Energy-conserving, Relativistic Corrections to Strong Shock Propagation
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Energy-conserving, Relativistic Corrections to Strong Shock Propagation
Eric R. Coughlin ∗ Columbia Astrophysics Laboratory, New York, NY 80980
ABSTRACTAstrophysical explosions are accompanied by the propagation of a shock wave through an ambientmedium. Depending on the mass and energy involved in the explosion, the shock velocity V canbe non-relativistic ( V ≪ c , where c is the speed of light), ultra-relativistic ( V ≃ c ), or moderatelyrelativistic ( V ∼ f ew × . c ). While self-similar, energy-conserving solutions to the fluid equationsthat describe the shock propagation are known in the non-relativistic (the Sedov-Taylor blastwave)and ultra-relativistic (the Blandford-McKee blastwave) regimes, the finite speed of light violates scaleinvariance and self-similarity when the flow is only mildly relativistic. By treating relativistic terms asperturbations to the fluid equations, here we derive the O ( V /c ), energy-conserving corrections to thenon-relativistic, Sedov-Taylor solution for the propagation of a strong shock. We show that relativisticterms modify the post-shock fluid velocity, density, pressure, and the shock speed itself, the latterbeing constrained by global energy conservation. We derive these corrections for a range of post-shockadiabatic indices γ (which we set as a fixed number for the post-shock gas) and ambient power-lawindices n , where the density of the ambient medium ρ a into which the shock advances declines withspherical radius r as ρ a ∝ r − n . For Sedov-Taylor blastwaves that terminate in a contact discontinuitywith diverging density, we find that there is no relativistic correction to the Sedov-Taylor solution thatsimultaneously satisfies the fluid equations and conserves energy. These solutions have implicationsfor relativistic supernovae, the transition from ultra- to sub-relativistic velocities in gamma-ray bursts,and other high-energy phenomena. Keywords: gamma-ray burst: general — hydrodynamics — methods: analytical — relativistic pro-cesses — shock waves — supernovae: general INTRODUCTIONA core-collapse supernova is, in the now-classic picture, initiated by the “bounce” of the overpressured, protoneutronstar that forms from the collapse of the iron core of a massive star (Colgate & White 1966). From the protoneutron starbounce is launched a shock wave, which propagates through and unbinds the stellar envelope to yield the supernova.If it is sufficiently energetic, the shock promptly plows through the star following the bounce and liberates the gas;alternatively, if it is not energetic enough to overcome the ram pressure of the infalling material and the dissociationof heavy nuclei in the core of the star (Arnett 1982), the shock “stalls” at small radii but can be revived by somemeans (e.g., neutrino heating, convective instabilities behind the shock, a dynamic instability of the standing shock, orthe magnetorotational amplification of magnetic fields; respectively, e.g., Bethe & Wilson 1985; Burrows et al. 1995;Blondin et al. 2003; M¨osta et al. 2015). If the shock fails to be revived, with sufficient angular momentum an accretiondisc forms around the natal black hole that can, through the combination of bipolar outflows and liberated accretionenergy, unbind the remaining stellar envelope in the collapsar picture of a long gamma-ray burst (Woosley 1993;MacFadyen & Woosley 1999; Woosley & Bloom 2006). Finally, even in the absence of sufficient angular momentum,a failed supernova generates a secondary, weak shock in the outer layers of the star from the mass lost to neutrinosduring the de-leptonization of the core (Nadezhin 1980; Lovegrove & Woosley 2013; Piro 2013; Coughlin et al. 2018a;Fern´andez et al. 2018; Coughlin et al. 2018b). [email protected] ∗ Einstein Fellow
Coughlin, E.R.
All of these explosion scenarios involve the formation and expansion of a shock wave into its surroundings, and thisshock leaves in its wake a “sea” of post-shock fluid (as, of course, do explosion scenarios not initiated by the collapseof a massive star, such as compact object mergers; e.g., Li & Paczy´nski 1998; Levinson et al. 2002; Nakar & Piran2011; Abbott et al. 2017). One of the most useful techniques for describing the spatial and temporal evolution of thepost-shock gas and of the shock itself is self-similarity. This mathematical technique exploits the scale invariance ofthe fluid equations and, in the absence of any temporal or spatial scales of the ambient medium, the necessary scaleinvariance of the solutions to those equations (e.g., Ostriker & McKee 1988).Among the best-known examples of a self-similar solution to the fluid equations is the Sedov-Taylor (ST) blastwave(Sedov 1959; Taylor 1950). The ST blastwave describes the propagation of an energy-conserving, strong (Mach numbermuch greater than one) shock into an ambient medium that possesses a power-law density profile. The conservation ofenergy implies that there is a unique shock speed V that can be directly related to the initial energy of the explosion,the impulsive injection of which initiated the explosion in the first place. The ST solution is also non-relativistic, inthat the shock speed is assumed to be much less than the speed of light and the energy is Newtonian, and hence termsof order V /c that enter into the relativistic fluid equations are ignored ( c being the speed of light). The ST blastwavecan be used to describe terrestrial explosions and can also constrain the age of supernova remnants (Chevalier 1976).In the other, extreme limit of an ultra-relativistic explosion – where the shock velocity is nearly equal to the speedof light – Blandford & McKee (1976) derived a distinct, energy-conserving, self-similar solution to the relativistic fluidequations. The Blandford-McKee (BMK) solution is the ultra-relativistic analog of the Sedov-Taylor blastwave, in thatthe conservation of energy implies that there is a unique shock Lorentz factor Γ = (1 − V /c ) − / that is relatableto the explosion energy. The BMK blastwave is ultra-relativistic in the sense that the solution only accounts forterms in the fluid equations to order O (1 / Γ ). While there are at present no (known) terrestrial applications of thissolution, gamma-ray bursts should exhibit some phase of shock propagation appropriate to ultra-relativistic speeds,and simulations have found evidence of the transition to a Blandford-McKee-type phase of explosion (Kobayashi & Sari2000; Duffell & MacFadyen 2013; Xie et al. 2018).In this paper we are interested in the behavior of an energy-conserving explosion that is between the limits of Newto-nian and ultra-relativistic. The specific question we ask is: when the flow is mildly relativistic, such that the shock speedis V ∼ f ew × . c , how do relativistic effects modify the Sedov-Taylor solution? There are particularly energetic super-novae and lower-luminosity gamma-ray bursts that can produce marginally relativistic shock speeds (Soderberg et al.2006; Drout et al. 2011; Corsi et al. 2014, 2016; Whitesides et al. 2017). For these modestly-relativistic flows, how dorelativistic corrections modify the shock velocity scaling predicted from the ST blastwave, and how are the post-shockvelocity, density, and pressure profiles altered from the self-similar solutions provided by the ST solution?In this paper, we answer these questions by considering special relativistic terms as perturbations to the non-relativistic fluid equations and, correspondingly, the solutions to those equations. In Section 2, we first give some basicconsiderations of the problem, and we derive order of magnitude estimates for the corrections to the shock velocitythat are induced by relativistic motion. In Section 3 we present a rigorous perturbation analysis of the shock jumpconditions and the fluid equations to leading relativistic order, and from those equations we derive relativistic, non-self-similar corrections to the post-shock velocity, density, and pressure that result from the velocity scale establishedby the finite speed of light. We also demonstrate that there is a unique, relativistic correction to the shock velocitythat results from the requirement that the energy – which includes relativistic terms – be exactly conserved behindthe shock front. We summarize and conclude in Section 5. GENERAL CONSIDERATIONS AND ORDER OF MAGNITUDE ESTIMATESWe characterize an explosion by the ejection of an amount of mass M ej with a corresponding energy E ej , which can becombined to yield a characteristic velocity V ej = p E ej /M ej . As this material encounters an ambient medium medium,a forward shock is generated that initially propagates at the same characteristic speed V ej . Once the shock entrainssufficient inertia from its surroundings that the initial mass of the explosion is forgotten, the forward shock propagationsettles into a self-similar state such that the boundary conditions at the shock (namely the jump conditions; see Section3.3) govern the entire post-shock evolution of the flow, while the characteristic ambient density ρ a and length scale r a and the shock energy E ej dictate the propagation of the shock itself. If the density of the ambient medium falls nergy-conserving, Relativistic Corrections to Strong Shock Propagation n , then, during this self-similar phase, energy conservation in thenon-relativistic limit implies that the shock velocity V is related to the shock position R via V ≃ s E ej ρ a r (cid:18) Rr a (cid:19) n − ≃ V ej (cid:18) Rr a (cid:19) n − . (1)The last line in this expression follows from the fact that the self-similar state is only reached once the mass entrainedfrom the ambient medium is comparable to the initial mass of the explosion, i.e., ρ a r ≃ M ej . This expression can berearranged and integrated to solve for the shock position as a function of time, which yields R ∝ t − n . (2)This temporal scaling of the shock position is the well-known, Newtonian-energy-conserving result derived indepen-dently by Sedov (1959) and Taylor (1950)We see from Equation (1) that when the mass involved in the explosion is small or the energy imparted to thatmass is large, both of which are possible in astrophysical contexts, the characteristic ejecta velocity can exceed thespeed of light. In this limit the Newtonian approach breaks down, and instead of being constrained by a characteristicvelocity, the shock can be parameterized by its Lorentz factor Γ = (1 − V /c ) − / . In the ultra-relativistic limit, thepost-shock inertia is dominated by the internal energy of the gas, which scales as ρ a Γ , and mass conservation impliesthat the post-shock gas is swept into a thin shell of width ∼ R/ Γ (Blandford & McKee 1976). To leading order inthe shock Lorentz factor, relativistic energy conservation then dictates that the Lorentz factor of the shock satisfiesΓ ≃ s E ej ρ a r c (cid:18) Rr a (cid:19) ( n − / ≃ V ej c (cid:18) Rr a (cid:19) ( n − / . (3)In this expression R ≃ c t , and it therefore follows that the Lorentz factor of the relativistic, self-similar shock evolvestemporally as Γ ≃ V ej c (cid:18) ctr a (cid:19) ( n − / . (4)In between these two limits – Sedov-Taylor when p E ej /M ej ≪ c and Blandford-McKee when p E ej /M ej ≫ c – howdoes the flow behave? In addition to the examples of hyperenergetic supernovae considered in Section 1, it can also beseen from Equation (4) that – even if the blastwave is extremely energetic and begins in the Blandford-McKee regime– the deceleration of the shock Lorentz factor (assuming n <
3, above which the Sedov-Taylor solution does not exist;we will always adopt n < n > t rel . Setting Γ = 1 in Equation (4) and rearranging showsthat this timescale is ∆ t rel ≃ r a c V − n ej ≃ r a c (cid:18) E ej M ej c (cid:19) − n . (5)For fiducial values of E ej ≃ erg, M ej ≃ . M ⊙ , n = 0, and setting r a ≃ N × R ⊙ , we find ∆ t rel ≃ × N sec.Therefore, even for large values of r a (or correspondingly small values of the ambient density), the ultra-relativisticphase of self-similar shock propagation can be very short lived. Relativistic terms that modify the solution fromthe non-relativistic, Sedov-Taylor phase – to which the flow eventually asymptotes – will then be present during thetransition from ultra- to non-relativistic shock expansion.At the order of magnitude level, the importance of relativistic corrections to the Sedov-Taylor blastwave can beunderstood by noting that the leading-order, relativistic modifications to the fluid equations appear as ∝ O ( v /c ),where v is the three-velocity of the fluid; we derive these corrections explicitly in Section 3, but such a scalingis reasonable from the observation that the four-velocity, which differs from the three velocity by a factor of Γ ≃ O ( v /c ), transforms in a covariant (i.e., tensor-like) sense and therefore enters manifestly into the relativistic fluidequations. There will therefore be corrections of this same order to the jump conditions at the shock front, and hencethe lowest-order, relativistic corrections to the post-shock velocity, density, and pressure will be of the order V /c .There will also be modifications to the conserved energy that enter as v /c (again, we show this explicitly below, Coughlin, E.R. but this feature follows naturally from the covariant nature of the four-velocity over the three-velocity). Therefore,in order to satisfy energy conservation, there must also be relativistic corrections of the order V /c that modify thepropagation of the shock itself. We thus expect that during the marginally-relativistic phase when V ej /c .
1, the shockspeed will be characterized by V ≃ V (cid:18) Rr a (cid:19) n − (cid:18) σ V c (cid:19) , (6)where σ is an unknown but otherwise pure number. Since the shock is assumed to be only mildly relativistic, it followsthat this relativistic term can be approximated as V c ≃ V c (cid:18) Rr a (cid:19) n − ∝ t n − − n , (7)where the last line follows from the expression for R ( t ) from the Sedov-Taylor solution (Equation 2). Thus, for n = 0,the relativistic corrections fall off as ∝ t − / , and become – as one would anticipate – less important as the shockdecelerates owing to the entrainment of mass from the ambient medium. Note, however, that this power-law decline inthe importance of relativistic effects is shallower than that predicted from the Blandford-McKee solution alone, beingΓ ∝ t − / for n = 0 (Equation 4); relativistic corrections to non-relativistic shock propagation can therefore be longerlived than might be anticipated by extrapolating the Blandford-McKee solution to the limit of Γ = 1. Furthermore,as the density of the ambient medium falls off more steeply, relativistic effects remain important for longer periods oftime (indeed, in the limit that n →
3, the relativistic corrections are permanent modifications to the shock speed).While these order of magnitude estimates provide useful diagnostics for probing the importance of relativistic effectsand the rate at which they should depreciate in time, they cannot be used, for example, to determine the constant σ thatenters into the correction for the shock velocity in Equation (6). Furthermore, while these simple estimates indicatethat the corrections to the post-shock fluid variables (the velocity, density, and pressure) enter at the level V /c , theytell us nothing about the spatial dependence of these corrections. To understand these aspects of the problem, we nowturn to a quantitative analysis of the relativistic fluid equations and treat the leading-order, relativistic corrections inthe perturbative limit. FLUID EQUATIONS AND PERTURBATIVE APPROACH3.1.
General continuity equations
The equations of hydrodynamics in covariant and differential form are given by ∇ µ T µν = 0 , (8)where ∇ µ is the covariant derivative and T µν = (cid:18) ρ ′ + γγ − p ′ (cid:19) U µ U ν + p ′ g µν (9)is the energy-momentum tensor of the fluid; here primes denote quantities measured in the comoving frame (i.e., inthe frame where the fluid is instantaneously at rest), ρ ′ is the fluid density, p ′ is the gas pressure, U µ is the fluidfour-velocity, and g µν is the inverse of the metric, and we also adopted an adiabatic equation of state such that theinternal energy e ′ is related to the pressure p ′ via e ′ = p ′ / ( γ − ∇ µ [ ρ ′ U µ ] = 0 , (10)and to maintain the invariance of the line element the contraction of the four velocity with itself must be conserved: U µ U µ = − . (11) nergy-conserving, Relativistic Corrections to Strong Shock Propagation U ν and Π βν = U β U ν + g βν , which respectively select out the time-like andspace-like components of the equations (note that U β Π βν = 0). Doing so and performing some simple manipulationsgives the entropy equation U µ ∇ µ s ′ = 0 , (12)where s ′ = ln [ p ′ / ( ρ ′ ) γ ] is the entropy, and the momentum equations ρ ′ U µ ∇ µ U ν + γγ − p ′ U µ ∇ µ U ν + Π µν ∇ µ p ′ = 0 . (13)3.2. Equations in spherical symmetry
Here we restrict solutions to the fluid equations to be spherically symmetric and irrotational, such that the metric isgiven by g µν = diag (cid:8) − , , r , r sin θ (cid:9) , where r is radial distance from the origin and θ is the standard polar angle inspherical coordinates, and the only non-vanishing components of the four-velocity are U r ≡ U and U t . From Equation(10) we can relate the time component of the four-velocity to the radial component via U t = p U , (14)and using this expression the continuity, entropy, and radial momentum equations respectively become ∂∂t h ρ ′ p U i + 1 r ∂∂r (cid:2) r ρ ′ U (cid:3) = 0 , (15) p U ∂s ′ ∂t + U ∂s ′ ∂r = 0 , (16) p U ∂U∂t + U ∂U∂r + p ′ Uρ ′ (cid:26)p U ∂∂t ln (cid:16) p ′ U γγ − (cid:17) + U ∂∂r ln (cid:16) p ′ U γγ − (cid:17)(cid:27) + 1 ρ ′ ∂p ′ ∂r = 0 . (17)Two other equations that will be useful in the following sections are the total energy and total radial momentumequations, which are given by the ν = t and ν = r components of Equation (8). These respectively read ∂∂t (cid:20)(cid:18) ρ ′ + γγ − p ′ (cid:19) (cid:0) U (cid:1) − p ′ (cid:21) + 1 r ∂∂r (cid:20) r (cid:18) ρ ′ + γγ − p ′ (cid:19) U p U (cid:21) = 0 , (18) ∂∂t (cid:20)(cid:18) ρ ′ + γγ − p ′ (cid:19) U p U (cid:21) + 1 r ∂∂r (cid:20) r (cid:18)(cid:18) ρ ′ + γγ − p ′ (cid:19) U + p ′ (cid:19)(cid:21) − p ′ r = 0 . (19)3.3. Strong shock jump conditions
Equations (15) – (17) govern the evolution of the post-shock fluid. Assuming that the shock generates neither mass,energy, nor momentum, the fluxes of these quantities must be conserved across the shock in the comoving frame of theshock itself. From Equations (15), (18), and (19), conserving these fluxes yields the following three jump conditions: ρ ′ U ′′ = ρ ′ a U ′′ a , (20) (cid:18) ρ ′ + γγ − p ′ (cid:19) U ′′ q U ′′ ) = (cid:18) ρ ′ a + γγ − p ′ a (cid:19) U ′′ a q U ′′ a ) (21) (cid:18) ρ ′ + γγ − p ′ (cid:19) ( U ′′ ) + p ′ = (cid:18) ρ ′ a + γγ − p ′ a (cid:19) ( U ′′ a ) + p ′ a . (22)Here U ′′ denotes the radial component of the four-velocity in the comoving frame of the shock, quantities with asubscript 2 are post-shock fluid quantities, and those with a subscript “a” pertain to the ambient medium. We will Coughlin, E.R. assume that the shock is sufficiently supersonic that the ambient pressure can be ignored, which is equivalent to statingthat the energy behind the blastwave is much greater than the internal energy of the ambient medium. In this case,these three equations can be combined into the following cubic to be solved for U ′′ in terms of U ′′ a : γ (cid:16) U ′′ ) (cid:17) ( U ′′ − U ′′ a ) + 2 γU ′′ a (cid:16) U ′′ ) (cid:17) − U ′′ − U ′′ a = 0 . (23)Of the three algebraic solutions to this equation, only one is purely real for all U ′′ a and hence physical. If we denotethe radial component of the lab-frame four-velocity of the shock by U s , then from Lorentz transformations it followsthat U ′′ a = − U s , while the radial component of the lab-frame post-shock velocity U is given by U = U s q U ′′ ) + U ′′ p U . (24)With the solution for the comoving, post-shock fluid velocity in terms of the lab-frame velocity of the shock fromEquation (23), Equation (24) yields the lab-frame post-shock fluid velocity for arbitrary shock speeds. In the nonrela-tivistic limit where U s ≪
1, the full expression reduces to U /U s = 2 / ( γ + 1), while in the ultra-relativistic limit where1 + U ≃ U it becomes U /U s = p /γ −
1. Figure 1 shows the exact solution for the ratio U /U s (purple, solid)when γ = 4 /
3, the non-relativistic limit of 6/7 (black, dashed), and the ultra-relativistic limit of 1 / √ U s U / U s ,p(cid:0) (cid:1) / U s (cid:2) / / / / U s U / U s U s ρ ’ (cid:3) / + U s (cid:4) Figure 1.
Left: The exact solution for the ratio of the post-shock, lab-frame velocity to the lab-frame shock four-velocity(purple), and the exact solution for the comoving pressure normalized by the square of the lab-frame shock velocity (green)as functions of the lab-frame shock four-velocity when γ = 4 /
3. The non-relativistic limit of both the velocity and pressureof 6 / ≃ .
86 is shown by the dashed, horizontal line, the ultra-relativistic limit of the velocity of U s / √ ≃ .
71 is given bythe dotted, horizontal line, and the ultra-relativistic limit of the pressure of 2 U / γ = 4 /
3. The non-relativistic limit of 7 is shown by the horizontal, dashed line, andthe ultra-relativistic limit of 2 √ The post-shock comoving density is simply found by inverting Equation (20) ρ ′ = − U s U ′′ ρ ′ a , (25)while the pressure is obtained by rearranging Equation (22) and using Equation (25) to remove the dependence on ρ ′ : p ′ = − γ − γ U s U ′′ p U q U ′′ ) − ρ ′ a . (26)As for the post-shock velocity, these jump conditions reduce to the known non-relativistic and ultra-relativistic limitswhen U s ≪ U s ≫
1. The solid, green curve in the left panel of Figure 1 shows the full solution for the post-shockpressure normalized by U , and the horizontal, dot-dashed line gives the ultra-relativistic limit of 2 − γ = 2 / γ = 4 / p ′ /U = 2 / ( γ + 1) = 6 / nergy-conserving, Relativistic Corrections to Strong Shock Propagation γ = 4 /
3, the dashed, horizontal line shows the non-relativistic limit of ( γ + 1) / ( γ −
1) = 7, and thedotted line depicts the ultra-relativistic limit of p γ − γ / ( γ −
1) = 2 √ U s /c . f ew (recall that U s is the radial component of the four-velocity,not the three-velocity). In this case, we can Taylor expand our expressions for the post-shock fluid quantities to thelowest, non-zero order in U s beyond the non-relativistic limit. Doing so, we find that Equations (24) – (26) become U ( R ) = 2 γ + 1 − γ − γ + 1( γ + 1) U ! U s , (27) ρ ′ ( R ) = γ + 1 γ − γ ( γ + 1) U ! ρ ′ a ( R ) , (28) p ′ ( R ) = 2 γ + 1 − γ ( γ − γ + 1) U ! ρ ′ a ( R ) U , (29)where for clarity we included in these relations the fact that they hold at the location of the shock position, R .In agreement with our arguments in Section 2, these expressions contain modifications to the non-relativistic jumpconditions at the order U /c . 3.4. Relativistic, Conserved energy
To initiate the outward motion of the fluid and the formation of the strong shock, there is an assumed-impulsiveinjection of a large amount of energy E ej (large relative to the ambient internal energy and any gravitational potentialenergy) into the ambient medium, which for typical supernovae is on the order of E ≃ − erg (though for failedand very weak supernovae, the shock energy is E ej ≃ − erg or less). Owing to the fact that the ambient mediumis assumed to be pressureless, there is no source of thermal energy as the shock moves out. Therefore, under theassumption that there are no sources or sinks of energy interior to the flow (or that, if there is a compact object suchas a black hole or neutron star accreting matter, the binding energy drained by the compact object is small comparedto the initial energy), this initial energy must be conserved as the shock propagates outward.It is tempting to relate this conserved energy to the integral of the energy density over the volume enclosed by theshock, i.e., E tot = 4 π Z RR c T r dr = 4 π Z RR c (cid:18)(cid:18) ρ ′ + γγ − p ′ (cid:19) (cid:0) U (cid:1) − p ′ (cid:19) r dr. (30)(The lower bound on this integral, R c , can be non-zero for certain combinations of the ambient density profile and theadiabatic index of the post-shock flow, where the solution terminates in a contact discontinuity; see Section 4 below.)However, this energy is not conserved , because the shock sweeps up rest mass energy from the ambient medium as itmoves outward. Indeed, this additional source of energy can be seen directly by integrating Equation (18) over thevolume enclosed by the shock and rearranging: ∂E tot ∂t = ρ ′ a R V. (31)The right-hand side is the change in the total energy budget due to the addition of rest mass energy from the ambientmedium. Thus, the total energy E tot is not equivalent to the initial, injected energy associated with the explosion.However, we see from integrating the continuity equation from R c to R ( t ) that ∂M∂t = ρ ′ a R V, (32)where M = 4 π Z RR c ρ ′ p U r dr. (33) Coughlin, E.R.
Using this expression for the right-hand side of Equation (31) and rearranging then gives ∂E ej ∂t = 0 , (34)where E ej = 4 π Z RR c (cid:18)(cid:18) ρ ′ + γγ − p ′ (cid:19) (cid:0) U (cid:1) − p ′ − ρ ′ p U (cid:19) r dr. (35)This energy is manifestly conserved as the shock advances into the ambient medium, and is the exact, relativisticanalog of the energy behind the blast that is relatable to the initial energy of the explosion.Including lowest-order, relativistic corrections to this integral for the conserved energy gives E ej = 4 π Z RR c (cid:18) ρ ′ U + 1 γ − p ′ (cid:19) r dr + 4 π Z RR c (cid:18) γγ − p ′ U + 18 ρ ′ U (cid:19) r dr. (36)The first term is the familiar, Newtonian expression for the total (kinetic plus internal) energy behind the blastwave, andthe Sedov-Taylor solution exactly conserves this quantity. The second term is the lowest-order relativistic correction,and scales as ∝ U (recall that the pressure behind the shock, by virtue of the shock jump conditions, is comparableto the ram pressure of the shock; specifically see Equation 29). These relativistic corrections to the energy imply that,if the energy is to be absolutely conserved, there must also be corrections to the shock velocity that account for theseadditional terms. 3.5. Self-similar equations and perturbed solutions to lowest relativistic order
When U ≪
1, Equations (15) – (17) reduce to the well-known Euler equations in spherical symmetry. Here, however,we are interested in the corrections to these equations that are induced by mildly relativistic flow, and hence we needto account explicitly for these corrections. The Sedov-Taylor blastwave also adopts the change of variables r → ξ = rR , (37)which removes the time dependence of the boundary conditions at the shock front and yields self-similar solutions ofthe form U = U s f ( ξ ) , ρ ′ = ρ ′ a (cid:18) Rr a (cid:19) − n g ( ξ ) , p = ρ ′ a (cid:18) Rr a (cid:19) − n U h ( ξ ) . (38)It is also useful to introduce the time-like variable χ = ln (cid:18) Rr a (cid:19) , (39)so the temporal and spatial derivatives then transform as ∂∂t = 1 R dRdt (cid:18) ∂∂χ − ξ ∂∂ξ (cid:19) = 1 R U s p U (cid:18) ∂∂χ − ξ ∂∂ξ (cid:19) , ∂∂r = 1 R ∂∂ξ , (40)where we used the fact that the shock three-velocity is related to the four-velocity via dR/dt = U s / p U . Thecontinuity, entropy, and radial momentum equations then become U s p U (cid:18) ∂∂χ h ρ ′ p U i − ξ ∂∂ξ h ρ ′ p U i(cid:19) + 1 ξ ∂∂ξ (cid:2) ξ ρ ′ U (cid:3) = 0 , (41) U s √ U p U (cid:18) ∂s ′ ∂χ − ξ ∂s ′ ∂ξ (cid:19) + U ∂s ′ ∂ξ = 0 , (42) nergy-conserving, Relativistic Corrections to Strong Shock Propagation U s √ U p U (cid:18) ∂U∂χ − ξ ∂U∂ξ (cid:19) + U ∂U∂ξ + p ′ Uρ ′ U s √ U p U (cid:18) ∂∂χ ln (cid:16) p ′ U γγ − (cid:17) − ξ ∂∂ξ ln (cid:16) p ′ U γγ − (cid:17)(cid:19) + U ∂∂ξ ln (cid:16) p ′ U γγ − (cid:17)! + 1 ρ ′ ∂p ′ ∂ξ = 0 . (43)We will now expand these equations to lowest, post-Newtonian order. Before doing so, however, we note that thereare exact, self-similar solutions (being the Sedov-Taylor solutions) in the Newtonian case, and hence we expect thatthe velocity, density, and pressure of the fluid should be approximately given by these solutions but with perturbationsthat are introduced from the velocity scale set by the speed of light. Moreover, investigating the shock jump conditions(27) – (29), we see that relativistic terms modify the post-shock fluid quantities at the order U . We therefore seeksolutions for the velocity, comoving density, and pressure that are of the form U = U s (cid:8) f ( ξ ) + U f ( ξ ) (cid:9) , (44) ρ ′ = ρ ′ a (cid:18) Rr a (cid:19) − n (cid:8) g ( ξ ) + U g ( ξ ) (cid:9) , (45) p ′ = ρ ′ a (cid:18) Rr a (cid:19) − n U (cid:0) h ( ξ ) + U h ( ξ ) (cid:9) . (46)From the shock jump conditions, the functions satisfy the following boundary conditions at the shock: f (1) = h (1) = 2 γ + 1 , g (1) = γ + 1 γ − , (47) f (1) = − γ − γ + 1 γ + 1( γ + 1) , g (1) = 2 γ ( γ −
1) ( γ + 1) , h (1) = − γ ( γ − γ + 1) . (48)In addition to satisfying these boundary conditions, energy must also be globally conserved. Returning to Equation(36) and using these forms for the velocity, density, and pressure, the conserved energy is given by E = 4 πρ ′ a r (cid:18) Rr a (cid:19) − n U Z ξ c (cid:18) g f + 1 γ − h (cid:19) ξ dξ + 4 πρ ′ a r (cid:18) Rr a (cid:19) n − U × U Z ξ c (cid:18) g f + g f f + 1 γ − h + γγ − h f + 18 g f (cid:19) ξ dξ. (49)If the second term in this expression were absent, then the energy would be conserved if4 πρ ′ a r (cid:18) Rr a (cid:19) − n U = E ∗ , (50)where E ∗ is a constant that scales with the total energy, and this is just the familiar velocity-radius relation for theSedov-Taylor blastwave. However, because the second, relativistic term modifies the energy, this scaling cannot holdexactly , as otherwise the relativistic corrections would violate energy conservation. There must therefore be relativisticcorrections to the shock velocity, and energy conservation dictates that these corrections must be of the form4 πρ ′ a r (cid:18) Rr a (cid:19) − n U = E ∗ (cid:0) σU (cid:1) , (51)where σ is a dimensionless number. Inserting this expression into the above equation for the total energy and requiringthat the relativistic terms cancel exactly yields σ Z ξ c (cid:18) g f + 1 γ − h (cid:19) ξ dξ + Z ξ c (cid:18) g f + g f f + 1 γ − h + γγ − h f + 18 g f (cid:19) ξ dξ = 0 . (52)0 Coughlin, E.R.
For a given σ , the functions f , g , and h are completely specified from the fluid equations. This fourth boundarycondition, which enforces global energy conservation, will therefore only be satisfied for a certain value of σ (for given n and γ ), making σ an “eigenvalue” from the standpoint that it is uniquely determined by this additional, integralconstraint.The governing equations for the self-similar functions can now be obtained by inserting Equations (44) – (46) and(51) into Equations (41) – (43) and keeping the lowest-order, surviving terms in U s . Doing so and performing somealgebra yields the following three equations for the non-relativistic quantities: − ng − ξ dg dξ + 1 ξ ddξ (cid:0) ξ g f (cid:1) = 0 , (53)12 ( n − f + ( f − ξ ) f ′ + 1 g dh dξ = 0 , (54) − nγ + ( f − ξ ) ddξ ln (cid:18) h g γ (cid:19) = 0 . (55)These three equations can be integrated three times exactly, with the pressure being analytically related to the velocityand density as h = γ − ξ − f γf − ξ g f . (56)The equations for the perturbed functions are given by − g − ξ ∂g ∂ξ + 1 ξ ∂∂ξ (cid:2) ξ ( g f + f g ) (cid:3) = 32 g f + 12 ξ ∂∂ξ (cid:2) g f (cid:3) −
12 1 ξ ∂∂ξ (cid:2) ξ f g (cid:3) , (57)( n − (cid:18) h h − γ g g (cid:19) + ( f − ξ ) ∂∂ξ (cid:20) h h − γ g g (cid:21) + f ∂s ∂ξ = − σ ( n − − f (cid:0) − f (cid:1) ∂s ∂ξ , (58)32 ( n − f − ξ ∂f ∂ξ + ∂∂ξ [ f f ] + 1 g (cid:18) ∂h ∂ξ − g g ∂h ∂ξ (cid:19) = − σ n − f − (cid:0) f − (cid:1) (cid:18)
12 ( n − f − ξ ∂f ∂ξ (cid:19) − γγ − f h g (cid:18) n −
92 + 3 γ + ( f − ξ ) ∂∂ξ ln (cid:20) f h γ − γ (cid:21)(cid:19) . (59)Equation (57) can be integrated, and using the boundary conditions at the shock front gives the following expressionfor the relativistic correction to the density: g = f ( f ξ − − f f − ξ g . (60)The integral constraint that determines the eigenvalue, Equation (52), can also be written as a fourth boundarycondition on the functions f , g , and h : subtracting the continuity from the energy equation and integrating from R c ( t ) to R ( t ) gives ∂E ej ∂t = R U ρ ′ a (cid:18) Rr a (cid:19) − n U ξ × (cid:26) (cid:18) g f ( f − ξ c ) + γγ − h (cid:19) f + γf − ξ c γ − h − (cid:18) g f + 1 (cid:19) ( f − ξ c ) g f + (cid:18) γ (cid:18) f − ξ c (cid:19) f + 12 ξ c (cid:19) h (cid:27) , (61)where we used Equation (60) to remove the dependence on g and all of the functions are evaluated at ξ c . By virtue ofEquation (56), this expression only contains a relativistic correction, and for energy-conserving solutions we thereforerequire that the term in braces (multiplied by ξ ) be equal to zero. Either this fourth boundary condition or theintegral constraint (52) can be used to determine σ . nergy-conserving, Relativistic Corrections to Strong Shock Propagation Shock position and unperturbed coordinates
Equation (51) can be rearranged and integrated numerically to yield the shock position as a function of time.However, we can also use the assumed-smallness of terms of order U /c to decompose the shock position and velocityinto their non-relativistic and relativistically-corrected parts; denoting the non-relativistic shock position and three-velocity as R and V = dR /dt and their relativistically-perturbed counterparts as R and V , we find4 πρ ′ a r (cid:18) R r a (cid:19) − n V (cid:18) − n ) R R (cid:19) (cid:18) V V (cid:19) (cid:18) V c (cid:19) = E ∗ (cid:18) σ V c (cid:19) , (62)where we introduced factors of c for clarity. This expression demonstrates, as expected, that the unperturbed shockvelocity and position are related via the standard, energy-conserving prescription for the Sedov-Taylor blastwave:4 πr ρ a (cid:18) R r a (cid:19) − n V = E ∗ ⇒ R r a = (cid:18) − n V i r a t (cid:19) − n , (63)where we defined V = E ∗ / (4 πρ a r ) as the unperturbed velocity of the shock when the shock position coincides with R = r a , while the relativistic corrections to the shock position and velocity satisfy V V + 3 − n R R = σ − V c . (64)This equation can be integrated to yield, if n = 1, R R = σ − n − V c (cid:18) R r a (cid:19) n − − (cid:18) R r a (cid:19) n − ! , (65) V V = σ − n − V c ( n − (cid:18) R r a (cid:19) n − − n − (cid:18) R r a (cid:19) n − ! , (66)while if n = 1 R R = σ − V c (cid:18) R r a (cid:19) − ln (cid:20) R r a (cid:21) , (67) V V = σ − V c (cid:18) R r a (cid:19) − (cid:18) − ln (cid:20) R r a (cid:21)(cid:19) . (68)The second term in parentheses in Equations (65) and (66) is a consequence of initial conditions, and arises from thefact that scale invariance allows us to define the relativistic corrections to the shock position to be zero at t = 0.Interestingly, if σ ≡
1, then the relativistic corrections to the shock velocity and position are exactly zero. This effectarises from a competition between the increase in the four-velocity generated by positive σ , and time dilation thatreduces the three-velocity from the four-velocity – when σ = 1 these two effects exactly balance to yield no relativisticcorrection to the shock velocity.Following Coughlin et al. (2019), we wrote our solutions for the relativistic corrections to the velocity, density, andpressure of the post-shock fluid in terms of the true shock position and velocity and the total self-similar variable ξ = r/R ( t ). While formally correct to order U /c , these expressions (specifically Equations 44 – 46) also containterms that are of a higher order than U /c . We can remove these additional terms by rewriting the solutions in termsof the “unperturbed” self-similar variable ξ = r/R ( t ) and using Equations (65) and (66) to write the correctionsto the shock position and velocity in terms of their non-relativistic counterparts; the resulting expressions, which areidentical to Equations (44) – (46) to order U /c , are U ( ξ , t ) = V (cid:18) f ( ξ ) + (cid:18) V V + 12 V (cid:19) f ( ξ ) − ξ f ′ ( ξ ) R R + V f ( ξ ) (cid:19) ≡ V { f ( ξ ) + f ∗ ( ξ , t ) } , (69) ρ ′ ( ξ , t ) = ρ ′ a (cid:18) R r a (cid:19) − n (cid:18) g ( ξ ) − ( ng + ξ g ′ ) R R + V g ( ξ ) (cid:19) ≡ ρ ′ a (cid:18) R r a (cid:19) − n { g ( ξ ) + g ∗ ( ξ , t ) } , (70)2 Coughlin, E.R. p ′ ( ξ , t ) = ρ ′ a (cid:18) R r a (cid:19) − n V (cid:18) h ( ξ ) − ( nh + ξh ′ ) R R + 2 (cid:18) V V + 12 V (cid:19) h ( ξ ) + V h ( ξ ) (cid:19) ≡ ρ ′ a (cid:18) R r a (cid:19) − n V { h ( ξ ) + h ∗ ( ξ , t ) } . (71)Finally, the three-velocity of the fluid is v = U √ U ≃ V (cid:26) f ( ξ ) + f ∗ ( ξ , t ) − V f ( ξ ) (cid:27) , (72)and the lab-frame density is given by ρ = ρ ′ p U ≃ ρ ′ a (cid:18) R r a (cid:19) − n (cid:26) g ( ξ ) + g ∗ ( ξ , t ) + 12 V f ( ξ ) g ( ξ ) (cid:27) . (73) SOLUTIONSHere we present the numerical solutions for the functions f , g , h , and the eigenvalue σ that satisfy the differentialEquations (58) and (59), with Equation (60) relating g to f , and the fourth, energy-conserving boundary condition,given either by the integral constraint (52) or Equation (61). Since the functions f , g , and h satisfy the boundaryconditions at the shock front, given by Equation (48), we can numerically integrate Equations (58) and (59) from ξ = 1for a given, initial guess for σ . We then perturb the guess for σ and calculate the change in the energy residual, i.e., wedetermine how well the new value of σ satisfies the fourth boundary condition given by Equation (52) or (61), whichmotivates the choice for the next σ that will better satisfy the fourth boundary condition. In this way, we iterativelydetermine the eigenvalue σ that globally conserves the energy behind the blastwave. ξ = r / R ( t ) f , g , h h g / g ( ) f - - - - = r / R ( t ) f , g , h h g / g ( ) f Figure 2.
Left: The Sedov-Taylor, self-similar velocity (blue), which is the four-velocity of the fluid normalized by the shockspeed; the density (orange), which is the comoving density normalized by the ambient density; and the pressure (red), which isthe gas pressure normalized by the ram pressure of the shock. Here we set γ = 4 / n = 0, such that the post-shock gas isradiation-pressure dominated and the ambient medium has a constant density. These fluid variables are plotted as functions ofthe self-similar variable ξ , which is just the spherical radius r normalized by the shock position at a given time. The pressure isalmost exactly constant, the velocity is effectively linear, and the density falls off extremely rapidly near the origin. Right: Theself-similar, relativistic correction to the fluid four-velocity (dark blue), comoving density (dark orange), and gas pressure (darkred) when γ = 4 / n = 0. These solutions satisfy global energy conservation and the jump conditions at the shock, andthe eigenvalue that ensures energy conservation is σ ≃ . The left panel of Figure 2 shows the self-similar velocity, density, and pressure for the Sedov-Taylor blastwave when γ = 4 / n = 0, corresponding to a radiation-pressure dominated, post-shock fluid and a constant ambient density.The linear velocity, constant pressure, and approximately zero density near the origin are familiar features of the Sedov-Taylor blastwave. The right panel of this figure gives the relativistic corrections to the velocity, density, and pressure for nergy-conserving, Relativistic Corrections to Strong Shock Propagation / r aR / (cid:5) (cid:6) V / V V / V R / R ξ = r / R ( t ) v ( ξ (cid:7)(cid:8) ) / V ( t ) Sedov - Taylorct / r a = / r a = / r a = / r a = / r a = / r a = Figure 3.
Left: The relativistic correction to the shock position (purple, solid) and the shock velocity (red, dashed), bothnormalized by their non-relativistic counterparts, for a constant ambient density ( n = 0) and a post-shock adiabatic index of γ = 4 /
3. Here we set the initial velocity of the shock to V i /c = 0 .
5. Right: The three-velocity of the fluid, normalized by theunperturbed shock velocity, as a function of radius r normalized to the unperturbed shock radius, for n = 0 (constant ambientdensity), γ = 4 /
3, and an initial shock velocity of V i /c = 0 .
5. The different curves are at the times shown in the legend, andthe black, dashed curve shows the Sedov-Taylor solution – which would be the solution if there were no relativistic terms – towhich the relativistic solution asymptotes at late times. The post-shock speed is slightly increased near the shock front, butfalls below the Sedov-Taylor solution for smaller radii. The solution near the origin also shows significant deviation from thenearly-linear behavior expected from the Sedov-Taylor solution alone. this combination of n and γ . The eigenvalue that results in the exact conservation of energy, including the relativisticterms, is σ ≃ . ξ ≃ .
95. This qualitative behavior implies that the material behind the shock becomes increasingly confined toa region very near the shock front, and that relativistic effects cause the material to be swept into an even thinnershell than the one predicted by the Sedov-Taylor blastwave alone. We also see that the perturbation to the pressure,while it does asymptote to a constant near the origin, shows much more variability than the unperturbed solution for ξ & .
1. This feature demonstrates that the total pressure behind the blast wave shows more spatial variation whenrelativistic effects are included, which is a familiar property of the ultra-relativistic, Blandford-McKee blastwave.The left panel of Figure 3 gives the correction to the shock position (solid, purple) and the correction to the shockspeed (dashed, red), each normalized by its non-relativistic counterpart, as functions of time in units of r a /c . Here weset n = 0, corresponding to a constant ambient density, the post-shock adiabatic index to γ = 4 /
3, and the initial shockvelocity to V i /c = 0 .
5. We see that the relativistic correction to the shock velocity is initially positive, correspondingto an increase in the shock position over the non-relativistic value, while at late times the correction to the shock speedchanges sign; this behavior is due to the competition between the effects of positive- σ , which increases the four-velocity(see Equation 51), and time dilation, which reduces the three-velocity over the four-velocity. In the asymptotic limitof t → ∞ , both of these corrections decay to zero and the flow settles into the non-relativistic regime.The right panel of Figure 3 shows the post-shock fluid three-velocity, normalized by the initial shock speed, as afunction of normalized radial position behind the shock front (see Equation 72). As for the left panel of this figure, herethe ambient density is constant ( n = 0), the adiabatic index is γ = 4 /
3, and the initial shock speed is V i /c = 0 .
5. Thedifferent, colored curves correspond to the times shown in the legend, and the black, dashed curve is the Sedov-Taylorsolution for this combination of n and γ (and is identical to the blue curve in the left panel of Figure 2). We seethat near the shock front the post-shock velocity is slightly increased, which is reasonable given the slight increasein the shock velocity itself, as demonstrated in the left panel of this figure. However, at small radii the velocity fallssignificantly below the non-relativistic prediction, and also displays more nonlinear behavior near the origin.The left panel of Figure 4 illustrates the normalized, lab-frame, post-shock density as a function of spherical radius r normalized by the shock radius, while the right panel of this figure gives the post-shock pressure normalized by theshock velocity (see Equations 73 and 71 respectively). As for the right panel of Figure 3, the different colored curves4 Coughlin, E.R. ξ = (cid:16) / R ( t ) ρ ( (cid:17)(cid:18) ) / ρ (cid:19) a Sedov - Taylorct / r a = / r a = / r a = / r a = / r a = / r a = ξ = (cid:20) / R ( t ) p ' ( ξ (cid:21)(cid:22) ) / ( (cid:23) a V ( t ) ) Sedov - Taylorct / r a = / r a = / r a = / r a = / r a = / r a = Figure 4.
Left: The lab-frame density of the post-shock fluid, normalized by the density of the ambient medium at the positionof the shock ρ ′ a ( t ) = ρ ′ a ( R ( t ) /r a ) − n , as a function of spherical radius r normalized to the non-relativistic shock position. Asfor Figure 3, here we set n = 0, corresponding to a constant ambient density, the post-shock adiabatic index to γ = 4 /
3, andthe initial shock speed is V i /c = 0 .
5. The colored curves correspond to the times in the legend, and the black, dashed curveis the Sedov-Taylor solution. Note that, for this figure, we restricted the range of ξ to be ξ > .
7, as below this range thecurves all rapidly approach zero (and show little variation from one another). The density is slightly increased relative to thenon-relativistic solution near the shock front, but falls below the Sedov-Taylor prediction at smaller radii, which demonstratesthat the mass behind the blastwave becomes further concentrated near the shock front as the solution becomes more relativistic.Right: The pressure behind the shock for the same parameters as in the left panel. We see that relativistic effects reduce thepost-shock pressure from the value predicted by the Sedov-Taylor solution near the origin, and the pressure also shows moresignificant variation than the nearly flat profile predicted in the non-relativistic limit. correspond to the times in the legend, the black, dashed curves are the Sedov-Taylor prediction, and we set n = 0(constant ambient density), γ = 4 /
3, and V i /c = 0 . x -axis in the left panel is compressed to highlightthe behavior near the shock; for ξ . .
7, all of the functions rapidly approach zero and show little deviation fromone another). We see that both the post-shock density and pressure increase above the Sedov-Taylor solution nearthe shock front, but, as is also true for the post-shock velocity, each of these quantities declines and falls below thenon-relativistic prediction at a radius not far behind the shock front. The post-shock pressure also shows significantdeviation from the nearly-constant value expected from the Sedov-Taylor blastwave. These findings confirm thatrelativistic effects tend to compress the fluid into a more confined region immediately behind the shock.The left Panel of Figure 5 gives the post-shock, relativistic correction to the velocity profile of the fluid, the rightpanel of this figure shows the relativistic correction to the post-shock, comoving density, and the left panel of Figure6 illustrates the post-shock correction to the pressure, and all of these panels set n = 0 (constant density ambientmedium). The different curves in each of these figures correspond to the adiabatic indices shown in the legend. We seethat, while all of these curves show the same qualitative trends, relativistic effects become amplified as the adiabaticindex of the gas decreases: the magnitude of the velocity reduction is more pronounced; the material behind the shockbecomes increasingly compressed to the shock itself; and the pressure has increased variation near the shock, possessesmore nonlinear behavior, and the decrease near the origin is enhanced. These findings – that relativistic effects becomemore important for smaller adiabatic indices – are consistent with the fact that the eigenvalue σ increases as γ decreases(see Table 1).The right panel of Figure 6 shows the relativistic correction to the shock velocity, normalized by the non-relativisticshock speed, as a function of time in units of r a /c . The different curves are appropriate to the adiabatic indices shownin the legend, the ambient density profile is constant ( n = 0), and we set the initial, non-relativistic shock speedto V i /c = 0 .
5. We see that, when γ is small, the initial velocity increases compared to the non-relativistic one, butbecomes negative after a time of ct/r a ≃ .
57 (the exact time at which this occurs can be derived analytically fromEquation 66). However, for values of γ & .
4, this behavior inverts, with the initial correction to the velocity beingnegative at early times and transitioning to positive after ct/r a ≃ .
57. This inversion occurs mathematically because σ crosses σ = 1 at this location, which is where the relativistic correction to the three-velocity equals zero (this feature nergy-conserving, Relativistic Corrections to Strong Shock Propagation - - - - ξ f γ = γ = γ = γ = γ = (cid:26)(cid:27)(cid:28) γ = γ = (cid:29)(cid:30)(cid:31) !" - - - - ξ g / g ( ) γ = γ = γ = γ = γ = &() γ = γ = Figure 5.
Left: The self-similar, relativistic correction to the velocity when n = 0 – corresponding to a constant-density ambientmedium – and the adiabatic index of the gas is given by those in the legend. Each curve shows the same, rough trend, and isnegative throughout the entire post-shock flow, reaches a minimum value near ξ ≃ .
7, and approaches zero near the origin;the latter feature ensures that the origin remains fixed for all of these solutions. The magnitude of the relativistic correctionto the velocity grows as the adiabatic index decreases, as does the eigenvalue σ that ensures that the solutions conserve theenergy behind the blastwave (e.g., γ = 1 . σ ≃
3, while γ = 1 . σ ≃ .
7; see Table 1 for a list of eigenvalues over arange of n and γ ). Right: The post-shock, relativistic correction to the density for the same parameters as the left panel. As γ approaches 1, the density becomes increasingly positive toward the shock front (note that g (1) ∝ ( γ − − ) but also reachesan increasingly negative value, and the transition to negative values approaches the location of the shock itself. This featuredemonstrates that the relativistic effects, which compress the post-shock fluid to a region that is more confined to the locationof the shock itself, become more important as γ decreases. - - * - - ξ h γ = +-. γ = γ = γ = γ = /45 γ = γ = - / r a V / V γ = γ = γ = γ = γ = γ = γ = Figure 6.
Left: The post-shock, relativistic correction to the pressure behind the blastwave for a constant-density ambientmedium ( n = 0) and the range of adiabatic indices shown in the legend. As the adiabatic index decreases, the reduction inthe post-shock pressure becomes more drastic near the origin, and the region over which the pressure experiences inflectionpoints becomes more localized to the shock itself. Right: The relativistic correction to the shock velocity, normalized by thenon-relativistic shock velocity, for the same parameters as the left panel; here we set the initial, unperturbed shock velocityto V i /c = 0 .
5. Because the eigenvalue increases as γ decreases, the initial correction to the velocity becomes correspondinglylarger. Interestingly, however, there is a value of γ at which σ drops below one, implying that the initial, relativistic correctionto the velocity changes sign. This change in sign is due to the fact that the lab-frame speed is affected by time dilation, whichcan outweigh the increase in the four-velocity (which is the three-velocity in the comoving frame of the non-relativistic shock)imparted by the positive value of σ . can be seen by inserting σ = 1 into Equation 51), and – as we noted above – arises physically because the three-velocityof the fluid is affected by time dilation.Figure 7 illustrates the self-similar correction to the post-shock four-velocity for γ = 4 / n shownin the legend (the density profile of the ambient medium falls off with spherical radius r as ρ ′ ∝ r − n ). We see that, as6 Coughlin, E.R. - - - - ξ = r / R ( t ) f n = = = = = = - - - - - - ξ = r / R ( t ) h n = = = = = = Figure 7.
Left: The relativistic, self-similar correction to the post-shock four-velocity for γ = 4 / n shown inthe legend, where n characterizes the power-law falloff of the ambient density with radius (i.e., ρ ′ a ∝ r − n ). For larger values of n ,the solution ends at a contact discontinuity at a finite ξ c , which is also where the Sedov-Taylor solution terminates. Right: Theself-similar, relativistic correction to the pressure of the post-shock fluid. As the density profile steepens from a constant densityto ρ ′ ∝ r − , the pressure corrections become less severe, and the reduction of the post-shock pressure immediately behind theshock becomes less pronounced. When n = 2 .
5, the pressure equals zero at a contact discontinuity, and the magnitude of thecorrection shows a slight increase. - - - ξ = r / R ( t ) g / g ( ) n = = = = = = ξ = r / R ( t ) g / g ( ) n = = = = Figure 8.
Left: The relativistic, self-similar correction to the comoving density behind the shock for γ = 4 / n shown in the legend, where the density of the ambient medium ρ ′ a falls off with spherical radius r as ρ ′ a ∝ r − n . When thenon-relativistic solution extends all the way to the origin, which occurs when n ≤
2, the relativistic correction to the densityapproaches zero near ξ = r = 0. However, when the solution terminates at a contact discontinuity, which occurs for n = 2 . n shown in the legend when the post-shockadiabatic index is γ = 4 /
3. The vertical, dashed lines indicate the locations of the contact discontinuity. When n = 2 . n = 2 .
7. Energy-conserving, relativistic corrections do not exist when the non-relativistic density either remainsfinite or diverges at the contact discontinuity. n increases, the relativistic correction to the velocity decreases in magnitude, and the solution with n = 2 . f ( ξ c ) = ξ c ,such that the fluid elements at these locations are stationary with respect to the shock. The right panel of this figureshows the correction to the pressure profile behind the shock, again with γ = 4 /
3. As n increases from 0 to 2, thepressure profile exhibits progressively less deviation behind the shock, and the overall magnitude of the correction isreduced. When n = 2 .
5, the post-shock correction to the pressure equals zero at the contact discontinuity, which isalso where the non-relativistic pressure equals zero. nergy-conserving, Relativistic Corrections to Strong Shock Propagation - - - - ct / r a V / V n = = = = = = γ σ n = = = = = = = Figure 9.
Left: The relativistic correction to the shock three velocity V , normalized by the non-relativistic (Sedov-Taylor)solution V , as a function of time. Here we set V i /c = 0 .
5, where V i is the velocity that the shock would have at t = 0 ifrelativistic effects were not included. Each curve corresponds to a different radial power-law index of the density of the ambientmedium n , so the ambient density declines with radius as ρ ′ a ∝ r − n , as shown in the legend. As n increases, relativistic effectsare longer-lived owing to the fact that the non-relativistic, Sedov-Taylor shock speed falls off as a shallower function of time.Right: The eigenvalue σ , which relates the shock velocity to the explosion energy and ensures the conservation of that energy,as a function of the adiabatic index γ . Different curves are appropriate to the value of the power-law index of the density of theambient medium given in the legend. Positive σ implies that the shock speed is increased relative to the non-relativistic valuein the comoving frame of the non-relativistic shock, i.e., observers moving with the Sedov-Taylor shock speed see an increaseto the shock speed owing to relativistic effects. When σ = 1, time dilation and the relativistic boost to the shock speed in thecomoving frame exactly balance to yield an observer-frame shock velocity that is identical to the Sedov-Taylor solution. For σ <
1, time dilation results in a three-velocity that is reduced compared to the Sedov-Taylor solution, and this can be seendirectly from the left panel of this figure.n γ σ = 6.05 3.08 2.10 1.63 1.35 1.17 1.08 0.950 0.882 0.831 0.790 0.758 0.724 0.7110.5 5.04 2.56 1.75 1.36 1.13 0.982 0.910 0.807 0.753 0.712 0.681 0.656 0.631 0.6211 4.02 2.05 1.40 1.09 0.911 0.796 0.741 0.663 0.623 0.593 0.571 0.554 0.537 0.5311.5 3.00 1.53 1.05 0.822 0.692 0.610 0.572 0.519 0.493 0.474 0.461 0.452 0.444 0.4422 1.98 1.01 0.698 0.553 0.474 0.427 0.405 0.378 0.367 0.360 0.357 0.357 0.360 0.3632.5 0.962 0.492 0.354 0.298 0.272 0.264 0.265 . . . . . . . . . . . . . . . . . . . . .2.7 0.559 0.299 0.235 0.224 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Table 1.
The eigenvalue σ that ensures energy conservation as a function of n , which parameterizes the density profile of theambient medium ( ρ ′ ∝ r − n ), and the post-shock adiabatic index of the gas γ . Values much greater than one indicate thatrelativistic effects are more important for less relativistic initial shock speeds, which are achieved for small n and γ . Cells withan ellipsis correspond to instances where the Sedov-Taylor (non-relativistic) solution possesses a finite or diverging density atthe contact discontinuity, for which we find no energy-conserving solution for the relativistic corrections. The left panel of Figure 8 gives the correction to the self-similar, comoving density behind the shock for γ = 4 / n , in the legend. As for the pressure, the relativistic contributions tendto be less pronounced as n increases from 0 to 2, with less of the variation being confined to the immediate vicinityof the shock and the magnitude of the variation reduced. However, when n = 2 .
5, we see that the magnitude of thecorrection increases again, and the function g actually diverges weakly at the location of the contact discontinuity.The right panel of this figure shows, for reference, the Sedov-Taylor solution for a subset of n , with the vertical, dashedlines coinciding with the location of the contact discontinuity. We see that, when n = 2 .
5, the non-relativistic densityat the contact discontinuity equals zero, while for n = 2 . g diverges at the contact discontinuity.8 Coughlin, E.R.
The relativistic correction to the lab-frame, three-velocity of the shock V , plotted relative to the non-relativisticvelocity V as a function of time, is shown in the left panel of Figure 9; here we set V i /c = 0 .
5, where V i is theNewtonian shock speed at t = 0. Different curves correspond to different radial power-law indices of the density ofthe ambient medium, n , such that ρ ′ a ∝ r − n . The correction is negative for larger values of n , which signifies thattime dilation actually reduces the shock speed below the Sedov-Taylor prediction in spite of the fact that the velocityis increased in the comoving frame of the non-relativistic shock. Relativistic effects remain important for longer timesas n increases, and this occurs because the Sedov-Taylor velocity – to the square of which, as shown in Equation (66),the relativistic correction is proportional – declines less slowly as the density profile of the ambient medium steepens(which arises physically from the reduced momentum flux across the shock for larger n ).The eigenvalue σ , which conserves the relativistic energy behind the shock and relates the shock speed to theexplosion energy, is shown as a function of γ in the right panel of Figure 9. Each curve corresponds to the power-lawindex of the ambient medium, n , shown in the legend, and the horizontal, dashed line simply indicates where σ = 1for clarity. The fact that σ is always positive implies that observers moving with the non-relativistic, Sedov-Taylorshock speed see a relativistically-boosted shock velocity in that frame. However, it is only for σ > σ < σ ( n, γ ) which were used to make this figure are given in Table 1.It can be shown from the integrals of Equations (53) – (55) that the self-similar, Sedov-Taylor density scales as g ( ξ ) ∝ (1 − f /ξ ) − n − nγn +3 γ − , (74)which implies that when the Sedov-Taylor solution ends in a contact discontinuity, the density diverges at that locationif n > / ( γ + 1) ≃ .
57 for γ = 4 / σ that satisfies globalenergy conservation for power-law indices that are steeper than this value. Mathematically, solutions in this regimedo not exist because we can combine the integral constraint on the energy (52) and Equation (60) to show that,if the relativistic correction to the energy is to remain finite, then we must have f ( ξ c ) = ξ c ( ξ − /
2. However,from Equation (61) and the fact that h ( ξ c ) = 0, we see that we must also have h ( ξ c ) = 0 to maintain energyconservation. The system is therefore over-constrained when the density diverges at the contact discontinuity, andthese two boundary conditions at ξ c will not, in general, be satisfied simultaneously for a single σ . In particular, wefind that the solution for which f ( ξ c ) = ξ c ( ξ −
1) possesses a finite, but non-zero pressure at the contact discontinuity,meaning that these solutions cannot simultaneously satisfy both the integral constraint (52) and Equation (61).It is also not surprising from a physical standpoint that these diverging-density solutions are problematic. For one, itwas shown by Goodman (1990) that such solutions are unstable to aspherical perturbations, as the decelerating natureof the fluid and the density inversion renders the contact discontinuity susceptible to the Rayleigh-Taylor instability.These Sedov-Taylor blastwaves therefore cannot be manifested in any physical (i.e., one with permissible angulardeviations from spherical symmetry) scenario. The diverging-density solutions also violate the self-similar hypothesisthat the flow is predominantly characterized by the physical conditions at a single point within the flow: for small n and γ , the vast majority of the mass is contained very near the shock front. However, when the density profilebecomes inverted, most of the mass is concentrated near the contact discontinuity, and the causal connectedness ofthe shocked fluid implies that the shock “knows” this property of the inner flow. It is therefore likely that, for thediverging-density solutions, the physical conditions both at the shock front and near the contact discontinuity remainimportant for establishing the long-term behavior of the shock, and the solution may be fundamentally non-self-similar.It is interesting to note that the ultra-relativistic, Blandford-McKee blastwave avoids this issue, as the comovingpressure behind the shock transforms as a higher power of the shock Lorentz factor than the comoving density. Thecontribution of the kinetic energy to the total energy behind the blastwave is therefore dropped in their solution, andonly the internal energy must be conserved. Thus, while the Blandford-McKee solution formally exists in this regime,it is likely that the integrated kinetic energy remains important for the dynamics (i.e., by virtue of the boundaryconditions at the shock, it is true in the ultra-relativistic limit that the kinetic energy is sub-dominant to the internalenergy, but that may not be true deeper within the flow; the non-existence of relativistic corrections to the Sedov-Taylorsolutions suggests that this may be the case for, at least, smaller values of the Lorentz factor). Note that this is not required when the density equals zero at the contact discontinuity, as in this situation the density scales as g ∝ (1 − ξ c ) − α near ξ c with α <
1; thus, while the correction to the comoving density diverges at ξ c , it does so in a way that yields afinite, relativistic correction to the energy. nergy-conserving, Relativistic Corrections to Strong Shock Propagation U = U s (cid:8) f ( ξ ) + U f ( ξ, χ ) (cid:9) , (75)and similarly for the other variables, and therefore maintained the derivatives with respect to χ in Equations (57)– (59) (recall that χ = ln R is a time-like variable). However, it is difficult to see how relativistic effects (to order V /c ) could modify the solution in a way other than one that scales as V /c , as this is the only physical smallnessparameter introduced in the problem. Therefore, any additional time dependence in the expression f ( ξ, χ ) should,instead, be regarded as a higher-order correction to the self-similar solution; we also note that this is precisely themotivation for expanding the self-similar functions as f ( ξ, χ ) ≃ f ( ξ ) + U f ( ξ ) – the second term accounts for thetime dependence that is induced by relativistic effects, and that time dependence can only physically be of the form U . While we acknowledge that this is not a rigorous proof of the non-existence of more general solutions, we find itsuggestive that no such solutions exist. SUMMARY AND CONCLUSIONSWhen the shockwave from an astrophysical explosion is strong (Mach number much greater than one) and non-relativistic, and therefore characterized by a shock speed V much less than the speed of light c , the Sedov-Taylor,energy-conserving blastwave provides an analytic, self-similar solution for the temporal evolution of the shock itself,and the time and space-dependent evolution of the post-shock velocity, density, and pressure. In the other, extreme-relativistic limit where V ≃ c , the Blandford-McKee blastwave gives the energy-conserving evolution of the shockLorentz factor and the post-shock fluid quantities. In between these two extremes, the finite speed of light introducesan additional velocity scale into the problem, which destroys the pure self-similarity of the solutions.In this paper, we analyzed the leading-order, relativistic corrections to the fluid equations – which enter as O ( V /c )– to understand the effects that such relativistic terms have on the non-relativistic, Sedov-Taylor solution for strongshock propagation. By treating such terms as perturbations (i.e., ignoring nonlinear terms that enter as higherpowers of V /c ), we showed that there are relativistic corrections to the Sedov-Taylor solutions for the post-shockfluid quantities that vary (consistent with expectations) as V /c . In particular, we demonstrated that the radialcomponent of the post-shock, fluid four-velocity can be written as U = U s (cid:8) f ( ξ ) + U /c × f ( ξ ) (cid:9) , where U s is theshock four-velocity, f is the Sedov-Taylor solution, ξ = r/R ( t ) with R ( t ) the shock position, and f is a functionthat is self-consistently determined from the fluid equations and the relativistic jump conditions at the shock. Thefunction f , and the analogous functions g and h for the density and pressure, respectively, induce more nonlinearbehavior to the post-shock velocity, further compress the post-shock material to the immediate vicinity of the shockitself, and generate greater variation in the post-shock pressure as compared to the Sedov-Taylor, non-relativistic limit(see Figures 2 – 4). These are all features of the ultra-relativistic, Blandford-McKee blastwave, where the pressuredeclines rapidly behind the shock and all of the material is swept into a shell of width ∆ R = R/U .In addition to the post-shock fluid quantities, we also determined the relativistic correction to the velocity of theshock itself. We denoted this additional correction by an “eigenvalue” σ , such that the relativistically-corrected shockfour-velocity is written implicitly as U R − n = E (cid:0) σU /c (cid:1) . When σ ≡
0, one recovers the familiar relationshipbetween the shock velocity and position that guarantees the conservation of the blast energy, E . However, owing tothe existence of relativistic corrections to the energy, σ cannot be exactly zero, and there must therefore be correctionsto the shock velocity that maintain total (i.e., including relativistic terms to order U /c ) energy conservation. Forall of our solutions, the value of σ was found to be positive, implying an increase to the shock four-velocity fromrelativistic effects; equivalently, observers moving at the non-relativistic shock speed measure a small, slight increaseto the shock velocity, and hence the true shock position leads the non-relativistic one in the comoving frame of the non-relativistic shock. Nevertheless, the lab-frame three-velocity is reduced from the four-velocity by time dilation, whichcounterbalances the relativistically-boosted effect of positive σ , and if σ ≡ γ = 4 / n > . Coughlin, E.R. density ρ ′ a falls off with spherical radius r as ρ ′ a ∝ r − n – the lab-frame velocity is reduced below the non-relativisticvalue (see Figure 9 and Table 1 for the values of σ for a range of n and γ ).Once the power-law index of the ambient density profile equals or exceeds the critical value n cr ( γ ) = 6 / ( γ + 1),the density at the contact discontinuity present in the Sedov-Taylor solution diverges. For Sedov-Taylor blastwaveswith n > n cr , we do not find any solution for σ that maintains a finite and conserved relativistic correction to theenergy, which mathematically follows directly from the nature of the solutions for f , g , and h and the integral thatmaintains the conservation of energy (see Equations 52 and 61). It is possible that more general, time-dependentsolutions (i.e., those that do not assume the form given by Equations 44 – 46) could be found in this regime that doconserve energy. However, based on the argument that additional time dependence from the self-similar, Sedov-Taylorsolutions is itself seeded by relativistic effects, it is difficult to see how any extra time dependence would not be inthe form of higher-order (than U /c ) terms. We therefore find it unlikely that such generalized solutions exist at theleading relativistic order.Here we described the leading-order, relativistic corrections to the fluid flow, which enter into the fluid equations andthe boundary conditions as U /c . One can regard our solutions as the first in a series expansion of the fluid equationsin U , and the next-order solution for (for example) the four-velocity would be U = U s (cid:8) f ( ξ ) + U f ( ξ ) + U f ( ξ ) (cid:9) ;one could then, by expanding the fluid equations to the next order, derive self-consistent equations for f , g , and h ,and the boundary conditions at the shock could be found by expanding the general jump conditions to the next-highestorder. In principle, one should also be able to construct the next-order solution for the Blandford-McKee solution, andapproach the problem from the other, ultrarelativistic direction by finding the next-highest-order correction in 1 / Γ.In this paper we focused on ambient density profiles less steep than ρ ′ a ∝ r − . For steeper density profiles, theshock enters an accelerating regime, and the self-similar solutions for the post-shock fluid quantities are providedby Waxman & Shvarts (1993) (see also Koo & McKee 1990). In this case, the self-similar flow is constrained to liebetween a sonic point within the interior of the flow and the shock front, and energy and mass are drained into anon-self-similar, inner region that is causally disconnected from the fluid near the shock. One could apply all of theformalism developed in this paper to derive the leading-order, relativistic corrections to such accelerating, self-similarflows, and the resulting equations for the functions f , g , and h would, in fact, appear almost identical to Equations(57) – (59), with the exception that various factors of n − dominate the self-similar solution at late times, owing to the increasing nature of U /c ,and one could interpret this result by saying that such self-similar solutions are “unstable” to relativistic corrections.These solutions for the relativistically-corrected shock speed and post-shock fluid quantities could be used for gen-erating more accurate models for the late-time lightcurves of long gamma-ray bursts as well as the lightcurves ofenergetic supernovae. In particular, the relativistic beaming induced by the marginally-relativistic velocity, and theprediction for the time over which the shock speed declines to sub-relativistic speeds, would yield correspondinglydifferent break timescales for the lightcurve of the event and peaks in the synchrotron spectrum (e.g., Sari et al.1998; De Colle et al. 2012). The event AT2018cow (Prentice et al. 2018; Rivera Sandoval et al. 2018; Ho et al. 2019;Kuin et al. 2019; Margutti et al. 2019; Perley et al. 2019), tentatively an extreme example of the class of fast-risingtransients (Drout et al. 2014), also provided evidence of a moderately relativistic outflow with speed ∼ . c ; the modelpresented here could be combined with current multiwavelength data to further constrain properties of the progenitorand surrounding medium.This work was supported by NASA through the Einstein Fellowship Program, Grant PF6-170170. I thank BrianMetzger, Eliot Quataert, and Jonathan Zrake for useful discussions.REFERENCES Abbott, B. P., Abbott, R., Abbott, T. D., et al. 2017,ApJL, 848, L12, doi: 10.3847/2041-8213/aa91c9Arnett, W. D. 1982, ApJL, 263, L55, doi: 10.1086/183923Bethe, H. A., & Wilson, J. R. 1985, ApJ, 295, 14,doi: 10.1086/163343 Blandford, R. D., & McKee, C. F. 1976, Physics of Fluids,19, 1130, doi: 10.1063/1.861619Blondin, J. M., Mezzacappa, A., & DeMarino, C. 2003,ApJ, 584, 971, doi: 10.1086/345812 nergy-conserving, Relativistic Corrections to Strong Shock Propagation21