---
title: "Two shocks and a separate point explosion"
description: "Build the ejecta-envelope similarity law, match the two shells at their contact and compare a separately integrated Sedov–Taylor benchmark."
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  zh-CN: "https://sandboxphysics.com/zh/stories/supernova-remnants/two-shocks-and-a-point-blast/"
last_modified: "2026-11-04"
---

[Supernova remnants, shocks and energetic particles](https://sandboxphysics.com/stories/supernova-remnants/)

Supernova remnants · L2

# Two shocks and a separate point explosion

Sandbox Physics · 2026-11-04

Two shocks arise because fast ejecta and ambient gas enter the interaction from different sides. Matching their shocked shells gives an ejecta-envelope solution. A Sedov–Taylor point explosion starts from a different idealization: zero ejecta mass and fixed energy in a uniform medium. Each solution is useful within its own assumptions.

## Before you begin

Read the strong-shock chapter. We introduce self-similarity and dimensionless radius here; integrals add mass or energy over spherical shells.

## Expanding ejecta carry a velocity label

In homologous expansion, each unshocked ejecta parcel moves freely, so its radius is its constant velocity times age: $v=r/t$. Consider a spherical envelope whose density falls with the inverse $n$th power of that velocity. Expansion dilutes a fixed velocity interval as $t^{-3}$. Writing the resulting density in radius coordinates gives the first expression below.

The constant $g^n$ fixes the envelope normalization; it is one named dimensional constant, not an extra numerical power to be adjusted independently. Outside lies stationary ambient gas of uniform density $\rho_0$. We assume cold upstream material and an adiabatic, nonradiative monatomic gas.

At a shock, pressure is of order upstream density times relative speed squared. If all three surfaces grow as $t^m$, both characteristic relative speeds scale as $t^{m-1}$. Their density ratio must therefore stay fixed for the two postshock pressures to maintain a self-similar match. The ejecta density at a surface scales as $t^{n-3-mn}$; setting its exponent to zero gives $m=(n-3)/n$.

$$
\rho_{\rm ej}=g^n t^{\,n-3}r^{-n},\qquad R_{\rm c}=\left(\frac{Ag^n}{\rho_0}\right)^{1/n}t^{(n-3)/n}
$$

The contact radius $R_{\rm c}$ has units of length. The dimensionless constant $A$ is determined by matching the two actual shocked-shell solutions.

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

## Pressure and velocity meet at the contact

Apply the previous chapter’s jump conditions separately at the forward shock and at the reverse shock. Integrate each shell toward the contact. The two sides must have a common contact radius, a common pressure and the velocity needed to move that contact. Their densities can differ; material does not freely cross this separating surface.

Our independent spherical Euler integration gives $R_{\rm f}/R_{\rm c}=1.18109$ and $R_{\rm r}/R_{\rm c}=0.934985$ for $n=7$. The same calculation for $n=12$ gives a different pair of shell thicknesses. The offline producer checks the rounded original Table 1 intervals and independently integrates the swept masses and energy.

Near the ideal contact, a dimensionless profile can become singular. The numerical integration stops a declared positive distance before that endpoint and compares two tolerances and endpoint offsets. No interpolation carries a density or temperature straight across the contact.

The interactive scene uses the n7 radii, immediate shock boundary values and known unshocked densities. The independent reader integrates the shocked shells for its scientific checks; the scene does not invent a full interior density map from the shell boundaries.

[![Own n7 and n12 Euler shock radii compared after normalizing the contact radius to one.](https://sandboxphysics.com/stories/supernova-remnants/materials/double-shock-radii.svg)](https://sandboxphysics.com/stories/supernova-remnants/materials/double-shock-radii.svg)

Own independent Euler output. R denotes reverse shock, C contact and F forward shock. The contact is one in each case; colors identify geometry. Open the SVG to enlarge its labels.

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

## A finite star cannot have an infinite power-law core

The envelope law describes the outer ejecta. Extending it down to zero velocity would give an unphysical central mass. For a dimensional example, join it continuously to a flat-density inner core at velocity $v_t$. Integrate mass and kinetic energy over velocity shells, using $r=vt$ and $dr=t\,dv$. The common time factors cancel.

If density is $Ct^{-3}$ in the core and $Ct^{-3}(v/v_t)^{-n}$ outside, the core and envelope mass terms are proportional to $1/3$ and $1/(n-3)$. The kinetic-energy terms contain $1/5$ and $1/(n-5)$. Both are finite for $n>5$, and their ratio yields the transition velocity below.

The envelope solution remains applicable only while $R_{\rm r}/t>v_t$. In the fixed-age Lab, this condition is checked over the short particle window as well as at its beginning. Increasing density eventually invalidates that condition; a different gas solution is then required and this input is refused.

$$
v_t^2=\frac{10(n-5)E}{3(n-3)M},\qquad C=\frac{M}{4\pi v_t^3\left[1/3+1/(n-3)\right]},\qquad g^n=Cv_t^n
$$

Here $E$ is the prescribed total initial kinetic energy and $M$ the finite ejecta mass. This normalization supplies the envelope. A later encounter with the core requires a new dynamical solution.

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

## Make a prediction at one fixed age

For $n=7$, radius scales as $\rho_0^{-1/7}$ when energy, ejecta mass and age stay fixed. Increase ambient density by a factor of 128. Because 128 is two to the seventh power, every surface radius halves. Its instantaneous speed also halves, since $V=mR/t$ at that same age.

Doubling age is a different experiment: the envelope radius grows by $2^{4/7}\simeq1.486$, whereas its speed changes by $2^{-3/7}\simeq0.743$. The App density presets keep the physical age at about 148.578 years; they do not move to another age to keep the reverse shock at a preferred location.

The factor 128 exercise illustrates the scaling algebra. It is not a claim that both environments lie inside the interactive control domain or remain in the envelope phase. Before applying any dimensional example, check the reverse-shock core condition.

Reading: Sandbox Physics independent supernova-remnant Python science reader

## A point blast has a different conserved starting quantity

In the separate Sedov–Taylor problem, a point deposits fixed energy $E$ into uniform ambient density $\rho_0$, with negligible ejecta mass. No ejecta core or reverse shock is supplied. Dimensions alone require the radius to scale as $(Et^2/\rho_0)^{1/5}$. A dimensionless coefficient remains.

To determine it, integrate the spherical Euler profile behind the strong forward shock, then integrate its kinetic and internal energy over volume. Requiring that integral to equal the prescribed energy fixes the coefficient. This is more information than dimensional analysis provides.

The independent producer uses three shrinking central cuts and reports the unresolved central thermal contribution separately. It obtains $\beta=1.1516664$ and $\beta^5=2.0259724$. Swept mass and a local energy-flux first integral provide additional checks. The public reader reruns the three-cut integrals and outputs these values.

The ejecta-envelope and point-blast laws have different initial assumptions. Their exponents can be compared, but placing one curve after the other does not solve the transition. A consistent transition through the ejecta core requires another dynamical calculation.

$$
R_{\rm ST}=\beta\left(\frac{Et^2}{\rho_0}\right)^{1/5},\qquad V_{\rm ST}=\frac{2R_{\rm ST}}{5t}
$$

This zero-ejecta, constant-energy benchmark is kept separate from the two-shock envelope solution.

Reading: Leahy & Williams (2017), A Python Calculator for Supernova Remnant Evolution, arXiv1701.05942v2 · Sandbox Physics independent supernova-remnant Python science reader

## Exercises and explanations

1. For $n=7$, what is $V_{\rm r}/(R_{\rm r}/t)$, and why does a reverse shock form?

   Read the explanation

   The ratio is $4/7$. Freely expanding ejecta arrive at $R_{\rm r}/t$, faster than the front moves outward, so their relative incoming speed is $3R_{\rm r}/(7t)$. This is the speed processed at that shock.

2. Can dimensional analysis determine the point-blast coefficient by itself?

   Read the explanation

   It fixes the powers of energy, density and time, but leaves a dimensionless constant. The pressure and velocity distribution, combined with the energy integral, fixes that constant.

3. Why is a smooth-looking join between envelope and point-blast radius curves insufficient?

   Read the explanation

   A radius alone does not specify the density, velocity, pressure or energy distribution. The state entering the ejecta core must satisfy the dynamical equations, not just a visually smooth radius curve.

## Try the prediction

### [Test the fixed-age scaling](https://sandboxphysics.com/labs/supernova-remnant-shocks/)

1. Choose two densities inside the permitted interval, predict the radius ratio using the inverse seventh power, then compare their accepted readouts.
2. Check that the same physical age is displayed and that all three boundary radii change together.
3. Run the independent point-blast replay in the method chapter. Compare the exponents on paper; keep its solution separate from the App snapshot.

[Previous: How a strong shock turns motion into heat](https://sandboxphysics.com/stories/supernova-remnants/strong-shocks-turn-motion-into-heat/) [Next: Energetic seeds gain momentum one mean cycle at a time](https://sandboxphysics.com/stories/supernova-remnants/energetic-seeds-and-finite-acceleration/)

## Sources and calculation scope

1. [Chevalier (1982), Self-similar solutions for the interaction of stellar ejecta with an external medium, ApJ258,790–797](https://ui.adsabs.harvard.edu/abs/1982ApJ...258..790C/abstract)

   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. [Leahy & Williams (2017), A Python Calculator for Supernova Remnant Evolution, arXiv1701.05942v2](https://arxiv.org/abs/1701.05942v2)

   Selected sections3.2.1/3.3.3 and printed coefficient on page16. The own Euler point-blast integral is independently recalculated; only these source sections, not the complete paper, are claimed read. TM99 full text was not acquired.

3. [Sandbox Physics independent supernova-remnant Python science reader](https://sandboxphysics.com/stories/supernova-remnants/materials/supernova-remnants-reader-science-v1.zip)

   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.