tokens: because, first, the amount of early injected electrons, which has already been well cooled, will be larger, and secondly, the cooling break will appear at lower energy for a larger t age . Figure 5. Influence of pulsar’s initial rotation period P 0 on the predicted nebula extent. See Section 3.1.2 for details. 3.1.3 Braking index The braking index n determines the injection profile of electrons and its influence on the energy-dependent morphology is complex. For a large braking index, e.g. n = 3, the age of the pulsar t age is much longer than the initial spin-down time-scale τ 0 (see Fig. 1 ). The spin-down luminosity in this case decreases as t −2 except at very early time (equation 3 ). For a smaller braking index, the spin-down luminosity decreases more quickly with time for t > τ 0 . However, τ 0 will become comparable or even larger than the age of the pulsar. As a result, the global decline slope of the spin-down luminosity from the birth of the pulsar to the present day is flatter for a smaller n . Note that radiation at smaller (larger) radius mainly arises from electrons injected at later (earlier) time, so a smaller n results in a steeper decrease of the electron density profile and hence leads to a less extended morphology and vice versa. Such a tendency can be seen from Fig. 6 except at the low-energy end and the high-energy end. At the low-energy end, the extent of the PWN is mainly determined by the age of the system, so it is larger for a smaller n . At the high-energy end, the cooling time-scale of electrons is very short (within a few thousand years) so the radiating electrons are just recently injected. In other word, the spatial distribution of high-energy electrons is not sensitive to the long-term temporal behaviour of the spin-down luminosity and hence the curves for all three values of n converge at high-energy end. Figure 6. Influence of pulsar’s braking index n on the predicted nebula extent. See Section 3.1.3 for details. 3.1.4 Spectral index We assume a single power law of index p for the injection spectrum (equation 5 ). The SED shows a peak around GeV that is naturally formed due to cooling of electrons and superposition of electrons injected at different epoch, without invoking a broken power-law function for the injection spectrum. The peak in the SED roughly corresponds to the transition in the slope of the present-day electron spectrum from −3 to >−3, so the injection spectral index p can influence the peak energy in the SED. We show the present-day electron spectrum ( E e versus ) with decomposing the contributions from different injection epoch in Fig. 7 for different p . We note that the spectrum shows a hump structure around the cooling break instead of a simple broken power law. This is because the broken power-law spectrum can be formed via synchrotron or IC cooling only when the injection rate of electrons is constant, while the latter decreases with time as the pulsar spins down. Figure 7. Electron
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