2. RELATIVISTIC SHOCK WAVES

Relativistic shock waves are considered in the context of relativistic fluid dynamics (Landau & Lifshitz, Reference Landau and Lifshitz1987). The equations of motion are described by the relativistic energy-momentum four-tensor T _μν given by

(1)$$T_{{\rm \mu} {\rm \nu}} = \lpar e + P\rpar U_{{\rm \mu}} U_{{\rm \nu}} + Pg_{{\rm \mu} {\rm \nu}}.$$

A Greek index takes the values: 0, 1, 2, 3 while we use the Latin index for the space values 1, 2, 3. U _μ = (γc, γv ₁, γv ₂, γv ₃) is the four-velocity where (v ₁, v ₂, v ₃) is the fluid element velocity in three dimension (x, y, z), e is the internal energy density of the fluid which includes the rest energy mass of the fluid particles, P is the pressure, the metric tensor g _μν is g ₀₀ = −1, g ₁₁ = g ₂₂ = g ₃₃ = 1, g _μν = 0 for μ ≠ ν and for a three dimension velocity v one defines

(2)$${\rm \gamma} = {1 \over \sqrt{1 - {\rm \beta}^2}}\semicolon \; {\rm \beta} = {{\rm \nu} \over c}.$$

The equations describing the relativistic fluid motion (neglecting internal friction and thermal conduction) are

(3)$$\eqalign{& {\rm Theenergy - momentumconservation}:\displaystyle{{\partial T_\mu ^\nu } \over {\partial x^v }} \equiv \partial _v T_\mu ^v = 0\, \cr & \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad for\,\mu = 0{\rm , }\>1{\rm , }2{\rm , }3. \cr & {\rm The}\;{\rm particle number conservation}:\displaystyle{{\partial (nU^\mu {\rm )}} \over {\partial x^\mu }} \equiv \partial _\mu \left( {nU^\mu } \right) = 0 \cr & {\rm The equation of State}:\quad \quad \quad \quad \quad \quad \quad \quad P = P\left( {e{\rm ,}\>\;n} \right)}$$

The Einstein summation convention (from 0 to 3) is assumed for identical indexes. Since Eq. (1) is the starting point of our calculation we write it more explicitly

(4)$$\eqalign{T_{00} &= {\rm \gamma}^2 \lpar e + P\rpar - P \cr T_{0i} &= T_{i0} = - {\rm \gamma}^2 \lpar e + P\rpar / c\, \,\hbox{for}\, \hbox{i} = 1\comma \; 2\comma \; 3 \cr T_{ij} &= {\rm \gamma}^2 \lpar e + P\rpar \left({\hbox{v}_i \over c} \right) \left({\hbox{v}_j \over c} \right) + P {\rm \delta}_{ij}\, \,\hbox{for i}, \hbox{j} = 1, 2, 3}$$

δ_ij = 1 for i = j and δ_ij = 0 for i ≠ j. T ₀₀ is the energy density [J/m³], −T _0i/c is the momentum component density [kg/m² × s] and T _ij is the momentum flux density.

We assume a one-dimensional (1D) shock wave where the shock propagates in the x direction and the shock wave surface singularity is in the y-z plane. Furthermore, the shock wave is time independent by looking into a coordinate system where the shock front is at rest. In this shock wave reference system (the shock system), the observer is at rest relative to the material on both sides of the shock front. We use the subscript 0 and 1 to denote the variables in front (where the shock wave did not arrive yet) and in the back (the shocked area) of the shock wave, respectively. The physical quantities: pressure P, particle density n, mass density ρ, energy density e, entropy density s, and temperature T are measured in their proper frame of reference, defined as the reference frame where the corresponding fluid element is at rest. The flow velocities v ₀ and v ₁, assumed in the x direction, are measured in the shock wave reference system. In this case, we define β₀ = v ₀/c, β₁ = v ₁/c, and γ₀, γ₁ according to Eq. (2).

The conservation laws across the shock wave singularity describe the flux densities conservation of energy −c[T _0x]₀ =  −c[T _0x]₁ and momentum [T _xx]₀ = [T _xx ]₁ as defined in Eq. (4) and the particle number flux defined by [nU _x]₀ = [nU _x]₁. These conservation laws are explicitly given by the following equations accordingly

(5)$$\eqalign{&{\rm \gamma}_0^2 {\rm \beta}_0 \left(e_0 + P \right)= {\rm \gamma}_1^2 {\rm \beta}_1 \left(e_1 + P_1 \right)\cr &{\rm \gamma}_0^2 {\rm \beta}_0^2 \left(e_0 + P_0 \right)+ P_0 = {\rm \gamma}_1^2 {\rm \beta}_1^2 \left(e_1 + P_1 \right)+ P_1 \cr &{\rm \gamma}_0 {\rm \beta}_0 n_0 = {\rm \gamma}_1 {\rm \beta}_1 n_1}$$

Regarding the particle conservation, the third of Eq. (5), it is important to clarify that for very high temperature T new particles can be created, such as electron-positron pairs. In this case, we consider the electric charge conservation (of the electron-positron system in this example).

The solution of Eq. (5) yields

(6)$$\eqalign{{\hbox{v}_0 \over c} ={\rm \beta}_0 =\sqrt{{\lpar P_1 - P_0\rpar \lpar e_1 + P_0\rpar \over \lpar e_1 - e_0\rpar \lpar e_0 + P_1\rpar }} \cr {\hbox{v}_1 \over c}= {\rm \beta}_1 =\sqrt{{\lpar P_1 - P_0\rpar \lpar e_0 + P_1\rpar \over \lpar e_1 - e_0\rpar \lpar e_1 + P_0\rpar }}.}$$

The relative velocity between the flows v ₀ and v ₁ on both sides of the singularity is given by v ₀₁

(7)$${\hbox{v}_{01} \over c}= {\rm \beta}_{01} = {{\rm \beta}_0 - {\rm \beta}_1 \over 1 - {\rm \beta}_0 {\rm \beta}_1} = \sqrt{{\lpar P_1 - P_0\rpar \lpar e_1 - e_0\rpar \over \lpar e_0 + P_1\rpar \lpar e_1 + P_0\rpar }}.$$

Substituting v₀ and v₁ into the third of Eq. (5) one gets the Hugoniot relation (Landau & Lifshitz, Reference Landau and Lifshitz1987; Taub, Reference Taub1948), first derived by Taub (Reference Taub1948),

(8)$${\lpar e_1 + P_1\rpar ^2 \over n_1^2} - {\lpar e_0 + P_0\rpar ^2 \over n_0^2} = \lpar P_1 - P_0\rpar \left[{\lpar e_0 + P_0\rpar \over n_0^2} + {\lpar e_1 + P_1\rpar \over n_1^2} \right].$$

Before proceeding with the relativistic calculations it is useful to check the non-relativistic (NR) limit of Eqs. (4), (5), and (8). The NR limit is defined by v ≪ c. The relativistic internal energy e [J/m³] includes the rest energy nmc ² = ρc ² of the fluid particles, where m is the rest mass of one particle. In going to the NR limit, one has to take into account the fluid motion and therefore before doing the limit properly one has to substitute ρ by ρ/γ = ρ(1 − v²/c ²)^1/2 ≈ ρ[1 − v²/(2c ²)]. One can define the internal energy per unit mass E(J/kg) by the relation e = ρc ² + ρE, so that the NR limit of e is ρc ² + ρE − (1/2)ρv². Thus, the non-relativistic energy-momentum four-tensor T _μν is given by: T ₀₀ → ρc ² + ρE+ (1/2)ρv², −cT _i0 → ρc ²v_i + [ρE + P + (1/2)ρv²]v_i, T _ij → ρv_iv_j + Pδ_ij. Furthermore, taking into account that in the NR limit P and ρE are much smaller than ρc ² then Eqs. (5) go to the standard Hugoniot equations in the shock wave frame of reference: Energy conservation, {[ρE + P + (1/2)ρv²]v}₀ = {[ρE + P + (1/2)ρv²]v}₁; momentum conservation, {P + ρv²}₀ = {P + ρv²}₁ and mass conservation, {ρv}₀ = {ρv}₁. Similarly, the Hugoniot Eq. (8) goes to its NR limit: E ₁ − E ₀ = (1/2)(P ₀ + P ₁)(1/ρ₀ − 1/ρ₁).

In order to solve the equations of motion, the knowledge of the EOS is required. We consider in this paper two EOS: The extreme relativistic (ER) EOS

(9)$$P = {1 \over 3} e\comma \;$$

and the ideal gas (IG) EOS

(10)$$\eqalign{e &= {\rm \rho} c^2 + {P \over \Gamma - 1} \cr P &= nk_B T}\comma \;$$

Γ is the specific heat ratio and k _B is the Boltzmann constant. The relativistic Hugoniot lines obtained from Eq. (8) for P ₀ = 0 and the EOS of Eqs. (9) and (10) are

(11)$$\eqalign{&ER - EQS\, \hbox{Hugoniot\colon } \displaystyle{{P_1 \over {\rm \rho}_0 c^2} = {1 \over 12}} \left[\matrix{\displaystyle\left({{\rm \rho}_1 \over {\rm \rho}_0} \right)^2 + \left({{\rm \rho}_1 \over {\rm \rho}_0} \right) \sqrt{\left({{\rm \rho}_1\over {\rm \rho}_0} \right)^2 + 48} } \right]\cr &IG - EOS\, \hbox{Hugoniot\colon } \left\{\matrix{\displaystyle{{P_1 \over {\rm \rho}_0 c^2}} = \left({\lpar \Gamma - 1\rpar ^2 \over \Gamma} \right)\left({{\rm \rho}_1 \over {\rm \rho}_0} \right)^2 - \left({\Gamma^2 - 1 \over \Gamma} \right)\left({{\rm \rho}_1 \over {\rm \rho}_0} \right) \cr \displaystyle{{k_B T_1 \over mc^2}} = \left({\lpar \Gamma - 1\rpar ^2 \over \Gamma} \right)\left({{\rm \rho}_1 \over {\rm \rho}_0} \right)- \left({\Gamma^2 - 1 \over \Gamma} \right)} \right.}$$

Note that for the ideal gas EOS our relativistic approach in Eq. (11) starts from a compression of (Γ + 1)/(Γ − 1), namely for Γ = 5/3 we have ρ₁/ρ₀ = 4, P ₁ = 0, T ₁ = 0 and the relativistic Hugoniot is for ρ₁/ρ₀ > 4. However, for the general case we cannot assume P ₀ = 0 and in this case we have

(12)$$\eqalign{&{\rm \kappa} \equiv {{\rm \rho}_1 \over {\rm \rho}_0}\semicolon \; {\rm \kappa}_0 \equiv {\Gamma + 1 \over \Gamma - 1}\semicolon \; \Pi_i \equiv {P_i \over {\rm \rho}_0 c^2}\, \,\hbox{for}\, \hbox{i} = 0\comma \; 1\semicolon \; \cr &\Pi_1^2 + b\Pi_1 + c = 0\cr \Pi_1 &= \left({1 \over 2} \right)\left(-b + \sqrt{b^2 - 4c} \right)\cr b &= {\left(\Gamma - 1 \right)^2 \over \Gamma} \left({\rm \kappa}_0 {\rm \kappa} - {\rm \kappa}^2 \right)+ \Pi_0 \left(\Gamma - 1 \right)\left(1 - {\rm \kappa}^2 \right)\cr c &= {\left(\Gamma - 1 \right)^2 \over \Gamma} \left({\rm \kappa} - {\rm \kappa}_0 {\rm \kappa}^2 \right)\Pi_0 - {\rm \kappa}^2 \Pi_0^2}$$

For Π₀ = 0 we get Eq. (11) for the ideal gas EOS. However, for a nonzero Π₀ we have two solutions but only the plus sign is appropriate in Eq. (12) since c < 0 and Π₁ > 0. Note also that Π₁ = 0 only if Π₀ = 0 and κ = 1 for Π₁ = Π₀ as physically requested. In general, Π₁ is a function of κ with Π₀ as a parameter, Π₁ = f(κ;Π₀) and Π₁ is positive for compressions larger than 1 as requested by a shock wave.

The relativistic Hugoniot of the (dimensionless) pressure P₁/(ρ₀c ²) is plotted in Figure 1 in the domain 1 < ρ₁/ρ₀ < 4 for initial pressures P ₀/(ρ₀c ²) = 10⁻² and 10⁻⁴ using an ideal gas EOS with Γ = 5/3. One can see that for initial low pressures that we are interested in this paper, P ₀/(ρ₀c ²) ≪ 10⁻⁴, the Hugoniot domain of interest is for ρ₁/ρ₀ > 4.

Fig. 1.

Hugoniot of the (dimensionless) pressure P ₁/(ρ₀c ²) in the domain 1 < ρ₁/ρ₀ < 4 for initial pressures P ₀/(ρ₀c ²) = 10⁻² and 10⁻⁴ using an ideal gas EOS with Γ = 5/3.

These Hugoniot lines for the dimensionless pressure P ₁/(ρ₀c ²) of Eq. (11) are plotted in Figure 2. The calculations were done in the relativistic approximation for the ideal gas EOS with Γ = 5/3 and 4/3. In Figure 1a, the compression up to 10 and in 1b compression up to 100 are plotted. The Hugoniot curves for the extreme relativistic EOS are shown for compression up to 10 in 1c and up to 100 in Figure 1d.

Fig. 2.

Hugoniot lines for the dimensionless pressure P₁/(ρ₀c²). The Hugoniot curves for the ideal gas EOS with Γ = 5/3 and 4/3 are shown in (a) for compression up to 10 and in (b) for compression up to 100. The Hugoniot curves for the extreme relativistic EOS are shown for compression up to 10 in (c) and up to 100 in (d).

3. COLLISION BETWEEN A FLYER AND A TARGET

The highest laser induced pressures, on the order of 10⁹ bars, have been obtained during the collision of a target with an accelerating foil. This acceleration was achieved by laser produced plasma or by X-rays from a cavity produced by laser plasma interactions. In this section, we solve this relativistic collision problem for a symmetrical collision, namely the target and the flyer have the same material properties (e.g., an aluminum-aluminum collision). In the relativistic collision, we derive pressures higher by many orders of magnitude than presently obtained.

The flyer has a known initial (before impact) velocity u _f in the laboratory frame of reference. This velocity is calculated theoretically in the next section and it seems possible to measure it experimentally using the relativistic Doppler effect (DE).

(13)$$\eqalign{longitudinal \;DE\colon {{\rm \lambda}_0 \over {\rm \lambda}_s} &= {{\rm \nu}_s \over {\rm \nu}_0} = \sqrt{{1 + {\rm \beta}_f \over 1 - {\rm \beta}_f}}\semicolon \; {\rm \beta}_f = {u_f \over c} \cr transversal \;DE\colon {{\rm \nu}_s \over {\rm \nu}_0} &= {\rm \gamma}_f \lpar 1 + {\rm \beta}_f \cos {\rm \theta}_o\rpar \semicolon \; {\rm \gamma}_f {1 \over \sqrt{1 - {\rm \beta}_f^2}}.}$$

The subscripts s and o refer to observer and source frames of reference accordingly, θ_o is the angle between the flyer motions and the observer, λ and ν are the appropriate wavelength and frequency of the electromagnetic wave. It is interesting and might be experimentally useful to see the ν_s/ν_o (θ_o = π/2) = γ_f relativistic effect.

We chose for simplicity the accelerated micro-foil (the flyer) and the target to be of the same material. After the collision a shock wave S1 will move into the target and another shock wave S2 will move into the flyer. The proper fluid quantities: density ρ = mn, where m is the mass of fluid particles and n the particle density, the pressure P, the energy density e, and temperature T before collision are denoted by subscript zero and after collision by subscript 1.

In Figure 3, a schematic description of the two shock waves is given with the appropriate notations in the laboratory frame of reference and in the two shock waves frame of reference S1 and S2. We denote by u the different velocities in the laboratory frame and by v the appropriate velocities in the S1 and S2 where the shock wave 1 and 2 are accordingly at rest. For the symmetrical collision we have v ₃ = v ₀ and v ₂ = v ₁.

Fig. 3.

Flow (u _p0, u _p1, u _p2) and shock waves (u _s1, u _s2) velocities after impact of flyer and target in the laboratory frame of reference. The flow velocities (v ₀, v_1,v ₂ = v _1,v ₃ = v ₀) are also defined in the shock wave reference frames S1 and S2.

If the 0 domain of the target is at rest in the laboratory then in this frame of reference the shock velocity u _s = −v₀. Note that v₀ > 0 and u _s < 0 means that the positive x coordinate is defined toward the back of the shock wave. Using Eqs. (6) with the previous definitions, β₀ = v₀/c, β₁ = v₁/c, and β_f = u _f/c where u _f is the flyer velocity in the laboratory frame of reference while v₀ and v₁ are the flow velocities in the shock wave frames of reference S1 and S2, we get the shock waves velocities in the laboratory frame of reference

(14)$$\eqalign{{u_{s1} \over c} &= -{\rm \beta}_0 = - \sqrt{{\left(P_1 - P_0\right)\left(e_1 + P_0\right)\over \left(e_1 - e_0\right)\left(e_0 + P_1 \right)}} \cr {u_{s2} \over c} &= {{\rm \beta}_1 + {\rm \beta}_f \over 1 + {\rm \beta}_1 {\rm \beta}_f} = {1 + {\rm \beta}_f \sqrt{\displaystyle{\left(e_1 - e_0\right)\left(e_1 + P_0 \right)\over \left(P_1 - P_0\right)\left(e_0 + P_1\right)}} \over {\rm \beta}_f + \sqrt{\displaystyle{\left(e_1 - e_0 \right)\left(e_1 + P_0 \right)\over \left(P_1 - P_0 \right)\left(e_0 + P_1 \right)}}}.}$$

In the second relation of Eq. (14), the appropriate relativistic addition of velocities was used.

In the non-relativistic case, the Galilean transformations yield for the particle velocity in the laboratory frame u _p1 = −(v₁ − v₀) at the SW1 surface singularity and u _p1 = (v₁ − v₀) − u _f at the SW2 surface singularity. From these two equations we get the well-known result: u _f = 2u _p1. We use now the same procedure by using the relativistic Lorentz transformation.

From Eq. (7) we get at the S1 surface singularity the particle flow velocity u _p1 in the shocked area in the laboratory frame of reference

(15)$$u_{\,p1} = \hbox{v}_{01} = -c \sqrt{{\left(P_1 - P_0\right)\left(e_1 - e_0 \right)\over \left(e_0 + P_1 \right)\left(e_1 + P_0 \right)}}$$

On the other hand, at the S2 surface singularity, the particle flow velocity u _p1 in the shocked area in the laboratory frame of reference is

(16)$$u_{\,p1} = {\hbox{v}_{01} - u_f \over 1 - \hbox{v}_{01} u_f /c^2}.$$

Using Eqs. (15) and (16) we get the relation

(17)$$\sqrt{{\left(P_1 - P_0 \right)\left(e_1 - e_0 \right)\over \left(e_0 + P_1 \right)\left(e_1 + P_0 \right)}} = {1 - \sqrt{1 - {\rm \beta}_f^2} \over {\rm \beta}_f}$$

These relations yield the known result u _f = 2u _p1 in the non relativistic limit.

The above analysis for the collision between a flyer and a target yields three equations: the Hugoniot Eq. (8), the impact relation given by Eq. (17), and the equation of state P = P(e, n). Therefore, we have three equations with three unknowns: e ₁, P ₁, and n ₁ assuming that we know the initial conditions e ₀, P ₀, and n _o and we measure experimentally the flyer impact velocity u _f. We shall solve these equations using the EOS of extreme relativistic and ideal gas as given by Eqs. (9) and (10), namely we calculate e ₁, P ₁, and ρ₁ as a function of u _f. The initial conditions for our suggested impact are: P ₀, e ₀ = ρ₀c ² + P ₀/(Γ − 1) for an initial density ρ₀. However, since in the impact under consideration the initial pressure is extremely small (on the relativistic scale) we can take P ₀ = 0 according to Eqs. (12) as described in Figure 1.

For the extreme relativistic EOS, the solution of the Hugoniot Eq. (8) and the EOS Eq. (9) gave our first relation of Eq. (11). The solution of this equation with the impact Eq. (17) yields the following solution

(18)$$\eqalign{&{P_1 \over {\rm \rho}_0 c^2} = {1 - {\rm \beta}_f^2 / 3 - \sqrt{1 - {\rm \beta}_f^2} \over -1 + {\rm \beta}_f^2 + \sqrt{1 - {\rm \beta}_f^2}} \cr &{{\rm \rho}_1 \over {\rm \rho}_0} = {P_1 \over {\rm \rho}_0 c^2} \sqrt{{6 \over 2 + \displaystyle{P_1 \over {\rm \rho}_0 c^2}}} \cr &\left({e_1 \over {\rm \rho}_0 c^2} \right)= 3 \left({P_1 \over {\rm \rho}_0 c^2} \right)}$$

For the ideal gas EOS, the solution of the Hugoniot Eq. (8) and the EOS Eq. (10) gave our second relation of Eq. (11). This equation with the impact Eq. (17) yields the following equation

(19)$$\eqalign{&{\left(\Gamma - 1 \right)\left[\left({{\rm \rho}_1 \over {\rm \rho}_0} \right)-1 \right]\left({P_1 \over {\rm \rho}_0 c^2} \right)+ \left({P_1 \over {\rm \rho}_0 c^2} \right)^2 \over \left[\left({P_1 \over {\rm \rho}_0 c^2} \right)+ 1 \right]\left[\left(\Gamma - 1 \right)\left({{\rm \rho}_1 \over {\rm \rho}_0} \right)+ \left({P_1 \over {\rm \rho}_0 c^2} \right)\right]} = B \cr &B \equiv {2 - {\rm \beta}_f^2 - 2 \sqrt{1 - {\rm \beta}_f^2} \over {\rm \beta}_f^2}}$$

B = 0 for β_f = 0, B = 1 for β_f = 1, and B(β_f) is a monotonic increasing function of β_f. Solving Eqs. (19) and the Hugoniot for the ideal gas (second relation of Eq. (11)) we get the following compression as a function of B(β_f) and Γ:

(20)$${{\rm \rho}_1 \over {\rm \rho}_0} = {\left(1 - B\right)\pm \Gamma \sqrt{1 - B} \over \left(1 - B \right)\left(\Gamma - 1 \right)}.$$

One can directly check that this solution satisfies Eqs. (19) and the second relation of Eq. (11). B is defined in the second Eq. (19), so that the compression as a function of the flyer velocity β_f is obtained. Substituting the compression from Eq. (20) into Eqs. (10) and (11) for the ideal gas EOS one gets the pressure, the temperature, and the internal energy for the shocked impact target. Since B < 1 and Γ > 1 and the compression is positive only for the solution with + sign in Eq. (20) is physical.

Numerical solutions for the density and pressure of the shocked impact target are given in Figures 4 and 5 for the extreme relativistic and ideal gas equations of states accordingly. Figure 4 shows the density and pressure of the shocked impact target as a function of the foil velocity β_f for extreme relativistic equation of states. The compression is shown Figure 4a for velocities from about 0.75c up to 0.9c and in Figure 4b for velocities from 0.9c to 0.999c. The pressure is shown Figure 4c for velocities from about 0.75c up to 0.9c, and in Figure 4d for velocities from 0.9c to 0.999c. Figure 5 gives the density and pressure of the shocked impact target as a function of the foil velocity β_f for the ideal gas equation of states with Γ = 5/3 (continuous line) and 4/3 (broken line). The compression is shown in Figure 5a for velocities up to 0.9c and in Figure 5b for velocities from 0.9c to 0.999c. The pressure is plotted in Figure 5c for velocities up to 0.9c, and in Figure 5d for velocities from 0.9c to 0.999c.

Fig. 4.

The density and pressure of the shocked impact target as a function of the foil velocity β_f for extreme relativistic equation of states. The compression is shown (a) for velocities from about 0.75c up to 0.9c and in (b) for velocities from 0.9c to 0.999c. The pressure is shown (c) for velocities from about 0.75c up to 0.9c and in (d) for velocities from 0.9c to 0.999c.

Fig. 5.

The density and pressure of the shocked impact target as a function of the foil velocity β_f for the ideal gas equation of states with Γ = 5/3 and 4/3. The compression is shown (a) for velocities up to 0.9c and in (b) for velocities from 0.9c to 0.999c. The pressure is shown (c) for velocities up to 0.9c and in (d) for velocities from 0.9c to 0.999c.

Note that unlike the non-relativistic limit where the compression is finite for an infinite asymptotic pressure (e.g., maximum compression in one shock wave is (Γ + 1)/(Γ − 1)) for an ideal gas, in the relativistic case, both the pressure and the density go to infinity with a scaling law P ~ ρ². This scaling law, like all other behavior of the thermodynamic variables, depends on the equation of state. It is interesting to point out that for the extreme relativistic EOS the asymptotic scaling law P ~ ρ² is also satisfied.

4. THE MICRO-FOIL ACCELERATION BY PONDEROMOTIVE FORCE

For very high laser irradiance a foil is accelerated by the electromagnetic radiation pressure. For our calculation, it is important to distinguish between physical quantities in the instantaneous rest frame of reference of the micro-foil (denoted by subscript F) and their laboratory frame of reference values (denoted without any subscript). For a laser irradiance I _F[erg/(cm² × s)] = cE _F²/(4π), where E _F is the laser electromagnetic field in the foil frame of reference, the radiation pressure is equal to the ponderomotive pressure given by

(21)$$P = P_F = {I_F \over c} \left(1 + R_F - T_F \right)= {2I_F R_F \over c}.$$

R _F is the reflected laser and T _F is the transmitted laser through the foil that has to be taken into account for thin foils and very high laser irradiances. The right hand side of Eq. (21) is obtained assuming the energy conservation equation: R _F + T _F = 1. Note that in Eq. (21) we have used the fact that the radiation pressure P = P _F is relativistic invariant. The laboratory laser irradiance I is related to the laser irradiance in the micro-foil rest frame I _F through the Lorentz transformation relations

(22)$$I = I_F \left({{\rm \omega} \over {\rm \omega}_F} \right)^2 = I_F \left({1 + {\rm \beta}_f \over 1 - {\rm \beta}_f} \right)\semicolon \; {\rm \beta}_f = {u_f \over c}\comma \;$$

u _f is the micro-foil velocity in the laboratory as was defined in previous sections. Using Eqs. (21) and Eq. (22) for R _F = 1 (for simplicity we take R _F = 1 and not as a parameter) we get the ponderomotive pressure as a function of laboratory frame quantities

(23)$$P = {2I \over c} \left({1 - {\rm \beta}_f \over 1 + {\rm \beta}_f} \right).$$

Using the momentum p ₁ = mγ_fu and energy E ₁= mγ_fc ² = (m ²c ⁴ + p ₁²c ²)^1/2 of a particle with mass m where γ_f = (1 − β_f²)^−1/2, one has

(24)$$\left({1 - {\rm \beta}_f \over 1 + {\rm \beta}_f} \right)= {\sqrt{m^2 c^2 + p_1^2} - p_1 \over \sqrt{m^2 c^2 + p_1^2} + p_1}.$$

Assuming that all particles of the micro-foil have the same mass m and momentum p ₁ then the momentum of the micro-foil is p _f = M _0fγ_fβ_fc where its rest mass is M _0f = ρ₀Sl, ρ₀ is the initial density, S the cross-section area and l the thickness of the micro-foil. The force accelerating the micro-foil is F = PS and using Eqs. (23) and (24) the Newton law of motion is given in the laboratory frame of reference by

(25)$$\eqalign{{dp_f \over dt} &= PS \Rightarrow {d {\rm \pi}_f \over dt} = \left({2I [ t - x\lpar t\rpar /c] \over c} \right)\left[{\sqrt{{\rm \pi}_f^2 + {\rm \sigma}_f^2} - {\rm \pi}_f \over \sqrt{{\rm \pi}_f^2 + {\rm \sigma}_f^2} + {\rm \pi}_f} \right]\cr {\rm \pi}_f &\equiv {\,p_f \over S}\semicolon \; {\rm \sigma}_f \equiv {M_{0f} \over S} = {\rm \rho}_0 l}$$

For our one-dimensional (1D) geometry x(t) is the foil location at a time t and the laser irradiance at the micro-foil at space-time (x, t) is caused by photons emitted by the laser at (t − x/c). For a constant laser irradiance I Eq. (25) was solved analytically, π_f = σ_fc[sinhψ + 1/(4sinhψ)] where ψ = (1/3)sinh⁻¹[6It/(ρ₀lc ²) + 2].

Instead of Eq. (25) we write the equation of motion where the velocity β_f is the variable and solve for β_f(t),

(26)$$\eqalign{&{dp_f \over dt} = PS \Rightarrow {d \over dt} \left[\left({\rm \rho}_0 lc \right){{\rm \beta}_f \over \sqrt{1 - {\rm \beta}_f^2}} \right]= {2I \over c} \left({1 - {\rm \beta}_f \over 1 + {\rm \beta}_f} \right)\Rightarrow \cr &{1 \over \left(1 + {\rm \beta}_f \right)^{1/2} \left(1 - {\rm \beta}_f \right)^{5/2}} {d {\rm \beta}_f \over dt} = {2I \over {\rm \rho}_0 c^2 l}}$$

For constant I, this equation is easily integrated

(27)$$\eqalign{\int_0^{\,{\rm \beta}_f} {dx \over \lpar 1 - x\rpar ^{5/2} \lpar 1 + x\rpar ^{1/2}} = {\left(2 - {\rm \beta}_f \right)\sqrt{1 - {\rm \beta}_f^2} \over 3 \left(1 - {\rm \beta}_f \right)^2} - {2 \over 3} \cr &= {2It \over {\rm \rho}_0 c^2 l} \equiv {t \over {\rm \tau}}.}$$

From this solution one can see that β_f(t/τ) → 1 for t/τ → ∞, namely the relativistic velocities are obtained if t/τ ≫ 1, which implies a laser pulse duration much larger than τ = ρ₀c ²l/(2I) or equivalently It ≫ ρ₀c ²l. For an initial density ρ₀ = 1 g/cm³, l = 0.1 µm one gets the dimensionless time τ ~ 50 fs and 0.5 fs for I = 10²² W/cm² and I = 10²⁴ W/cm² accordingly.

Figure 6 describes the micro-foil velocity β_f as a function of the laser pulse duration t in units of τ = ρ₀c ²l/(2I), where ρ₀ is the initial density, l is the foil thickness and I is the laser intensity [W/cm²].

Fig. 6.

Micro-foil velocity β_f as a function of the laser pulse duration t in units of τ = ρ₀c ²l/(2I), where ρ₀ is the initial density, l is the foil thickness and I is the laser intensity [W/cm²].

The kinetic energy of the foil W _Kf and the ratio between W _Kf and the laser energy W _L = ItS are given by

(28)$$\eqalign{{w_{Kf} \over S} &= {\rm \rho}_0 lc^2 \left({1 \over \sqrt{1 - {\rm \beta}_f^2}} - 1 \right) \cr {W_{Kf} \over W_L} &= {\left({1 \over \sqrt{1 - {\rm \beta}_f^2}} - 1 \right)\over \left[\displaystyle{\lpar 2 - {\rm \beta}_f\rpar \sqrt{1 - {\rm \beta}_f^2} \over 6 \left(1 - {\rm \beta}_f \right)^2} - {1 \over 3} \right]}}$$

The solution of Eq. (27) imply the asymptotic relativistic behavior W _Kf/S ~ t ^1/3 for t → ∞ or equivalently for β → 1. On the other domain the non-relativistic limit W _Kf/S ~ t ² is obtained for t → 0. According to the ratio W _Kf/W _L one can see that the maximum efficiency to accelerate a foil is 30% for a flyer velocity of about 0.6c.