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Supernova remnants, shocks and energetic particles

Supernova remnants · L1

How a strong shock turns motion into heat

Sandbox Physics ·

A strong shock slows gas in the shock frame and raises its pressure. For a cold monatomic ideal gas, conservation of mass, momentum and energy fixes the density jump at four. A temperature follows only after we also specify which particles share the thermal energy.

Stand still beside the shock

Choose the local xx axis to point from upstream to downstream. Let the incoming and outgoing gas speeds u1,u2u_1,u_2 be positive along it. Write mass density ρ\rho in kilograms per cubic metre and pressure PP in pascals. A tiny planar patch is sufficient when its thickness is much smaller than the curvature radius.

Follow a fixed area of this stationary front. In a short time, upstream gas fills a cylinder whose length is speed times time. Multiplying its volume by density gives the mass crossing the area. A steady shock cannot accumulate an ever-growing pile of mass inside that narrow transition, so the incoming and outgoing mass fluxes agree.

Call their common value jj. Define compression r=ρ2/ρ1r=\rho_2/\rho_1. Mass conservation immediately gives u2=u1/ru_2=u_1/r. Compression therefore requires a slower downstream flow in this frame. This does not say that shocked ambient gas moves inward in the remnant-centre frame.

j=ρ1u1=ρ2u2,r=ρ2ρ1j=\rho_1u_1=\rho_2u_2,\qquad r=\frac{\rho_2}{\rho_1}
Mass flux has units of mass per area per time. The same front is used on both sides.

Reading: Chevalier (1982), Self-similar solutions for the interaction of stellar ejecta with an external medium, ApJ258,790–797

Keep pressure in the momentum balance

Moving gas transports momentum, and pressure also pushes on the boundary of a control volume. Their sum is the momentum flux. Take the upstream gas to be cold enough that its pressure is negligible compared with its ram pressure. This is the strong-shock assumption used here.

Substitute u2=u1/ru_2=u_1/r into the momentum balance. The downstream pressure is P2=ρ1u12(1−1/r)P_2=\rho_1u_1^2(1-1/r). The gas has slowed, but pressure now carries the missing part of the momentum flux.

Pressure alone has not fixed the compression. We still need to track energy. Take a nonradiative, nonmagnetic ideal gas with no energetic-particle feedback. Its internal energy per unit mass is pressure divided by density and by one less than the adiabatic index. Transporting this gas also requires pressure work; internal energy plus that work is the enthalpy.

The adiabatic index γ\gamma is the ratio of heat capacities at constant pressure and at constant volume. For a monatomic gas whose thermal energy occupies the three translational directions, this ratio is 5/35/3.

P1+ρ1u12=P2+ρ2u22,h=γP(γ−1)ρP_1+\rho_1u_1^2=P_2+\rho_2u_2^2,\qquad h=\frac{\gamma P}{(\gamma-1)\rho}
Here hh is enthalpy per unit mass, in joules per kilogram. It includes the work needed to push gas into and out of the control volume.

Reading: Chevalier (1982), Self-similar solutions for the interaction of stellar ejecta with an external medium, ApJ258,790–797

The energy equation selects the compression

Divide the energy flux by the common mass flux. What remains is kinetic energy plus enthalpy per unit mass. Neglect upstream enthalpy, insert the downstream pressure from momentum conservation and divide by the incoming kinetic energy.

The resulting quadratic has a root r=1r=1, which means no shock, and the compressive root shown below. A monatomic gas has γ=5/3\gamma=5/3, yielding r=4r=4. The outgoing speed is one quarter of the incoming speed, and the downstream pressure is three quarters of the incoming ram pressure.

These values are linked consequences of the same three balances. Choosing a fourfold density jump and an unrelated downstream speed would lose mass conservation. A radiative shock can compress much more; magnetic stresses or a dynamically important energetic-particle population also change this ideal-gas problem.

u122=u222+h2⟹1=1r2+2γγ−1r−1r2\frac{u_1^2}{2}=\frac{u_2^2}{2}+h_2\quad\Longrightarrow\quad 1=\frac{1}{r^2}+\frac{2\gamma}{\gamma-1}\frac{r-1}{r^2}
The first term is surviving bulk kinetic energy; the second is downstream enthalpy.
(r−1)(r−γ+1γ−1)=0,r=4,P2=34ρ1u12(r-1)\left(r-\frac{\gamma+1}{\gamma-1}\right)=0,\qquad r=4,\quad P_2=\frac34\rho_1u_1^2
The compressive branch is used for a cold monatomic ideal gas.

Reading: Chevalier (1982), Self-similar solutions for the interaction of stellar ejecta with an external medium, ApJ258,790–797 · Sandbox Physics independent supernova-remnant Python science reader

Calculate a conditional temperature

To translate pressure into temperature, write P2=ρ2kT/(μmp)P_2=\rho_2kT/(\mu m_p). Here kk is Boltzmann’s constant, mpm_p is the proton mass and μmp\mu m_p is the mean mass per thermal particle, counting both ions and electrons. For fully ionized hydrogen, one proton and one electron share each hydrogen atom’s mass, so μ≃0.5\mu\simeq0.5.

For a worked example, use u1=3000 km s−1u_1=3000\,\mathrm{km\,s^{-1}} and an equilibrated fully ionized hydrogen gas. The jump gives T≃1.02×108 KT\simeq1.02\times10^8\,\mathrm{K}, corresponding to kT≃8.81 keVkT\simeq8.81\,\mathrm{keV}. The factor of four in density is essential in this calculation.

A collisionless astrophysical shock need not give electrons and ions the same temperature immediately. Its mean pressure jump can be well defined while the electron temperature relevant to an X-ray spectrum differs from this single-temperature estimate. Predicting that spectrum also needs ionization and emission physics.

kT=316μmpu12kT=\frac{3}{16}\mu m_pu_1^2
The temperature scales with speed squared. The stated mean particle mass and thermal-sharing assumption are part of the result.

Reading: Sandbox Physics independent supernova-remnant Python science reader

Return to the expanding remnant

A forward shock moving outward at VfV_{\rm f} into stationary ambient gas has u1=Vfu_1=V_{\rm f}. Its downstream gas moves outward at 3Vf/43V_{\rm f}/4 in the remnant-centre frame. Subtracting the shock speed recovers the downstream shock-frame speed with the appropriate inward-pointing local axis.

The reverse shock meets ejecta that are already expanding. Their unshocked speed at the reverse radius is Rr/tR_{\rm r}/t. The upstream relative speed is this ejecta speed minus the reverse-shock speed. Using the forward shock’s speed there would heat the wrong incoming flow. This difference leads naturally to the two-shell problem in the next chapter.

Reading: Chevalier (1982), Self-similar solutions for the interaction of stellar ejecta with an external medium, ApJ258,790–797

Exercises and explanations

  1. At fixed incoming speed and composition, upstream density doubles. What happens to downstream density, pressure and the conditional temperature?

    Read the explanation

    Downstream density and pressure both double. Their ratio is unchanged, so the conditional temperature stays the same. More gas is being processed, but each unit mass has the same incoming kinetic energy.

  2. A shock speed is doubled while the assumptions remain valid. By what factor does the conditional temperature change?

    Read the explanation

    It rises by four because T∝u12T\propto u_1^2. The density compression remains four in the strong-shock ideal-gas limit; it is not multiplied by the same temperature factor.

  3. Why does the energy flux contain enthalpy instead of internal energy alone?

    Read the explanation

    Pressure pushes fluid through the surfaces of the control volume and performs work. Adding that pressure work to internal energy gives enthalpy. Leaving it out yields the wrong compression.

Try the prediction

Change density while holding age fixed

  1. Predict how increasing ambient density changes the remnant radius and shock speed at one age, then compare dilute and dense presets.
  2. Open the boundary details. Compare upstream and immediate downstream mass densities at each shock; each local strong jump is four.
  3. Use the displayed forward speed to calculate the conditional pure-hydrogen temperature offline. The App colors label surfaces and known unshocked density; the conditional temperature belongs to this offline exercise.

Sources and calculation scope

  1. Chevalier (1982), Self-similar solutions for the interaction of stellar ejecta with an external medium, ApJ258,790–797

    Original complete text and printed equations/Table1. Separate cold spherical Euler shells, n>5 envelope phase; no continuation through the core or arbitrary shocked-density interpolation.

  2. Sandbox Physics independent supernova-remnant Python science reader

    Own offline Euler n7/n12, independent zero-ejecta point blast, mean cohorts, angle/first-passage/Jensen checks and synthetic plots. Recalculates fresh coefficients and all default steps; original papers/data and app/Engine code excluded.

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