In which shell-type SNRs should we look for gamma-rays and neutrinos from P-P collisions?

We present a simple analytic model for the various contributions to the non-thermal emission from shell-type SNRs and show that this model's results reproduce well the results of previous detailed calculations. We show that the <= 1 TeV gamma ray emission from the shell type SNRs RX J1713.7-3946 and RX J0852.0-4622 is dominated by inverse-Compton scattering of CMB photons (and possibly infrared ambient photons) by accelerated electrons. Pion decay (due to proton-proton collisions) is shown to account for only a small fraction, less than or similar to 10(-2), of the observed flux, as assuming a larger fractional contribution would imply non-thermal radio and x-ray synchrotron emission and thermal x-ray bremsstrahlung emission that far exceed the observed radio and x-ray fluxes. Models where pion decay dominates the >= 1 TeV flux avoid the implied excessive synchrotron emission (but not the implied excessive thermal x-ray bremsstrahlung emission) by assuming an extremely low efficiency of electron acceleration, K-ep less than or similar to 10(-4) (where Kep is the ratio of the number of accelerated electrons to the number of accelerated protons at a given energy). We argue that observations of SNRs in nearby galaxies imply a lower limit of K-ep greater than or similar to 10(-3), and thus rule out K-ep values less than or similar to 10(-4) (assuming that SNRs share a common typical value of K-ep). It is suggested that SNRs with strong thermal x-ray emission, rather than strong non-thermal x-ray emission, are more suitable candidates for searches of gamma rays and neutrinos resulting from proton-proton collisions. In particular, it is shown that the neutrino flux from the SNRs above is probably too low to be detected by current and planned neutrino observatories. Finally, we note that the magnetic field value implied by the comparison of x-ray to gamma-ray emission, similar to 10 mu G, can be used to constrain magnetic field amplification.