Two shocks and a short acceleration window

Supernova Remnant Shocks and Particle Cycles

How do surrounding gas and a local field change a remnant and its energetic seeds? Fast ejecta drive a shock into surrounding gas while a reverse shock processes the ejecta. At the same age, denser surroundings make the remnant smaller and slower. Repeated scattering can raise the momentum of already energetic seeds; the short saved cycle model does not calculate their initial injection or the remnant’s radiation.

Drag the shock to change ambient density at one age. Change a local parallel field and follow the same saved mean-cycle particle record.

Forward shock radius $R_{\rm f}=1.52\,\mathrm{pc}$

Reverse shock radius $R_{\rm r}=1.2\,\mathrm{pc}$

Retained representative

pc

$327\,\mathrm{GeV}$

Escaped share of seed weight $f_{\rm esc}=69.7\%$

Change the medium

Fixed gas age: $148.578\,\mathrm{yr}$ .

Upstream hydrogen density

1.00

n_{\rm H}/(\mathrm{cm^{-3}})

Local normal magnetic field

10.0

B/(\mu\mathrm G)

Read a saved mean cycle

62

\ell

Drag gold for density, purple for field. Drag space to orbit.

Computing the complete default mean-cycle record.

Snapshot, assumptions and budgets

Fixed energy and ejecta mass: $E=10^{44}\,\mathrm J,\quad M=3M_\odot,\quad n=7$ .

The same quarter-wedge is opened in all three spherical surfaces for display; their radii are unchanged and their colours identify geometry. The gas still fills the physical sphere. Soft points sample known unshocked density inside this view cut, with each jittered position evaluated in that same physical field. No density is filled into the unknown shocked layers, and no X-ray emissivity is calculated.

Static sample colours use a clipped logarithmic mass-density scale, $\log_{10}[\rho/(\mathrm{kg\,m^{-3}})]\in[-22.5,-18.5]$ .

The patch is locally parallel. Both arrows have the same normal field; their length uses a square-root strength scale, not distance. The patch handle is enlarged.

Seeds are already energetic protons. Bohm scattering and fluid-convected centres are prescribed; thermal injection, waves, drift, feedback and radiation are not solved.

The external proton seed is fixed: $(pc)_0=100\,\mathrm{GeV}$ .

The gas remains a snapshot. Playback follows only the saved mean cycles of a short window; there are no tracked spatial particle orbits.

A finite record endpoint is not a measured maximum proton energy. An exact mean-cohort ledger does not certify the physical stochastic spectrum.

Registered corners, endpoints and withheld cases passed the original numerical/domain gates. This is finite evidence, not a continuous-domain error theorem.

Bell (1978), 147–156 · Chevalier (1982), 790–797

Model assumptions

This is one specified ejecta-envelope snapshot, using the own n7 spherical adiabatic double-shock solution and a fixed physical age. It is separate from the zero-ejecta Sedov benchmark and has no computed transition into the ejecta core. The three surfaces are the forward shock, contact and reverse shock; they are geometry, not X-ray brightness. Samples show only known unshocked density. No invented post-shock density gradient is filled in. A small parallel patch uses equal normal fields upstream/downstream and prescribed Bohm scattering with fluid-convected scattering centres. The field grip changes strength along that normal; it does not create an oblique shock or a global radial-field solution. Seeds are already ultrarelativistic protons. Positive expected cohorts receive the leading Bell mean log gain and downstream escape probability. Playback reads only saved mean cycles within one percent of the specified age; the gas stays fixed. There are no individual particle trajectories, thermal injection, wave growth/damping, Alfvén drift, pressure feedback, radiative loss or gamma-ray spectrum. Seed energy is external; gained energy, escaped and retained weights have explicit ledgers and a passive-work gate. A finite mean-clock/diffusion-length endpoint is not a true maximum proton energy. Finite joint corners/endpoints/withheld inputs passed the original gates; no uniform continuous-domain error theorem or real stochastic spectrum is claimed. The nonzero mean-log closure discrepancy is retained in the independent scientific reference.

Start with the essentials

Focus question: Which changes come from gas dynamics, and which come from the prescribed scattering?
One-sentence intuition: Density changes the fixed-age shock state. Magnetic strength changes the prescribed cycle clock. The particle model remains passive and does not move the gas snapshot.

Core mathematical model

One envelope family at one age

R_{\rm c}=(Ag^n/\rho_0)^{1/n}t^{(n-3)/n},\qquad n=7

The unshocked envelope falls as the inverse seventh power of radius at this age; its normalization and the dimensionless coefficient are fixed. The computed forward and reverse radii are fixed multiples of the contact radius. Increasing ambient density at fixed energy, mass and age reduces all three radii.

Leading mean gain and escape

G=\frac{4(u_1-u_2)}{3c},\qquad P=\frac{4u_2}{c},\qquad u_2=\frac{u_1}{4}

Speeds are measured in the shock frame and belong to scattering centres moving with the gas. The gain is a mean log-momentum increment of a full cycle; the escape probability gives its downstream lost fraction.

Prescribed Bohm clock

\kappa=\frac{pc}{3eB},\qquad\frac{dt}{d\ln p}=\frac{3}{u_1-u_2}\left(\frac{\kappa_1}{u_1}+\frac{\kappa_2}{u_2}\right)

The field is parallel, so its normal component is equal on the two sides. Wave growth lies outside this prescription. Within it, a larger field shortens the time needed for the same log-momentum change.

Follow every weight and energy contribution

N_{\rm ret}+N_{\rm esc}=N_0,\quad K_{\rm ret}+K_{\rm esc}=K_{\rm seed}+W_{\rm scat}

External seed energy is separate from newly gained energy. Their sum is conserved exactly for the energies assigned to these mean cohorts. The physical accuracy still depends on how well that mean represents the distribution.

Common difficulties

Reverse relative to which material?

Typical misconceptionA reverse shock must shrink toward the centre in the remnant-centre frame.

Better mental modelIt propagates backward relative to the freely expanding ejecta that enter it. Its radius in the remnant-centre frame can still increase. Here only the fixed instantaneous radius is shown.

A mean cycle is not a tracked particle

Typical misconceptionEvery proton follows the displayed deterministic energy and a known spatial orbit.

Better mental modelPositive weights represent expected cohorts with one mean-log momentum. The independent angular test retains a nonzero difference between mean energy and exponentiated mean log energy. No individual orbit or calibrated spectral width is computed.

Run the experiment

01
Hold age fixed, change density
Drag the gold point on the forward shock outward. Read the ambient density and the three surface sizes, then choose Dense gas slows the front.
What to observe: At this same age, a larger radius corresponds to lower ambient density. The shaded boundaries locate the two shocks and the contact; their colors label surfaces rather than X-ray brightness.
02
Change the local field, not its direction
Drag the purple arrow tip along the patch normal, or compare the weak and strong presets at the same density.
What to observe: The gas radii remain the same. The prescribed scattering clock changes, so the short window can contain different numbers of complete cycles.
03
Calculate one baseline cycle
For the baseline shock, use the displayed speeds to estimate the escape probability and mean log gain, then advance one saved cycle.
What to observe: Both are about 0.0191. Roughly 1.91 percent of the current weight escapes, while survivors gain about 1.93 percent in representative momentum. Those are distinct changes; escaped weights keep their previous cycle momentum.
04
Read the saved short window
Play the saved cycles, pause on a frame and export it. Compare retained and escaped weights, then open the energy details.
What to observe: All plots and export use this accepted record and selected frame. The endpoint belongs to a short mean-clock window, not an inferred maximum energy for a remnant of this age.

Supernova Remnant Shocks and Particle Cycles

Read the same saved mean cycles

Two shocks can belong to one expanding remnant

Core mathematical model

One envelope family at one age

Leading mean gain and escape

Prescribed Bohm clock

Follow every weight and energy contribution

Common difficulties

Reverse relative to which material?

A mean cycle is not a tracked particle

Run the experiment

Hold age fixed, change density

Change the local field, not its direction

Calculate one baseline cycle

Read the saved short window

Related experiments and articles

Supernova remnants, two shocks and energetic seeds