The synchrotron energy loss of a particle with energy $\varepsilon_e$ is described as $$\frac{d\varepsilon_e}{dt}=-\frac{4}{3}\sigma_{\rm T}c\beta^2\frac{B^2}{8\pi}\left(\frac{\varepsilon_e}{m_ec^2}\right)^2,$$ where $\sigma_{\rm T}$ is the Thomson scattering cross section, $\beta$ is the particle velocity normalized by the speed of light and $m_e$ is the mass of electron/positron. Then, The typical energy loss of the electrons/positrons $\Delta\varepsilon_e$ with energy $\varepsilon_e$ can be estimated as $$\frac{\Delta\varepsilon_e}{\varepsilon_e}\sim 0.3\left(\frac{B_0}{10^{8.5}{\rm G}}\right)^2 \left(\frac{\Omega}{10^3{\rm s}^{-1}}\right)^4\left(\frac{R}{10^6{\rm cm}}\right)^6.
\label{eq:2.8}$$ This means that the high-energy electrons/positrons injected into the shocked region lose roughly 30% of the energy by the synchrotron radiation before diffusing out into ISM. Therefore, as in the case of white dwarf pulsars (KIK11), we can conclude that the radiative energy loss of electrons/positrons in the pulsar wind nebula is not so large.

The above expressions for the estimate of the energy losses are only applicable to the case that the velocity of a MSP is subsonic in ISM. The observed velocity of MSPs is less than that of canonical pulsars in average sense [@Ho05]. However, some MSPs have the large velocity and a few MSPs actually forms bow shock nebulae [@St03; @HB06]. In this case, the size of the bow shock is described as (e.g., Wilkin 1996) $$R_{\rm bow}=\left(\frac{L_{\rm sd}}{4\pi c\rho V^2}\right)^{1/2}\sim 