STABILITY OF THE FORWARD/REVERSE-SHOCK SYSTEM FORMED BY THE IMPACT OF ARELATIVISTIC FIREBALL ON AN AMBIENT MEDIUM

XiaohuWang and Abraham Loeb

AstronomyDepartment, HarvardUniversity, 60 Garden Street, Cambridge,MA 02138; xwang@cfa.harvard.edu, aloeb@cfa.harvard.edu

and

Eli Waxman

Department of CondensedMatter Physics,Weizmann Institute, Rehovot 76100, Israel; waxman@wicc.weizmann.ac.ilReceived 2001 September 14; accepted 2001 December 10

ABSTRACT

We analyze the stability of a relativistic double (forward/reverse)-shock system that forms when the fire-ball of a gamma-ray burst (GRB) impacts on the surrounding medium.We find this shock system to be stableto linear global perturbations for either a uniform or a wind (r�2) density profile of the ambient medium. Forthe wind case, we calculate analytically the frequencies of the normal modes that could modulate the earlyshort-term variability of GRB afterglows. We find that perturbations in the double-shock system couldinduce damped oscillatory fluctuations in the observed flux on short timescales during the early phase of anafterglow.

Subject headings: gamma rays: bursts — shock waves

1. INTRODUCTION

Gamma-ray bursts (GRBs) and their afterglows are mostnaturally described by the relativistic ‘‘ fireball ’’ model (see,e.g., Paczynski & Rhoads 1993; Katz 1994; Meszaros &Rees 1993, 1997;Waxman 1997a, 1997b; Sari, Piran, &Nar-ayan 1998). In this model, a compact source releases a largeamount of energy over a short time and produces a fireballthat expands relativistically as a thin shell. When the shellencounters the circumburst medium, two shocks areformed: a forward shock that propagates into the circum-burst medium and accelerates it, and a reverse shock thatpropagates into the relativistic shell and decelerates it (Rees& Meszaros 1992; Katz 1994; Meszaros, Rees, & Papatha-nassiou 1994; Sari & Piran 1995; Sari, Narayan, & Piran1996). Later on, after a significant mass of circumburstmedium is accumulated, the shell approaches a self-similarbehavior, as originally described by Blandford & McKee(1976), in which there is only one forward shock propagat-ing into the circumburst medium. The circumburst mediumcould be either the interstellar medium (ISM) or a progeni-tor wind.

The stability of the Blandford-McKee (1976) solution hasbeen demonstrated recently by Gruzinov (2000). Here weanalyze the stability of a forward/reverse relativistic shocksystem. This double-shock system exists during an impor-tant phase in the evolution of GRBs, and its stability hasobservational consequences. In particular, oscillations orinstabilities could translate to specific patterns of temporalvariability in the light curves of GRB afterglows.

In our linear perturbation analysis, we generalize the‘‘ thin-shell ’’ method first introduced by Vishniac (1983) inthe nonrelativistic regime. This method simplifies the equa-tions describing the stability of a spherical shock when thewavelength of the perturbation is much larger than thethickness of the shocked shell. In our relativistic treatment,we focus on global perturbations for which the wavelengthis much larger than the thickness of the forward/reverse-shock system. We consider the regime of GRB parametersin which the reverse shock is relativistic (although in reality

it may also be nonrelativistic). In x 2 we derive the perturba-tion equations for the forward/reverse-shock system. In x 3we show the analytical results for the wind case and thenumerical results for both the wind and ISM cases. Finally,in x 4 we summarize our main conclusions.

2. LINEAR PERTURBATION EQUATIONS

The interaction between a relativistically expanding shelland the circumburst medium results in a double-shock sys-tem, as shown in Figure 1. The system includes four distinctregions: the circumburst medium (region 1) and the shockedcircumburst medium (region 2) are separated by the for-ward shock (shock 1), while the shocked material in the shell(region 3) and the unshocked material in the shell (region 4)are separated by the reverse shock (shock 2). Regions 2 and3 are separated by a contact discontinuity. Our analysis isdone in the GRB source frame, where the circumburstmedium is at rest. We use a spherical coordinate systemwhose origin is located at the center of the explosion. Theradii of the contact discontinuity, shock 1, and shock 2 aredenoted by R0, R1, and R2, respectively. We refer to thecombination of regions 2 and 3 as the layer whose stabilitywe consider. Similarly to Vishniac (1983), we make the thin-shell approximation, i.e., we assume that

R1 � R2

R0< kðR1 � R2Þ5 1 ; ð1Þ

where k is the wavenumber of the perturbations. Note thatalthough shock 1 is relativistic, shock 2 could be either rela-tivistic or nonrelativistic. In this paper, we only consider thesituation in which shock 2 is relativistic.

The equations of motion for an ideal relativistic fluid are

@

@tð��Þ þ D

x ð��uÞ ¼ 0 ; ð2Þ

�2

c2ðeþ pÞ @u

@tþ ðu x

DÞu� �

þ D

pþ u

c2@p

@t

� �¼ 0 ; ð3Þ

The Astrophysical Journal, 568:830–844, 2002 April 1

# 2002. The American Astronomical Society. All rights reserved. Printed in U.S.A.

830

where �, u, e, p, and � are the fluid density, velocity, energydensity, pressure, and Lorentz factor, respectively. Wedefine the surface density �, bulk radial velocity Vr, andaverage tangential velocityVT of region 2 as

�ð�; �Þ ¼ R�20

Z R1

R0

��r2 dr ; ð4Þ

Vrð�; �Þ ¼ ð�R20Þ

�1

Z R1

R0

��urr2 dr ; ð5Þ

VTð�; �Þ ¼ ð�R20Þ

�1

Z R1

R0

��uTr2 dr : ð6Þ

The time evolution of these variables can be obtained byintegrating equations (2) and (3) across region 2, using theboundary conditions at shock 1 and at the contact disconti-nuity, and neglecting terms of higher order in ðR1 � R2Þ=R0

and kðR1 � R2Þ. We get

@t� ¼ �2Vr

R0�þ �1c� �

D

T xVT ; ð7Þ

@t�ðR0Þ ¼ � 2��1�ðR0Þ�1cþ33=4

21=2��1 p

3=4ðR0Þ�1=41

�1=2ðR0Þc1=2

þ �ðR0Þ2c

Vr

D

T xVT ; ð8Þ

@tVT ¼ 1

3��1c2 �1 � 33=421=2

p3=4ðR0Þ�1=41

�3=2ðR0Þc3=2

" #

D

TR0

� ��1�1cVT � Vr

R0VT � ��1 c2

3�2ðR0Þ

D

T�

þ c2

3�3ðR0Þ

D

T�ðR0Þ ; ð9Þ

where �1 is the density of the unshocked circumburstmedium just in front of shock 1, �ðR0Þ and pðR0Þ are theLorentz factor and pressure at the contact discontinuity,and

D

T denotes the tangential derivatives. In deriving theabove equations, we also assumed that the radial velocities

are dominated by the bulk motion of regions 2 and 3(denoted hereafter as the ‘‘ shock layer ’’), so that _RR0 � Vr.The full derivation of the above equations is given in theAppendix.

From equations (7) and (8), we obtain the unperturbedequations

@t�ð0Þ ¼ �2

vc

Rð0Þ0

�ð0Þ þ �1c ; ð10Þ

@t�c ¼ �2 �ð0Þ� ��1

�c�1cþ33=4

21=2�ð0Þ

� ��1p3=4c �

1=41

�1=2c c1=2

; ð11Þ

where vc ¼ _RRð0Þ0 ¼ V

ð0Þr and �c ¼ �ð0ÞðR0Þ ¼ 1= 1� v2c=ð

c2Þ1=2 are the velocity and Lorentz factor of the unperturbedcontact discontinuity, pc ¼ pð0ÞðR0Þ is the unperturbed pres-sure at the contact discontinuity, and we use a superscript(0) to denote unperturbed values.

For the shocked shell in region 3, we define the surfacedensity, bulk radial velocity, and average tangential velocityto be

�3ð�; �Þ ¼ R�20

Z R0

R2

��r2 dr ; ð12Þ

Vr3ð�; �Þ ¼ ð�3R20Þ

�1

Z R0

R2

��urr2 dr ; ð13Þ

VT3ð�; �Þ ¼ ð�3R20Þ

�1

Z R0

R2

��uTr2 dr ; ð14Þ

where a subscript 3 denotes quantities in region 3. The timederivatives of the above variables can be derived similarly tothose in region 2,

@t�3 ¼ �2Vr

R0�3 þ

�4�2ðR0Þ

�4c� �3

D

T xVT3 ; ð15Þ

@t�ðR0Þ ¼��13

�4�ðR0Þ

�4c� 33=4��13

p3=4ðR0Þ�1=44 �1=2ðR0Þ�1=24 c1=2

þ �ðR0Þ2c

Vr

D

T xVT3 ; ð16Þ

@tVT3 ¼1

3��13 c2

�2 33=4� � p3=4ðR0Þ�1=44

�1=24 �1=2ðR0Þc3=2

� �42�2ðR0Þ

�4

�

D

TR0 � ��13

�4�2ðR0Þ

�4cVT3

� Vr

R0VT3 �

1

4�2ðR0Þ@t�4�4

� �VT3

� ��13

c2

3�2ðR0Þ

D

T�3 þc2

3�3ðR0Þ

D

T�ðR0Þ ; ð17Þ

where �4 and �4 are the density and Lorentz factor of theunshocked shell (region 4) just in front of shock 2. We havealso used the relation Vr3 � Vr, as appropriate in the thin-shell approximation.

The unperturbed equations for region 3 are

@t�ð0Þ3 ¼ �2

vc

Rð0Þ0

�ð0Þ3 þ �4

�2c�4c ; ð18Þ

RR R

region 2region 3region 4

shockedshell

shockedshellcircumburst

medium

contactshock 1shock 2 discontinuity

2 0 1

circumburstmedium

region 1

Fig. 1.—Structure of the forward/reverse-shock system

SHOCK STABILITY FROM IMPACT OF FIREBALL 831

@t�c ¼ �ð0Þ3

� ��1�4�c

�4c� 33=4 �ð0Þ3

� ��1p3=4c �

1=44 �

1=2c

�1=24 c1=2

: ð19Þ

Equations (7), (8), (9), (15), (16), and (17) make a com-plete set of equations for the evolution of the forward/reverse-shock system. The perturbation equations dependon the density profile of the circumburst medium. In the fol-lowing subsections, we derive the perturbation equationsfor a uniform medium (such as the ISM) and for a progeni-tor wind.

2.1. UniformMedium (ISM)

For a uniform circumburst medium, �1 ¼ const. Wedefine the perturbation variables � � �=�ð0Þ � 1, �3 ��3=�

ð0Þ3 � 1, DR � R0 � R

ð0Þ0 , and Dp � pðR0Þ=pð0ÞðR0Þ �1.

From equations (7), (8), and (9), we obtain the followingperturbation equations:

@t� ¼ � 2

Rð0Þ0

@tDRþ 2vc

Rð0Þ0

� �2DR� �ð0Þ

� ��1

�1c�

� D

T xVT ; ð20Þ

@2t DR ¼ 2 �ð0Þ

� ��1�1c2

�2c� 33=4

21=2�ð0Þ

� ��1p3=4c �

1=41 c1=2

�7=2c

" #�

þ"4 �ð0Þ� ��1

�1c

� 7 33=4ð Þ23=2

�ð0Þ� ��1p

3=4c �

1=41

�3=2c c1=2

#@tDR

þ 37=4

25=2�ð0Þ

� ��1p3=4c �

1=41 c1=2

�7=2c

Dp þvc

2�2c

D

T xVT ;

ð21Þ

@tVT ¼ 1

3�ð0Þ

� ��1

c2 �1 � 33=4 21=2� � p

3=4c �

1=41

�3=2c c3=2

" #

D

TDR

� �ð0Þ� ��1

�1cVT � vc

Rð0Þ0

VT � c2

3�2c

D

T�

þ c

3

D

T ð@tDRÞ : ð22Þ

Assuming that �4 is a constant and that there is no shellspreading, we get �4 / R�2. Using this scaling, we derive thefollowing perturbation equations from equations (15), (16),and (17):

@t�3 ¼ � 2 �ð0Þ3

� ��1

�4�4@tDR

þ 2vc

Rð0Þ0

� �2� 2 �

ð0Þ3

� ��1�4

�2c

�4c

Rð0Þ0

264

375DR

� �ð0Þ3

� ��1�4

�2c�4c�3 �

D

T xVT3 ; ð23Þ

@2t DR ¼

"� �

ð0Þ3

� ��1�4�4c

�4c2

þ 33=4 �ð0Þ3

� ��1p3=4c �

1=44 c1=2

�1=24 �

5=2c

#�3

þ"� 4 �

ð0Þ3

� ��1�4�2c

�4c

þ5 33=4� �2

�ð0Þ3

� ��1 p3=4c �

1=44

�1=24 �

1=2c c1=2

#@tDR

þ"� 2 �

ð0Þ3

� ��1�4�4c

�4c2

Rð0Þ0

þ 33=4

2�ð0Þ3

� ��1p3=4c �

1=44 c1=2

�1=24 �

5=2c

1

Rð0Þ0

#DR

� 37=4

4�ð0Þ3

� ��1p3=4c �

1=44 c1=2

�1=24 �

5=2c

Dp þvc2�2c

D

T xVT3 ;

ð24Þ

@tVT3 ¼1

3�ð0Þ3

� ��1

c2

� 2 33=4� � p

3=4c �

1=44

�1=24 �

1=2c c3=2

� �42�2c

�4

" #

D

TDR

� �ð0Þ3

� ��1�4�2c

�4cVT3 �vc

Rð0Þ0

VT3

� 1

4�2c

@t�4�4

� �VT3 �

c2

3�2c

D

T�3 þc

3

D

T ð@tDRÞ :

ð25Þ

In total, we have six perturbation equations (20)–(25)insix variables: �, �3, DR, VT , VT3, and Dp. In order to solvethese equations, we first need to find the unperturbedvalues �ð0Þ, �

ð0Þ3 , �c, and pc from equations (10), (11), (18),

and (19). The time dependence of �c has been derived bySari & Piran (1995) and Sari et al. (1996). When both theforward shock and the reverse shock are ultrarelativisticand strong,

�c / �1=24 f 1=4 ; ð26Þ

where f ¼ �4=�1. For �1 ¼ const, �4 ¼ const, and �4 / R�2,we get

�c / R�1=2 / t�1=2 : ð27Þ

With vc � c andRð0Þ0 � ct, equation (10) yields

�ð0Þ � 1

3�1ct : ð28Þ

For �4 / t�2 and �c / t�1=2, equation (18) gives

�ð0Þ3 � 1

2

�4

�2c�4ct ¼ const : ð29Þ

By substituting equation (28) into equation (11), we find

pc �114=3

37=322=3�2c�1c

2 ¼ 1:187�2c�1c2 <

4

3�2c�1c

2 ; ð30Þ

832 WANG, LOEB, & WAXMAN Vol. 568

where ð4=3Þ�2c�1c2 is the pressure just behind the forwardshock. Similarly, by substituting equation (29) into equation(19), we get

pc �54=3

44=33

�24�2c

�4c2 ¼ 0:449

�24�2c

�4c2 >

4

3��3�3

2�4c2 ; ð31Þ

where ��3�3 � �4=ð2�cÞ is the Lorentz factor of the shockedshell (region 3) with respect to the unshocked shell (region4) and ð4=3Þ ��3�32�4c2 is the pressure just behind the reverseshock. The pressure difference between the two sides of thelayer causes it to decelerate. By combining equations (30)and (31), we find

�c � 0:784�1=24 ð�4=�1Þ1=4 : ð32Þ

Substitution of the values of �ð0Þ, �ð0Þ3 , and pc into the per-

turbation equations (20)–(25) yields

@t� ¼ � 2

ct@tDRþ 2

ct2DR� 3

t� � D

T xVT ; ð33Þ

@2t DR ¼ 1

2

c

�2c t� � 29

4

1

t@tDRþ 33

8

c

�2c tDp þ

c

2�2c

D

T xVT ;

ð34Þ

@tVT ¼ � 8

3

c

t

D

TDR� 4

tVT � c2

3�2c

D

T� þc

3

D

Tð@tDRÞ ;

ð35Þ

@t�3 ¼ �4�2cct

@tDR� 2

ct2DR� 2

t�3 �

D

T xVT3 ; ð36Þ

@2t DR ¼ 1

2

c

�2c t�3 �

7

4

1

t@tDR� 11

4

1

�2c t2DR

� 15

8

c

�2c tDp þ

c

2�2c

D

T xVT3 ; ð37Þ

@tVT3 ¼4

3

c

t

D

TDR� 3

tVT3 �

c2

3�2c

D

T�3 þc

3

D

Tð@tDRÞ :

ð38Þ

Combining equations (34) and (37) and eliminating Dp, weget

@2t DR ¼ 5

32

c

�2c t� þ 11

32

c

�2c t�3 �

111

32

1

t@tDR� 121

64

1

�2c t2DR

þ 5

32

c

�2c

D

T xVT þ 11

32

c

�2c

D

T xVT3 : ð39Þ

Next we expand the spatial dependence of the perturba-tion variables in spherical harmonics. We choose to normal-ize these variables so as to make them dimensionless and ofa similar magnitude through the definitions

DR ¼Xl;m

DRðl; m; tÞ Rð0Þ0 =�2c

� �Ylmð�; �Þ ; ð40Þ

� ¼Xl;m

�ðl; m; tÞYlmð�; �Þ ; ð41Þ

�3 ¼Xl;m

�3ðl; m; tÞYlmð�; �Þ ; ð42Þ

VT ¼Xl;m

VT ðl; m; tÞ cRð0Þ0 =�c

� � D

TYlmð�; �Þl

; ð43Þ

VT3 ¼Xl;m

VT3ðl; m; tÞ cRð0Þ0 =�c

� � D

TYlmð�; �Þl

: ð44Þ

Equations (33), (35), (36), (38), and (39) can be rewrittenas

dtDR ¼ F ; ð45Þ

dt� ¼ � 2

�2cF � 2

�2c tDR� 3

t� þ ðl þ 1Þ

�ctVT ; ð46Þ

dtVT ¼ l

3�cF � 2l

�ctDR� l

3�ct� � 9

2tVT ; ð47Þ

dt�3 ¼ �4F � 8

tDR� 2

t�3 þ

ðl þ 1Þ�ct

VT3 ; ð48Þ

dtVT3 ¼l

3�cF þ 2l

�ctDR� l

3�ct�3 �

7

2tVT3 ; ð49Þ

dtF ¼ � 239

32tF � 143

16t2DRþ 5

32t2� þ 11

32t2�3

� 5

32

ðl þ 1Þ�ct2

VT � 11

32

ðl þ 1Þ�ct2

VT3 : ð50Þ

The perturbation variables in the above six equations aredimensionless and only functions of time. We have addedthe variable F so that all the equations have the form offirst-order differential equations.

2.2. WindMedium

If the circumburst medium is a progenitor wind,�1 / R�2. Accordingly, the perturbation equations (20) and(21) need to be changed to

@t� ¼ � 2

Rð0Þ0

@tDRþ 2vc

Rð0Þ0

� �2� 2 �ð0Þ

� ��1 �1c

Rð0Þ0

264

375DR

� �ð0Þ� ��1

�1c� �

D

T xVT ; ð51Þ

@2t DR ¼ 2 �ð0Þ

� ��1�1c2

�2c� 33=4

21=2�ð0Þ

� ��1p3=4c �

1=41 c1=2

�7=2c

" #�

þ"4 �ð0Þ� ��1

�1c

� 7 33=4ð Þ23=2

�ð0Þ� ��1p

3=4c �

1=41

�3=2c c1=2

#@tDR

þ"4 �ð0Þ� ��1 �1c2

�2cRð0Þ0

� 33=4

23=2�ð0Þ

� ��1p3=4c �

1=41 c1=2

�7=2c R

ð0Þ0

#DR

þ 37=4

25=2�ð0Þ

� ��1p3=4c �

1=41 c1=2

�7=2c

Dp þvc

2�2c

D

T xVT :

ð52Þ

No. 2, 2002 SHOCK STABILITY FROM IMPACT OF FIREBALL 833

Equations (22)–(25) remain the same as in the uniformmedium case.

Since �1 / R�2 and �4 / R�2 in the wind case, equation(26) implies that �c is constant over time. Equations (10),(11), (18), and (19) then yield the unperturbed parameters

�ð0Þ � �1ct ; ð53Þ

�ð0Þ3 � �4

�2c�4ct ; ð54Þ

pc �4

3�2c�1c

2 � 1

3

�24�2c

�4c2 ; ð55Þ

�c �1ffiffiffi2

p �1=24

�4�1

� �1=4

: ð56Þ

Substitution of equations (53)–(55) into the perturbationequations (51), (52), and (22)–(25) yields

@t� ¼ � 2

ct@tDR� 1

t� � D

T xVT ; ð57Þ

@2t DR ¼ � 3

t@tDRþ 3

�2c t2DRþ 3

2

c

�2c tDp þ

c

2�2c

D

T xVT ;

ð58Þ

@tVT ¼ � c

t

D

TDR� 2

tVT � c2

3�2c

D

T� þc

3

D

Tð@tDRÞ ; ð59Þ

@t�3 ¼ �2�2cct

@tDR� 1

t�3 �

D

T xVT3 ; ð60Þ

@2t DR ¼ � 3

2

1

t@tDR� 3

2

1

�2c t2DR� 3

4

c

�2c tDp þ

c

2�2cD

T xVT3 ;

ð61Þ

@tVT3 ¼1

2

c

t

D

TDR� 2

tVT3 �

c2

3�2c

D

T�3 þc

3

D

Tð@tDRÞ :

ð62Þ

Combining equations (58) and (61) and eliminating Dp, weget

@2t DR ¼ � 2

t@tDRþ c

6�2c

D

T xVT þ c

3�2c

D

T xVT3 : ð63Þ

Using the same normalized perturbation variables asdefined in equations (40)–(44), we can rewrite equations(57), (59), (60), (62), and (63) as

dtDR ¼ F ; ð64Þ

dt� ¼ � 2

�2cF � 2

�2c tDR� 1

t� þ ðl þ 1Þ

�ctVT ; ð65Þ

dtVT ¼ l

3�cF � 2l

3�ctDR� l

3�ct� � 2

tVT ; ð66Þ

dt�3 ¼ �2F � 2

tDR� 1

t�3 þ

ðl þ 1Þ�ct

VT3 ; ð67Þ

dtVT3 ¼l

3�cF þ 5l

6�ctDR� l

3�ct�3 �

2

tVT3 ; ð68Þ

dtF ¼ � 4

tF � 2

t2DR� ðl þ 1Þ

6�ct2VT � ðl þ 1Þ

3�ct2VT3 : ð69Þ

These six first-order differential equations are the final per-turbation equations for the wind case.

3. SOLUTIONS OF THE PERTURBATION EQUATIONS

For the wind case, �c ¼ const, and we can solve the per-turbation equations analytically. If we define F ¼ F 0=t, thenequations (64)–(69) can be rewritten in a matrix form:

dt

F 0

DR

�

VT

�3

VT3

0BBBBBBBB@

1CCCCCCCCA

¼ 1

t

�

�3 �2 0 �ðl þ 1Þ6�c

0 �ðl þ 1Þ3�c

1 0 0 0 0 0

� 2

�2c� 2

�2c�1

ðl þ 1Þ�c

0 0

l

3�c� 2l

3�c� l

3�c�2 0 0

�2 �2 0 0 �1ðl þ 1Þ�c

l

3�c

5l

6�c0 0 � l

3�c�2

0BBBBBBBBBBBBBBBBBB@

1CCCCCCCCCCCCCCCCCCA

�

F 0

DR

�

VT

�3

VT3

0BBBBBBBB@

1CCCCCCCCA

: ð70Þ

In matrix notation, the above equation is

dty ¼ 1

tAy ; ð71Þ

where y is a vector and A is a 6� 6 time-independentmatrix.

The matrix A can be diagonalized through the transfor-mation

X�1AX ¼ diagð�1 � � ��6Þ ¼ D ; ð72Þ

where �1, . . ., �6 are the eigenvalues of the matrix A, and Xis the matrix formed by columns from the eigenvectors (i.e.,the kth column of X is the eigenvector corresponding to theeigenvalue �k). Equation (71) can then be transformed to

dty ¼ 1

tXDX�1� �

y¼)dt X�1y� �

¼ 1

tD X�1y� �

: ð73Þ

By defining a new vector y0 ¼ X�1y, we get the equation

dty0 ¼ 1

tDy0 ; ð74Þ

which has six components:

dty0k ¼ 1

t�ky

0k; k ¼ 1; . . . ; 6 : ð75Þ

834 WANG, LOEB, & WAXMAN Vol. 568

Since �k can be a complex number, we write �k ¼ ak þ ibk.The solution to equation (75) is then

y0k ¼ cktakþibk ¼ ckt

akeibk ln t ; ð76Þ

where ck is a constant dictated by the initial conditions.Each y0k defines a mode of the shock system. There are sixmodes in total corresponding to six eigenvalues of thematrix A. The real part of each eigenvalue dictates the over-all temporal behavior of the mode, while the imaginary partdetermines its oscillation frequency. The vector y can bederived from the relation

y ¼ Xy0 : ð77Þ

Hence, each component of the vector y is a linear combina-tion of the six different modes. The vector c whose compo-nents are ck can be obtained from the initial conditions

c ¼ y0ðt ¼ 1Þ ¼ X�1yðt ¼ 1Þ ; ð78Þ

namely,

c1

c2

c3

c4

c5

c6

0BBBBBBBB@

1CCCCCCCCA

¼ X�1

F 0ðt ¼ 1ÞDRðt ¼ 1Þ�ðt ¼ 1ÞVTðt ¼ 1Þ�3ðt ¼ 1ÞVT3ðt ¼ 1Þ

0BBBBBBBB@

1CCCCCCCCA

: ð79Þ

The eigenvalues of the matrix A can be calculated for dif-ferent values of �c and l. Figure 2 shows the real and imagi-nary parts of the six eigenvalues as functions of l for�c ¼ 500. Each row in the figure contains two panels, corre-sponding to the real and imaginary parts of a particulareigenvalue. The results show that for l � 320, all six eigen-values are real numbers, and so there are no oscillations.For 320 < l � 432, two eigenvalues are complex numbersand they are a pair of complex conjugates, implying thattwo modes are oscillating with the same frequency. Forl > 432, there are two pairs of complex conjugates. Thetransition from real eigenvalues to complex eigenvaluesoccurs when l � �c, as expected from the fact that oscilla-tions are possible only when causality allows communica-tion across the scale of a wavelength for modes with l& �c.

For the thin-shell approximation to be valid, we requirethat the wavelength of the perturbation be much larger thanthe thickness of the forward/reverse-shock system in theshock frame. The thickness of the shock system is.2R0=��2cin the observer frame, and thus .2R0=��c in the shockframe. Here � is a constant that ranges between �4 and�12for the wind and ISM profiles, respectively. The wavelengthof the perturbation is �2R0=l in the shock frame. There-fore, we enforce an upper limit on l of�10�c. Figure 2 showsthat for all values of l, the real parts of the six eigenvaluesare ��1. This implies that all modes are decaying fasterthan or are proportional to t�1. Since each perturbationvariable is a linear combination of the six modes, we con-clude that all perturbation variables should also decay fasterthan or be proportional to t�1. Thus, the system is stable.Note that for large �c, we expect the results to depend onlyon l=�c (see eq. [70]), and so our particular choice of�c ¼ 500 can be scaled appropriately to other values of �c.

For l4�c, the eigenvalues admit the following analyticalsolutions,

�1 ¼ �1 ; �2 ¼ � 3

2� 1ffiffiffi

3p l

�ci ; �3 ¼ � 3

2þ 1ffiffiffi

3p l

�ci ;

ð80Þ

�4 ¼ � 19

9; �5 ¼ � 13

9þ 1ffiffiffi

2p l

�ci ; �6 ¼ � 13

9� 1ffiffiffi

2p l

�ci ;

ð81Þ

while in the limit of l5 �c,

�1 ¼ �1 ; �2 ¼ �2þ 1

3

l2

�2c; �3 ¼ �1� 1

3

l2

�2c; ð82Þ

�4 ¼ �1� 1

2

l2

�2c; �5 ¼ �2� 1

3ffiffiffi2

p l

�c; �6 ¼ �2þ 1

3ffiffiffi2

p l

�c:

ð83Þ

We have calculated the corresponding eigenvectorsnumerically as shown in Figure 3 (for �c ¼ 500). Each rowin the figure contains two panels that show the real andimaginary parts of one of the six components of the eigen-vectors as functions of l. Different line types correspond tothe six different eigenvectors. The complex eigenvectors areall scaled to have a unit magnitude. Since each eigenvectorcorresponds to a mode, the relative values of the six compo-nents of the eigenvector measure the physical significance ofperturbations in different physical parameters for thatmode. For example, the mode corresponding to the eigen-value �1 ¼ �1 has the temporal behavior of t�1; the eigen-vector for this mode is (�0.275, 0.275, �0.824, 0, 0.412, 0),implying that this mode does not involveVT andVT3 pertur-bations. Also note that this mode does not depend on l,while all other modes change with l.

Equations (64)–(69) can also be solved numerically. Wenormalize all initial values of the perturbation variables tounity. The temporal interval of the calculation is from t ¼ 1to 15, and �c is chosen to be 500. Here t ¼ 1 marks the timewhen the perturbations are added to the forward/reverse-shock system. To a distant observer, this corresponds toTt¼1 � Rt¼1=2�2c c,

1 where Rt¼1 is the radius of the double-shock system at t ¼ 1. Figure 4 shows our results. The sixpanels show the time evolution of DR, dtDR, �, VT, �3, andVT3. We show results for four different l values, namelyl ¼ 5, 50, 500, and 5� 103. These plots indicate that all per-turbation variables decay quickly with time. For small val-ues of l (e.g., l ¼ 5 and 50), there are no oscillations. Forlarge values of l (e.g., l ¼ 5� 103), the oscillations exist butdamp away quickly. These results are consistent with ouranalytical derivations. For l ¼ 500, the oscillations start toappear, although they are not apparent in the plot becauseof their low frequency. For l4�c, we can calculate the fre-quencies of the oscillations using the eigenvalues listed inequations (80) and (81). The oscillations have the formexp i 1=

ffiffiffi3

p� �l=�cð Þ ln t

�or exp i 1=

ffiffiffi2

p� �l=�cð Þ ln t

�so that

the oscillation period increases with time. For l ¼ 10�c, the

1 Note that for a GRB located at a cosmological redshift z, one shoulduse Tt¼1 � ð1þ zÞRt¼1=2�2c c instead. We omit the cosmological redshiftfactor in the text in order to keep the expressions simpler.

No. 2, 2002 SHOCK STABILITY FROM IMPACT OF FIREBALL 835

shortest period is �2.4, corresponding to 2:4Tt¼1 to anobserver. Hence, the oscillations could produce fluctuationsin the observed flux on timescales as short as a few timesTt¼1. Because the decay of the modes is not exponentiallyfast but rather moderate, the observed flux might showmeasurable fluctuations if the initial perturbations are suffi-ciently large.

In the ISM case, �c is not constant, and so we can notwrite equations (45)–(50) in a matrix form that admits ananalytic solution. Instead, we have to solve these equationsnumerically. In order to test the validity of the perturbation

equations and the numerical code, we compared the numeri-cal results for a spherical perturbation with l ¼ 0 with theanalytic solution derived by directly perturbing the radialequations of motion, and found an excellent agreementbetween the two calculations. Similarly to the wind case, wenormalized all initial values of the perturbation variables tounity and chose an initial �c ¼ 500. Our numerical resultsare shown in Figure 5 and qualitatively resemble the windcase. Overall, the perturbations decay rapidly with time,and oscillations appear only for large values of l. Similarlyto the wind case, for l ¼ 10�c the shortest period of the oscil-

0 200 400 600 800 1000−2

−1.5

−1

−0.5

0

l

Re[

λ 1]

0 200 400 600 800 1000−1

−0.5

0

0.5

1

l

Im[λ

1]

0 200 400 600 800 1000

−2

−1.8

−1.6

−1.4

l

Re[

λ 2]

0 200 400 600 800 1000

−1

−0.5

0

l

Im[λ

2]0 200 400 600 800 1000

−1.6

−1.4

−1.2

−1

l

Re[

λ 3]

0 200 400 600 800 1000

0

0.5

1

lIm

[λ3]

0 200 400 600 800 1000

−2

−1.5

−1

l

Re[

λ 4]

0 200 400 600 800 1000−0.4

−0.3

−0.2

−0.1

0

0.1

l

Im[λ

4]

0 200 400 600 800 1000−2.2

−2

−1.8

−1.6

−1.4

l

Re[

λ 5]

0 200 400 600 800 1000−0.5

0

0.5

1

1.5

l

Im[λ

5]

0 200 400 600 800 1000

−2

−1.8

−1.6

−1.4

l

Re[

λ 6]

0 200 400 600 800 1000−1.5

−1

−0.5

0

0.5

l

Im[λ

6]

Fig. 2.—Real and imaginary parts of the six eigenvalues in the wind case as functions of l, with �c ¼ 500

836 WANG, LOEB, & WAXMAN Vol. 568

lations is �2, corresponding to �2Tt¼1 to an observer.Again, the damped fluctuations in the flux are detectable ifthe initial perturbations are sufficiently large.

For the double-shock system we are considering, Sari &Piran (1995) have defined four critical radii, RN ¼‘3=2=D1=22, where the reverse shock becomes relativistic;RD ¼ ‘3=4D1=4, where the reverse shock crosses the shell;R� ¼ ‘=2=3, where the forward shock sweeps up a massM0=; and Rs ¼ D2, where the shell begins to spread if the

initial Lorentz factor varies by the order of . Here‘ � ðE=n1mpc2Þ1=3 is the Sedov length, D is the width of theshell, E is the equivalent isotropic energy of the fireball, M0

is the mass of the initial baryonic load of the fireball, is theinitial thermal Lorentz factor of the fireball, and n1 is thenumber density of the ISM. These four radii are simplyrelated as follows: RN=� ¼ R� ¼ �1=2RD ¼ �2Rs, withthe dimensionless quantity � � ðl=DÞ1=2�4=3. When � > 1,we have the Newtonian case, with Rs < RD < R� < RN ; the

0 200 400 600 800 1000−0.4

−0.2

0

0.2

0.4

l

Re[

v 1]

0 200 400 600 800 1000

−0.02

−0.01

0

0.01

0.02

l

Im[v

1]

0 200 400 600 800 1000

−0.2

0

0.2

0.4

l

Re[

v 2]

0 200 400 600 800 1000

−0.05

0

0.05

l

Im[v

2]0 200 400 600 800 1000

−1

−0.5

0

0.5

1

l

Re[

v 3]

0 200 400 600 800 1000

−0.5

0

0.5

lIm

[v3]

0 200 400 600 800 1000−1

−0.5

0

l

Re[

v 4]

0 200 400 600 800 1000

−0.2−0.1

00.10.2

l

Im[v

4]

0 200 400 600 800 1000−1

−0.5

0

0.5

l

Re[

v 5]

0 200 400 600 800 1000

−0.5

0

0.5

l

Im[v

5]

0 200 400 600 800 1000

0

0.2

0.4

0.6

l

Re[

v 6]

0 200 400 600 800 1000−1

−0.5

0

0.5

1

l

Im[v

6]

Fig. 3.—Real and imaginary parts of the six components of different eigenvectors (modes) in the wind case, with �c ¼ 500. Each row corresponds to onecomponent of the eigenvector. Different line types correspond to six different eigenvectors: the thick solid line refers to the eigenvector of �1, the thick dashedline to �2, the thin solid line to �3, the thin dashed line to �4, the dotted line to �5, and the dash-dotted line to �6.

No. 2, 2002 SHOCK STABILITY FROM IMPACT OF FIREBALL 837

reverse shock is still Newtonian when it crosses the shell.When � < 1, we have the relativistic case, withRN < R� < RD < Rs; the reverse shock becomes relativisticbefore it crosses the shell. In this paper, we only considerthe latter case. For typical GRB parameters, E ¼ 1052 ergs, ¼ 103, n ¼ 1 cm�3 , and D ¼ 3� 1011 cm, we have‘ � 2� 1018 cm, � � 0:25, RN � 5� 1015 cm, andRD � 4� 1016 cm. For our calculation, the perturbationscan be added to the system between RN and RD. Thus t ¼ 1corresponds to the time between RN=c and RD=c. To an

observer, this corresponds to the time Tt¼1 betweenRN=2�2c c and RD=2�2c c, or �0.3–2.5 s. Thus, the timescalesof the fluctuations in the observed flux could be as short as afew seconds.

4. DISCUSSION

We have solved the perturbation equations describing thedouble (forward/reverse)-shock system that forms duringthe impact of a highly relativistic fireball on a surrounding

0 5 10 150

0.2

0.4

0.6

0.8

1

1.2

t

∆R

l=5

l=50

l=500

l=5000

0 5 10 15

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

1.2

t

d t∆R

0 5 10 15−4

−3

−2

−1

0

1

2

t

δ

0 5 10 15

−1.5

−1

−0.5

0

0.5

1

t

VT

0 5 10 15−1

0

1

2

3

4

t

δ 3

0 5 10 15−0.5

0

0.5

1

1.5

t

VT

3

Fig. 4.—Evolution of the perturbation variables in the wind case, with �c ¼ 500. Four different line types correspond to l ¼ 5, 50, 500, and 5� 103, asmarked.

838 WANG, LOEB, & WAXMAN Vol. 568

medium. For both a uniform and a wind ð1=r2Þ density pro-file of the ambient medium, we have found the shock systemto be stable to global perturbations. We therefore do notexpect the shock to fragment. Our results are limited to rela-tivistic reverse shocks, and appear to differ qualitativelyfrom previous results in the nonrelativistic regime (Vishniac1983).

Our results also apply to collimated outflows as long asthe double-shock system is formed at a time when the Lor-

entz factor of the outflow is larger than the collimationangle.

We derived the frequencies of the normal modes thatcould modulate the short-term variability at the early phaseof GRB afterglows. The results imply that perturbations inthe double-shock system could produce fluctuations in theobserved flux on timescales as short as a few seconds for�c � 500 in the ISM case. These damped short-term fluctua-tions are detectable if the initial perturbations are suffi-

0 2 4 6 8 10−0.2

0

0.2

0.4

0.6

0.8

1

1.2

t

∆R

l=5

l=50

l=500

l=5000

0 2 4 6 8 10

−1

−0.5

0

0.5

1

t

d t∆R

0 2 4 6 8 10−6

−5

−4

−3

−2

−1

0

1

t

δ

0 2 4 6 8 10−3

−2.5

−2

−1.5

−1

−0.5

0

0.5

1

1.5

t

VT

0 2 4 6 8 10

−1

0

1

2

3

4

5

t

δ 3

0 2 4 6 8 10−0.5

0

0.5

1

1.5

2

2.5

t

VT

3

Fig. 5.—Evolution of the perturbation variables in the ISM case, with �c ¼ 500

No. 2, 2002 SHOCK STABILITY FROM IMPACT OF FIREBALL 839

ciently large. The fluctuations could be supplemented byvariability on much longer timescales due to density inho-mogeneities in the ISM; such inhomogeneities can lead tovariability on timescales of tens of minutes in the opticalband and days in the radio (Wang & Loeb 2000).

This work was supported in part by grants from theIsrael-US Binational Science Foundation (BSF 98-00343),NSF (AST 99-00877; AST 00-71019), and NASA (NAG 5-7039; NAG 5-7768).

APPENDIX

Here we provide full details for the derivation of equations (7)–(9) in x 2.We start by listing the equations of motion for a rel-ativistic fluid in spherical coordinates. The continuity equation reads

@

@tð��Þ þ 1

r2@

@rðr2��urÞ þ

1

r sin �

@

@�ðsin ���u�Þ þ

1

r sin �

@

@�ð��u�Þ ¼ 0 ; ðA1Þ

and the three components of the momentum equation are

�2

c2ðeþ pÞ @ur

@tþ ur

@ur@r

þ u�1

r

@ur@�

þ u�1

r sin �

@ur@�

� 1

rðu2� þ u2�Þ

� �þ @p

@rþ urc2

@p

@t¼ 0 ; ðA2Þ

�2

c2ðeþ pÞ @u�

@tþ ur

@u�@r

þ u�1

r

@u�@�

þ u�1

r sin �

@u�@�

þ 1

ruru� þ cot �u2�

� �� �þ 1

r

@p

@�þ u�

c2@p

@t¼ 0 ; ðA3Þ

�2

c2ðeþ pÞ @u�

@tþ ur

@u�@r

þ u�1

r

@u�@�

þ u�1

r sin �

@u�@�

þ 1

rðuru� þ cot �u�u�Þ

� �þ 1

r sin �

@p

@�þ u�

c2@p

@t¼ 0 ; ðA4Þ

where �, e, p, and � are the fluid density, energy density, pressure, and Lorentz factor, respectively, and ur, uh, and u� are thethree components of the fluid velocity.

For the forward/reverse-shock system under consideration (see Fig. 1), we define the following shell–averaged variables forregion 2,

�ð�; �Þ ¼ R�20

Z R1

R0

��r2 dr ; ðA5Þ

Vrð�; �Þ ¼ ð�R20Þ

�1

Z R1

R0

��urr2 dr ; ðA6Þ

VTð�; �Þ ¼ ð�R20Þ

�1

Z R1

R0

��uTr2 dr ; ðA7Þ

where uT is the tangential velocity vector.Since the shocked material is relativistic, we adopt the relativistic equation of state, p ¼ e=3, in region 2. Equation (A1)

yields the evolution of the surface density

@t� ¼ �2_RR0

R0�þ R1

R0

� �2

�ðR1Þ�ðR1Þ _RR1 � �ðR0Þ�ðR0Þ _RR0 þ R�20

Z R1

R0

@

@tð��Þr2 dr

¼ �2_RR0

R0�þ R1

R0

� �2

�ðR1Þ�ðR1Þ½ _RR1 � urðR1Þ� þ �ðR0Þ�ðR0Þ urðR0Þ � _RR0

�� R�2

0

Z R1

R0

D

x ð��uTÞ½ �r2 dr : ðA8Þ

Since r ¼ R0 defines the contact discontinuity between the shocked shell and the shocked circumburst medium and there is nomass flow across the contact discontinuity, we get urðR0Þ ¼ _RR0. Because the forward shock is a strong relativistic shock, wehave the following shock jump conditions at shock 1,

�2ðR1Þ ¼ �2s1=2 ; ðA9Þ

�ðR1Þ=�1 ¼ 4�ðR1Þ ; ðA10Þ

where �ðR1Þ and �ðR1Þ are the Lorentz factor and density of the fluid just behind the shock front, �s1 is the Lorentz factor ofthe shock front, and �1 is the density of the unshocked circumburst medium just in front of the shock front. In the highly rela-tivistic regime,

_RR1 � c 1� 1

2�2s1

� �; ðA11Þ

urðR1Þ � c 1� 1

2�2ðR1Þ

� �: ðA12Þ

840 WANG, LOEB, & WAXMAN Vol. 568

From equations (A9)–(A12), we obtain

�ðR1Þ�ðR1Þ½ _RR1 � urðR1Þ� � �1c : ðA13Þ

Thus, equation (A8) can be rewritten as

@t� ¼ �2_RR0

R0�þ R1

R0

� �2

�1c� R�20

Z R1

R0

½ D

x ð��uTÞ�r2 dr : ðA14Þ

To linear order, the last integration term in the above equation can be approximated by

�R�20

Z R1

R0

½ D

x ð��uTÞ�r2 dr ¼ ��

D

T xVT þ R�20

Z R1

R0

½ D

x ð��uTÞ�r

R0� 1

� �r2 dr ; ðA15Þ

where the operator

D

T � ðhh=R0Þð@=@�Þ þ ð}}=R0Þð@=@�Þ acts as follows on a scalar� and a vector f :

D

T� ¼ 1

R0

@�

@�hh þ 1

R0 sin �

@�

@�}} ; ðA16Þ

D

T x f ¼ 1

R0 sin �

@

@�ðsin �f�Þ þ

1

R0 sin �

@f�@�

: ðA17Þ

Note that the second term on the right-hand side of equation (A15) is of higher order in ðR1 � R0Þ=R0 than the preceding term,and hence can be ignored in the thin-shell approximation. Thus, equation (A14) can be rewritten as

@t� ¼ �2_RR0

R0�þ R1

R0

� �2

�1c� �

D

T xVT : ðA18Þ

Similarly to the above derivation, we obtain for the bulk radial velocity

@tVrð�; �Þ ¼ � @t�

�Vr � 2

_RR0

R0Vr þ ��1 R1

R0

� �2

�ðR1Þ�ðR1ÞurðR1Þ½ _RR1 � urðR1Þ� þ ��1�ðR0Þ�ðR0ÞurðR0Þ½urðR0Þ � _RR0�

� ð�R20Þ

�1

Z R1

R0

½ D

x ð��uTÞ�urr2 dr� ð�R20Þ

�1

Z R1

R0

�c2

4�p

@p

@rþ urc2

@p

@t

� �r2 dr

� ð�R20Þ

�1

Z R1

R0

�� uT x

D

urð Þr2 drþ �R20

� ��1Z R1

R0

��u2Tr dr : ðA19Þ

The last two terms on the right-hand side of the above equation are nonlinear. By substituting equations (A13) and (A14) intothe above equation and keeping terms to the linear order, we get

@tVrð�; �Þ ¼ ��1 R1

R0

� �2

�1c urðR1Þ � Vr½ � þ �R20

� ��1Vr

Z R1

R0

½ D

x ð��uTÞ�r2dr� �R20

� ��1Z R1

R0

ur½

D

x ð��uTÞ�r2 dr�

� �R20

� ��1Z R1

R0

�c2

4�p

@p

@rþ urc2

@p

@t

� �r2 dr : ðA20Þ

In order to evaluate the integral in the last term of the above equation, we need the relation between p and � inside region 2.Since entropy is conserved in this region,

d

dt

p

�4=3

� �¼ 0 ; ðA21Þ

implying that p=�4=3 remains constant for a given fluid particle. Hence, a fluid layer that is at a distance xðtÞ from the contactdiscontinuity (r ¼ R0) inside region 2 maintains a constant p=�4=3 over time, and its value is decided by the Lorentz factor ofshock 1 at the time when this layer first crosses shock 1. However, at a particular time, different layers across region 2 have dif-ferent values of p=�4=3. Assuming that region 2 is decelerating with � / r�1=2 (for the uniform ISM case), we get

pðxÞ�4=3ðxÞ

¼ c2

3 41=3ð Þ�1=31

�aR1=2a

ð8�2aRaxþ R2aÞ1=4

" #2=3

; ðA22Þ

where �a and Ra are the Lorentz factor and radius of region 2 at the initial time. Apparently, the dependence of pðxÞ=�4=3ðxÞon x is very weak, and so within the context of the thin-shell approximation, we simply assume that p=�4=3 is constant acrossregion 2 at any given time. This assumption is indeed satisfied in the numerical simulations performed by Kobayashi, Piran, &Sari (1999). In equation (A22), the term 8�2aRax can be at most comparable to R2

a (this happens in the very last stage of theevolution when the reverse shock crosses the shell), and so the error introduced by our approximation is small. For the windcase, � � const, and we can also treat p=�4=3 as a constant across region 2.

No. 2, 2002 SHOCK STABILITY FROM IMPACT OF FIREBALL 841

We can now calculate the integralZ R1

R0

�c2

4�p

@p

@rr2 dr ¼

Z R1

R0

c2

3�

@�

@rr2 dr ¼ c2

3�ðR0Þ

Z R1

R0

@�

@rr2 dr � c2

3�ðR0Þ�ðR1ÞR2

1 � �ðR0ÞR20 �

Z R1

R0

2r�dr

� �: ðA23Þ

In the thin-shell approximation, all radial velocities are dominated by the overall radial motion of the shock layer. Hence, � istreated as a constant and is taken out of the integral. For the uniform ISM case, the density difference between the two edgesof region 2 is not small; hence, the last term inside the square brackets is of order ðR1 � R0Þ=R0 times the difference betweenthe previous two terms, and so it can be neglected. For the wind case, there is no pressure gradient across region 2, and so theabove integral vanishes. However, even in this case we can ignore the last term and keep only the first two terms, because lateron we replaceR1 withR0, and so the difference between the first two terms vanishes.

Another relevant integral isZ R1

R0

�c2

4�p

urc2

@p

@tr2 dr ¼

Z R1

R0

c2

3�

urc2

@�

@tr2 dr ¼ Vr

3�ðR0Þ

Z R1

R0

@�

@tr2 dr ¼ Vr

3�ðR0Þ@

@t

Z R1

R0

�r2 dr� _RR1�ðR1ÞR21 þ _RR0�ðR0ÞR2

0

� �

¼ Vr

3�ðR0Þ�

�ðR0Þ2R0

_RR0 þR2

0

�ðR0Þ@t�� R2

0�

�2ðR0Þ@t�ðR0Þ � _RR1�ðR1ÞR2

1 þ _RR0�ðR0ÞR20

� �: ðA24Þ

In the above derivation, we pulled ur out of the integration assuming that it equals Vr, as appropriate in the thin-shell approxi-mation. By substituting equation (A18) into equation (A24) and making use of the following two relations,

c2 � Vr_RR0 �

c2

�2ðR0Þ; ðA25Þ

c2 � Vr_RR1 �

3c2

4�2ðR0Þ; ðA26Þ

we getZ R1

R0

�c2

4�p

@p

@rþ urc2

@p

@t

� �r2 dr ¼ 4R2

0

3�2ðR0Þ�1c

2 � R20

3�3ðR0Þ�ðR0Þc2 � �

R20

3�2ðR0ÞVr

D

T xVT � �R2

0

3�3ðR0ÞVr@t�ðR0Þ : ðA27Þ

Thus, equation (A20) is now changed to

@tVr ¼ ��1 R1

R0

� �2

�1c½urðR1Þ � Vr� þ �R20

� ��1Vr

Z R1

R0

½ D

x ð��uTÞ�r2 dr� �R20

� ��1Z R1

R0

ur½

D

x ð��uT Þ�r2 dr� �

� ��1 4

3�2ðR0Þ�1c

2 þ ��1 1

3�3ðR0Þ�ðR0Þc2 þ

1

3�2ðR0ÞVr

D

T xVT þ 1

3�3ðR0ÞVr@t�ðR0Þ : ðA28Þ

In order to close the final equations, we can only have one free variable for the radial velocities. We use the approximation thatur is constant across region 2 with Vr ¼ urðR1Þ, as appropriate under the thin-shell approximation. Hence, the first two termsin equation (A28) both vanish, and we end up with the following equation:

@tVr ¼ ���1 4

3�2ðR0Þ�1c

2 þ ��1 1

3�3ðR0Þ�ðR0Þc2 þ

1

3�2ðR0ÞVr

D

T xVT þ 1

3�3ðR0ÞVr@t�ðR0Þ : ðA29Þ

Since

@t�ðR0Þ�3ðR0Þ

¼ Vr

c2@tVr ; ðA30Þ

equation (A29) can be rewritten as

@t�ðR0Þ ¼ �2��1�ðR0Þ�1cþ1

2��1�ðR0Þcþ

�ðR0Þ2c

Vr

D

T xVT : ðA31Þ

Because p=�4=3 is a constant across region 2, we obtain the following relation:

�ðR0Þ ¼ 33=4 21=2� � p3=4ðR0Þ�1=41

�1=2ðR0Þc3=2: ðA32Þ

Using this result, we can rewrite equation (A31) as

@t�ðR0Þ ¼ �2��1�ðR0Þ�1cþ33=4

21=2��1 p

3=4ðR0Þ�1=41

�1=2ðR0Þc1=2þ �ðR0Þ

2cVr

D

T xVT : ðA33Þ

842 WANG, LOEB, & WAXMAN Vol. 568

For the tangential velocity, we have

@tVTð�; �Þ ¼ � @t�

�VT � 2

_RR0

R0VT þ ��1 R1

R0

� �2

�ðR1Þ�ðR1ÞuTðR1Þ _RR1 � urðR1Þ �

þ ��1�ðR0Þ�ðR0ÞuTðR0Þ urðR0Þ � _RR0

�� �R2

0

� ��1Z R1

R0

��uruTr

r2 dr� �R20

� ��1Z R1

R0

�c2

4�p

D

TpþuTc2

@p

@t

� �r2 dr

� �R20

� ��1Z R1

R0

��ðuT x

D

uT Þr2 dr� �R20

� ��1Z R1

R0

uT

D

x ð��uTÞr2 dr : ðA34Þ

Apparently, the last two terms in the above equation are nonlinear and can be ignored. By substituting equations (A13) and(A18) into the above equation, we get

@tVT ¼ ��1 R1

R0

� �2

�1c½uTðR1Þ � VT � þ VT

D

T xVT � �R20

� ��1Z R1

R0

��uruTr

r2 dr� �R20

� ��1Z R1

R0

�c2

4�p

D

TpþuTc2

@p

@t

� �r2 dr :

ðA35Þ

The second term on the right-hand side of equation (A35) is nonlinear and can be neglected. In the thin-shell approximation,the third term on the right-hand side of equation (A35) can be approximated as�VrVT=R0. Using the shock jump conditionsat shock 1 and making use of the fact that the tangential velocities must be continuous across the shock front, we obtain

uTðR1Þ ¼ �urðR1Þð

D

TR1Þ : ðA36Þ

Based on these considerations, equation (A35) can be rewritten as

@tVT ¼ ��1 R1

R0

� �2

�1c½�urðR1Þð

D

TR1Þ � VT � �Vr

R0VT � �R2

0

� ��1Z R1

R0

�c2

4�p

D

TpþuTc2

@p

@t

� �r2 dr : ðA37Þ

Next we consider the integration term in the above equation, which includesZ R1

R0

�c2

4�pð D

TpÞr2 dr ¼Z R1

R0

c2

3�ð D

T�Þr2 dr ¼Z R1

R0

c2

3�2½ D

Tð��Þ�r2 dr�Z R1

R0

�c2

3�2ð D

T�Þr2 dr

¼ c2

3�2ðR0Þ

D

T

Z R1

R0

��r2 dr� �ðR1Þ�ðR1ÞR21ð

D

TR1Þ þ �ðR0Þ�ðR0ÞR20ð

D

TR0Þ� �

� c2

D

T�ðR0Þ3�3ðR0Þ

Z R1

R0

��r2 dr

¼ c2

3�2ðR0ÞR2

0

D

T�þ 2�R0ð

D

TR0Þ � �ðR1Þ�ðR1ÞR21ð

D

TR1Þ þ �ðR0Þ�ðR0ÞR20ð

D

TR0Þ �

� c2

3�3ðR0ÞR2

0�

D

T�ðR0Þ : ðA38Þ

In the last pair of square brackets of equation (A38), the second term is much smaller than the fourth term, and so it can beneglected. Another integration term isZ R1

R0

�c2

4�p

uTc2

@p

@tr2 dr ¼

Z R1

R0

1

4�2ð@t ln pÞ��uTr2 dr �

1

4�2ðR0Þ½@t ln pðR1Þ�

Z R1

R0

��uTr2 dr

¼ @t�ðR0Þ2�3ðR0Þ

�R20VT þ 1

4�2ðR0Þ@t�1�1

� ��R2

0VT : ðA39Þ

Because @t ln p does not change much across region 2, we took it out of the integration in the above derivation.By substituting equation (A31) into equation (A39), we getZ R1

R0

�c2

4�p

uTc2

@p

@tr2 dr � � R2

0

�2ðR0Þ�1cVT þ R2

0

4�3ðR0Þ�ðR0ÞcVT þ 1

4�2ðR0Þ@t�1�1

� ��R2

0VT : ðA40Þ

Now, by substituting equations (A38) and (A40) into equation (A37), we get

@tVT ¼ 1

3��1c2 �1 �

�ðR0Þ�ðR0Þ

� �

D

TR0 � ��1�1cVT � Vr

R0VT � ��1 c2

3�2ðR0Þ

D

T�þ c2

3�3ðR0Þ

D

T�ðR0Þ �1

4�2ðR0Þ@t�1�1

� �VT :

ðA41Þ

In deriving the above equation, we made the assumption that surface irregularities due to variations in the thickness of the

No. 2, 2002 SHOCK STABILITY FROM IMPACT OF FIREBALL 843

shock layer are of higher order than irregularities due to the bulk displacement of the shock layer (regions 2 and 3 in Fig. 1), soD

TR0 ¼

D

TR1 ¼

D

TR2. This is appropriate under the thin-shell approximation. Also, the last term in the above equation ismuch smaller than the third term for the wind case and is equal to zero for the ISM case, and so can be neglected. If we nowsubstitute equation (A32) into the above equation, we get equation (9) in x 2.

The derivation of the perturbation equations of region 3 is very similar to that of region 2.

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844 WANG, LOEB, & WAXMAN