The Cosmic-Ray Knee and Ankle as Bookkeeping: The Galactic Cosmic-Ray Power Budget Recomputed from the Measured Spectrum

The energy budget of cosmic rays is recomputed by direct numerical integration of a standard broken-power-law parameterization of the measured spectrum, with breaks at the knee (3 x 10^15 eV) and the ankle (5 x 10^18 eV). The integrated local energy density is 1.73 x 10^-12 erg cm^-3, or 1.078 eV cm^-3, in agreement with the commonly quoted value of about 1 eV cm^-3. Of this total, 99.998% is carried below the knee, 0.0019% between the knee and the ankle, and less than one part in 10^8 above the ankle. Sustaining the Galactic population below the ankle in a confinement volume of 2.08 x 10^67 cm^3 (radius 15 kpc, full thickness 1 kpc) with a residence time of 10^7 yr requires 1.14 x 10^41 erg s^-1. Compared with the kinetic power of supernovae releasing 10^51 erg each, this is 10.8% at a rate of one supernova per 30 yr, 14.3% at one per 40 yr, and 17.9% at one per 50 yr, recovering the classic result that roughly one tenth of the supernova kinetic power suffices. The energy density above the ankle is 1.40 x 10^-20 erg cm^-3; replenished over a Hubble time it corresponds to an extragalactic emissivity of 9.5 x 10^35 erg Mpc^-3 s^-1, a strict lower limit that counts only the observed particles above the ankle. The knee and the ankle are therefore not power features: the power is set almost entirely by GeV particles and by confinement, while the breaks price maximum accelerator energy and the transition between populations. This is a model calculation; every assumption is stated explicitly and every derived number is reproduced by the accompanying script.