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 \geq 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 infra-red ambient photons) by accelerated electrons. Pion decay (due to proton-proton collisions) is shown to account for only a small fraction, \lesssim10^-2, of the observed flux, as assuming a larger fractional contribution would imply nonthermal 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 \geq 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 \lesssim 10^-4 (K_ep is the ratio of the number of accelerated electrons and 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 \gtrsim 10^-3, and thus rule out K_ep values \lesssim 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 (Abridged).