Rosalba Perna and Mario Vietri- A Self-Similiar Solution for the Propagation of a Relativistic Shock in an Exponentiial Atmosphere

8/3/2019 Rosalba Perna and Mario Vietri- A Self-Similiar Solution for the Propagation of a Relativistic Shock in an Exponentiial Atmosphere

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The Astrophysical Journal, 569:L47L50, 2002 April 10 2002. The American Astronomical Society. All rights reserved. Printed in U.S.A.

A SELF-SIMILAR SOLUTION FOR THE PROPAGATION OF A RELATIVISTIC SHOCKIN AN EXPONENTIAL ATMOSPHERE

Rosalba Perna1,2

and Mario Vietri3

Received 2002 February 6; accepted 2002 March 13; published 2002 March 18

ABSTRACTWe derive a fully relativistic, self-similar solution to describe the propagation of a shock along an exponentially

decreasing atmosphere, in the limit of a very large Lorentz factor. We solve the problem in planar symmetryand compute the acceleration of the shock in terms of the density gradient crossed during its evolution. We applyour solution to the acceleration of shocks within the atmosphere of a hypernova and show that velocities consistentwith the requirements of gamma-ray burst models can be achieved with exponential atmospheres spanning awide density range.

Subject headings: gamma rays: bursts shock waves stars: atmospheres

1. INTRODUCTION

In a supernova (SN) explosion, the shock wave must ulti-

mately emerge from the body of the star and begin to propagatedown the exponential density gradient of the stellar atmosphere.In Newtonian fluid dynamics, the propagation of such a shockis described by a self-similar solution (Raizer 1964; Grover &Hardy 1966; Hayes 1968). This self-similar solution is un-avoidable (Raizer 1964). In fact, it turns out that as the shockaccelerates, a sonic point is formed, separating matter locatedimmediately behind the shock from the flows initial condi-tions; this prevents pressure waves (i.e., causal information) toreach postshock material from the area where the flows initialconditions are set. Thus, all flows will converge to the samesolution, irrespective of their initial conditions. The lack ofdependence on initial conditions severely constrains the en-suing flow, by restricting the number of parameters on which

the shock evolution may depend; in fact, this constraint is sostrong that only a single (self-similar) solution exists.

In the Newtonian solution, it is found that the shock velocityincreases exponentially, in such a way that the shock reachesspatial infinity in a finite time. This is clearly impossible whenaccount is taken of special relativity, and therefore the New-tonian solution must break down at high shock velocities, asexpected. The question of precisely how the shock evolves inrelativistic conditions has not been investigated so far.

This problem has acquired a new urgency within the hy-pernova model (Paczynski 1998; MacFadyen & Woosley 1999)for gamma-ray bursts (GRBs). In fact, Meszaros & Rees (2001)have shown that, despite the large assumed energy release ofthe central engine, the outwardly moving shock in the large

star hypothesized to give rise to the GRB can only reach aLorentz factor of at the end of the H envelope, andG 10ithey had to invoke (without explicit computations) shock ac-celeration down the star exponential atmosphere to reach therequired Lorentz factors of . The existence of aG 100300fself-similar temporal structure in GRBs is also suggested byan analysis of their power spectra (Beloborodov, Stern, &Svensson 2000). It is the purpose of this Letter to derive a fully(special) relativistic self-similar solution for the problem of

1 Harvard Junior Fellow, Harvard Smithsonian Center for Astrophysics, 60Garden Street, Cambridge, MA 02138.

2 Osservatorio Astronomico di Roma, via dellOsservatorio, 2, I-00040Monte Porzio Catone, Rome, Italy.

3 Universita di Roma 3, Via della Vasca Navale 84, 00147 Roma, Italy.

shock propagation in an exponential atmosphere ( 2) and todiscuss its use in the hypernova model for GRBs ( 3).

2. RELATIVISTIC SELF-SIMILAR FLOW

We shall assume that a cold ( ) material is stratifiedT p 0with a density distribution ; the shock is sup-r p r exp (kx )0posed to move toward , thus with positive velocityx p

. The symmetry is assumed to be planar since both thev 1 0length scale of stellar atmospheres and the total extension ofthe atmosphere are much smaller than the stellar radius. In theNewtonian analog, the shock speed is set by a purely dimen-sional argument (e.g., Chevalier 1990): , with a beingV p a/ktan adimensional constant to be determined. In the special rel-ativistic problem, dimensional arguments alone fail because the

presence of the light speed c allows the construction of a newadimensional quantity ( ), thus spoiling the above argument.kctHowever, we can recover from this impasse by appealing toboth dimensional and covariance arguments. The timelike partof the shock four-speed is of course determined by theU

m

identity , while the spacelike part must be built inmU U p 1m

covariant fashion from the available quantities. Defining a four-vector , we can then write for the spatial partk p (0, k, 0, 0)

m

of U

a adX akaU p p , (1)

mds k k sm

where . This is the only sensible solution to ourm 2k k p k 1 0m

dimensional/covariant problem. We see from this that, as longas we use proper time, the structure of the problem is identicalto that of the Newtonian analog. In particular, the shock reachesspatial infinity within a finite proper time. We may choose

for the moment when this occurs, so that the flow iss p 0restricted to . Incidentally, note that this implies .s ! 0 a ! 0Physically, this makes perfect sense: as the shock accelerates,its proper time is contracted by its Lorentz factor (G) withrespect to the fluid time. The shock speed in terms of the fluidtime can be obtained by remembering that , anddX/ds p vGsubstituting in the above equation, one finds ( from nowc p 1


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Fig. 1.Numerical solution of eqs. (17), (18), and (19) with a given byeq. (22).

with respect to y),

Rp

R

g(2/a 1)(1 4y/a 1/g) (10/a 3),

3/2 3/2g 6y/a (1 g/2 2yg/a)(1 4y/a 1/g)

(17)

g Rp (1/g 1/2 2y/a) (2/a 1), (18)

2g R

N 2g[(2 a)/a] g/gp . (19)

N g(1 4y/a) 1

From these, we see what fixes a: the denominator of the right-hand side of equation (17) goes to zero at a critical point ycand, unless the numerator simultaneously does the same (which

will only occur for a special value of a), a nonintegrable sin-gularity will ensue. The same phenomenon occurs in the New-tonian analog, where it has been shown that this critical pointis a sonic point. It is exactly because there is a sonic point thata self-similar solution can develop; in fact, material betweenthe shock and the sonic point is not in causal contact withpostsonic-point material, and thus its properties cannot bedetermined by the problems initial conditions.

To keep the problem nonsingular, we impose that the nu-merator and denominator of equation (17) vanish simulta-neously, obtaining

8 2ag(y )(1 4y /a) p (20)c c

2 a

for the numerator and

4 12g(y )(1 4y /a) p (21)c c

2

for the denominator. The conditions above lead to two solutionsfor a, one positive and the other negative. The positive solutionmust be discarded because otherwise the shock would not reachspatial infinity for (eq. [1]). We are therefore left withs r 0only one sensible solution,

12 192

a p 4.309401. (22)6

The location of the sonic point, , where both the numeratorycand the denominator vanish, is determined through the nu-merical integration of equations (17), (18), and (19). This yields

. Also note that , andy 0.46 g(y)(1 4y/a) g(0) p 0.5ctherefore the denominator of equation (19) is always well-behaved. The full numerical solution for , , andg(y) R(y) N(y)is illustrated in Figure 1.

3. APPLICATION TO GRBs

It can be seen from equation (6) that thenecessaryasymptoticLorentz factor required for a proper modeling of the properties

of GRBs, i.e., (remember that the Lorentz factor ofG 150fthe matter is only that of the shock) can be reached by1/ 2crossing a modest factor of 105 density decrease in the expo-nential atmosphere (assuming ). Yet the total energy ofG 10imaterial moving at these large Lorentz factors is only modest.In fact, let us compute the distribution of kinetic energy withbaryon number. Consider a cylindrical fluid element withsurface area A (with the normal along the direction of fluidmotion x) and length along the direction x. This elementdxwill have a bulk kinetic energy of 2dE p Amc n(x)g(x)dx p

. What we want to know is the2 1a a 1/2 1Ar c G G N(y)g (y)k dy0 i

energy distribution at the moment in which the shock gets outof the exponential atmosphere. Let us indicate with the shockGfLorentz factor at that moment. Then

3/2dE N(y)g (y)2 1 a 1ag p 2Ar c k G G , (23)i i f dg g

which can be coupled to the solution to yield a para-g p g(y)metric representation of , the distribution of kineticg dE/dgenergy with respect to the final Lorentz factor. The adimen-sional part of the function in the previous equation is plottedin Figure 2. The numerical factor can be estimated using equa-

tion (6):

2 1dE r k rH i48g p 10 ergs ( )13 12 9 3dg 10 cm 10 cm 10 cma a1 3/210 G 2N(y)g (y)f

# . (24)( ) ( ) G 150 giIn this expression, the values for rH (the edge of the H shell),ri (the matter density at the end of the H shell, and thus pre-sumably at the beginning of the exponential atmosphere), and

(the shock Lorentz factor at the end of the H shell thatGicoincides with the beginning of the exponential atmosphere)are taken from Meszaros & Rees (2001). Taking the adimen-


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L50 PROPAGATION OF RELATIVISTIC SHOCK Vol. 569

Fig. 2.Quantity (the adimensional part of ; eq. [23])3/2 2N(y)g(y) /g g dE/dgplotted as a function of . The ticks in boldface indicate the present1/2g(y)position of matter that, before being shocked, was located at 5, 10, and 15exponential scale lengths .1k

sional factor from Figure 2, we see that the kinetic energy fallsshort of the average isotropic burst energy (Schmidt 1999) byabout 3 orders of magnitude (note that here what we are com-puting is the isotropic energy).

Before showing how to come out of this impasse, we needto consider which part of Figure 2 can actually be obtained ina realistic model. The reason for this limitation comes fromthe fact that the idealized model presented here has been prop-

agating down an exponential atmosphere forever, while in arealistic model only a finite amount of matter could have con-verged onto this self-similar solution, given the finite dimensionof the star. For this reason, we have plotted in Figure 2 threeticks, which correspond to the present position of matter thatwas located, before being reached by the shock, at 5, 10, and15 exponential scale lengths from the shocks present location.If we think that the initial atmosphere extends for 5, 10, and15 exponential scale lengths, then we can believe the part of

Figure 2 located to the right of their respective tick marks.Leftward of them, the true, physical solution will depart fromthe one shown here, and the total kinetic energy at the cor-responding values of the Lorentz factor will be much smallerthan the values plotted (obviously, since the physical solutionhas finite mass and energy, which is not true for the idealizedone).

Let us now suppose that the putative burst progenitor possesses

a wide exponential atmosphere, spanning some 9 orders ofmagnitude in density. Then, for an initial Lorentz factor G i

, the shock Lorentz factor at the end of the atmosphere is101000 (eq. [6]); once again, the overall energy of matter movingat this speed would be modest (1046 ergs), but Figure 2 showsthat most of the energy (before the cutoff implied by the finiteextent of the atmosphere) will come out at a Lorentz factor 0.2of the shocks factor, i.e., at . With the values above,G 200matterthe total energy then amounts to ergs, in reasonable526# 10agreement with Schmidts (1999) isotropic estimates.4 It shouldbe noticed that inspection of the numerical solution for largevalues of the distance parameter y shows the material to be cold( ), so that there will be negligible further acceleration ofe nthese slower shells by work, and remains ap dV G 200matter

good estimate of the coasting Lorentz factor.To summarize, the application of our solution to shock ac-

celeration in the atmosphere of a massive star has shown thatin atmospheres with small density range, the amount of energycarried by material accelerated to the typical GRB Lorentzfactors falls short of the GRB required energetics. However,for stars with a wide density range in their atmospheres (109

orders of magnitude), a sufficient quantity of energy is carriedby later shells of material moving at the typical GRB Lorentzfactors. In this model, the early emission would be dominatedby an ultrahard component because of the very large G shellsslowing down in the interstellar medium. This is at least qual-itatively consistent with virtually all bursts studied in somedetail by BeppoSAX (see Frontera et al. 2000, especially their

Fig. 2), which exhibit, in their first few seconds only, spectrapeaking beyond the Gamma-Ray Burst Monitors observinglimit of 700 keV.

R. P. thanks the Osservatorio Astronomico di Roma for itskind hospitality during the time that this work was carried out.

4 Spectral properties (i.e., line production) would be similar to those pre-dicted by Kallaman, Meszaros, & Rees (2001).

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Beloborodov, A. M., Stern, B. E., & Svensson, R. 2000, ApJ, 535, 158Blandford, R. D., & McKee, C. F. 1976, Phys. Fluids, 19, 1130Chevalier, R. A. 1990, ApJ, 359, 463

Frontera, F., et al. 2000, ApJS, 127, 59Grover, R., & Hardy, J. W. 1966, ApJ, 143, 48Hayes, W. D. 1968, J. Fluid. Mech., 32, 305Heavens, A. F., & Drury, L. OC. 1988, MNRAS, 235, 997Iwamoto, N. 1989, Phys. Rev. A, 39, 4076

Kallman, T. R., Meszaros, P., & Rees, M. J. 2001, preprint (astro-ph/0110654)

MacFadyen, A., & Woosley, S. E. 1999, ApJ, 524, 262

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Rosalba Perna and Mario Vietri- A Self-Similiar Solution for the Propagation of a Relativistic Shock in an Exponentiial Atmosphere