ray sources without a strong X-ray counterpart is not affected by the broken power-law assumption. This is because the low energy particles do not contribute to X-ray and high energy γ -ray emissions. 2009 Fermi Symposium, Washington, D.C., Nov. 2-5 4 10 40 10 42 10 44 10 46 10 48 10 50 10 0 10 2 10 4 10 6 10 8 10 10 2  m e  c 2  N( )[ergs] Lorentz Factor No Cooling(10kyr) 0.3kyr 1kyr 3kyr 10kyr B. Conclusions In this paper, we built a model of the spectral evo- lution of PWNe and applied this model to the Crab Nebula as a calibrator of our model. Especially, the magnetic field evolution model is unique and can be close to the reality. FIG. 3: Evolution of the particle distribution. The thin solid line is the distribution at 300yr from the birth. The thick solid, thin dotted and the thin dashed lines are those at 1kyr, 3kyr, and 10kyr, respectively. The dot-dashed line is the total injected particles until the age 10kyr. Used parameters are the same as in Figure 1. The flux decrease of the γ -rays is more moderate than radio to X-rays, because the magnetic field de- creases rapidly. This result means that old PWNe can be observed as γ -ray sources without a strong X- ray counterpart. De Jager & Djannati-Ata¨ı (2008) [3] also suggested that old PWNe can be observed as TeV unidentified sources, because the X-ray emitting par- ticles are cooled more rapidly than TeV γ -ray emitting particles. This is not the same reason as our conclu- sion. The current observed spectrum of the Crab Neb- ula is reconstructed when the fraction parameter has a small value η = 0 . 003. This is consistent with the prediction of the magnetization parameter σ ≪ 1 ob- tained by Kennel & Coroniti (1984a)[9]. They ob- tained σ ≪ 1 from the viewpoint of the current dy- namical structure of the Crab Nebula, while we deter- mine η ≪ 1 from the viewpoint of the spectral evolu- tion. ticle distribution in four energy ranges. First, for γ > 10 $^{8}$, the particle number increases with time. Owing to this increase of the high energy particles, the IC/CMB flux at 10kyr is larger than that at 1kyr above 10 TeV. Secondly, for 10 5 < γ < 10 $^{8}$, the evolu- tion of the particle distribution is complex. The distri- bution is softer than the injection distribution because of the synchrotron cooling in this range. Thirdly, for 10 2 < γ < 10 $^{5}$, the change of the particle distribu- tion is small, the difference between the dashed line (10kyr) and dot-dashed line (10kyr without the cool- ing effects) is only a factor of two. This leads to an important conclusion that the radio flux decrease is mainly because of the decrease of the magnetic field. Combining with the observations of the radio flux de- crease, our model of the magnetic field evolution can be near the truth. Lastly, there exist the particles whose energy is lower than γ$_{min}$ . This is because the adiabatic cooling is still effective at low energy. The smaller value of the current magnetic field B ( t$_{age}$ ) = 87 µ G than inferred value ≈ 300 µ G in most of other papers is needed to reconstruct the observed spectrum of the Crab Nebula. Recent study by Volpi et al. (2008) [16] indicated that the spatially averaged magnetic field strength B$_{mean}$ ≈ 100 µ G in their rela- tivistic MHD simulation. This is close to our value of the magnetic field strength. Our model can predict the spectral evolution of the Crab Nebula, the observed flux decrease of the Crab Nebula at radio wavelengths can be explained by our magnetic field evolution model. The observed flux de- crease rate of the Crab Nebula at optical wavelengths is somewhat larger than our model, but the trend that the decreasing rate increases with frequency is the same as observations. Finally, because we can reproduce the current spec- trum of the typical PWN with the spectral evolution model, we can apply this to other PWNe. We will be able to study whether the observed variations of the PWNe can be understood by the spectral evolution or other effects should be included. Our model of the magnetic field evolution is some- what ad hoc. The time dependence of the magnetic field strength B ∝ t $_{−}$1 . 5 for t > τ$_{0}$ is assumed. This is close to other theoretical considerations [5, 12]. More- over, because our result of the radio flux decrease of the Crab Nebula is almost consistent with the obser- vation, our model of the magnetic field evolution can be near the truth. For the injection spectrum of the particle distribu- tion, the acceleration of the particles is an unsolved problem and we adopt the broken power-law injection. It should be noted that one of the important conclu- sion in our study that old PWNe can be observed as γ - A. Discussion IV. DISCUSSIONS AND CONCLUSIONS eConf C091122