Repeated shock crossings produce a conditional power law

Consider a steady planar nonrelativistic shock. In its rest frame the upstream plasma enters at speed u₁ and leaves downstream at u₂, with compression ratio r=u₁/u₂. A strong shock in a monatomic ideal gas has r≈4 if magnetic pressure, radiative losses and energetic-particle backreaction are negligible. The accelerated particles are relativistic, moving much faster than either flow. Their distributions are assumed nearly isotropic in the respective local plasma scattering frames, not simultaneously isotropic in the shock frame.

Scattering approximately preserves particle energy in each local scattering frame, but the upstream and downstream frames approach one another. A full crossing and return cycle therefore yields a mean fractional gain proportional to their speed difference. Downstream advection removes some particles from further cycles. For small gain and escape probability, the competition gives a power law: surviving particles become rarer as their energy rises. Injection into this scattering population is an additional plasma-physics problem, not determined by the slope argument alone.

For r=4, the differential relativistic energy spectrum has index two. The isotropic phase-space density instead scales as f(p) proportional to p⁻⁴ because the momentum-space volume measure contributes p² and dE/dp is approximately constant. A nonrelativistic particle energy does not equal pc, so converting that same momentum distribution gives a different energy spectrum. Finite shock strength, scattering-center drift, obliquity and nonlinear backreaction can also change the elementary result. An infinite stationary E⁻² distribution would require an unbounded logarithmic energy budget.

The local acceleration timescale is energy divided by its mean gain rate. For equal upstream and downstream diffusion coefficients D and compression four, it is 20D/u₁². If D depends on energy, the time to accelerate through an extended energy range requires integrating this local time over logarithmic energy. An acceleration history must fit within the age and confinement time and beat the relevant losses. A gyroradius smaller than the accelerator is a necessary geometric condition for magnetic confinement, but does not establish that sufficient cycles occur before the source evolves.

Electrons can lose energy through synchrotron radiation and inverse Compton scattering before the shock age is reached. The simple Compton loss formula applies only in the Thomson regime; recoil suppresses scattering in the Klein–Nishina regime. Proton and nuclear losses have different target and threshold requirements. Escape also depends on the boundaries: an upstream free-escape scale and a downstream absorbing surface are different models. Thus a measured photon cutoff does not directly equal a proton maximum energy without an emission model, particle composition and transport assumptions.

The probability lab checks repeated independent loss tests only. Reset and set both controls to p=0.01,n=20: predict zero-success probability 0.99²⁰≈0.818. Reset and set p=0.02,n=20: it falls to 0.98²⁰≈0.668. Interpret success as a hypothetical loss flag. The zero-flag bar models survival through the tests; higher-count bars are not repeated escapes of one already removed particle. The instrument contains no energy gain, shock structure or feedback, and its exact bars are not sampled particle trajectories.