tial spin-down time of the pulsar. Note that this mag- netic field evolution model may be ad hoc, but its be- havior is very similar to those adopted in other works [5, 12]. For example, Rees & Gunn (1974) [12] con- sider the stretching and winding of the magnetic field line inside the uniform PWN and gives B ( t ) ∝ t $_{−}$1 . 5 for n = 3 and B ( t ) ∝ t $_{−}$5 / 3 for n = 2 . 5 for t > τ$_{0}$ , where n is the braking index of the pulsar. Another example of de Jager et al. (2009) [5] gives B ( t ) ∝ t $_{−}$1 . 3 in their calculation of non-relativistic MHD equations. in this range. For the components inside the PWN, we assume that the PWN is composed of only magnetic field and relativistic electron-positron plasma, both of which are injected from the pulsar inside the PWN. We divide the energy injection from the pulsar into the magnetic field energy ˙ E$_{B}$ and the relativistic particle energy ˙ E$_{e}$ using the time independent parameter η (0 ≤ η ≤ 1). The fraction parameter η is the injection ratio of the magnetic field energy to the spin-down en- ergy. The fraction parameter η in our model is similar to the magnetization parameter σ in the KC model, although they are not the same. The magnetization parameter σ is the ratio ˙ E$_{B}$/ ˙ E$_{e}$ at the pulsar wind region immediately upstream the termination shock. On the other hand, the fraction parameter η pertains to ˙ E$_{B}$/ ( ˙ E$_{B}$ + ˙ E$_{e}$ ) into the PWN region uniformly. For the particle injection, we also need to deter- mine the injection spectrum of the relativistic par- ticles. Following Venter & de Jager (2006) [14], we assume that the injection spectrum of the relativistic particles Q$_{inj}$ ( γ, t ) obeys a broken power-law We assume that the distribution of the particles in the PWN is isotropic, and then the particle distribu- tion function can be easily volume integrated and is described by N ( γ, t ). The evolution of the particle dis- tribution N ( γ, t ) is given by the continuity equation in the energy space 2009 Fermi Symposium, Washington, D.C., Nov. 2-5 2 C. Evolution of Particle Distribution ∂ ∂t N ( γ, t ) + ∂ ∂γ ( ˙ γ ( γ, t ) N ( γ, t )) = Q$_{inj}$ ( γ, t ) . (4) Q$_{inj}$ ( γ, t ) = { Q$_{0}$ ( t )( γ/γ$_{b}$ ) − p 1 for γ$_{min}$ ≤ γ ≤ γ$_{b}$ , Q$_{0}$ ( t )( γ/γ$_{b}$ ) − p 2 for γ$_{b}$ ≤ γ ≤ γ$_{max}$ , (1) We consider the cooling effects of the relativistic particles ˙ γ ( γ, t ) including the synchrotron radiation, the inverse Compton scattering off the Cosmic Mi- crowave Background Radiation (CMB), and the adia- batic expansion, i.e., where γ is the Lorentz factor of the relativistic elec- tron and positron. We introduce time independent parameters γ$_{min}$ , γ$_{b}$ , γ$_{max}$ , p$_{1}$ and p$_{2}$ which are the minimum, break and the maximum Lorentz factors and the power-law indices at the low and high en- ergy ranges of the injection spectra, respectively. We require that the normalization Q$_{0}$ ( t ) satisfies the fol- lowing equation Note that the inverse Compton cooling ˙ γ$_{IC}$ ( γ ) does not depend on time because we consider that the tar- get photon field is only the CMB. As the cooling ef- fects, we do not include the inverse Compton scatter- ing off the synchrotron radiation field for simplicity because it never be a more important cooling process than the synchrotron cooling. We include it in the calculation of the radiation spectrum. where m$_{e}$ and c are the mass of an electron (or positron) and the speed of light, respectively. Because the magnetic field lines are stretching and winding, it is difficult to model the evolution of the magnetic field in the context of the uniform PWN. We have to solve the relativistic magnetohydrodynamics (MHD) equations to determine the realistic magnetic field evolution [6]. For simplicity, we assume that the magnetic field evolution are determined in the form of the magnetic energy conservation, In this section, we apply our model to the Crab Neb- ula as the standard calibrator of our model. The Crab Nebula is one of the best observed PWN at almost all observable wavelengths. The Crab Pulsar has the period 3 . 31 × 10 $^{$_{−}$2}$s, its time derivative 4 . 21 × 10 $^{$_{−}$13}$s · s $_{−}$1 and the braking in- dex 2.51. The progenitor supernova is SN1054, which means the age of the Crab Nebula t$_{age}$ ≈ 950yr. Note that τ$_{0}$ ≈ 760yr. Here, we adopt that the Crab Neb- ula is a sphere of the diameter ≈ 3pc. Combining with t$_{age}$ ≈ 950yr, the constant expansion velocity is v$_{PWN}$ ≈ 1500km / s, which is close to the observed ex- pansion velocity of the Crab Nebula. In this model, the magnetic field approximately evolves as B ( t ) ∝ t $_{−}$1 . 5 for t > τ$_{0}$ , where τ$_{0}$ is the ini- 4 π 3 ( R$_{PWN}$ ( t )) 3 · ( B ( t )) 2 8 π = ∫ t 0 ηL ( t $^{′}$) dt $^{′}$. (3) (1 − η ) L ( t ) = ∫ γ max γ min Q$_{inj}$ ( γ, t ) γm$_{e}$c $^{2}$dγ, (2) ˙ γ ( γ, t ) = ˙ γ$_{syn}$ ( γ, t ) + ˙ γ$_{IC}$ ( γ ) + ˙ γ$_{ad}$ ( γ, t ) . (5) B. Evolution of Magnetic Field III. APPLICATION TO THE CRAB NEBULA eConf C091122