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Thermalization of e , , p plasma
Aksenov A.G.
I n s t i t u t e f or C o m p u t e r - A i d e d D e s i g n R A S

Moscow, December 2010

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Motivation of the work
We study kinetic properties of , e , p plasma, relevant, for example, for GRB phenomenon (or ...reshell) and compact astrophysical sources (the wind). We check assumptions used for the investigation of such phenomenon: the hydrodynamical approximation, timescales, optical depths. In some cases the plasma is not in the thermal equilibrium, and we need to use the kinetic description. In some cases it is possible to use the HD description, but not in the whole region of the solution. In publications 2004, 2005, 2007 we considered , e plasma evolution in the frame of kinetic Boltzmann equations. In 2009, 2010 we expanded the kinetic approach for plasma with the baryonic loading. Baryons in progenitors of GRB' (massive stars, NS) are s important for the bursts time duration and for the spectrum. In future we want to consider 1D nonuniform plasma transport in GRBs. Di¤erent optical depth for di¤erent particles require kinetic approach. Spectra di¤er from thermal. Large gamma factors.
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Parameters of the plasma
To exclude the other particles except , e the temperatures ranges at the thermal equilibrium are (mildly relativistic and relativistic plasma) 0.1 MeV . Tth . 10 MeV. From the GRB parameters 104
8

erg

E

0

1054 erg, 106 cm

R0

108 cm,

one can estimate the temperature in the thermal equilibrium given by ...rst formula. Large depth for the pair production = T n R 1 and local space consideration. We consider the uniform and isotropic plasma at this stage of investigations. The baryonic loading B =
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np M c

2

.
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Parameters important for calculations
3 The non ideality plasma parameter nrD 1 10 3 , where q p k c d = 4 B Tne = e is the Debye length. The plasma can be e2 described by 1-particles distribution functions fi (t , p). 1

The classicality parameter { = e 2 /( hvr ) = / r . { < 1 -- quantum description of plasma. The Coulomb logarithm = Mdvr / h, M is reduced mass. Intensity of interactions between protons and other particles = n R . 1. h i 1 /2 h2 The degeneracy F = m c (3 2 ne )2 /3 1. The plasma is non-degenerate > F except the upper bound of the temperature region. Parameters for perturbative expansions are and m /M .
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Timescales

Pair production, Compton and electron-electron scattering: te e te (T nc ) 1 ;

= t3 p = 1 tc ; Expansion timescale: thyd c /R0 ; Proton-proton: pm pm 1 , vp (np tpp ) 1 M ( n te e ) M ve , 1 1, Electron-proton: tep t p; M c 2 ee Proton Compton scattering: 2 mc 2 . (np tp ) 1 (n te ) 1 , Mc 2
Cooling: t
br

ve

c;

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The task to be solved
In this talk we consider homogenous and isotropic plasma 1 fi ( , t ) = c t


q

(

q i

q fi ) , i

where "i " is the particle kind, "q " is the number of the reaction. We take arbitrary initial data fi ( , 0). We consider the time evolution to the steady state.
1

We want to know is it possible to describe the GRB plasma in the approximation of the thermal equilibrium? (Goodman 1986 proposed.) Are di¤erent scenarios for the GRB proposed by Cavallo and Rees 1978 valid? Is the prediction about the relaxation to the thermal equilibrium by Pilla, Shaham 1997 true? Which timescales do we have?
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2

3

4


Reactions rates
Textbook Berestetskii et al 1982, Swensson, 1984, Haug, 1985. Binary interactions MÜller, Bhabha e1 e2 ! e1 0 e2 0 e1 e2 ! e1 0 e2 0 Single Compton e ! e 0 Pair production and annihilation 0 $e e Radiative and pair producing variants Bremsstrahlung e1 e2 $ e1 e2 e1 e2 $ e1 0 e2 0 Double Compton e $e 0 0 0 0 Radiative pair production and three photon annihilation 0 $e e 0 0 e e $ 0 00

Table: Reactions with e

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Reactions rates

Binary interactions Coulomb scattering 00 p 1 p2 ! p1 p2 p e !p 0 e 0

Radiative and pair producing variants Bremsstrahlung 00 p 1 p2 $ p1 p2 p e $p 0 e 0

Table: Reactions with protons

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Numerical Method
1

2

3

We introduced the computational grid for the phase space , µ, (instead of p). We replaced the integrals by sums. We obtained the set of ODE' to solve. s There are several characteristic times for di¤erent processes in the problem. The obtained system of ODE' is sti¤. (Eigenvalues of the s Jacobi matrix di¤ers signi...cantly, and the real parts of eigenvalues are negative.) We used Gear' method (Hall & Watt 1976) to integrate s ODE' numerically. The method is high-order stable implicit scheme. s We do not use Monte Carlo simulations (Pilla, Shaham reached only kinetic equilibrium). Our code is conservative for the energy. Also the method conserves the particles number. We prefer to use, instead of distribution functions fi , spectral energy densities Ei ( ) = 4 3 i ( ) fi , c3
i fi

(p, t )d rd p =
Pa i r p l a s m a

4 3 i fi d rd c3

i

= Ei d r d . (1)
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Collisional integrals: two-particle interactions
Di¤erential probability for all processes per unit time and unit volume ( h = c = 1, V = 1) dw = (2 )4
(4 )

(Pf

Pi )

jM... j 2
b

2 b


a

0 d pa , (2 )3

0 where pa are momenta of outgoing particles, b are energies of particles before and after interaction, M... are corresponding matrix elements. As example consider absorption coe¢ cient for Compton scattering Z f d k 0 d pd p 0 w k 0 ,p 0 k ,p = t (2 )9 [ f ( k 0 , t ) f ( p 0 , t ) ( 1 + f ( k , t ) ) ( 1 f ( p, t ) )

f ( k , t ) f ( p, t ) ( 1 + f ( k 0 , t ) ) ( 1

f (p0 , t ) ) ] ,

where p and k are momenta of electron (positron) and photon 0 respectively, d p = d do 2 /c 3 , d k0 = d 0 02 do /c 3 w = (2 )4 (p + q
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p

0

q0 )

16

jM j

2

00 ee Pa i r p l a s m a

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Collisional integrals: three-particles interactions
As example of 3-particle reaction consider relativistic bremsstrahlung 0 0 e1 + e2 $ e1 + e2 + 0 . For the time derivative, for instance, of the distribution function f2 one has
Z
0 0 d p2 d k0 d p1 d p1 (2 )4 (2 )1 2

f2 =

(4 )

(Pf

Pi )

25

000 1212

jM... j

2

fk0 f10 f20 (1

f1 ) ( 1

f2 )

f1 f2 ( 1

f10 )(1

f20 )(1 + fk0 ) .

In the case of kinetic equilibrium in nondegenerate case we have multipliers proportional to exp kB T in front of the integrals. The calculation is then reduced to the known thermal equilibrium case.

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Detailed balance conditions (pair plasma)
Consider distribution functions f = 1 exp






,


f= exp

1


,

1

+1

where = kT /(mc 2 ) and = µ/(mc 2 ). Suppose e $ e 0 0 is in detailed balance. This means reaction rate vanishes f (1
0 f 0 )f (1 + f ) = f 0 (1 0 f )f (1 + f ),

which leads to = k . Analogous results for e e to = k . Only triple reactions give k = 0!

$ 1 2 leads

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Balance conditions

Detailed balance conditions with respect to a given direct and inverse interaction leads to the following constraints on temperatures and chemical potentials: Interaction e e scattering e p scattering e scattering pair production Triple interactions
+

I II III IV V

Parameters of DFs + = , 8 + , p = , 8 ,p = , 8 , + + = 2 , if = , = 0, if =

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Mass scaling for ep ! e 0 p

0

We can calculate ep0 , (E )ep0 for the particle with mass M0 e e instead of M and to obtain the transformation
ep e

m,

t

M0 M

ep0 e

, ( E )

ep e

t

M0 ( E ) M

ep0 e

.

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Nondegenerate case. Initial conditions

Flat initial spectral densities Ei ( i ) = const. Total energy density = 1024 erg/cm3 . Plasma is dominated by photons with small amount of electron-positron pairs, the ratio between energy densities in photons and in electron-positron pairs = / = 10 5 . Baryonic loading parameter B = 10 p = 2.7 1018 erg/cm3 .
3

, corresponding to

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Case I: concentrations

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Energy densities

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Temperatures

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Chemical potentials

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Spectra

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Spectra t = 4 10

14

s

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Spectra t = 5 10

13

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Timescales for set of parameters , B
We want to use our computational tool to know timescale for the plasma relaxation to the thermal state from the arbitrary initial state for all sets of parameters , B : 1023 1033 erg/cm3 , 10
3

B

3.
Zt
t

The relaxation timescale is de...ned t
th t !

lim

(t )

() d dt

1 t...n t
in

...n

(t 0 )

in

(t d dt

m ax

)

dt 0 .

The estimations give tt
h

max(t3p , min(tep , tpp )).

We selected initial spectra FD with initial temperatures T = 2T0 , T = T0 /2, Tp = T0 /4, and = 0, = + .
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Timescales of photons, electrons and positrons

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Timescales of protons

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Normalized by timescales of photons, electrons and positrons

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Normalized by timescales of protons

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Conclusions
1

2

3

We get two types of equilibriums in e , plasma from ...rst principles. 1. kinetic equilibrium with T , 6= 0, timescale (n0 c ) 1 due to 2 particles reactions. 2. thermal equilibrium with T , = 0 due to detailed balance in 3 particles reactions timescale (ne 0 c ) 1 . For protons we see timescale can M 1 at low B , and (n0 c ) 1 at high B . The be m (n 0 c ) timescale for thermal equilibrium . 10 10 s is much shorter comparing to the timescale of the expansion (& ms). Thermalization timescales for e , coincide. The timescale depends from , B . The estimations of the order of the magnitude for the timescale by t3p and min(tep , tpp ) sometimes is wrong. Neglected weak interactions, n and become important in GRBs sources at high temperatures. The moving of a charge particle in the nonuniform ...eld (...elds uctuations) also generates particles, but the e¤ect is small in compare with triples interactions.
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Conclusions

1

In the non degenerate case we can reduce 3-particles reaction to the equilibrium reactions rates. Otherwise we should calculate QED integrals t achieve thermal equilibrium. We expanded the method on spherical symmetric case. We can integrate kinetic Bolzmann equations for large optical depths. But huge lorenz factors require special consideration.

2

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Large energy density. Degenerate case

We considered only 2-particles interactions. We obtained FD and BE distribution functions in the ...nal state with nonzero chemical potential. We can talk only about a kinetic equilibrium. The timescale for 2-particles reactions in the degenerate case is larger than in the case without taking into account the degeneracy due to the blocking of electrons states at low energy. Starting from small di¤erence at th = 2 the di¤erence in timescale become 3 order at th = 10. The dependence of timescale from the energy density is not monotonic.

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High density ( = 10) pair plasma evolution

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Kinetics of the pair plasma. Concentrations pro...les.

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.

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Luminosities

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Average energies

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Photon spectra








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