0.6 Pre0.4 a a==52 e =0.8
ua a <3 e =0.6
q 0.2
S0.2 e <0,397
e <0,397
0 0
1 2 3 4 5 6 1 2 3 4 5 6
c c
e =0.8
3 a =10
1 e =0.6
a =5
a =10
0.8
sity ure2
e den0.6 perat ea==02.2
cl0.4 a =5 m
rti e =0.8 Te1
Pa e =0.6 a =2
0.2 e =0.2
0 0
1 2 3 4 5 6 1 2 3 4 5 6
c c
Figure1. ThedistributionsoftheLorentzfactor,pressureandparticledensityoftheshocked
gasasfunctionsofself-similarvariableχ. Allquantitiesarenormalizedtounityattheshock
front where χ = 1. Long-dashed lines correspond tothe losses uniformly distributedin the
downstreammediumanddecreasingwithtimeast−1. Short-dashed linesareforthecaseof
localizedlosseswithδ=η=0.Thesolidline—solutionofBlandfordandMcKee(1976).
Inthecaseoflocalizedlosseswetreatthemasdiscontinuities oftheenergy,
momentumandparticlenumberfluxesattheshockfront,whicharecharacter-
ized by three parameters ε, δ, η equal to the fractions of corresponding fluxes
lost at the shock front in the front comoving frame. We obtain the following
expressions for the pressure p2, number density n2 and the Lorentz factor γ2
immediatelybehindtheshockfront:
p2 = 2ξ1Γ2w1, γ22 = 1ξ2Γ2, n2 = 2√2ξ3Γn1, whereξ1 = 3 (21−δ)ξ2 ,
3 2 2ξ2 ξ2+1
−
3(1 (1 ε)/(1 δ)) (1 η) √ξ2
ξ2 = 1+ − − − , ξ3 = − .
1+p4 3(1 ε)2/(1 δ)2 √2 1 ξ2
− − − −
TheydifferfromtheTaubadiabatonlybynumericalfactorsξ ,whichbecome
i
2
unityintheabsence oflosses. Herew1 = n1mpc istheenthalpy density and
n1 thenumberdensityoftheexternal gas.
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Tofindaself-similar solution weassume Γ2=t m andintroduce thesimi-
−
larity variable χ=ct r, where r isdistance from the center and R the current
ct−R
radiusoftheshockf−ront. Wefindself-similarsolutionsinthecaseoflocalized
losses and in the case of ϕ=α, α=const, but they also exist for non-uniform
ct
distributions of losses if α=α(χ). The velocity, pressure and particle density
arefoundintermsofvariables Γ,χ.
From the energy balance equation we find that the power law index m is
in the range 3 m 6. There are two regions in the parameter space where
≤ ≤
the solutions are qualitatively different. In the case of small losses, α<3 or
ε+2 √109δ<1+2 √109, the index m rises from 3 to 6as the losses increase.
−14 −14
The pressure, Lorentz factor and number density of the downstream are pro-
portional topowers ofthesimilarity variable whose indices are different from
those in the solution of Blandford and McKee (1976). On the contrary, large
lossesleadtotheuniversaldeceleration lawoftheshock: misequalto6. The
problem is fully integrable but solutions can not be written as explicit. In the
high-losssolutionsthereappearsanexpandingsphericalcavityboundedbythe
contactdiscontinuity andthetemperatureatitsedgetendstoinfinity.
Thesolutions obtained arepresented inFig.1.
Conclusion
We have analyzed the dynamics of relativistic shock wave with losses due
toescapeofneutralparticlesfromplasmaflow. Bothforlocalizedandfordis-
tributed losses there are self-similar solutions, which are different from those
foundpreviouslyforlosslesscase. Wefindthatincreasingofthelosseschange
the dynamics of the shock deceleration qualitatively. In the case of small
losses, the role of ejected material asymptotically vanishes and the Lorentz
factor of the shock decreases as t m with m varying from 1,5 (no losses) to
−
3. In the opposite case of large losses, the shock decelerates in accordance
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with universal law Γ t and the energy content in the ejecta constitutes a
−
∼
significant fraction of the total energy budget. Also, in the presence of large
losses,thetemperatureandtheLorentzfactorofthefluidbehindtheshockcan
benon-monotonic functions ofdistance fromtheshock.
Acknowledgments
ThisworkwassupportedbytheRFBRgrantsnos. 02-02-16236 and04-02-
16987,thePresidentoftheRussianFederationProgramforSupportofLeading
ScientificSchools(grantno. NSh-1744.2003.2),andtheprogram"Nonstation-
ary Phenomena in Astronomy" of the Presidium of the Russian Academy of
Science. E.V. Derishev acknowledges the support from the Russian Science
SupportFoundation.
RELATIVISTICSHOCKWAVESWITHLOSSESINGRBSOURCES 5
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