We investigate the time-dependent behavior of Crab-like pulsar wind nebulae (PWNe) generating a set of models using 4 different initial spin-down luminosities ($L_0 ={1,0.1,0.01,0.001} times L_{0, {rm Crab}}$), 8 values of magnetic fraction ($eta =$ 0.001, 0.01, 0.03, 0.1, 0.5, 0.9, 0.99, and 0.999, i.e., from fully particle dominated to fully magnetically dominated nebulae), and 3 distinctive ages: 940, 3000, and 9000 years. We find that the self-synchrotron Compton (SSC) contribution is irrelevant for $L_{SD}$=0.1, 1, and 10% of the Crab power, disregarding the age and the magnetic fraction. SSC only becomes relevant for highly energetic ($sim 70%$ of the Crab), particle dominated nebulae at low ages (of less than a few kyr), located in a FIR background with relatively low energy density. Since no pulsar other than Crab is known to have these features, these results clarify why the Crab Nebula, and only it, is SSC dominated. No young PWN would be detectable at TeV energies if the pulsars spin-down power is 0.1% Crab or lower. For 1% of the Crab spin-down, only particle dominated nebulae can be detected by H.E.S.S.-like telescopes when young enough (with details depending on the precise injection and environmental parameters). Above 10% of the Crabs power, all PWNe are detectable by H.E.S.S.-like telescopes if they are particle dominated, no matter the age. The impact of the magnetic fraction on the final SED is varied and important, generating order of magnitude variations in the luminosity output for systems that are otherwise the same (equal $P$, $dot P$, injection, and environment).
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