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The terminal bulk Lorentz factor of relativistic electron-positron jets

The terminal bulk Lorentz factor of relativistic electron-positron jets
The terminal bulk Lorentz factor of relativistic electron-positron jets

a r X i v :a s t r o -p h /9804178v 1 17 A p r 1998

Mon.Not.R.Astron.Soc.000,Ln–Ln (1998)Printed 1February 2008

(MN plain T E X macros v1.6)

The terminal bulk Lorentz factor of relativistic electron-positron jets.

N.Renaud and G.Henri

Laboratoire d’Astrophysique,Observatoire de Grenoble,BP 53X F-38041Grenoble Cedex France Accepted —-.Received —-

ABSTRACT

We present numerical simulation of bulk Lorentz factor of relativistic electron-positron jet driven by Compton rocket e?ect from accretion disc radiation.The plasma is as-sumed to have a power-law distribution n e (γ)∝γ?s whith 1<γ<γmax and is con-tinuously reheated to compensate for radiation losses.We include full Klein-Nishina (hereafter KN)cross section,and study the role of energy upper cut-o?γmax ,spectral index s ,and source compactness.We determine terminal bulk Lorentz factor in the case of supermassive black holes relevant to AGN and stellar black holes relevant to galactic microquasars.In the latter case,Klein-Nishina cross section e?ect are more important,and induce terminal bulk Lorentz factor smaller than in the former case.Our result are in good agreement with bulk Lorentz factors observed in galactic sources (GRS1915+105,GROJ1655-40)and extragalactic ones.Di?erences in scattered radi-ation and acceleration mechanism e?ciency in AGN environment can be responsible for the variety of relativistic motion in those objects.We also take into account the in?uence of the size of the accretion disc;if the external radius is small enough,the bulk Lorentz factor can be as high as 60.

Key words:galaxies:active -galaxies:jets -radiation mechanism:miscellaneous -stars:individual:GRS1915+105,GROJ1655-40

1INTRODUCTION

Superluminal motion observed in Active Galactic Nuclei (AGN),especially in the blazars class,seems to be closely linked with high-energy emission.Such motion was recently observed in the Galaxy (Mirabel &Rodriguez 1994,Hjellm-ing &Rupen 1995,Tingay et al.1995)in the so-called mi-croquasars.Nevertheless di?erences are noticeable in those two cases.The latter systems were observed with small value of bulk Lorentz factor (around 2.5),while in the former ones values of about 10-20are frequent.It is well known that the radiation pressure acting on electron-positron plasma in the vicinity of a near Eddington accreting object is very e?-cient to accelerate the plasma outwards,since the gravita-tional force is around 1000times weaker than for an electron-proton plasma.However,Phinney (1987)has shown that for a realistic accretion disc emission,only moderate values of bulk Lorentz factors can be reached.Li &Liang (1996),have recently proposed that this mechanism could explain the rel-atively small γb ~2.5observed in galactic objects.They considered plasma composed with both e +e ?and e ?p and obtained the equation of motion using the Thomson cross section and including gravitation force.To explain higher values of γb ,O’Dell (1981)proposed the so-called ’Comp-ton rocket’e?ect,i.e.anisotropic Inverse Compton e?ect on a highly relativistic plasma.It was then argued (Phinney

1982)that because Compton cooling is always much more

rapid than bulk acceleration,only small value of γb ∞could be reached by this mechanism.However,taking into account that in the frame of the ’two-?ow’model (Sol,Pelletier &Ass′e o 1989),a pair plasma could be reheated by the tur-bulence triggered by a surrounding jet (Henri &Pelletier 1991),Marcowith,Henri &Pelletier (1995)showed that the Compton rocket becomes much more e?cient and acceler-ates pair plasma to Lorentz factor γb ~10.In another work Sikora et al.(1996)studied the radiation drag in AGN jets.They included relativistic electron-positron plasma acceler-ated in situ and relativistic protons which contribute to the inertia of the ?ow.They considered radiation emitted from an accretion disc,partially reprocessed by the outer part of the disc or by spherically distributed matter at a given distance from the central object.All Compton interactions were computed using Thomson cross section.They showed that in most cases jets should undergo radiation drag,and that the e?ciency of this mechanism becomes important for purely pair plasma dominating the jet luminosity.The aim of this paper is to study how Compton rocket e?ect can accelerate a pure pair plasma in the vicinity of accretion disc,taking into account the full KN cross section.Follow-ing Marcowith et al.(1995),we consider a blob of e +e ?pair plasma with an isotropic energy distribution in the comov-ing frame n ′e (γ′)∝γ

′?s

,where s is the spectral index.We c

1998RAS

2N.Renaud and G.Henri

assume that the acceleration process is e?cient enough in the jet to get a stationary energy distribution.We assume that the dynamics of the pair plasma is decoupled from the electron proton component and we do not include gravita-tion force.The radiation?eld is coming from an accretion disc.We include KN corrections in the description of the In-verse Compton interactions.We study the in?uence of both compactness of the radiation source,spectral index and up-per cut-o?γmax,and make then comparisons between AGN and galactic microquasars.We also consider the in?uence of scattered radiation by a Broad Line Region(BLR)and dusty torus around the central black hole.Finally we discuss the in?uence of accretion disc’s size that could be relevant to the high value of bulk Lorentz factors.

2COMPTON ROCKET EFFECT WITH

KLEIN-NISHINA CORRECTIONS

2.1Notations

All energies are measured in unit m e c2.We refer all quanti-ties expressed in the blob rest frame by a prime′,all quanti-ties in the particle rest frame by a star?and quantities in the disc frame are not labelled.Photon energies will be labelled byε,and the unit direction vector by k.We use the KN di?erential cross section(Rybicki&Lightman1979)given by:

16 ε?1ε?+ε?

1+ε?(1?cosφ?) .

σT=

4πε0m e c2is the electron classical radius.This expression

applies to the scattering of a photon with energyε?and di-

rection k?in a photon with energyε?1and direction k?

1,and

cosφ?=k?.k?

1

.

2.2The general picture

Figure1shows the general con?guration of the model.

The pair plasma is assumed to be described in the bulk rest frame by an energy distribution n′e(z,γ′)∝γ′?s for γmin<γ′<γmax,with s,γmin andγmax independent of z (see paragraph3.2.1for a further discussion of this assump-tion).The radiation force is due to soft photons coming from a standard accretion disc(Shakura&Sunyaev1973)around a Schwarzschild black hole.The inner radius of the accretion disc is r i=3r g(where r g is the Schwarzschild radius of the black hole).The outer radius r e is a free parameter.We use the black-body approximation for the disc emission so that the speci?c intensity at a radius r is:

Iν(r)=Bν(T ef f(r)),(1) where Bνis the Planck function and the e?ective tempera-ture T ef f is given by:

T ef f(r)= 3GM˙M

x 1+x+16x3+1

3M⊙

1017g.s?1

4π2 r2r1r1?αd r g(ν?νLyα),(5) The radiation is re-emitted by a ring for r1

c 1998RAS,MNRAS000,Ln–Ln

The terminal bulk Lorentz factor of relativistic electron-positron jets.3 at a distance z on the jet axis is:

Iν(z,θ)=

χL d

1+ z z2+r20and for z>r0,

4πσ r4r3T4(r)r d r Bν((T(r)),(7)

T is the dust temperature,σis the Stefan-Boltzmann con-

stant,andχ′is the fraction of the disc luminosity that is

reprocessed.

2.3Computation of the radiative force

Let

d p′

1?1/γ′2c its

velocity.The force exerced upon the pair plasma is then in

the blob rest frame:

F′z= d?′e dγ′n′e(γ′,?′e)d p′

F grav ~

m p

c 4πH′ (1+2

d t?

=? d n?dσ

d t?

=?c d n?dσ

d t′

=

d p?x

c

d E?

d t′

=?σT d n?(f p(ε?)+β′f E(ε?)).(14)

The two functions f p and f E are given by:

f p(ε?)=(1+ε?)f(ε?),

f E(ε?)=?ε?f(ε?),

where the function f(ε?)corresponds to the ultrarelativistic

case(β′=1)and is given by:

f(ε?)=

1

ε?3(1+2ε?)3

+ln(1+2ε?)×

1

ε?3(1+2ε?)3

.

We then use these expressions in equation(8).

We can estimate the errors in the two extreme regimes

described above.In the head-on approximation the?rst cor-

rections are roughly

1

γ2crit

~γcrit ε ~ ε 2/3.For AGN ε ~10?4and we

?ndγcrit~20?30with a maximum error~0.2%while for

a microquasar ε ~10?2andγcrit~5with a maximum

error~5%.

2.4Equation of motion

Following Phinney(1982)we determine the acceleration of

pair plasma by considering the conservation of stress-energy

c 1998RAS,MNRAS000,Ln–Ln

4N.Renaud and G.Henri

tensor leading to Phinney’s(7)and(8)equations in the bulk rest frame:

?

?t′

[(ρ′+p′)γbβb]=F′z+βb F′0.(16) Combining these two equations and for a reheated relativis-tic plasma(with p′=ρ′/3)one?nds the equation of motion (with dz′=βb dt′and z=z′):

dγb

ρ′

1

3γ2

b

+1

.(17)

To compute the radiative force by equation(14)we need the

di?erential photon distribution in each electron rest frame.

For this we use the Lorentz invariant

d n

γ′ 1/7,while

for z crit>r e,γb∞∝ r i γ′ 1/4(see Appendix

A).l=

σT L

The terminal bulk Lorentz factor of relativistic electron-positron jets.

5 Figure3.Equilibrium Lorentz factorγbeq as a function of z in

the Thomson limit.γbeq is given for r e=10r g,r e=102r g,

r e=103r g and r e=3.103r g.We also represent the two asymp-

totic regimesγbeq∝z1/4for zr e,with

r e=3.103r g

.

Figure4.The Equilibrium Lorentz factor calculated including

KN corrections compared to Thomson solution.The?gure cor-

responds to a stellar black hole M=5M⊙withγmax=107

for spectral index s=1.5and s= 5.The external radius is

r e=3103r g and L=L Edd.

in this paragraph still holds even including KN corrections.

The critical point z crit is reached rather close to the central

engine(before104Schwarzchild radii).For z>z crit the mo-

tion is nearly balistic and so independent of the radiation

force which has become too weak.It is so independent of

any variation of the pair distribution unless these variations

strengthen the radiation force.This scenario would require

a more e?cient acceleration mechanism when moving away

from the central source,which is very unlikely.Therefore

our assumption of a stationary pair energy distribution on

a large range of z does not in?uence strongly the terminal

value ofγb,or in other words this value is essentially deter-

mined by the local parameters at the critical distance.

3.2.2In?uence of the energy upper cut-o?and spectral

index

Figure6illustrates the in?uence of the spectral index(for

1.5≤s≤5)and the energy cut-o?(for103≤γmax≤

107)on the terminal Lorentz factor.We chose M=5M⊙

representative for stellar black holes and M=109M⊙for

supermassive black holes.The calculations were carried out

for L=L Edd and L=0.1L Edd,where L is the luminosity

of the accretion disc.

Results are very sensitive to the spectral index value.

There are3di?erent behaviours according to the value of s.

(i)for s<2there exists a maximum terminal Lorentz

factor as a function ofγmax and the curve intercepts the

other ones.

(ii)for2

less pronounced.The variation as a function ofγmax are

smoother.

(iii)for s>3no variation withγmax.

We?nd that for low value ofγmax,our solutions agree with

the Thomson regime solutions.Nevertheless KN corrections

reduce the e?ciency of the Compton rocket e?ect.As a

matter of fact,in the Thomson regime,an increase ofγmax

leads to an increase of γ′2 (for s<3)and so ofγb∞.This

mechanism is valid until KN corrections begin to dominate,

roughly whenγmax ε ~1.So whenγmax is greater than

ε ?1the radiation force does not increase anymore whereas

the plasma inertiaρ′is much more important.This leads to

a less e?cient acceleration mechanism.This e?ect is larger

for small indexes explaining the inversion of the curve for

highγmax.Indeed we?nd that acceleration is much more

e?cient for s=2andγmax=107than for s=1.5and the

same value ofγmax,in the case of stellar black hole.When

steepening the pair distribution,the radiation force is dom-

inated by the low energy part of the distribution.This fact

explains why no variation is apparent withγmax for s>3.

The plasma behaves dynamically as a cold one,and we?nd

small value of terminal Lorentz factor.Finally including KN

corrections in the calculations gives rise to an absolute upper

limit to maximal Lorentz factor for a given luminosity.

3.2.3In?uence of the black hole mass

As discussed above the in?uence of the mass of the central

black hole is predominant.Stellar black holes with soft X-ray

emission( ε ~10?2)are less e?cient in accelerating blob

of pair plasma than supermassive black hole with softer ra-

diated emission( ε ~10?4),because KN saturation e?ects

occur at much lower energy.A more realistic description of

the accretion disc around stellar black holes reinforces this

discrepency.As shown in?gure7the radiation emitted from

a two-temperature disc(Shapiro et al.1976)leads to smaller

γb∞than in the case of standard accretion disc radiation.

Because it is well established that accretion disc around stel-

lar black holes should radiate up to a few keV(as in a two-

temperatures disc),KN corrections play an important role

in this case.

c 1998RAS,MNRAS000,Ln–Ln

6N.Renaud and G.

Henri

Figure5.Solutions of the equation of motion in the case of a

stellar black hole(M=5M⊙).We chose two di?erent initial con-

ditionsγb init=2andγb init=5.r e=3103r g and L=L Edd.

3.2.4In?uence of the luminosity

As ε ∝˙M1/4M?1/2,the luminosity of the disc also directly

in?uences the maximum of the functionγb∞as a function

ofγmax.As shown in?gure6,the maximum takes place

at higherγmax when the luminosity decreases.Besides less

luminous systems contribute to a lower radiation force and

so to a less e?cient acceleration.In the Thomson regime one

has a dependanceγb∞∝L1/7.

3.2.5E?ect of scattered radiation

All the results described above are obtained when studying

the disc radiation alone.We also include in our calculation

BLR radiation?elds corresponding to two cases:

a)re-emission from a ring located between r1and r2and

with an emissivity given by equation(5)

b)re-emission from spherically distributed matter at a

distance r0from the central black hole with an emissivity

given by equation(6).

Figure8displays the equilibrium Lorentz factor in the pres-

ence of a BLR located between r1=104r g and r2=105r g

(case a),?gure8)in the Thomson regime.We also plotted

the equlibrium Lorentz factor including KN corrections as

well as the solution of the equation of motion for a plasma

with s=2andγmax=105.As one can see,the e?ect of

BLR onγbeq is very weakened by KN corrections.This can

be understood because the photons coming from the BLR

are blueshifted by the relativistic motion in the blob rest

frame whereas the photons coming from the disc are red-

shifted.So the dragging force from the BLR is much more

reduced by KN corrections than the accelerating one due

to disc photons.Due to the weakness of the radiation?eld,

the dynamical solutionγb(z)is still less a?ected than the

equilibrium value.The case of a spherical shell located at

r0=104r g(case b))is illustrated in?gure9.One can see

that the di?used radiation?eld strongly a?ects the motion

that is almost stopped at the crossing of the shell,where ra-

diation density is dominated by the isotropically scattered

photons.However,the plasma is quickly reaccelerated after

the crossing,and high Lorentz factor can be reached again.

Sikora et al.(1996)argued that scattered radiation

should lead to e?cient radiation drag.However,the in?u-

ence of scattered radiation is strongly governed by the posi-

tion of scattering clouds with respect to the critical distance

z crit:if they lie below this critical distance,the plasma

will be temporarily braked during the crossing of scatter-

ing region,but will be quickly reaccelerated after it.If the

clouds lie above z crit,the terminal Lorentz factor can indeed

be strongly a?ected.Figure10and11display the terminal

Lorentz factor as a function ofγmax for di?erent distances

of the BLR.It can be seen that if it is close enough,the BLR

can give still higher Lorentz factor than for the disc alone

for the highest value ofγmax.This is because the solid angle

substended by it at z crit is so small that its radiation?eld

has an accelerating rather than decelerating e?ect.But even

for a BLR between104and105r g,terminal Lorentz factor

around10are clearly reachable.We conclude that the pres-

ence of broad lines can a?ectγb∞,but does not prevent in

general highly relativistic motions.

We also discuss the e?ect of a possible emission from a

hot dusty torus surrounding the central black hole outside

the BLR region.The e?ect of such a scattered radiation

?eld is much more important because KN corrections are

here almost negligible.Figure12shows the terminal Lorentz

factor in the presence of an IR emitting ring heated at T=

1500K,located between r3and r4.It is obvious that the

terminal Lorentz factor is strongly reduced to values around

3for the less favourable case(r3=104r g,r4=105r g).The

situation is a little less dramatic for more distant sources

(r3=105r g,r4=106r g)because the isotropic radiation

density is lowered,andγb∞of5can be reached.

3.2.6In?uence of the accretion disc size

For small value of r e the radiation is more anisotropic

and so more e?cient to accelerate the pair plasma(see?gure

13).In this?gure r e=10r g and we show the solutions for

which we obtain the largestγb∞in extragalactic and galac-

tic cases.In this con?guration the radiation force is more

e?cient andγb∞can be as high as60in the extragalac-

tic case.The dependance on s andγmax is shown in?gure

14,where we extend the previous calculation to the case

r e=10r g.We?nd the same global behaviour ofγb∞with

spectral index andγmax.

4COMPARISONS WITH OBSER V ATIONS

Our model gives good agreement with observations of rela-

tivistic blob ejection in AGN and microquasars for the’disc

alone’solution.Figure6shows that in the most favourable

con?guration(s=1.5,γmax~106and L=L Edd)Compton

rocket e?ect is able to accelerate pair plasma up to Lorentz

factorγb∞≥20.This value is only gradually reached,the jet

being much slower at small distances.One can obtain higher

c 1998RAS,MNRAS000,Ln–Ln

The terminal bulk Lorentz factor of relativistic electron-positron jets.

7

Figure 6.Terminal Lorentz factor γb ∞as a function of γmax for di?erent value of spectral index s .The two top pannels correspond to M =109M ⊙black hole with r e =3.103r g ,L =L Edd (left)and L =0.1L Edd (right).The two bottom pannels correspond to M =5M ⊙black hole with r e =3.103r g ,L =L Edd (left)and L =0.1L Edd

(right).

Figure https://www.doczj.com/doc/5214242785.html,parisons of terminal Lorentz factor for a standard accretion disc (right)and a two-temperature disc (Shapiro et al.1976)extended up to 15r g (left).In both case r e =3103r g ,M =5M ⊙and the accretion rate is the Eddington one.

value for super-Eddington systems.One can notice that ob-servations of the faster extragalactic superluminal motion correspond to such value of bulk Lorentz factor (Vermeulen

&Cohen 1994).

This value is however strongly dependent on the spec-tal index s and the high energy particle distribution cut-o?

c

1998RAS,MNRAS 000,Ln–Ln

8N.Renaud and G.

Henri

Figure8.Equilibrium Lorentz factor as a function of z for radia-

tion coming from an accretion disc and BLR radiation from a ring

located between r1=104r g and r2=105r g.The emissivity is de-

scribed by equation(5)withχ=0.1andα=2.We also show

the solution of the equation of motion for this case.The solid

curve showsγbeq in Thomson regime.The plasma parameters are

s=2andγmax=105.The black hole mass is M=109M⊙and

L=L Edd

.

Figure9.Same as?gure8for a BLR radiation from a shell

located at r0=104r g.The emissivity is described by equation

(6)withχ=0.1.

γmax.The precise value of s is not obvious to derive from

observations.High energy spectra show typical X-ray spec-

tral indexes around0.5,which correspond to s?2.There is

very often a spectral break in the MeV range:this could be

attributed to a break in the particle distribution,that would

correspond toγmax=102.However,Marcowith et al.(1995)

have shown that this break could be very well

reproduced

Figure10.Terminal Lorentz factorγb∞as a function ofγmax

for the spectral index s=2.The solid line shows the solution

obtained including KN corrections for a standard accretion disc.

The scattered radiation is described by equation(5)withχ=0.1,

α=2and di?erent locations of the BLR

ring.

Figure11.Same as?gure10for scattered radiation described

by equation(6)withχ=0.1and di?erent locations of the BLR

shell.

byγ?γabsorption with an actual particle distribution giv-

ing the primary photon spectrum that can extend to much

higher energy.So high energy spectra may not be good indi-

cators of the upper cut-o?γmax.Moreover,the high energy

emission probably takes place at relatively small distances

(around102r g)from the center,well below z crit:the?nal

bulk Lorentz factor is not yet reached at this distance.On

the other hand,the detection of photons with at least30

GeV,and even above TeV for some BL Lacs,implies an

upper cut-o?γmax≥105.

c 1998RAS,MNRAS000,Ln–Ln

The terminal bulk Lorentz factor of relativistic electron-positron jets.

9 Figure12.The in?uence of a dusty ring combined with a BLR

on the terminal Lorentz factorγb∞as a function ofγmax,for the

spectral index s=2.The solid line shows the solution obtained

including KN corrections for a standard accretion disc.The BLR

ring emissivity is described by equation(5)withχ=0.1,α=2.

The dusty ring emissivity is given by equation(7)withχ′=0.1or

χ′=0.05.We chose di?erent locations for these two

components.

Figure13.Solutions of the equation of motion for a supermassive

black hole(M=109M⊙)and a stellar black hole(M=5M⊙)

with a compact accretion disc.r e=10r g and L=L Edd.γb init

is the initial condition of the solution.We also represent the equi-

librium Lorentz factor in Thomson regime.

Radio spectral indexes are also di?cult to assess be-

cause of synchrotron self-absorption,especially for radio-

?at quasars where the observed spectrum results probably

from the superposition of many self-absorbed spectra.The

optically thin index ranges mostly from0.5to1,which

corresponds to2≤s≤ 3.With reasonable parameters

(2≤s≤3andγmax≤105),our model predicts typical

values4≤γb∞≤10which are the most frequently ob-

served(Vermeulen&Cohen1994).Moreover,as shown in

Sect.3.2.5,KN e?ects prevents from strong Compton drag

induced by BLR radiation in the vicinity of the central black

hole.The fastest superluminal motions may be attributed to

those objects for which the BLR is weak and/or closest to

the central object.

On the other hand,the slowest motions(γb∞≤4)can

be obtained in the presence of a rich and extended environ-

ment of scattering material,such as BLR clouds and dusty

torus.The presence of dust is inferred from an enhance-

ment in the near infrared in some quasars spectra(Bairvai-

nis1987).Nevertheless a nonthermal infrared component,

attributed to synchrotron radiation from the relativistic jet,

is also usually observed in radio-loud AGN.This compo-

nent is generally predominant in?at spectrum radio quasars

(Neugebauer et al.1986)and is necessary to explain rapid

variations observed in infrared?aring objects.We can specu-

late that these objects with the largest superluminal motion

are those where the scattered thermal component is par-

ticularly weak.Such’nonthermal’objects should not su?er

strong radiation drag,while’thermal’ones should have the

lowest terminal Lorentz factors.In conclusion the diversity

among observed superluminal motions can be reproduced

by our model by considering both in?uence of the plasma

acceleration mechanism and AGN environment.

The observed value of Lorentz factor of about2.5for

the two microquasars(GRS1915+105,Mirabel&Rodriguez

1994and GROJ1655-40,Hjemlling&Rupen1995)with

large spectral indexes(respectively s~4,Finoguenov et

al.1994and s~4.6,Harmon et al.1995)are consistent

with our results.We show?gure15the dependance ofγb∞

on compactness of the source for s=4and s=5.We

?nd that observations require L~0.2?0.3L Edd,which is

very close to the result by Li&Liang(1996).This result

is not strongly a?ected by the mass of the central black

hole if the compactness of the source remains the same.The

steep spectrums observed in these two objects ensure us that

the terminal Lorentz factor is just dependant on the low

energy cut-o?of the electron distribution.In this case a more

realistic description of the accretion disc emission does not

change our results for such spectral indexes(?gure7).Yet

the maximal value ofγb∞is model dependant.The part of

high energy emission contributes to decrease this value from

the one obtained for a standard accretion disc.We?nd that

value of about6can be reached in the most favourable case.

Finally,γb∞is much higher for a small sized accretion

disc.We?nd value of about60in the case of an external

radius of10r g in the case of supermassive black hole.

5CONCLUSION

We have considered the bulk acceleration of an electron-

positron pair plasma in the vicinity of a central black hole.

The acceleration is due to Compton rocket e?ect on the

plasma and the radiation force originates from a standard

accretion disc emission.The pair plasma is continuously re-

heated by an e?cient turbulent mechanism wich takes place

in the frame of the’two-?ow’model.Thus we assumed in

our calculations a stationary power-law energy distribution

for the pair.We included KN corrections in the computation c 1998RAS,MNRAS000,Ln–Ln

10N.Renaud and G.

Henri

Figure 14.Terminal Lorentz factor γb ∞as a function of γmax for di?erent value of spectral index s .The two pannels correspond to r e =10r g ,L =L Edd ,M =109M ⊙(left)and M =5M ⊙

(right).

Figure 15.The terminal Lorentz factor as a function of L/L Edd

and for two spectral indexes (s =4and s =5).M =5M ⊙,γmax =103.

of the radiation force,and solved numerically the equation of motion.We studied con?gurations relevant to relativistic motion in AGN and galactic microquasars.The main results of our calculations can be summarized as follows:

(i)For a given luminosity,the terminal Lorentz factor γb ∞admits an absolute maximum due to KN corrections.Val-ues of about 20can be reached in the extragalactic case,for su?ciently ?at spectrum (s ~1.5)and accretion at the Ed-dington rate which may correspond to the highest observed relativistic motion.For more reasonable plasma parameters (s ≤2and γmax ≤106)the Compton rocket e?ect can ac-count for the typical value of terminal Lorentz factor inferred from observations (γb ∞).

(ii)Scattered radiation from extended BLR or dusty torus can brake e?ciently relativistic motion,even for a high en-ergy plasma.This Compton drag and a weak plasma heat-ing can be responsible for the lowest terminal Lorentz fac-tor observed.The highest superluminal motion could be at-tributed to objects with particularly weak di?use component and very e?cient heating.

(iii)For stellar black holes KN corrections are important leading to smaller values than for a supermassive black hole.Recent observations of relativistic ejection in galactic micro-quasar are consistent with our results.ACKNOWLEDGMENTS

We would like to thank the anonymous referee for his helpful remarks,especially for the description of scattered radiation.REFERENCES

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1998RAS,MNRAS 000,Ln–Ln

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APPENDIX A:EQUILIBRIUM LORENTZ

F ACTOR AND TERMINAL LORENTZ F ACTOR

IN THOMSON REGIME

Eddington parameters are de?ned as follows:

J=

1

4π μIν(?)d?dν,(A1)

K=

1

r1/4

i

(A4)

In the case z>r e we haveεu=r/z<1for1

we can expand the term(1+ε2u2)?βinto series:

(1+ε2u2)?β= k(?1)kβ(β+1)...(β+k?1)

k!

× u2k?1e?12k?3/2 .(A6)

The?rst coe?cients inεare then:

I(2,β,1)=A0?βA1ε2+

β(β+1)

3?u?1e+

2

3

,

A1=u e?2u1/2

e

+1~u e,(A8)

A2=u3e

5

+

1

3

.

The equivalents are given for u e?1.This gives for the

Eddington parameters:

J=(L/8π2r2i)(1?2b/3)?1ε2(A0?3/2A1ε2+15/8A2ε4),

H=(L/8π2r2i)(1?2b/3)?1ε2(A0?2A1ε2+3A2ε4),(A9)

K=(L/8π2r2i)(1?2b/3)?1ε2(A0?5/2A1ε2+35/8A2ε4).

The equilibrium Lorentz factor isγbeq=(1?β2beq)?1/2where

βbeq=x?(x2?1)1/2,(A10)

with

x=

J+K

8

u3eε4,(A12)

βbeq=1?

u3/2

e

r3/4

e

r1/4

i

.(A14)

The asymptotic solution of equation(17)is approxima-

tivelyγb∞=γbeq(z crit),where z crit is the location where

the radiative force becomes to weak to maintainγb(z)=

γbeq(z).This occurs when the evolution ofγbeq,i.e.?z0=

γbeq/(dγbeq/d z)=dlog(z)/(z dlog(γbeq)),is larger than the

evolution ofγb towardsγbeq,i.e.?z1=(dγb/d z)/(γb?γbeq).

Using equation(17)one?nds:

z crit~ 1dγb γbeq ?1.(A15)

In the Thomson regime we can use equation(9)and(10)

which give in the relativistic case:

d F′z

c

dγb

=

2H

m e c3

γ′2

r e 3/4 γ′2

m e c3r i

γ′2

γ′ 1/7.(A20)

This paper has been produced using the Royal Astronomical

Society/Blackwell Science T E X macros.

c 1998RAS,MNRAS000,Ln–Ln

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6、那脑袋里的智慧,就像打火石里的火花一样,不去打它是不肯出来的。 ——莎士比亚 7、多数人都拥有自己不了解的能力和机会,都有可能做到未曾梦想的事情。 ——戴尔·卡耐基 8、勿问成功的秘诀为何,且尽全力做你应该做的事吧。 ——美华纳 9、苦难有如乌云,远望去但见墨黑一片,然而身临其下时不过是灰色而已。 ——里希特 10、幸运并非没有恐惧和烦恼;厄运也决非没有安慰和希望。 ——培根

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16、不要慨叹生活的痛苦!慨叹是弱者。 ——高尔基 17、不应当急于求成,应当去熟悉自己的研究对象,锲而不舍,时间会成全一切。 凡事开始最难,然而更难的是何以善终。 ——莎士比亚 18、如烟往事俱忘却,心底无私天地宽。 ——陶铸 19、生活真象这杯浓酒,不经三番五次的提炼呵,就不会这样可口!——郭小川 20、人生的价值,即以其人对于当代所做的工作为尺度。 ——徐玮

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