THE RELATION BETWEEN POST-SHOCK TEMPERATURE, COSMIC-RAY PRESSURE, AND COSMIC-RAY ESCAPE FOR NON-RELATIVISTIC SHOCKS

Our starting point is the basic equations expressing conservation of mass, momentum, and energy flux. We evaluate these expressions for three distinct regions: (0) far upstream, where the presence of accelerated particles can be neglected, (1) in the shock precursor just upstream of the subshock, and (2) behind (downstream of) the shock. Mass flux conservation gives

$$\begin{equation} \rho _0v_0 = \rho _1v_1 = \rho _2 v_2, \end{equation} \tag{ 1 }$$

with v being the velocity of the gas with respect to the shock. Note that v₀ = V_s, the shock velocity in the observer's frame of reference. Momentum flux conservation can be expressed as

$$\begin{equation} P_0 + \rho _0 V_s^2 = P_1 + \rho _1 v_1^2 = P_2 + \rho _2 v_2^2, \end{equation} \tag{ 2 }$$

with P being the pressure.

At this point it is convenient to introduce the Mach number M₀ for the shock structure far upstream,

$$\begin{equation} M_0^2 \equiv \frac{1}{\gamma _g} \frac{\rho _0V_s^2 }{P_0}, \end{equation} \tag{ 3 }$$

with γ_g being the adiabatic index of the thermal particles. In addition, we introduce the compression ratios across the different regions 0, 1, and 2:

$$\begin{equation} \chi _1 \equiv \frac{\rho _1}{\rho _0},\; \chi _2 \equiv \frac{\rho _2}{\rho _1},\; \chi _{12} \equiv \chi _1\chi _2 = \frac{\rho _2}{\rho _0}. \end{equation} \tag{ 4 }$$

We now use the assumption that the thermal pressure in the precursor is only due to adiabatic heating in the precursor, i.e., $P_{1,\rm{th}}= P_0 \chi _1^{\gamma _g}$ (cf. Drury et al. 2009) and that across the subshock the pressure associated with the accelerated particles does not change (see Drury & Voelk 1981; Achterberg 2004). The cosmic-ray terms therefore cancel each other in Equation (2), when considering the momentum flux from region 1 to 2. In other words, the non-thermal particle pressure at the subshock is only relevant for reducing the Mach number, but leads otherwise again to the standard, one-fluid relation for shock compression with a reduced Mach number given by

$$\begin{equation} M_1^2 = \frac{\rho _1v_1^2}{\gamma _gP_1} = \frac{\rho _0V_s^2/\chi _1}{\gamma _g P_0\chi _1^{\gamma _g}} =M_0^2 \chi _1^{-(\gamma _g+1)}. \end{equation} \tag{ 5 }$$

Combining Equations (2)–(4), one finds for the total downstream pressure P₂ and the thermal pressure P_th:

$$\begin{equation} \frac{P_2}{\rho _0V_s^2} = \frac{P_{2,\rm{cr}} + P_{2,\rm{th}}}{\rho _0V_s^2} = \frac{1}{\gamma _g M_0^2} +\left (1 - \frac{1}{\chi _{12}}\right), \end{equation} \tag{ 6 }$$

$$\begin{equation} \frac{P_{2,\rm{th}}}{\rho _1v_1^2} = (1 - w) \frac{P_2}{\rho _1v_1^2} = \frac{\chi _1^{\gamma _g+1}}{\gamma _gM_0^2} + \left (1 - \frac{1}{\chi _2}\right), \end{equation} \tag{ 7 }$$

with the subscript $\rm{cr}$ referring to the cosmic rays. We have introduced here the symbol w describing the fraction of the downstream pressure contributed by cosmic rays (cf. Vink 2008; Helder et al. 2009):

$$\begin{equation} w\equiv \frac{P_{2,\rm{cr}}}{P_{2,\rm{th}}+P_{2,\rm{cr}}}. \end{equation} \tag{ 8 }$$

Using ρ₁v²₁ = ρ₀V²_s/χ₁ we can now derive an expression for the fractional cosmic-ray pressure for a given Mach number M₀:

$$\begin{equation} w = \frac{\big(1-\chi _1^{\gamma _g}\big) + \gamma _g M_0^2\big(1 - \frac{1}{\chi _1}\big)}{1 + \gamma _g M_0^2 \big(1 -\frac{1}{\chi _{12}}\big)}. \end{equation} \tag{ 9 }$$

For high Mach number shocks, this equation simplifies to

$$\begin{equation} w = \frac{1 - \frac{1}{\chi _1}}{1 -\frac{1}{\chi _{12}}}= \frac{\chi _{12} - \chi _2}{\chi _{12} -1}. \end{equation} \tag{ 10 }$$

The compression ratio of the subshock is given by the standard shock relation:

$$\begin{equation} \chi _2 = \frac{(\gamma _g + 1)M_1^2}{(\gamma _g-1)M_1^2 + 2}, \end{equation} \tag{ 11 }$$

with M₁ being given by Equation (5).

Equations (5), (9), and (11) together show that there is a one-to-one relation between the compression ratio in the precursor and the downstream fractional cosmic-ray pressure for a given upstream Mach number M₀.

In order to determine the escaping energy flux carried away by particles diffusing away far upstream, we need to use the expression for conservation of energy flux across the shock, but with a modification in order to express the fact that energy flux may be lost from the system (e.g., Berezhko & Ellison 1999):

$$\begin{equation} \left \lbrace P_2 + u_2 + \frac{1}{2}\rho _2 v_2^2\right \rbrace v_2 = \left \lbrace P_0 + u_0 + (1-\epsilon _{\rm esc})\frac{1}{2}\rho _0V_s^2\right \rbrace V_s, \end{equation} \tag{ 12 }$$

with u = P/(γ − 1) being the internal energy, and

$$\begin{equation} \epsilon _{\rm esc} = \frac{F_{{\rm cr}}}{\frac{1}{2}\rho _0 V_s^3} \end{equation} \tag{ 13 }$$

being the escaping cosmic-ray energy flux, normalized to the total kinetic energy flux of the shock. Note that the escaping energy flux can only be taken out of the kinetic energy flux, as this is the only source of free energy. If we would have considered radiative losses, then the factor (1 − _esc) should have been in front of P₀ as well, as the upstream thermal energy can also be radiated away.

Following Vink (2008) and Helder et al. (2009), we introduce for convenience

$$\begin{equation} G_0 \equiv \frac{\gamma _g}{\gamma _g -1},\; G_2 \equiv w\frac{\gamma _{{\rm cr}}}{\gamma _{{\rm cr}}-1} + (1-w)\frac{\gamma _g}{\gamma _g -1}. \end{equation} \tag{ 14 }$$

Reordering the terms and using Equation (6) gives

$$\begin{equation} \epsilon _{\rm esc} = 1 + \frac{2G_0}{\gamma _g M_0^2} - \frac{2G_2}{\gamma _g M_0^2 \chi _{12}} - \frac{2G_2}{\chi _{12}}\left (1 - \frac{1}{\chi _{12}}\right) - \frac{1}{\chi _{12}^2}. \end{equation} \tag{ 15 }$$

This equation completes the thermodynamic relation between the precursor compression ratio χ₁ and downstream non-thermal pressure and overall energy flux escape. The resulting relation between _esc and w can be seen in Figure 1. For high Mach number shocks, the second and third terms can be omitted.

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In the limit of γ_cr = 5/3 and M₀, M₁ → ∞, one can show that the relation between _esc and w is well approximated by

$$\begin{equation} \frac{\epsilon _{\rm esc}}{w} = \left (1 - \frac{1}{\chi _{12}}\right)^2. \end{equation} \tag{ 16 }$$

Since realistically γ_cr < 5/3 (see Section 3), Equation (16) serves as an upper bound on the escape flux.

For low Mach number shocks, there is a critical Mach number, M₀ ≈ 6, below which cosmic-ray acceleration cannot be thermodynamically supported for γ_cr = 4/3, as can be seen in Figure 1. For γ_cr = 5/3 (Figure 2), cosmic-ray acceleration can be supported for lower Mach numbers, but there is still a limit, because _esc < 0 for M₀ ≈ 2.5.

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To summarize, taking χ₁ as the principal variable and M₀ as an input parameter, one can calculate the subshock Mach number M₁ (Equation (5)) from which the subshock compression ratio follows (Equation (11)), and therewith the overall compression ratio χ₁₂ = χ₁χ₂. This can then be used to calculate w (Equation (9)) and _esc (Equation (15)). We illustrate this in Figure 1.

Note that although for a given w the escape flux _esc can be calculated, the opposite is not true. The reason is that _esc has a maximum. In fact, mathematically the curves beyond the maximum declines further than those indicated in Figure 1 (left), but this would correspond to an unphysical compression factor of χ₂ < 1 at the subshock. The peak value in the overall compression ratio corresponds to a maximum in _esc. The peak value can be found by differentiating the expression for the total compression ratio (see Equation (11)):

$$\begin{equation} \chi _{12}= \frac{(\gamma _g+1)M_1^2\chi _1}{(\gamma _g-1)M_1^2+2} = \frac{(\gamma _g+1)M_0^2\chi _1^{-\gamma _g}}{(\gamma _g-1)M_0^2\chi _1^{-(\gamma _g+1)}+2} \end{equation} \tag{ 17 }$$

with respect to χ₁, and setting dχ₁₂/dχ₁ = 0. This gives

$$\begin{equation} \chi _1 =\left (\frac{\gamma _g-1}{2\gamma _g}M_0^2 \right)^{1/(\gamma _g+1)}. \end{equation} \tag{ 18 }$$

Inserting this in Equation (11) using Equation (5) shows that χ₁₂ reaches a maximum for a subshock compression ratio of

$$\begin{equation} \chi _{2,\rm{max}}=\frac{\gamma _g}{\gamma _g-1}=5/2, \end{equation} \tag{ 19 }$$

with the numerical value valid for γ_g = 5/3.

From Equations (6) and (9) or (7) it is possible to find an expression for the downstream temperature:

$$\begin{eqnarray} k_{\rm B}T_2 &= &P_{2,\rm{th}}/n_2 \nonumber\\ \nonumber &=&(1 - w)\frac{1}{\chi _{12}} \left [ \frac{1}{\gamma _g M_0^2} + \left (1 - \frac{1}{\chi _{12}}\right) \right ] \mu m_{\rm p}V_s^2\\ & =& \frac{1}{\chi _1\chi _{12}} \left [ \frac{\chi _1^{\gamma _g+1}}{\gamma _g M_0^2} + \left (1 - \frac{1}{\chi _{2}}\right) \right ] \mu m_{\rm p}V_s^2, \end{eqnarray} \tag{ 20 }$$

with k_B being the Boltzmann constant and μ being the mean mass per particle in units of the proton mass m_p. This expression should be compared to the temperature expected behind a strong single-fluid shock:

$$\begin{equation} k_{\rm B}T_2 =\frac{1}{\chi _{12}} \left (1 - \frac{1}{\chi _{12}}\right) \mu m_{\rm p}V_s^2 = \frac{3}{16}\mu m_{\rm p}V_s^2, \end{equation} \tag{ 21 }$$

which can be obtained from Equation (20) by setting χ₁₂ = 4 (valid for a strong shock with γ_g = 5/3), w = 0, and M₀ → ∞.

It is useful to define the ratio

$$\begin{equation} \beta = \frac{k_{\rm B}T_2}{\frac{3}{16}\mu m_{\rm p}V_s^2} \end{equation} \tag{ 22 }$$

between the downstream temperature in the presence of cosmic-ray acceleration and the expected temperature for a pure gas shock, as this is a quantity that can be related to existing measurements (cf., Helder et al. 2009, 2010). The behavior of β is shown in Figure 3 (left). Note that Equation (20), like the expression for w (Equation (9)), only relies on momentum conservation. As a result, there is a unique relation between β and w. Since w does not depend on the adiabatic index γ_cr of the accelerated particles, β does not depend on γ_cr, and, therefore, β does not depend on the energy distribution of the accelerated particles.

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In addition, we define the ratio σ = k_BT₂/k_BT₀ between downstream and upstream temperatures, as it is a quantity used by Blasi et al. (2005) and Drury et al. (2009). The expression for this quantity is

$$\begin{eqnarray} \sigma = \frac{k_{\rm B}T_2}{k_{\rm B}T_0} &= &(1-w)\frac{P_2}{\rho _0V_s^2} \frac{\gamma _g M_0^2}{\chi _{12}} \nonumber\\ \nonumber & =& (1-w) \frac{1}{\chi _{12}}\left [ 1 + \gamma _g M_0^2\left (1 - \frac{1}{\chi _{12}}\right) \right ]\\ & =& \frac{1}{\chi _1\chi _{12}}\left [ \chi _1^{\gamma _g+1} + \gamma _g M_0^2 \left (1 - \frac{1}{\chi _2}\right)\right ]. \end{eqnarray} \tag{ 23 }$$

The behavior of σ as a function of w is shown in the right-hand panel of Figure 3. In addition, we show in Figure 4 the downstream Mach number M₂. This shows that for high w the downstream Mach number rapidly approaches M₂ = 1. For χ₂ = 1, i.e., a continuous shock, we have M₂ = 1, which leads to an unstable situation, as was already established by Drury & Falle (1986).

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As mentioned in Section 1, there is now observational evidence for magnetic field amplification by shocks in young SNRs. In our two-fluid approach, we ignore the effects of magnetic fields on the flow parameters, although we hope to come to this issue in the near future. For now, we offer an assessment of the magnitude of magnetic amplification on the cosmic-ray-dominated shocks.

We start by noting that the downstream magnetic field in several young SNRs is consistent with (B²₂/8π)/ρ₀V²_s ≈ 1% (Völk et al. 2005; Vink 2008). The question now is what is the perpendicular magnetic field pressure on the subshock compared to the ram pressure, i.e., what is (B²_1,⊥/8π)/(ρ₁v²₁)? This can be calculated using the fact that B_1,⊥ = B_2,⊥/χ₂:

$$\begin{equation} \frac{B_{1,\perp }^2/(8\pi)}{\rho _1v_1^2}= \frac{B_{2,\perp }^2/(8\pi)}{\rho _0V_s^2} \left (\frac{\chi _1}{\chi _2^2}\right)= \frac{B_{2,\perp }^2/(8\pi)}{\rho _0V_s^2} \left (\frac{\chi _{12}}{\chi _2^3}\right). \end{equation} \tag{ 24 }$$

Since B²_2,⊥ < B²₂ we can say that observations of young SNRs indicate that

$$\begin{equation} \frac{B_{1,\perp }^2/(8\pi)}{\rho _1v_1^2} \lesssim 1\% \left (\frac{\chi _{12}}{\chi _2^3}\right). \end{equation} \tag{ 25 }$$

From Figure 1 (right) we can see that for a large range of w the factor χ₁₂/χ³₂ is smaller than 1. Around the maximum of χ₁₂ we have χ₂ = 2.5. So in order to have (B²₂/8π)/ρ₀V²_s ≳ 10%, one needs χ₁₂ ≳ 158, which is larger than for any of the models depicted in Figure 1.

However, there has been a debate as to whether B² ∝ ρ₀V²_s (Völk et al. 2005) or B² ∝ ρ₀V³_s (Vink 2008; Tatischeff 2009). For the young SNRs, the precise proportionality does not change our conclusion very much, but in very young SNRs and/or for radio supernovae this may be relevant. In particular, for SN 1993J, which was a bright radio supernova, magnetic fields as high as 50 G have been inferred, and B²/(8π) ∼ 0.1P_cr (e.g., Tatischeff 2009). At such a level, the magnetic fields may start to become dynamically important.

Apart from amplifying the magnetic field upstream of the shock, cosmic rays in the precursor may also give rise to non-adiabatic gas heating (e.g., Vladimirov et al. 2008; Caprioli et al. 2009). The effects of non-adiabatic heating will be similar to having a lower Mach number shock, i.e., non-adiabatic heating in the precursor limits the maximum overall compression ratio and limits the maximum possible fractional cosmic-ray pressure in the downstream region.

The entropy jump across a shock is given by the relation (e.g., Zeldovich & Raizer 1966)

$$\begin{equation} \Delta S= \frac{3}{2}k_{\rm B} \ln \left (\frac{P_2\rho _2^{-\gamma _g}}{P_0\rho _0^{-\gamma _g}}\right). \end{equation} \tag{ 26 }$$

This neglects the entropy increase associated with the accelerated particles. Brown et al. (1995) calculated the entropy of non-thermal distributions, which, not surprisingly, always have entropy values below a thermal distribution. Moreover, they normalize the distribution to the number of particles involved. However, in reality, the cosmic-ray component may contribute a dominant fraction of the pressure, but the number of particles is always much smaller than the number of thermal particles. So neglecting the entropy of the cosmic rays is a good approximation.

Figure 5 (left) shows that when the fractional cosmic-ray pressure w increases the jump in entropy decreases. The right-hand panel is more interesting as it shows that for certain values of the escaping energy flux _esc there are two possibilities for the jump in entropy. The maximum value of _esc as a function of entropy change ΔS again occurs for the value for which χ₁₂ has a maximum. One can speculate that whenever there are possible solutions to obtain a certain value for _esc the nature chooses the one that offers the highest entropy jump. On the other hand, the escape has to be facilitated by the details of the acceleration process and particle spectrum, so one should not discount the low entropy branch too easily.

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In the present model, the quantities χ₂, χ₁₂, w, β, and σ are independent of γ_cr and encompass the solutions for particle acceleration based on non-linear kinetic models (Blasi et al. 2005; Vladimirov et al. 2008; Kang et al. 2009; Reville et al. 2009), in which the same equations of the mass flux and the momentum flux conservation are adopted. Typically, the kinetic models depend on two free parameters: the injection efficiency and the maximum momentum, beyond which particles will escape from the system.

As can be seen in Figure 6, our formalism agrees with the results of Blasi et al. (2005), but the solutions by Blasi et al. (2005) all cluster around w ≈ 0.9 and χ₁₂ around the maximum possible compression factor. The two-fluid approach, however, allows for a broader range of solutions.

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In the two-fluid model, _esc depends on γ_cr, which is essentially a free parameter. In the kinetic models, _esc is determined self-consistently by the overall hydrodynamic structure of the shock, including the dynamical effects of the accelerated particles. The energy distribution of the accelerated particles then determines γ_cr = 1 + P_cr/u_cr, where P_cr is the cosmic-ray pressure and u_cr is the cosmic-ray energy density. Hence, the kinematic models predict that the _esc − w diagram should deviate from ours with γ_cr = 4/3 and should lie between our curves of γ_cr = 4/3 and 5/3. In practice, however, the cosmic-ray adiabatic index, as obtained by the kinetic models, is very close to γ_cr = 4/3 for p_max = 10⁵mc, as can be seen in Figure 6. For p_max = 10²mc, we found that a modest increase to γ_cr = 1.4 resulted in a good match between the two-fluid solutions and the kinetic model of Blasi et al. (2005).

Interestingly, the two-fluid solutions presented here indicate that a higher energy flux escape is necessary for γ_cr>4/3, but from the point of view of the spectral energy distribution, the energy escape flux is more difficult to achieve for spectra with spectral energy indices Γ>2 (N(E) ∝ E^−Γ), corresponding to γ_cr>4/3. The reason is that for Γ>2 most of the cosmic-ray energy is contained by mildly relativistic particles, whereas most of the escaping particles will be near the maximum of the energy distribution (p_max in the Blasi model). Hence, the escaping particles carry away only a small fraction of the internal energy. For Γ < 2 most energy is indeed concentrated around p_max and the necessary escape flux is easily generated.

So from a thermodynamic point of view, a high fractional cosmic-ray pressure w requires large values of _esc, which in turn is best achieved with γ_cr ≈ 4/3.

One of the motivations for this work has been the measurements of the downstream temperature of SNRs with reasonably well-known velocities (Helder et al. 2009, 2010). These temperatures were measured using the broad component of the Hα line. This component is due to charge exchange between neutral hydrogen atoms entering the shock and the downstream population of shock-heated protons. Hence, the width of the line is caused by thermal Doppler broadening and reflects the downstream proton temperature. There is some ambiguity as to how to relate the proton temperature to the overall downstream plasma temperature, as the different plasma constituents (electrons, protons, helium, other ions) may not be in thermal equilibrium. However, the proton temperature is always expected to be within a factor of two of the mean plasma temperature.

Using Equation (20) (see also Figure 3) one can easily estimate from a measured temperature and shock velocity what the downstream fractional cosmic-ray pressure is and what the required escape flux is. The only ambiguity that is left for estimating _esc is what the effective cosmic-ray adiabatic index is. However, γ_cr is not important for determining w. For the SNRs under consideration, the shock velocities are well in excess of 1000 km s⁻¹. For a typical sound speed in the interstellar medium of 10 km s⁻¹ we have M₀>100. This means that the high Mach number approximation is valid as long as w ≲ 0.9. This appears to be the case for the SNRs considered below.

For the northeastern region of the TeV gamma-ray-emitting remnant RCW 86, Helder et al. (2009) measured a downstream temperature of k_BT_p = 2.3 ± 0.3 keV for a measured shock velocity of V_s = 6000 ± 2800 km s⁻¹. Nominally, this corresponds to β = 0.055 (see Equation (20)), but given the systematic uncertainties and the ambiguity due to non-equilibration of temperatures β could be as high as β = 0.31. Using now Equations (9) and (15) in the limit for high Mach numbers, the measured values of β correspond to downstream fractional cosmic-ray pressures and escape fractions of w = 0.81, _esc = 0.59 and w = 0.51, _esc = 0.20, for γ_cr = 4/3 and β = 0.055 and β = 0.31, respectively. For γ_cr = 5/3 this is _esc = 0.72 and _esc = 0.38, respectively, with w unchanged.

For the young Large Magellanic Cloud remnant 0509-67.5, Helder et al. (2010) determined the post-shock temperatures in two regions. The most constraining measurement was for the southwestern region for which k_BT_p = 28.7 keV for a conservative velocity estimate of V_s = 5000 km s⁻¹, corresponding to β = 0.58. This translates into a downstream fractional cosmic-ray pressure of w = 0.29, _esc = 0.06 for γ_cr = 4/3 or _esc = 0.19 for γ_cr = 5/3.

Interestingly, these new estimates of w based on Equations (9) and (15) are not far from the lower limits given by Helder et al. (2009, 2010). The reason is that the less constraining relations used by Helder et al. (2009, 2010) allow in principle for a higher fractional cosmic-ray pressure by reducing the cosmic-ray escape flux. The relations derived here do not allow for this possibility. Moreover, the relation that was used to determine the lower limit by Helder et al. (2009, 2010) is a limiting case of the shock relations presented here (see Equation (16)).

This brings us back to one of the principal issues currently discussed in cosmic-ray physics, namely, do very efficiently accelerating shocks exist? The post-shock temperatures measured by Helder et al. (2009, 2010) indicate high values for the cosmic-ray pressure, but not as high as usually found by particle acceleration models, where the efficiency is often found to be close to 90%, with total compression factors as high as χ₁₂ = 70 (e.g., Blasi et al. 2005). It has been argued by Drury et al. (2009) that such extreme conditions may exist in the prominent TeV gamma-ray source SNR RX J1713.7-3946 (Aharonian et al. 2004). Apart from high gamma-ray luminosity, this idea is based on the lack of detectable thermal X-ray emission. According to Drury et al. (2009), this could be due to a very low post-shock plasma temperature, possibly as low as six times the upstream. Alternatively, RX J1713.7-3946 may evolve in a low-density medium, resulting in a low thermal luminosity, as the thermal luminosity scales with the square of the density. The equations derived here allow for very low downstream temperatures. However, the claim by Drury et al. (2009) that the downstream temperature may only be six times the upstream temperature seems unrealistic, as Equations (9) and (15) show that this is only true for unrealistically low Mach numbers. For more realistic Mach numbers for young SNRs, say M>75, the downstream temperature is at least 80 times the upstream temperature.

It could be that the young SNRs that we observe are already past their prime as cosmic-ray accelerators, although the cosmic-ray escape in RCW 86 appears still high enough to explain the cosmic-ray energy production of SNRs. Specifically, it has been argued that in the very early SNR phase/supernova phase cosmic-ray acceleration may be very efficient (e.g., Bell & Lucek 2001; Vink 2008), see, for example, the case of SN 1993J (Tatischeff 2009). Note that a steady escape of cosmic rays during the evolution of an SNR is favored over a more sudden release of cosmic rays at the end of the SNR evolution, as in the latter case adiabatic losses have lowered the energy of the cosmic rays (Voelk et al. 1984). If one wants to explain the cosmic-ray energy density in the Galaxy by cosmic-ray escaping from young SNR shocks (i.e., _esc = 5%, see Section 1), then on average over the lifetime of the SNR a fractional cosmic-ray pressure is necessary of at least w ≈ 30% for γ_cr = 4/3.