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Low X-ray emission challenges supernovae remnants as the source of cosmic-ray electrons

The X-ray synchrotron emission of each of the young supernova-remnants (SNRs) SN1006, Kepler, Tycho, RCW86 and Cas A, is roughly given by $νL_ν\sim 10^{45}\rm{erg}/t$, where $t$ is the remnant's age. The electrons emitting the X-ray emission cool fast, implying that the X-ray emission is calorimetric and equal to half of the cosmic ray (CR) electron acceleration efficiency (per logarithmic interval of particle energies, at multi TeV energies). Assuming Sedov-Taylor expansion, the resulting CR electron yield per SNR is estimated to be $E^2dN_e/dE\approx 6νL_νt \sim 10^{46}\rm erg$. This is about two orders of magnitudes below the required amount for explaining the observed electron CRs at $E\sim 10\rm GeV$. Possible resolutions are 1. a soft acceleration spectrum allowing much more energy at $E\sim 10\rm GeV$ compared to $E\sim 10\rm TeV$, 2. an increased acceleration efficiency at later phases of the SNR evolution (unlikely), or 3. SNRs are not the source of CR electrons.