$\gamma$-ray Radiation from the Pulsar Magnetosphere

High energy radiation from various classes of galactic and extragalactic objects has been observed for nearly 30 years. A large fraction of galactic sources is associated with neutron-stars: rotation powered pulsars, accretion powered pulsars and so on. Neutron stars are very small by astronomical standards. Our own Sun's radius is 100 times bigger than the radius of the Earth. However, the typical radius of a neutron star is thought to be only about 10 kilometers (6.25 miles). At the same time, a neutron star contains up to 1.5 times as much matter as the Sun, making the density of these objects tremendous. A teaspoon of neutron star material weighs about a billion (1,000,000,000) tons. This much matter in such a small space creates an enormous gravitational field, so powerful, in fact, that it can bend light!

Figure 8:

Pulsars are highly magnetized rapidly spinning stars schematically shown in FIG. 8. The strength of magnetic field is estimated by the assumption that the rotation energy of the neutron star powers the radiation originally via the magnetic dipole radiation. The rotation energy $E_{R}$ of a neutron star of inertia moment $I$ rotating with angular velocity $\Omega =2\pi/P$ and its time derivative read

$$E_{R} = {1\over 2}I\Omega^2\simeq 2\times 10^{46}\ ({I\over 10^{45} {\rm g cm^2}})({P\over 1\sec})^{-2} {\rm erg}$$ (48)

$$\dot{E_{R}} = I\Omega\dot{\Omega}\simeq -4\times 10^{31}\ ({I\over 10^{45} {\r... ...cm^2}})({P\over 1\sec})^{-3}\ ({\dot{P}\over 10^{-15}}) {\rm erg\ sec^{-1}}.\$$ (49)

Where $P$ is a pulsar period. The amount of energy transfered from rotation spin-down, Eq. 49, is channeling into the dipole radiation

$$I_d = {2\over 3c^2}\ddot{m}^2 = {2\over 3c^2} m^2\sin^2\Theta\Omega^4,$$ (50)

where $m$ is the magnetic moment. $m$ is related to the magnetic field at the stellar surface $B_s$ by

$$B_s\sim 2{m\over R_s}$$ (51)

Then the radiation rate reads

$$I_d \simeq {{B_s}^2R_s^6\sin^2\Theta \over 6c^3}\Omega^4,$$ (52)

where $R_s$ is the stellar radius. Therefore $I_d=-\dot{E_R}$ gives

$$({B\over 10^{12} {\rm Gauss}})^2 \simeq 6\times 10^{15} ({R_... ...^{45} {\rm g cm^2}})({P\over 1\sec})({\dot{P}\over 10^{-15}}).$$ (53)

Figure: $P-\dot{P}$ diagram for Rotation Powered Pulsars.The pulsars detected exclusively in radio are indicated with dots.

Observations of $P$ and $\dot{P}$ indicate $B_s\sim 10^{12}$ Gauss as plotted in FIG. 9.

Detection of radio, X-ray, and $\gamma$-rays from pulsars has indicated that high energy radiation takes places in pulsar magnetosphere. Process relevant for production and transfer of high energy radiation in a pulsar are:

• Curvature Radiation

• Magnetic Pair Creation $\gamma B\to e^+e^-$

• Pair Creation $\gamma\gamma\to e^+e^-$

• Synchrotron Radiation

• Inverse Compton Scattering

Electrons are thought to be accelerated to relativistic energies and very high energy (VHE) $\gamma$-rays are produced via curvature radiation. Generated photons pair-produce $e^{\pm}$ pairs by either magnetic pair creation (such as Crab Pulsar) or photon-photon pair creation (like Vela pulsar). These secondary electrons are subject to the synchrotron radiation.

Now we can estimate the energy of $\gamma$-rays radiated by the accelerated electrons in the magnetosphere. Equation 47 gives the typical photon energy of synchrotron photons in the magnetosphere of $B_s=10^{12}$ Gauss. Using $\omega_B/B = e/m_ec = 1.76\times 10^7$ rad/sec/Gauss, $\hbar = 1.16\times 10^{-22}$ MeV s and that $m_ec^2=511$ keV, we obtain

$$\epsilon_{\gamma}\sim 46 ({E_e\over 10{\rm MeV}})^2 ({B_s\over 10^{12} \rm Gauss})\quad {\rm MeV}.$$ (54)

Therefore, electrons accelerated to $\sim$ 10 MeV are able to radiate photons with roughly same order of their energies, which is consistent with the Pulsar observation.