J. C. Hidalgo and S. Mendoza- Self–similar imploding relativistic shock waves

8/3/2019 J. C. Hidalgo and S. Mendoza- Selfsimilar imploding relativistic shock waves

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arXiv:astro-ph/0406603v1

27Jun2004

Selfsimilar imploding relativistic shock waves.

J. C. Hidalgo1,2 and S. Mendoza11Instituto de Astronoma, Universidad Nacional Autonoma de Mexico, AP 70-264, Distrito Federal 04510, Mexico

2Current address: Churchill College, Cambridge CB3 ODS, United Kingdom(Dated: June 29, 2004)

Selfsimilar solutions to the problem of a strong imploding relativistic shock wave are calcu-lated. These solutions represent the relativistic generalisation of the Newtonian GouderleyLandau

Stanyukovich problem of a strong imploding spherical shock wave converging to a centre. The so-lutions are found assuming that the preshocked flow has a uniform density, and are accurate forsufficiently large times after the formation of the shock wave.

PACS numbers: 47.75.+f 47.40.Nm

I. INTRODUCTION

When a large quantity of energy is injected on a very small volume over an almost instant interval of time, matterand energy diverge from the deposited volume or detonation region. This phenomenon generates a very high pressureover the surrounding gas, producing an explosion. When the pressure difference between the detonation region and

the matter at rest is sufficiently large, a strong explosion is produced and the discontinuity gives rise to a divergentshock wave from the detonation region. This explosive shock wave has a perfect spherical shape if the detonationvolume is sufficiently small and there are no anisotropies or barriers in the surrounding medium. It was first Sedov[1] who formulated and solved analytically this problem [see also 2]. The problem was also treated by in his PhDdissertation [see 4, for a full description of the general solution] and in less detail by Taylor [5]. Both authors derivedthe corresponding selfsimilar equations and obtained numerical solutions. As shown by Sedov [2], the selfsimilarityindex of the problem is obtained by requiring the energy inside the shock wave to be a fundamental constant parameter.Under these circumstances, the famous SedovTaylor similarity index of 2/5 is obtained [6, 7] for a similarity variable = r/t2/5, where r is the radial distance measured from the point of explosion (the origin) and t represents time.

The inverse physical situation is represented by a strong imploding spherical shock wave that moves through auniform density medium. It was first investigated by Guderley [8] and fully solved by Landau & Stanyukovich [seefor example 4, 6, 7]. In this case, a detailed construction of the selfsimilar solution was found by demanding thesolution to pass through a critical point admitted by the hydrodynamical equations. The similarity variable is then

given by = r/t

with a similarity index = 0.68837 for a monoatomic gas.The first attempt to build similarity solutions for relativistic hydrodynamics was introduced by the pioneering workof Eltgroth [9, 10] with the identification of a selfsimilar variable J v/c, where v is the velocity of the flow andc the speed of light. Soon after that, Blandford and McKee [11] solved the relativistic similarity problem for a verystrong explosive shock wave. On their analysis, they introduced a similarity variable = (1 r/ct) 1 + 2(m + 1)2 ,where represents the Lorentz factor of the explosive shock wave measured in the rest frame of the unshocked gas andm is the similarity index. Their analysis represents a relativistic generalisation of the work made by SedovTaylorStanyukovich.

In this article we construct selfsimilar solutions for the case of a very strong imploding shock wave converging toa centre. This constitutes a relativistic generalisation to the GuderleyLandauStanyukovich problem. In order to doso, as it happens for the nonrelativistic case, we make the solution to pass through an admissible singularity point,which in turn fixes the value of the similarity index of the flow.

The next section describes briefly the main equations for relativistic hydrodynamics on a flat spacetime and brieflymention the main results for shock waves on any system of reference. Later in the article, we write down the self

similar equations behind the flow of a strong relativistic shock wave. We calculate the similarity variable and theselfsimilar index of the problem and solve numerically the equations of motion. Finally, we describe the propertiesof the solution.

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II. SHOCK WAVES IN RELATIVISTIC HYDRODYNAMICS

The equations that govern the motion of a relativistic flow on a flat spacetime are well described by Landau andLifshitz [6]. First, the conservation of energy and momentum are represented by the condition that the divergence ofthe energymomentum tensor T must vanish, i.e.

T

x

= 0, (1)

with

T = wuu pg . (2)In here and in what follows Greek indices have values 0, 1, 2, 3 and Latin ones have values 1, 2, 3. As usual, weuse Einsteins summation convention over repeated indices. The coordinates x are such that x = (ct, r), wherer is the spatial radius vector, t represents the time and c is the velocity of light. For a flat spacetime, the metrictensor g is given by g00 = 1, g11 = g22 = g33 = 1 and g = 0 for = . The fourvelocity u is defined bythe relation u = dx/ds, where the interval ds is given by ds2 = gdxdx . All thermodynamical quantities, inparticular the pressure p, the enthalpy per unit volume w = e +p, the internal energy density e and the temperatureT are all measured in the proper frame of the fluid. The internal energy e constructed in this way contains the rest

mass energy of a particular fluid element and so, it is related to the purely thermodynamical energy density bythe relation

e = c2 + , (3)

where is the proper mass density.In the absence of sources and sinks of particles, the particle number density n satisfies the continuity equation

n

x= 0, (4)

In this equation, n represents the particle number density fourvector defined by n nu.For one dimensional flow, equation (1) can be rewritten as two equations, the equation of conservation of energy

(T0/x = 0) and the equation of conservation of momentum (T1/x = 0) respectively as

1

c

t

e + 2p

1 2

+1

rk

r

rk

p + e

1 2

= 0, (5)

1

c

t

p + e

1 2

+1

rk

r

rk2

p + e

1 2 p

r= 0, (6)

The previous two equations were written in a way to account for a specific symmetry of the problem, so that k =0, 1 and 2 correspond to motions with planar, cylindrical and spherical symmetry respectively.

The relativistic theory of shock waves was first investigated by Taub [12, 13] and is well described by Landau andLifshitz [6]. Traditionally the analysis is made on a system of reference in which the shock wave is at rest. In orderto choose a more general system of reference, McKee and Colgate [ 14] wrote the necessary equations that describe a

shock wave on any inertial system of reference. For the case of an imploding shock wave, it is convenient to choose asystem of reference in which the unshocked gas is at rest. In this particular system of reference the jump conditionsacross a strong shock wave are simple and take the form [ 11, 14]

e

=

1

e1 + p1

s

, (7)

1=

1 +1

1 , (8)


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=

1 2 ( 1) / . (9)

In the previous equations and in what follows, quantities without subindexes and those with subindex 1 label postshocked and preshocked hydrodynamical quantities respectively. The Lorentz factor of the postshocked flow andthat of the shock wave are represented by and respectively. The corresponding values of the velocity measuredin units of the speed of light are and s. On both sides of the discontinuity, we have assumed that the gas obeys a

polytropic relation, that is [4]

p n, (10)where is the polytropic index.

III. SELFSIMILAR EQUATIONS AND BOUNDARY CONDITIONS

Let us consider a strong spherical shock wave converging to a centre. The pre-shocked material is assumed to have aconstant density 1. We analyse the problem in a state such that the radius R of the shock wave is much smaller thanthe initial radius R0 it had when it was produced. Under these circumstances, as it happens for the nonrelativisticcase [4, 6, 7], the flow is largely independent of the specific initial conditions. One can thus imagine a spherical pistonthat pushes the flow to the centre from very large distances in such a way that it produces a strong shock wave. Thatis,

p

n p1

n1. (11)

Using equation (7) and equation (8), the inequality (11) for a polytropic gas implies that

1 1 1

p1c21

1 +

11 1

p1c21

1, (12)

where 1 and are the polytropic indexes for the preshocked and post-shocked gas respectively. The Lorentz factor

for the shocked gas that satisfies the previous inequality is so large that the postshocked flow velocity is comparableto the velocity of light.

In order to simplify the problem, in what follows we assume the gas to be ultrarelativistic, so that

p = ( 1)e = 13

e. (13)

A shocked gas with this equation of state satisfies the following jump velocity condition

2 =1

22. (14)

according to equation (9). From this last relation and inequality (12) it follows that 2

1 for a strong shock wave.

This justifies the restriction of our further analysis to zeroth order quantities in 2

. With this, it follows that thedensity jump condition given by equation (8) takes the form

n = 42n1,

in which n represents the particle number density of the shocked gas as measured with respect to the preshockedgas, i.e. n = n. In terms of the Lorentz factor of the shock wave, this last equation can be written as

n = 22n1. (15)


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On the other hand, using the equation of state (13), the energy jump condition (7) takes the form

e =n

n11 = 2

21.

and so, the pressure jump condition is given by

p =

2

3 2

1, (16)

where 1 = 1c2 for a cold pre-shocked gas and 1 = 4p1 for an ultrarelativistic one. The three conditions givenby (14), (15) and (16) represent postshocked boundary conditions to the problem. These conditions are exactly thesame ones used by Blandford and McKee [11] for the case of a relativistic explosive shock wave.

Let us write the equations of motion for the relativistic flow behind the shock wave. Due to the symmetry of theproblem, let us consider a one dimensional flow with spherical symmetry. With this, the energy conservation equation(5) and the momentum equation (6), take the form

d

d

p4

=

p

, (17)

d

dlnp34

= 4

r2

r r2

. (18)

The third and last equation can be either the continuity equation, or the entropy conservation equation. We selecthere the entropy conservation equation which, for an ultrarelativistic gas is simply given by [6]

d

dt

pn4/3

= 0. (19)

Equations (17), (18) and (19) together with the boundary conditions (14), (15) and (16) completely determine themathematical problem.

IV. SELFSIMILAR SOLUTION

Let us now search for a similarity variable to the problem concerned. In order to do so, let us recall that for thecase of a relativistic strong explosion as presented by Blandford and McKee [11], if R represents the radius of theshock measured from the origin, then the ratio R/2 is a characteristic parameter of the problem. With this ratio,Blandford and McKee [11] built a similarity variable for the problem. Unfortunately, it is not possible for the case of arelativistic strong implosion to follow that approach. The Appendix shows an alternative way in which the similarityvariable obtained by Blandford and McKee [11] can be calculated. The advantage of the approach presented in theAppendix, is that we can generalise the method for the case we are dealing in this article.

The similarity variable for the case of a strong relativistic implosion is found as follows. The radius of the shockwave R converges to the origin as time increases, so that its velocity dR/dt must be negative. Thus,

dR

c dt=

1 1

2

1/2

1 1

22

, (20)

for a Lorentz factor of the shock wave measured from the system of reference of the unshocked gas. In orderto integrate this equation, we must know the time dependence of the Lorentz factor . For the case of a strongrelativistic explosion, this is given by equation (A3). We can use a similar power law time dependence by taking intoconsideration the fact that the shock wave accelerates as it converges to a point. In other words, the Lorentz factor must increase as time advances, that is

2 = A (t)m , (21)for times t 0. The value of the constant factor A is determined by the specific initial conditions of the problemand m is the similarity index. The time interval t 0 has been chosen in complete analogy to the Newtonian case


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[cf. 6, 7]. Under these circumstances, the time t = 0 corresponds to a time in which the shock wave collapses to theorigin, i.e. R = 0.

Substitution of equation (21) on relation (20) yields the integration

R = c |t|

1 12

1

(m + 1)2

. (22)

In exactly the same form in which we choose the similarity variable for the case of a strong relativistic implosion (cf.Appendix), let us choose here the similarity variable as

=1 + 2(m + 1)2 (1 r/R)

=1 + 2(m + 1)2

1 +r

ct

1

(1 1/2(m + 1)2)

.

Thus, to order O(2), this similarity variable takes the form

=

1 +r

ct

1 + 2(m + 1)2

. (23)

For the problem we are concerned, we are interested on the region that lies behind the shock wave, which correspondsto r R. Thus, the variable can take values such that 1 .

In principle, the analysis extends to infinite distances from the origin, where the shock wave was produced. Thismeans that the solution to the problem demands boundary conditions for two extremes. One of the extremes corre-sponds to R = r, for which = 1 (the radius of the shock wave). The other boundary corresponds to the oppositeside r = , where = .

Buckinghams theorem of dimensional analysis [2] motivated by the boundary conditions given by equations (14),(15) and (16), demands the existence of three dimensionless hydrodynamical functions f(), g() and h() given by

p =2

31

2f(), (24)

n =2n12h(), (25)

2

=

1

2 2

g(). (26)

These dimensionless functions satisfy the following boundary condition at the shock wave

f( = 1) = g( = 1) = h( = 1) = 1. (27)

The introduction of the similarity variable means that the hydrodynamical functions change their independentvariables to 2 and , instead of r and t. The derivatives of r and t in terms of these new variables are thus given by

ln t=m

ln 2+

(m + 1)(22 )

, (28)

ct

r =

1 + 2(m + 1)2

. (29)

Using these two equations, the total time derivative d/dt takes the form

d

d ln t= m

ln 2+ (m + 1)

2

g

. (30)

With the aid of equations (24), (25) and (26) we can rewrite equations (17) and (18) in terms of 2 and . In fact,to O(2) we obtain


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(m + 1)(2 g) f

f g+ 4(m + 1)(2 g) g

g2= 3m, (31)

f

f g[(m + 1)(8 2g)] 8(m + 1) g

g2= 2m 8, (32)

where the prime denotes derivative with respect to . These two equations can be rewritten in matrix form as

(2 + g) 2(2 g)8 2g 8

f/f gg/g2

=

3m/(m + 1)

(2m 8)/(m + 1)

, (33)

which implies, according to Cramers rule, that

1

f g

df

d=

8(m 1) + g(4m)(m + 1) [4 + 8g 4g22] , (34)

1

g2dg

d=

4 7m + g(2 + m)(m + 1) [4 + 8g 4g22] . (35)

Equations (34) and (35) are accompanied by the boundary conditions for f() and g() given in equation (27).The unique value for the similarity index m and the uniqueness of the solution is determined by the following

physical consideration, analogous to the one presented in the Newtonian case [cf. 4].For a given time, the values of the energy and velocity for a particular fluid element must decrease as the radius r

increases. For the case of an ultrarelativistic gas, these quantities can be written in terms of the pressure p and theLorentz factor of the flow respectively. A necessary condition that guarantees the decrease of these quantities isgiven by

p

r< 0,

2

r< 0, r R.

With the aid of equations (24), (26) and (29) these inequalities take the form

df()

d> 0,

dg()

d> 0, 1. (36)

The integral functions f() and g() depend on the choice of the parameter m and so, a family of integral curves canbe found for different selfsimilar flows. However, with the restrictions imposed by inequalities (36), it is possible tointegrate the differential equations (34) and (35) in a unique form. This happens because the condition ( 36) imposesa unique value for the similarity index m.

The derivative dg()/d in equation (35) has a positive value at the surface of the shock wave, i.e. when = g = 1.In order to guarantee a positive value for all < 1, this derivative must never vanish. In the same form, the continuityof the function g() means that the denominator on the right hand side of equation ( 35) should not vanish. In orderto avoid these problems, we restrict the integral curve to pass through the point (, g()) for which the numeratorand denominator on equation (35) vanish simultaneously.

The denominator on equation (35) vanishes when the following condition is satisfied

g =

1 2

11

, (37)

which represents a hyperbola on the g plane. On the other hand, the numerator on equation (35) vanishes when

4 7m + g(2 + m) = 0, (38)which also represents a hyperbola on the same plane. The point (, g()) where both hyperbolas intersect iscalculated by direct substitution of the positive value g on equation (38). This implies that the similarity index mmust necessary have the unique value

m = 0.78460969. (39)


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r[AU]

p[Pa]

4000035000300002500020000150001000050000

4

3.5

3

2.5

2

1.5

1

0.5

0

FIG. 1: The figure shows pressure profiles for a strong relativistic imploding shock wave of a gas cloud with initial radiusR0 = 1 pc, a temperature T = 10 K and a particle number density of 100 cm

3. The relativistic shock wave is generated byan external pressure 0.1 Pa. These values are appropriate for cold molecular clouds in the interstellar medium of a typical

galaxy. From left to right, the figure shows the pressure distribution as a function of the radius r for time values of 50, 100 and150 days before the full collapse occurs. The envelope curve represents the pressure values that the shocked gas reaches justbehind the shock wave. The pressure grows without limit as the shock wave moves towards the origin of the cloud.

If we now follow the same procedure for the derivative df()/d given by equation (35), then it follows that thesimilarity index m obtained when the numerator and denominator of equation (35) vanish simultaneously is exactlythe same as the one given by equation (39).

The value of the similarity index given by equation (39) guarantees that pressure and velocity gradients havenegative values for the shocked gas. Using a 4th rank RungeKutta method it is then possible to obtain the integralcurve g() from equation (35). The function f() is then calculated by the same numerical method applied to equation(34).

Let us now obtain a differential equation for the function h(). To do this, we must substitute the differentialequations (34) and (35) in equation (19) in order to obtain

1

gh

dh

d=

2(3m 4) + 2g [6 3m] 2g22 [2m/4](2 g)(m + 1) [4 + 8g 4g22] . (40)

The integral curve h() of this equation is calculated with the already integrated values of g() and the unique valueof m given by equation (39). When the solution h() is obtained, it is observed that the curve does not intersectany singular point of its derivative in the range 1. Furthermore, the derivative dh/d evaluated on = 1 has apositive value. This means that the particle number density n as presented by relation (25) decreases monotonicallyas r increases for fixed values of time t.

Once the solutions for the functions f(), g() and h() are found, it is then possible to calculate the dimensionalhydrodynamical quantities using equations (24), (25) and (26).

We have found solutions to the problem by considering initial conditions of a typical interstellar gas cloud (for futureapplications to astrophysical situations) with a uniform particle number density n1 = 100cm3 and temperature

T1 = 10 K. The imploding shock wave is induced from an external pressure which has an initial value of 0.1Pa. Thesolutions obtained are shown in Figures 1 to 4. The important physical property to distinguish from all figures is thatthe values of the hydrodynamical quantities (pressure, density, Lorentz factor and velocity respectively) increase onthe surface of the shock wave as it converges to the centre.

V. DISCUSSION

The selfsimilar solution found in this article, shows that the values of the hydrodynamical quantities decrease ina given fluid element as it moves behind the shock wave. For example, the variation of the pressure in time as a fluid


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r[AU]

n[cm3]

4000035000300002500020000150001000050000

80000

70000

60000

50000

40000

30000

20000

10000

0

FIG. 2: Particle number density n profiles are shown in the figure as a function of the radial distance r measured in AstronomicalUnits (AU). The original cloud was taken to be an interstellar cold molecular cloud such as the one described in Figure 1. Fromleft to right, the plotted curves show the particle number density distribution for times of 50, 100 and 150 days previous to

the collapse of the shock wave respectively. The envelope curve shows the particle number density for the gas that flows justbehind the shock wave. The particle number density n tends to infinity as the shock wave reaches the origin.

r[AU]

2

4000035000300002500020000150001000050000

200

150

100

50

0

FIG. 3: Different Lorentz factor profiles as function of the radial distance r measured in Astronomical Units (AU) areshown in the figure. The profiles are calculated for the shocked gas behind a strong imploding relativistic shock wave travellingthrough a cold molecular interstellar cloud with the same conditions as the one shown in Figure 1. This plot shows, from leftto right, Lorentz factor values corresponding to times of 50, 100 and 150 days before the collapse of the imploding shock wave.The envelope curve shows the values of the Lorentz factor for particles that have just crossed the strong shock wave. As theplot shows, the Lorentz factor diverges as the shock wave reaches the centre of the cloud.

element moves behind the shock wave is given by


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r[AU]

|

|

4000035000300002500020000150001000050000

1

0.99

0.98

0.97

0.96

0.95

0.94

0.93

0.92

FIG. 4: The figure shows velocity profiles measured in units of the speed of light for a shocked gas behind a strong relativisticimploding shock wave. The shock wave converges to the centre of a typical interstellar cold cloud with properties described inFigure 1. The plotted curves show, from left to right, velocity distribution profiles 50, 100 and 150 days before the shock wave

fully collapses to the origin of coordinates. The absolute value of the velocity decreases behind the shock wave as the radialdistance r (measured in Astronomical Units) increases. The envelope curve represents the velocity of the flow just behind theshock wave as a function of radius.

dp

dt= 1|t|

dp

d ln t

=1

|t|m

ln 2+ (m + 1)

2

g

p,

= 1|t|mp + p (m + 1)(2 g)

1

gf

f

,


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E =814 |t|3 1 + 2(m + 1)21

01

f g

1

1 + 2(m + 1)2

2d,

812 |t|3 (0). (43)In these relations, the quantity (0) is a constant and all quantities were approximated to O(2). Substitution ofequation (21) in equation (43) yields

E = 81A |t|3m (0). (44)This equation, together with the value of the similarity index given by equation ( 39), means that the energy containedinside the shell is proportional to a positive power of time. In other words, the energy inside this selfsimilar shelltends to zero as the imploding shock wave collapses to the origin, which occurs when t 0.

The constant A was introduced in equation (A3) and has not been determined so far. In order to fix the value for theconstant A, let us give a similar argument as the one provided for the nonrelativistic case as presented by Zeldovichand Raizer [7]. The values of A are determined by the specific initial conditions that generate the shock wave. Thus,through a specified value of this constant, different flows behind the imploding shock wave are generated for differentvalues of the initial pressure difference. For the particular case mentioned in this article, we have presented the laststages before the collapse of the implosion of a typical interstellar cloud, which was generated by a very strong externalpressure and where the treatment requires a relativistic analysis. At these stages of the implosion, the flow pattern isnot too sensitive of the specific initial conditions of the problem.

Finally, we mention the regime under which the discussed solution remains valid. This is simply determined by thevalue of the postshocked flow velocity. For sufficiently large distances behind the shock wave, the velocity of the gas in units of the speed of light reduces so that its Lorentz factor 1. In this velocity regime, the approximationsmade on this article are no longer valid and the relativistic selfsimilar solution has no meaning.

VI. ACKNOWLEDGEMENTS

Acknowledgments

J. C. Hidalgo acknowledges financial support from Fundacion UNAM and CONACyT (179026). S. Mendozagratefully acknowledges financial support from CONACyT (41443).

[1] L. Sedov, Prikl. Mat. Mekh. 9, 2 (1946).[2] L. Sedov, Similarity and Dimensional Methods in Mechanics (CRC Press, 1993), 10th ed.[3] K. P. Stanyukovich, Ph.D. thesis, BAuman Institute of Technology, Moscow, U.S.S.R. (1947).[4] K. P. Stanyukovich, Non-steady motion of continuous media (Oxford University Press, Oxford, U.K., 1960).[5] G. I. Taylor, Proc.Roy. Soc. Ser. A 201, 175 (1950).[6] L. Landau and E. Lifshitz, Fluid Mechanics, vol. 6 of Course of Theoretical Physics (Pergamon, 1987), 2nd ed.[7] I. B. Zeldovich and Y. P. Raizer, Physics of Shock Waves and High-Temperature Hydrodynamic Phenomena (Dover, New

York, 2002).[8] G. Guderley, Luftfahrtforschung 19, 302 (1942).[9] P. G. Eltgroth, Phys. Fluids 14, 2631 (1971).

[10] P. G. Eltgroth, Phys. Fluids 15, 2140 (1972).

[11] R. D. Blandford and C. F. McKee, Physics of Fluids19

, 1130 (1976).[12] A. H. Taub, Physical Review 74, 328 (1948).[13] A. H. Taub, Annual Reviews of Fluid Mechanics 10, 301 (1978).[14] C. R. McKee and S. A. Colgate, Astrophys. J. 181, 903 (1973).

APPENDIX A: RELATIVISTIC SIMILARITY INDEX FOR A STRONG EXPLOSION

Let us consider a relativistic strong explosion, diverging from the origin of coordinates. As mentioned in the article,similarity solutions were found by Blandford and McKee [11]. Let us show in here how to obtain their similarityvariable using an alternative argument that can be extended naturally for the case of a strong relativistic implosion.


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The rate of change of the radius R(t) of the shock wave in units of the speed of light is given by

R/c =

1 1

2

1/2, (A1)

where represents the Lorentz factor of the shock wave. A Taylor expansion of the right hand side of this equationto order 2 gives

R = c

1 1

22

. (A2)

Let us consider that 2 is a power law function of time only, so that

2 =A

tm, (A3)

in which A is an unknown constant (but can be found from the specific initial conditions of the problem) and theconstant m represents the similarity index. Substitution of equation (A3) on (A2) yields the integral

R = ct

1 1m + 1

122

. (A4)

The product ct has dimensions of length. Thus, we can write the dimensionless ratio

R

r 1

22 (m + 1),

where we have introduced a new dimensionless quantity . To first order approximation, the inverse of the previousrelation is given by

r

R= 1 +

22

(m + 1),

so that the dimensionless variable is

=

1 rR

2(m + 1)2. (A5)

Note that is such that = 0 when r = R.Selfsimilar motion means that the hydrodynamical functions written in dimensionless form must be powers of the

similarity variable. It is then convenient to take as similarity variable the function + 1, and not , i.e.

= 1 +

1 rR

2(m + 1)2. (A6)

The range of this similarity variable should be taken for 1. The value = 1 corresponds to the shock wave. Asdescribed by Blandford and McKee [11], the hydrodynamical equations simplify to great extent when this variable isused. Substitution of equations (A3), (A4) in (A6) the similarity variable at O(2) is given by

=

1 rc t

1 + 2(m + 1)2

. (A7)

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J. C. Hidalgo and S. Mendoza- Self–similar imploding relativistic shock waves