The role of kink instability in Poynting-flux dominated jets

0.54 MB·English

by  Dimitrios Giannios

Astronomy & Astrophysics manuscript no. 4107 February 5, 2008
(DOI: will be inserted by hand later)
The role of kink instability in Poynting-flux dominated jets
Dimitrios Giannios and Henk C. Spruit
Max Planck Institutefor Astrophysics, Box 1317, D-85741 Garching, Germany
Received / Accepted
6
0 Abstract. The role of kink instability in magnetically driven jets is explored through numerical one-dimensional
0
steady relativistic MHD calculations. The instability is shown to have enough time to grow and influence the
2
dynamics of Poynting-flux dominated jets. In the case of AGN jets, the flow becomes kinetic flux dominated at
n distances >∼1000rg because of the rapid dissipation of Poynting flux. When applied to GRB outflows, the model
a predictsmoregradualPoyntingdissipation and moderately magnetized flowat distancesof∼1016 cmwherethe
J
deceleration of the ejecta due to interaction with the external medium is expected. The energy released by the

3 instability can power the compact “blazar zone” emission and the prompt emission of GRB outflows with high
2 radiative efficiencies.

2
Key words.Magnetohydrodynamics (MHD) – Instabilities – Gamma rays: bursts– Quasars: general
v
2
7
1 1. Introduction Beskinetal.1998).Further accelerationofthe flow is not
1 straightforward within ideal MHD. It can be shown, for
0 Relativistic collimatedoutflows havebeen extensivelyob-
example,thataradialflowisnotacceleratedafterthefast
6 servedin active galactic nuclei (AGN) and X-ray binaries
point (e.g. Beskin 1998). This is a result of the fact that
0 (XRB). Gamma-raybursts (GRB) are alsobelieved to be
/ the magnetic pressure and tension terms of the Lorentz
h connected to ultrarelativistic and collimated outflows to
force almost cancel each other (Begelman & Li 1994). A
p overcome the “compactness problem” (e.g. Piran 1999)
limited degree of acceleration of the flow is possible if it
-
and to explain the achromatic afterglow breaks (Rhoads
o has a decollimatingshape (i.e. the magnetic field diverges
r 1997; Sari et al. 1999). It is also believed that all these
faster than radial; Li et al. 1992; Begelman & Li 1994).
t
s sources are powered by accretion of matter by a compact
Magnetized jets suffer from a number of instabilities.
a object.
: Interactionwiththeenvironmentcausesinstabililtyofthe
v The widely accepted mechanism for jet acceleration
Kelvin-Helmholtztypeandkinkinstabilitycausesinternal
Xi and collimation in the context of AGN and XRB jets rearrangementofthefieldconfiguration.Herewefocuson
is that of magnetic driving. According to this paradigm,
r kink instability, since it internallydissipates magneticen-
a magnetic fields anchored to a rotating object can launch
ergy associated with the Poynting flux. As demonstrated
an outflow. The rotating object can be a star (Weber
elsewhere (Drenkhahn 2002; Drenkhahn & Spruit 2002;
& Davis 1967; Mestel 1968), a pulsar (Michel 1969;
Spruit & Drenkhahn 2003) such internal energy dissipa-
Goldreich & Julian 1970), an accretion disk (Bisnovatyi-
tion directly leads to acceleration of the flow. Dissipation
Kogan & Ruzmaikin 1976; Blandford 1976; Lovelace
steepenstheradialdecreaseofmagneticpressure,thereby
1976; Blandford & Payne 1982) or a rotating black hole
lifting the cancellation between outward pressure force
(Blandford & Znajek 1977). The material is accelerated
and inward magnetic tension, and allowing the magnetic
thermallyuptothesonicpointandcentrifugallyuntilthe
pressure gradient to accelerate the flow.
Alfv´en point, defined as the point where the flow speed
equals the Alfv´en speed. After the Alfv´en point the in-
ertia of mater does not allow corotation of the magnetic 1.1. “AC” versus “DC” outflows
field.Asaresult,themagneticfieldlinesbend,developing
While dissipation of magnetic energy can thus hap-
a strong toroidal component.
pen through kink instability in an initially axisymmetric
Further out the flow passes through the fast magne-
(“DC”)flow,itcanalsohappenmoredirectlybyreconnec-
tosonicpointwheremostoftheenergyoftheflowremains
tionintheoutflowgeneratedbyanon-axisymmetricrota-
inthe formofPoyntingfluxinthe caseofrelativisticout-
tor(“AC”flow).Thetwocasesbehavedifferentlyinterms
flows(Michel1969;Goldreich&Julian1970;Sakurai1985;
of the acceleration profile, and the location and amount
Send offprint requests to: [email protected] of radiation produced by the dissipation process. A non-
2 Dimitrios Giannios and HenkC. Spruit:The role of kink instability in jets
axisymmetric rotator produces a “striped” outflow (as in component of the magnetic field while accelerating the
the case of a pulsar wind) with reconnectable changes of flowatthe sametime.Thedissipationofmagneticenergy
direction of the field embedded in the flow, and energy is almost complete and fast in the case of AGN jets, so
releaseindependent ofthe opening angleofthe jet.Inthe that on parsec scales the flow has become kinetic energy
DC case, where energy release is instead mediated by an dominated, in agreement with current interpretations of
instability,therateofenergyreleaseislimitedbythetime theobservations(e.g.Sikoraetal.2005,wherethepossible
it takes an Alfv´en wave to travel across the width of the effects of magnetic dissipation are also discussed briefly).
jet. This makes it a sensitive function of the jet opening The DC modelwith kink instability alsoproducessig-
angle. nificantflowaccelerationintheGRBcase,butconversion
The “AC” case has been studied in detail by ofthe Poyntingflux islesseffective thanthe AC modelin
Drenkhahn (2002), Drenkhahn and Spruit (2002), with this case.
application to Gamma-ray bursts. In the case of AGN The structure of the paper is as follows. In Sec. 2 we
and XRB, on the other hand, the collimated jet is ar- discuss MHD instabilities in jets and focus on the kink
guably best understood if the field in the inner disk is instability and its growth rate. The model is described in
of uniform polarity, resulting in an initially axisymmetric Sec.3includingtheassumptions,thedynamicalequations
flow. Another difference is the lower bulk Lorentz factors and the parameters at the base of the flow. In Sec. 4, we
in the AGN/XRB case, resulting in faster energy release applythemodeltothecaseofAGNjetsandGRBs,while
(in units of the dynamical time of the central engine). the last two Sections present the discussion and conclu-
sions.
The purpose of this paper is to explore the conse-
quences of magnetic dissipation by internal instability in
such axisymmetric (or DC) cases, and its observational 2. The kink instability
signatures. We also apply the calculations to the GRB
Magnetized outflows are subject to a variety of in-
case,wherewecomparetheresultswiththeACcasestud-
stabilities. These can be classified as pressure driven,
ied before.
Kelvin-Helmholtz and current driven instabilities (see,
e.g., Kadomtsev 1966; Bateman 1978). Pressure driven
1.2. Energy release and field decay by the instability instabilities (Kersal´e et al. 2000; Longaretti 2003) are re-
lated to the interplay between the gas pressure and the
We limit ourselves to a flow with constant opening angle.
curvatureofmagneticfieldlines.Theyarerelevantcloseto
Thatis,weleaveasidethecollimationprocess.Kinkinsta-
thelaunchingregionoftheoutflowsandmaybeimportant
bility is modeled by adding a sink term to the induction
as long as the outflow is still subsonic. Kelvin-Helmholtz
equationto accountforthe non-idealMHD effects arising
(KH) instabilities (Ray 1981; Ferrari et al. 1981; Bodo et
from it.
al. 1989; Hardee & Rosen 1999) arise from velocity gra-
Linear stability theory of kink instability yields a dients in the flow and may be important in the shear-
growth time of the order tk = rθ/vA,φ where r is the ing layer between the outflow and the external medium.
radius of the jet, θ its opening angle and vA,φ the Alfv´en KH instabilities have been extensively studied and be-
speed based on the azimuthal component Bφ of the field. come strongestin the regionbeyondthe Alfv´enpoint but
This is independent of the poloidal field component, at still within the fast magnetosonic point. Current driven
least for the so-called internal kink modes (which do not (CD; Eichler 1993; Spruit et al. 1997; Begelman 1998;
disturb the outer boundary ofthe field, cf. Bateman 1978 Lyubarskii 1999; Appl et al. 2000) instabilities have re-
for details). For stability analysis and numerical simula- ceived much less attention but are the most relevant ones
tions with astrophysical applications see, e.g. Begelman for Poynting-fluxdominatedoutflows,since they cancon-
(1998);Appletal.(2000);Leryetal.(2000);Ouyedetal. vert the bulk Poynting flux into radiation and kinetic en-
(2003);Nakamura&Meier(2004).Basedonlineartheory, ergyofthe flow(forthe roleofCDinstabilitiesinanelec-
wewouldpredictthatthepoloidalfieldcomponentcanbe tromagnetic model for GRBs see Lyutikov & Blandford
ignored for the rate of energy release. 2003). Among the CD instabilities, the m =1 kink in-
It is not clear, however, that the poloidal component stability is generally the most effective. In this work, we
can be ignored for the nonlinear development of the in- focus onthe effect ofthe kink instability onthe dynamics
stability, which is what actually determines the energy of these outflows.
release. As a way to explore the effect of possible nonlin-
earstabilizationbyapoloidalcomponentwecomparetwo
2.1. The growth rate of the instability
casesinthecalculations:onewithenergyreleaseandfield
decay given by the Alfv´en time acrossthe jet ( tk above), While magnetized outflows can be accelerated “centrifu-
andoneinwhichthisrateisassumedtobereducedbythe gally” by large scale poloidal fields (Blandford & Payne
poloidal component. This mainly affects the early phases 1982; Sakurai 1985, 1987), at the radius of the light-
of the acceleration of the flow beyond the light cylinder. cylinder inertial forces become significant and the mag-
We find that kink instability has time to grow in the netic field cannot force corotation. At this radius the
AGN and XRB cases, dissipating energy in the toroidal strength of the toroidal and the poloidal components are
Dimitrios Giannios and HenkC. Spruit:The role of kink instability in jets 3
comparable. Further out, the induction equation dictates 3. The model
that, within ideal MHD, the toroidal component domi-
A magnetically launched outflow passes through three
nates over the poloidal one since the strength of the for-
mer scales as 1/r while that of the latter as 1/r2. This characteristic points where the speed of the flow equals
the speed of slow mode, the poloidal Alfv´en wave and
magnetic configuration of a strongly wound-up magnetic
the fast mode and are called the slow magnetosonic, the
field like this is known, however, to be highly unstable to
Alfv´en and the fast magnetosonic points respectively. For
thekinkm=1modefromtokamakexperiments(see,e.g.,
flows where the energydensity of the magnetic fielddom-
Bateman 1978). Linear stability analysis has shown that
inates that of matter, the Alfv´en point lies very close to
the growth time of the instability is given by the Alfv´en
the light-cylinder
crossingtimeacrosstheoutflowinaframecomovingwith
it (Begelman 1998; Appl et al. 2000).
Thestudyofthenon-linearevolutionoftheinstability RL =c/Ω, (4)
demands three dimensional relativistic MHD simulations
over many decades of radii and it is, therefore, not sur- where Ω is the angular velocity of the foot point (e.g.
prisingthatthe issueisnotsettled.Leryetal.(2000)and Camenzind 1986). At the Alfv´en radius most of the cen-
Baty & Keppens (2002) argued in favor of the dynamical trifugalaccelerationhasalreadytakenplaceandthemag-
importanceofthe instabilityinreorganizingthemagnetic netic field cannotforce corotationof matter.At this loca-
configuration inside the jet. It has been argued, however, tion,thetoroidalandthepoloidalcomponentsofthemag-
that the jet creates a “backbone” of strong poloidal field netic field are comparable in magnitude. Further out, the
whichslowsdownthedevelopmentofinstabilities(Ouyed flow passes through the fast magnetosonic point at a dis-
et al. 2003; Nakamura & Meier 2004). In view of these tance a few RL (Sakurai1985;Lietal.1992;Beskin
∼
works and since the growth rate of the instability is im- etal.1998).Atthelocationofthefastmagnetosonicpoint
portantforthisstudy,weconsidertwoalternativesforthe the speed of the four-velocity of the flow equals µ1/3,
∼
non-linearstageofthe instability.Inthe firstcase,the in- where µ is the Michel magnetization parameter (i.e., the
stability proceeds at the Alfv´en crossing time across the energyfluxperunitrestmass;Michel1969).ForPoynting-
outflow(assuggestedbylinearstabilityanalysis)andrear- flux dominated flows (i.e., µ 1), most of the energy is
≫
ranges the magnetic field configuration to a more chaotic stillinmagnetic formatthis pointsincethe ratioofmag-
one. In this case the instability time scale is given by the netic to matter energy flux is µ2/3. There is thus a
∼
expression (in the central engine frame) choice between flows with high Lorentz factors (but inef-
ficient conversion of Poynting flux to kinetic energy), or
rθγ efficient conversion at the price of low terminal Lorentz
t = . (1)
k
vA,φ factors. Better conversion within ideal MHD appears to
be hard to achieve except by decollimation of the flow
We will refer to this as the “fast kink” case.
(Li et al. 1992; Begelman and Li 1994, Bogovalov 2001;
For the second case, we reduce the dissipation rate
Daigne and Drenkhahn 2002; but see claims to the con-
by a suitable (but arbitrary) function of the poloidal-to-
trary by Vlahakis and Konigl 2003a,b; Fendt and Ouyed
toroidal field ratio.
2004).Evenwithsuchdecollimation,theadditionalaccel-
tk = rθγ eBpco/Bφco. (2) earnadtiDonreinskrhaathhner2m00o2d)e.st (Begelman and Li 1994; Daigne
v
A,φ
We set the initial conditions of our calculation at the
Wewillrefertothisasthe“slowkink”case.Thisrecipeis fastmagnetosonicpointr .Tomaketheproblemtractable
0
meant only as a means to explore the possible effect that we make a number of simplifying assumptions. First, we
the poloidal field component could have on the net accel- limit ourselves to a radial, static flow. Evidently, this ap-
erationofthe flow,if it affects the dissipation rate,and is proach does not allow us to explore the important issue
notmeanttobequantitative.Numericalsimulationsofthe ofjetcollimation(see,howeverSection5.1).Furthermore,
instability would be needed to determine which of these theflowisassumedone-dimensionalbyignoringthestruc-
prescriptions (if any) of the growth time scale is close in ture of the jet in the θ direction. Also, we ignore the
describingitsnon-lineardevelopment(seealsoSection5). azimuthal component of the velocity. This component is
In the last expressions,Bφco,Bpco, are the toroidalandthe not dynamically important beyond the fast magnetosonic
poloidalcomponentsofthemagneticfieldasmeasuredby point(e.g.Goldreich&Julian1970)andcanbeneglected
an observer comoving with the flow, θ is the jet opening from the dynamic equations. On the other hand, the
angle, γ is the bulk Lorentz factor of the flow and vA,φ is poloidalcomponent(takentoberadialforsimplicity)still
the φ component of the Alfv´en speed given by has to be taken into account when modeling the effect of
thekinkinstabilitysinceitinfluencesitsgrowthtimescale
u Bco
v =c A,φ , u = φ . (3) [seeEqs.(1),(2)].Thesesimplifyingassumptionsminimize
A,φ 1+u2 +u2 A,φ (4πw)1/2 the number of the free parameters of the model, allowing
A,φ A,p
us to study the effect of each on the jet dynamics, as will
q
Here, w is the internal enthalpy, to be defined below. become clear in the next sections.
4 Dimitrios Giannios and HenkC. Spruit:The role of kink instability in jets
3.1. Dynamical equations and
B
To determine the characteristics of the flow as a function Bco = φ. (13)
of radius, one needs the conservation equations for mass, φ γ
energy and angular momentum. These equations can be
broughtinthe form(if,forthe moment,weneglectradia- 3.2. Modeling the kink instability
tive losses; e.g. Lyutikov 2001; Drenkhahn 2002)
The set of equations presented in the previous section
∂rr2ρu=0, (5) is not complete. There is one more equation needed to
βB2 determine the problem at hand, which is the induction
∂ r2 wγu+ φ =0, (6) equation. For ideal MHD, the induction equation yields
r
4π
∂ βrB = 0 and can be integrated to give the scaling
(cid:16) (cid:17) r φ
(1+β2)B2 B 1/r for relativistic flows. One can immediately see
∂ r2 wu2+p+ φ =2rp, (7) φ ∝
r 8π that the Poynting-flux term in equation (9) is approxi-
(cid:16) (cid:17) mately constant and no further accelerationof the flow is
where w = ρc2 +e+p is the proper enthalpy density, e
possiblewithinidealMHDforaradialflow.Thisisaresult
andparetheinternalenergyandpressurerespectivelyand
of the fact that the magnetic pressure and tension terms
u=γβ is the radialfour-velocity.We stillneed toassume
of the Lorentz force almost cancel each other (Begelman
an equation of state that will provide a relation between
& Li 1994).
the pressurep andthe internalenergy.Assuming anideal
Weargue,however,thatwhenthe toroidalcomponent
gas,wetakep=(γ 1)e,whereγ istheadiabaticindex.
a− a of the magnetic field becomes dynamically dominant the
Mass conservation (5) can be integrated to yield mass
kink instability sets in. The instability drives its energy
flux per sterad
from B2 on the instability growth time scale. This effect
φ
M˙ =r2uρc, (8) can be crudely modeled by the addition of one sink term
onthe righthandside ofthe induction equationfollowing
while energy conservation gives the total luminosity per
Drenkhahn (2002), Drenkhahn & Spruit (2002)
sterad
βc(rB )2 rBφ
L=wr2γuc+ φ . (9) ∂rβrBφ = . (14)
− ct
4π k
The first term of the last expression corresponds to the The kink instability time scaleis givenby the expressions
kineticenergyfluxandthesecondtothePoyntingflux.A (2)or(1)dependingonwhetherthepoloidalcomponentof
key quantity is the “magnetic content” of the flow which the magnetic field is assumed to have a stabilizing effect.
wewillrefertoasthemagnetizationparameterσ,defined When the instability sets in, Bφ drops faster than 1/r
as the ratio of the radial Poynting to matter energy flux and acceleration of the flow is possible at the expense of
its magnetic energy.
L B2
σ = pf = φ . (10)
L 4πγ2w
kin
3.3. Radiative losses
For a flow to reach large asymptotic Lorentz factors
(observations indicate γ 10 20 for quasars, and theo- The dynamical equations (6) and (7) are derived under
∼ −
reticalargumentsarisingfromthe“compactnessproblem” theassumptionthatnoenergyormomentumescapefrom
such as Piran 1999 constrain γ>100 for GRBs), it must the outflow.This is accurate when the instability releases
start with a high energy to ma∼ss ratio. Within the fire- energy in the optically thick region of the flow. On the
ball model for GRBs (Paczyn´ski 1986; Goodman 1986) the other hand, in the optically thin regime energy and
this means that e ρc2. In this work, we focus on the momentummaybetransferedintotheradiativeformthat
≫
opposite limit where most of the energy is initially stored escapes and does not interact with matter. Let Λ be the
in the magnetic field (σ0 1), while we treat the flow emissivity of the medium in the comoving frame, that is,
as cold (i.e., e<ρc2). Obvi≫ously, there can exist an inter- the energy that is radiated away per unit time and per
mediate regim∼e of a “magnetized-fireball” models where unit volume. If the emission is isotropic in the comoving
both e/ρc2 and σ0 1 at the base of the flow. frame the energy and momentum Eqs. (6), (7) including
≫
Finally, the strength of the radial component is given the radiative loss terms are (Ko¨nigl & Granot 2002)
by flux conservation by the expression
βB2 Λ
Br =Br,0 rr0 2. (11) ∂rr2 wγu+ 4πφ!=−r2Γc , (15)
(cid:16) (cid:17)
For a flow that is moving radially with a bulk Lorentz
B2 Λ
factor γ, the expressions that relate the comoving com- ∂ r2 wu2+p+ 1+β2 φ =2rp r2u . (16)
r
ponents of the magnetic field to those measured in the 8π! − c
(cid:0) (cid:1)
central engine frame are
The importance of the cooling term depends on the
Bco =B (12) coolingtimescale.Ifitisshortcomparedtotheexpansion
r r
Dimitrios Giannios and HenkC. Spruit:The role of kink instability in jets 5
time scale, the matter stays cold during the dissipation be cold at r , i.e. e=0 and using the previous expression
0
process. In this limit, all the dissipated energy is locally with Eqs. (9), (10) one finds for ρ and B
0 φ,0
radiated away. The dissipative processes that appear in
L
the non-linear stage of the instability are poorly under- ρ = (19)
0
stood. It could be the case that the released energy leads r02c3 σ0(σ0+1)3
tofastmovingparticles(i.e.electronsandions)and/orto
and p
Alfv´en turbulence (Thompson 1994). Synchrotron emis-
sionis a plausible fastcoolingprocessforthe electrons.It 4π σ 1/4 L
0
B = . (20)
isparticularlyeffectiveinourmodel,becausethemagnetic φ,0 r σ +1 c
0 0 r
fieldstrengthsarehighinaPoyntingflux dominatedout- (cid:16) (cid:17)
The role of the different model parameters becomes
flow.Ions,however,are,due totheir highermasses,much
clearinthenextsectionwherethemodelisappliedtothe
less efficient radiators.
caseofAGNjetsandGRBs.Outofthefreeparametersof
The form of this cooling term we assume here is
the model, σ and θ are of special importance. The mag-
0
netizationσ determinesthe“magneticdominance”ofthe
ecu 0
Λ=κ (17) flow, i.e., the speed of the flow at the fast magnetosonic
r
point and at a large distance from the central engine. On
the other hand, the opening angle θ is directly related to
where k is an adjustable cooling length parameter. The
the growth rate of the instability [see Eqs. (1), (2)]. The
cooling length is the distance by whichthe matter travels
instability has enoughtime to growif it is faster than the
outward while the internal energy e is lost. When κ 1
≫ expansion time r/c. The ratio of the two time scales at
the cooling length is very short, only a small fraction of
the base of the flow is (using prescription(1) for the time
the expansion length scale r and thus qualifies for the
scale of the kink instability)
description of a fast cooling flow. This, in more physical
terms, corresponds to the case where most of the energy θγ0c θ√σ0c
is dissipated to fast moving (and therefore fast cooling) tk/texp = . (21)
v ≃ v
A,φ A,φ
electrons. On the other hand, setting κ 1, the cooling
≪
lengthismuchlongerthantheexpansionlengthandmost If t /t 1, the kink instability does not have enough
k exp
≫
of the energy stays in the flow leading to more efficient time to grow and the evolution is close to that predicted
adiabatic expansion. This is the case when the dissipated by idealMHD (where not muchaccelerationtakes place).
energy is mostly shared among the ions. Ontheotherhand,ift /t 1,theinstabilitygrowsfor
k exp
≪
many e-foldings andturns almostallthe magnetic energy
in the flow into radiation and kinetic flux. Keeping the
3.4. Initial conditions, model parameters
openinganglefixed,thishappensmuchmoreefficientlyin
lower σ flows (provided that σ >1 so that the Alfv´en
The characteristics of the flow are determined when a 0 0
speed is a significant fraction of t∼he speed of light). We
numberofquantitiesarespecifiedatthefastmagnetosonic
return to this point in the next sections.
pointr whichistakentobe afewtimesthelightcylin-
0
∼
der radius (Sakurai1985;Begelmanet al. 1994;Beskinet
al. 1998), or expressed in terms of the gravitational ra- 4. Applications
dius r =GM/c2 of the central engine r 100r . These
g 0 g
∼ Although at first sight different, jets in AGNs (and mi-
quantitiesarethe initialmagnetizationσ ,the luminosity
0
croquasars) and GRBs probably have central engines of
L, the opening angle θ, the ratio B /B and the cool-
r,0 φ,0
similar characteristics. AGN jets are launched in the in-
ing length scale κ. The quantities one has to solve for so
ner regions of magnetized accretion disks (Blandford &
as to determine the characteristics of the flow are ρ, e, u
Payne 1982),or drive their power by magnetic fields that
and B as functions or radius r. This is done by integrat-
φ
are threading the ergosphere of a rotating black hole
ingnumericallythemass,energy,momentumconservation
(Blandford&Znajek1977).InthecaseofGRBs,thesame
equations and the modified induction equation. The pa-
centralenginemaybeatwork,ortheenergyistappedbya
rameters of the model determine the initial values of ρ, e,
millisecond magnetar (Usov 1992; Klu´zniak & Ruderman
u and B at r .
φ 0
1998; Spruit 1999). In all of these situations, strong mag-
Theinitialfour-velocityforourcalculationsisassumed
netic fields playanimportantroleandmostofthe energy
to be
is released in the form of Poynting flux. 1
u =µ1/3 =√σ , (18) 1 An exception is the possibility of creation of a fireball by
0 0
neutrino-antineutrino annihilation at the poles of a hyperac-
creting compact object (Jaroszyn´ski 1993; Mochkovitch et al.
in accordance with previous studies (Michel 1967;
1993), an idea applied to long bursts within the collapsar sce-
Goldreich & Julian 1970; Camenzind 1986; Beskin et al. nario (Woosley 1993; MacFadyen & Woosley 1999) and short
1998) which show that at the fast point the ratio of burstswithinthebinarymergerscenario(Blinnikovetal.1984;
Poynting to kinetic flux is µ2/3. The flow is assumed to Eichler et al. 1989; Janka et al. 1999; Aloy et al. 2005)
6 Dimitrios Giannios and HenkC. Spruit:The role of kink instability in jets
All the above scenarios may give rise to magnetized
outflows, whose evolution depends, to a large extent, on
the dominance of the magnetic energy or on the ratio of
the Poynting-fluxto matter energy flux atthe base ofthe σ=5
0
σ=8
flow. By varying this ratio, one can apply the model to 0
σ=10
30 0
jets in both the cases of GRBs and AGNs. σ=25
0
)
r
4.1. AGN jets (
γ
10
Relativistic jets are commonly observed in AGNs to have
bulk Lorentz factors in the range γ 10 20. Such ter-
∼ −
minal Lorentz factors can be achieved for the ratio σ
0
of Poynting to matter energy flux of the order of several 3
at the fast magnetosonic Point r0. The location of the 100 1000 10000 1e+05 1e+06 1e+07 1e+08
fast point is most likely at a few light cylinder radii (e.g. r/r
Sakurai1985;Camenzind 1986)and is taken to be 100r . g
g
Actually,σ isaveryimportantparameteroftheflow. Fig.1. The bulk Lorentz factor of the flow as a function
0
Its effect on the acceleration of the flow is clearly seen in of radius for different values of σ . The black, red, blue
0
Fig. 1, where the bulk Lorentz factor is plotted as a func- and green curves correspond to σ = 5, 8, 10 and 25 re-
tion radius r for different σ . The rest of the parameters spectively. The solid curves correspond to the case where
0
have the values θ = 10o, B /B = 0.5, while the en- Eq. (1) is used for the instability growth time scale (fast
r,0 φ,0
ergy released by the instability is assumed to be locally kink) and the dashed to the one where Eq. (2) is used
radiatedaway(thisisdonebytakingthe“coolinglength” (slow kink case).
parameterκ 1).Theresultsdonotdependonthelumi-
≫
nosity L of the flow in the case of AGN jets, while r sets
0
the scale of the problem (since it is the only length scale)
which means that the results can be trivially rescaled in To keep this study fairly general, we have calculated the
the case of a different choice of r . bulk Lorentz factor of the flow in the two extreme cases.
0
In the “fast cooling” case, all the released energy is radi-
The solid lines in Fig. 1 correspond to the case where
atedawayveryefficiently,whileinthe“slowcooling”case,
Eq.(1)isusedforthetimescaleofthekinkinstability(i.e.,
the energy is assumed to stay in the flow (practically this
the fast kink case)andthe dashed lines to the case where
means that we set the cooling length parameter κ 1).
the instability is slowed down by the poloidal component
≪
ofthemagneticfieldandthetimescaleisgivenbyEq.(2) In Fig. 2, the bulk Lorentz factor of the flow is plot-
(i.e.,theslowkinkcase).FromFig.1,onecanseethatthe ted for σ0 = 8. The red curves correspond to the “fast
instabilityactsquicklyandacceleratestheflowwithin1-2 cooling” case and the black to the “slow cooling” one.
orders of magnitude in distance from the location of the The asymptotic bulk Lorentz factor differs substantially
fast magnetosonic point. The acceleration is faster in the inthesetwocases,showingthatalargefractionoftheen-
“fastkink”caseandmuchmoregradualinthe“slowkink” ergy of the flow can in principle be radiated away due to
one. This is due to the fact that close to the base of the the instability-related dissipative processes. Furthermore,
flow the ratio Bco/Bco =γB /B 1 and the instability the acceleration of the flow depends on the jet opening
r φ r φ ∼
is slowed down [see Eq. (2)]. Further out, however, the angle and is faster for narrower jets (see green curves in
toroidalcomponentofthefieldalsodominatesintheframe Fig.2).This isexpected,sincefor anarroweropening an-
comovingwiththeflowandtheinstabilityproceedsfaster. gle, the Alfv´en crossing time across the jet is shorter and
Atlargerdistances,practicallyallthemagneticenergyhas so is the instability growth timescale.
been dissipated and the terminal Lorentz factors are very Another quantity ofspecialinterestis the Poyntingto
similar in the slow and fast kink cases. matter energy flux ratio σ as a function of radius. While
The acceleration of the flow and the terminal Lorentz the flow is initially moderately Poynting flux dominated,
factor depend also on what fraction of the instability- theσdropsrapidlyasafunctionofdistanceandtheflowis
released energy is radiated away. If the dissipative pro- matter-dominatedatdistancesr>103rg independently of
cesses that appear in the non-linear regime of the evolu- theprescriptionoftheinstability∼orthecoolingtimescales.
tion of the instability lead to fast moving electrons, then Far enough from the fast magnetosonic point, practically
itiseasytocheckthattheywillradiateawaymostofthis allthemagneticenergyhasbeentransferedtothematter,
energythroughsynchrotron(and/orinverseCompton)ra- and the bulk Lorentz factor saturates.
diation on a time scale much shorter than the expansion The thick lines in Fig. 3 show the ratio of the radial
timescale. If, on the other hand, most of the energy is to the toroidal components of the magnetic field. This
dissipated to the ions, then most of it stays in the sys- ratio drops rapidly as a function of distance showing that
tem as internal energy and accelerates the flow further. B B . So, despite the fact that the instability grows
φ r
≫
Dimitrios Giannios and HenkC. Spruit:The role of kink instability in jets 7
tional reason to assume Poynting-flux dominated jets on
scaleslargerthanafewpcandthattheobservedemission
on these scales can be understood as energy dissipated in
shocks internally in the flow (Sikora et al. 1994; Spada et
30 al.2001)orduetointeractionoftheflowwiththeexternal
medium.
Our model predicts that most of the energy is in the
r) form of kinetic flux at distances say >103 104r
γ ( 10 1017 1018m cm, where m is a black∼hole−of 109gso≃-
− 9 9
lar masses. So, on pc scales the magnetic fields are dy-
θ=6ο, fast kink, fast cooling
θ=6ο, slow kink, fast kink namically insignificant, in agreement with observations.
fast kink, fast cooling
Further information on the dynamics of AGN jets comes
fast kink, slow cooling
from the shortest variability timescale in the optical and
3
gamma-ray bands in blazars. This timescale can be as
100 1000 10000 1e+05 1e+06 1e+07 1e+08
short as a few days, indicating that most of the non-
r/r
g thermal radiation comes from a compact region of size
R < 1017 cm (the so-called blazar zone). On the other
Fig.2.ThebulkLorentzfactordependenceonthecooling
han∼d, polarimetry measurements of the variable optical,
efficiency of the flow and jet opening angles. The solid
infrared and mm radiation are consistent with a toroidal
curves correspond to the fast kink case while the dashed
magnetic field geometry on sub-pc scales (e.g. Impey et
to the slow kink one. The black, red and green curves
al. 1991; Gabuzda & Sitko 1994; Nartallo et al. 1998).
correspond to fast cooling, slow cooling and jet opening
angle of 6o respectively. Since most of the magnetic energy is dissipated on
these scales, it is quite probable that the observed ra-
diation is the result of the instability-released energy,
provided that the dissipative processes lead to wide
enoughparticle distributions (see alsoSikora et al. 2005).
However, one cannot exclude the possibility that, within
slow kink, slow cooling
slow kink, fast cooling this model, the “blazar zone” emission is a result of in-
fast kink, fast cooling
fast kink, slow cooling ternal shocks. On scales of 1017 cm, the magnetization
1 Br/Bφ, fast cooling, slow kink parameter of the flow is of the order of unity and it is in-
Br/Bφ, fast cooling, fast kink
) terestingtostudytheoutcomeofinternalshocksofmoder-
r
(
σ ately magnetizedplasma.The richblazarphenomenology
may indicate that both these mechanisms (i.e. magnetic
0.1
dissipation and internal shocks) are at work.
Further constraints on where the acceleration of
the flow takes place come from the lack of bulk-flow
ComptonizationfeaturesinthesoftX-rays.Thisindicates
0.01
100 1000 10000 1e+05 1e+06 1e+07 1e+08 thatγ<10at 103r (Begelman&Sikora1987;Moderski
g
r/r et al. ∼2003). T∼his shows that the acceleration process is
g
still going on at these distances. In view of our results,
Fig.3. The dependence of the magnetization parameter
this could in principle rule out the “fast kink” case since
on the radius for the different prescriptions of radiative
theaccelerationappearstobetoofastandγ 10already
cooling and the instability timescale.Notice that the flow ∼
at 300r orso.Atthispoint,however,theuncertainties
g
becomes matter-dominated at distances greater than ∼
in the model are too high to make a strong statement on
∼
10r . The thick lines show the ratio of the radial to the
0 thisissue.If,forexample,thefastpointislocatedatafac-
toroidalcomponentsofthemagneticfieldintheflow.The
tor of, say, 3 largerdistance,our results arecompatible
dominant component is clearly the toroidal one. ∼
withthelackofsoftX-rayfeatures.Numericalsimulations
of the instability are needed so that these issues can be
settled (see also discussion in Sect. 5).
quicklyfromthetoroidalcomponentofthemagneticfield,
this component still dominates over the radial one.
Having solved for the dynamics of the flow predicted 4.2. Gamma-ray bursts
byourmodelingofthekinkinstability,weturntotheim-
plications of these findings for observations of AGN jets. The analysis we follow so as to apply the model to GRBs
One highly debated issue is whether the AGN jets are is very similar to that described in the previous sections.
Poynting-fluxdominatedonpcandkpcscalesornot.The The only new ingredient that has to be added is related
case-dependent arguments are reviewed in Sikora et al. to the very high luminosities that characterize the GRB
(2005), where it is shown that there is no strong observa- jets. As a result, the inner part of the flow is optically
8 Dimitrios Giannios and HenkC. Spruit:The role of kink instability in jets
thick to electron scattering and matter and radiation are fromthetotalenergydensity:e e e .Theintegration
bb
≡ −
closely coupled. At the photospheric radius the optical proceeds with an adiabatic index of γ = 5/3. The tem-
a
depth drops to unity and further out the flow is optically peratureθ isthetemperatureoftheemittedblack-body
ph
thin.So,thehighluminosityintroducesanewlengthscale radiation which has a luminosity per sterad of
to the problem that has to be treated in a special way
4
described in the next section. Lph =rp2h3ebbuphγphc for r ≥rph. (25)
The integration continues until large distances from
4.2.1. Below and above the photosphere the source (taken as 1016 cm, where the afterglow phase
starts). There, the radiative luminosity is determined by
At the photosphere, the equation of state changes from
one dominated by radiation to one dominated by the gas L =L L L . (26)
rad pf mat
− −
pressure.Toconnectthetwo,theradiationemittedatthe
Thismeansthattheradiativeluminosityisthesumofthe
photosphere has to be takeninto account.The amount of
energyinvolvedcanbesubstantial,andappearsasan(ap- photosphericluminosity plus the componentcomingfrom
theinstability-releasedenergyabovethephotosphere.The
proximate)black body component in the GRB spectrum.
role of the photospheric component and its connection to
It depends on the temperature of the photosphere.
The temperature at the photosphere is kT m c2 the observed spectral peaks of the GRB prompt emission
e
≪ in internal shock and slow dissipation models (like this
for all parameter values used so that pairs can be ne-
one)hasbeenstudiedinanumberofrecentpapers(Ryde
glected. The photosphere is then simply defined as (e.g.
2005; Rees & M´esza´ros2005; Pe’er et al. 2005).
Abramowicz et al. 1991)
∞
4.2.2. Results
(1 β)γκ ρdr 1, (22)
es
− ≡
Zrph Followingtheproceduredescribedintheprevioussection,
we have calculated the bulk Lorentz factor of the flow for
whereκ istheelectronscatteringopacity.Thedynamics
es
different values of σ at the base of the flow and for the
oftheflowdependonthelocationofthephotospheresince 0
two prescriptions for the timescale of the kink instabil-
above the photosphere all the dissipated energy can in
ity [Eqs. (1), (2)]. The fast magnetosonic point is set to
principle be radiated away while this is not possible at
R = 108 cm, the jet opening angle to θ = 10o, the ini-
large optical depths. To solve Eq. (22) for r , we have 0
ph
tial ratio B /B = 0.5 and the luminosity of the flow
followed an iterative method. First, we guess a value for r,0 φ,0
L=1051 erg/sec sterad.
r and integrate the dynamical equations assuming no
ph ·
The results are given in Fig. 4, where it is shown
radiative losses below the photosphere and fast cooling
that the flow reaches terminal Lorentz factors γ>100 for
above it. Then we calculate the optical depth τ from r
ph
σ >30.Thesolidcurvescorrespondtothecasew∼herethe
to and,ifτ differsfromunitybymorethanathreshold 0
∞ tim∼escale for the kink instability is given by Eq. (1) (fast
value( 0.01inthese calculations),anew guessforr is
ph
∼ kinkcase)andthedashedtothecasewhereEq.(2)isused
made.The procedurecontinuesuntilthe definition(22)is
forthetimescaleofthegrowthoftheinstability(slowkink
satisfied.
case). Notice that the initial acceleration of the flow dif-
At the photosphere one has to subtract the energy
fers in the two cases, being much faster in the fast kink
and momentum carried away by the decoupled radia-
case. This is expected since this case is characterized by
tion. To calculate these quantities one needs the temper-
rapiddissipationofmagneticenergyandaccelerationfrom
ature at the photosphere. The dimensionless temperature
θ = kT/(m c2) in the optically thick region is given by the base of the flow, while in the slow kink case the non-
e
negligible poloidal component of the magnetic field close
the solution of
tor slowsdowntheinstability.TheterminalLorentzfac-
0
m 8π5m c2 torsare,however,similarinthetwocases.Noticealsothat
e=3 eρc2θ+ e θ4 (23)
m 15 λ3 there is a discontinuity in the slope of the curves γ(r) at
p e
the location of the photosphere which is a result of our
whereλ istheelectronComptonwavelength.Theterms simplistic approach (for details see previous section).
e
in(23)correspondtothematterandradiationenergyden- A second key parameter of the model is the opening
sity.Fromthissolution,wehavealsofoundthattheinter- angle of the jet. For smaller opening angles, the insta-
nal energy is always dominated by radiation (for param- bility timescale becomes shorter and the flow is acceler-
eters relevant for GRBs), so we take γ = 4/3 below the ated faster and to higher terminal Lorentz factors as is
a
photosphere. showninFig.5.This implies that forsmaller opening an-
At the photosphere we calculate the temperature θ gles, more magnetic energy is dissipated and the flow is
ph
andsubtractthe radiationenergydensity ofablackbody lessstronglymagnetizedatlargedistances.Thisisclearly
showninFig.6,wherethePoyntingtomatterenergyflux
8π5m c2 ratioisplottedasafunctionofradiusr(comparethethin
e = e θ4 (24)
bb 15 λ3 ph curvewiththethickblackdashedcurves).Noticethatthe
e
Dimitrios Giannios and HenkC. Spruit:The role of kink instability in jets 9
1000 1000
)100 )
r r
( (100
γ σ0=100, fast kink γ θ=2ο
σσσσ0000====13330000000, ,,,f asfssallotos twwk ki nkkinkiinnkkk θθθθ====7123ο000οοο
10 σ0=30, slow kink
10
1e+08 1e+09 1e+10 1e+11 1e+12 1e+13 1e+14 1e+15 1e+16 1e+08 1e+09 1e+10 1e+11 1e+12 1e+13 1e+14 1e+15 1e+16
r (cm) r (cm)
Fig.4. The bulk Lorentz factor of the flow for different Fig.5. The bulk Lorentz factor of the flow for different
σ and for the fast and slow kink case. Notice that larger jet opening angles θ. For smaller opening angles, the ter-
0
values for σ result in faster outflows. minalLorentzfactorsoftheflowbecomelargerbecauseof
0
moreefficientdissipationofthemagneticenergy.Theblue
curve corresponds to the non-axisymmetric case studied
σ(r) curves are discontinuous at the location of the pho- by Drenkhahn & Spruit (2002).
tosphere. This is caused by our simplified treatment at
the location of the photosphere, where we subtract the
energydensity ofthe radiationfield(seeprevioussection) andreverseshockemissionincaseswherequickfollowups
and reduce the internal enthalpy of the flow, increasing were possible suggeststhe existence of frozen-inmagnetic
theratioofPoyntingtomatterenergyflux.Moredetailed fields in the ejecta (Kumar & Panaitescu 2003) that are
radiative transfer models of the transition from optically dynamicallyimportant,withσ>0.1(Zhang& Kobayashi
thick to optically thin condition, predict a rather sharp 2005).Rapidfollow-upsintheX∼-rays,UVandopticalare
transition which indicates that our simple approach does now possible thanks to Swift satellite and ground based
not introduce large errors. telescopesandcantestourmodelwhichpredictsamagne-
In Figs.5 and 6 we havealso plotted (see blue curves) tizationparameteroftheorderofunityfortheoutflowing
the bulk Lorentz factor and the magnetization σ as func- material in the afterglow region.The XRT instrument on
tionsofr forthe “typicalvalues”ofthe parametersofthe board Swift has already provided several early X-ray af-
model proposedby Drenkhahn & Spruit (2002;the “AC” terglows(e.g.Tagliaferrietal.2005;Campanaetal.2005;
flow).Inthecontextofthatmodelthemagneticfieldlines Burrowset al.2005).Many of these observationsindicate
change direction on small scales and magnetic reconnec- a slow fading component at times 102 104 sec after
−
tion dissipates magnetic energy and accelerates the flow. the GRB trigger (Nowsek et al. 2005; Zhang et al. 2005;
Notice that the non-axisymmetric model predicts more Panaitescuetal.2005)whichmay be expected by the de-
gradual acceleration and rather higher terminal Lorentz celerationof ejecta with σ>1 (Zhang & Kobayashi2005;
factors (for the same initial magnetization of the flow) Zhang et al. 2005) in agre∼ement with our model predic-
than the current model. Furthermore, it is characterized tions.
by efficient dissipation of the Poynting flux, resulting in The ratio σ is even higher for the range of distances
negligible magnetization sufficiently far from the central r 1013 1015 cm where internal shocks are expected
∼ −
engine (at least in the case where the non-decayable ax- to happen in the internal shock scenario for GRBs (Rees
isymmetric component is negligible). & M´esza´ros1994;Piran1999)and is expected the reduce
One important point deduced from Fig. 6 is that, for their radiative efficiency. However, allowing for non-ideal
σ >100, the flow remains Poynting-flux dominated even MHDeffectsintheshockedregion,Fanetal.(2004)show
0
at ∼large distances away from the source where deceler- that the radiative efficiency of σ 1 plasma may not be
∼
ation of the outflow because of its interaction with the muchlowerthanthe σ =0case.Onthe otherhand,since
interstellarmedium or the stellar wind is expected, which the efficiency of internal shocks to convert kinetic energy
means that the instability is not fast enough to convert into gamma rays is already low (typically of the order
most of the magnetic energy into bulk motion of matter. of a few percent; Panaitescu et al. 1999; Kumar 1999)
Afterglowobservationscaninprincipleprobeto the mag- andobservationsindicatemuchhigherradiativeefficiency
netic content of the ejecta through early observations of (Panaitescu& Kumar 2002;Lloyd & Zhang 2004),we in-
the reverse shock emission (Fan et al. 2002; Zhang et al. vestigate the possibility that the energy released by the
2003;Kumar&Panaitescu2003).Modelingoftheforward instability powers the prompt gamma-ray emission.
10 Dimitrios Giannios and HenkC. Spruit:The role of kink instability in jets
1000 0.5
σο=30, fast kink
100 σσσοοο===311000,00 s,, lfsoalwostw kk ikinniknkk y 0.4 0.3
σο=300, fast kink nc
σ (r) 10 σσοο==310000,, sθl=o2wο ,k silnokw kink e efficie 0.3 0.2
v 0.2
1 ati
di
a
r 0.1
0.1
0.1
0
1e+08 1e+09 1e+10 1e+11 1e+12 1e+13 1e+14 1e+15 1e+16 10 100 1000 1 10 100
σ θ (deg)
r (cm) ο
Fig.6. The magnetization σ(r) of the flow as a func- Fig.7. The radiative efficiency of the flow, defined as the
tion of distance for different σ and jet opening angles. ratioofthe radiatedluminosity overthe luminosityofthe
0
Keeping θ = 10o, the jet is still magnetically dominated flow for different σ and θ. The black and red stars cor-
0
at large distance from the source for σ > 100. Smaller respond to the fast kink and slow kink cases respectively.
0
jet opening angles lead to lower values of∼σ∞. The blue For opening angles of 6o (in accordance with the val-
∼
curve corresponds to the non-axisymmetric case studied ues deduced by achromatic breaks of the afterglows) the
by Drenkhahn & Spruit (2002). The discontinuity at the efficiency reachesvalues of 20%.The circles correspond
∼
location of the photospheric radius is a result of the sub- to the non-axisymmetric case studied by Drenkhahn &
traction of the radiation energy density from the internal Spruit (2002).
energy of the flow.
shown for different values of σ . The non-axisymmetric
0
modelcanhaveahigherradiativeefficiency whichisclose
In Fig. 7, we plot the radiative efficiency -defined as to 50% for σ >100 (See also Giannios & Spruit 2005
0
the radiatedluminosityLrad divided by the flow luminos- for∼moredetailed∼study onthe spectraexpected fromthis
ity L- for different values of σ0 and θ. Fixing the angle θ model).
to 10o, one can see that the radiative efficiency peaks at
16%for σ 100.For smaller values of σ , most of the
0 0
∼ ∼ 5. Discussion
magnetic energy is dissipated below the photosphere and
is lost to adiabatic expansion, resulting in lower radiative This work suggests that the kink instability plays a sig-
efficiencies.For largervalues of σ0, the flow remains mag- nificant role in the dynamics of magnetized outflows.The
neticallydominatedatallradii,keepingthe radiativeeffi- instabilitysetsinoncethetoroidalcomponentofthemag-
ciencylower.Furthermore,the“slowkink”casehasrather netic field becomes dominant and drives its energy by B
φ
higher efficiencies and this comes from the fact that dis- on a short time scale. The energy dissipated by the in-
sipation happens at larger radiiand therefore in optically stability accelerates the flow and turns it into kinetic flux
thinenvironments.So,the modelcanhavelargeradiative dominated flow for AGN jets at distance >1000r and to
g
efficiencies for σ0 10 500. Notice that one also needs moderately magnetized flow for GRB jet∼s in the the af-
σ0>30toovercome∼the“−compactnessproblem”(e.g.Piran terglowregion.If the dissipatedmagnetic energyis trans-
199∼9). ferred to fast moving electrons with wide enough energy
Fixing σ0 = 100, one can now calculate the radiative distribution, then it can power the blazar zone emission
efficiency for different opening angles of the flow. Smaller and the prompt GRB emission with high radiative effi-
opening angles result in more magnetic energy dissipated ciency.
(by shortening the instability timescale) and therefore to These results have been compared with those that
smaller values of Poynting to matter flux at large dis- are predicted by other dissipative models (Coroniti 1990;
tances.Thisalsomeansthatmoreenergyisradiatedaway. Spruit et al. 2001; Lyubarsky & Kirk 2001; Drenkhahn
Althoughverymodeldependent,theopeninganglesofthe 2002;Drenkhahn&Spruit2002).Accordingtothesemod-
GRB jets can be estimated by the achromatic breaks of els, if the magnetic field lines change direction on small
the afterglow lightcurves (Rhoads 1997, Sari et al. 1999). scales, magnetic energy can be dissipated through re-
For θ 6o (a value typically inferred), the efficiency is connection processes. Drenkhahn (2002) and Drenkhahn
∼
quite high and of the order of 20%. & Spruit (2002) applied this idea to GRB outflows and
In Fig. 7, the radiative efficiency of the non- showed that efficient acceleration and radiation (as high
axisymmetric model (Drenkhahn & Spruit 2002) is also as 50%) is possible. In the context of this model, most of