constant. In the Thomson limit $\epsilon^{\prime}\ll1$, and thus
the scattered photon energy in PRF can be approximated as

\begin{equation}
\epsilon{}_{s}^{\prime}\simeq\epsilon^{\prime}+\epsilon{}^{\prime2}(\mu^{\prime}-\mu{}_{s}^{\prime})^{2}/2\approx\epsilon^{\prime}.\label{cascade.ics_simple_scateng}
\end{equation}

Equations <ref> and <ref>
show that a differential magnetic Compton scattering cross section
when $\gamma\gg1$ is similar in form to a nonmagnetic Thomson cross
section. The important difference is that the magnitude of the cross
section is enhanced when $\epsilon^{\prime}$ approaches $\epsilon_{_{B}}$
and is depressed at energies $\epsilon^{\prime}<\epsilon_{_{B}}$.
The total cross section for magnetic Compton scattering, obtained
by integrating Equation <ref> over $\mu{}_{s}^{\prime}$,
was calculated by 50, 190 and is given by

\begin{equation}
\sigma^{\prime}=\frac{\sigma_{_{{\rm IC}}}}{2}\left[\frac{u^{2}}{\left(u+1\right){}^{2}}+\frac{u^{2}}{\left(u-1\right){}^{2}+a^{2}}\right],\label{eq:cascade.ics_cross}
\end{equation}

where $\sigma_{_{{\rm IC}}}=\sigma_{{\rm _{T}}}$, $\sigma_{{\rm _{T}}}$
is the Thomson cross section, $u=\epsilon^{\prime}/\epsilon_{_{B}}$,

§.§.§ ICS cross section in the Klein-Nishina regime

The Klein-Nishina regime includes quantum effects due to the relativistic
nature of scattering, and it requires that $\gamma\epsilon\left(1-\mu\right)\gtrsim1$.
The principal effect is to reduce the cross section from its classical
value as the photon energy in PRF becomes large. In the Klein-Nishina
regime instead of $\sigma_{{\rm _{IC}}}=\sigma_{_{{\rm T}}}$ we can
use the following relationship

\begin{equation}
\sigma_{_{{\rm IC}}}=\sigma_{_{{\rm KN}}}=\frac{3}{4}\sigma_{_{{\rm T}}}\left\{ \frac{1+\epsilon^{\prime}}{\epsilon^{\prime3}}\left[\frac{2\epsilon^{\prime}\left(1+\epsilon^{\prime}\right)}{1+2\epsilon^{\prime}}-\ln\left(1+2\epsilon^{\prime}\right)\right]+\frac{1}{2\epsilon^{\prime}}\ln\left(1+2\epsilon^{\prime}\right)-\frac{1+3\epsilon^{\prime}}{\left(1+2\epsilon^{\prime}\right)^{2}}\right\} .
\end{equation}

In an extreme relativistic regime $\epsilon^{\prime}\gg1$ the Klein-Nishina
formula can be simplified to

\begin{equation}
\sigma_{{\rm _{KN}}}\approx\frac{3}{8}\sigma_{_{{\rm T}}}\epsilon^{\prime-1}\left[\ln\left(2\epsilon^{\prime}\right)+\frac{1}{2}\right].\label{eq:cascade.kn_cross}
\end{equation}

The above formula clearly shows that Inverse Compton Scattering is
less efficient for photons with energy in PRF significantly exceeding
particle rest energy.

§.§.§ QED Compton Scattering cross section

Previous studies on upscattering and energy loss by relativistic particles
have used the non-relativistic, magnetic Thomson cross section for
resonant scattering or the Klein-Nishina cross section for thermal-peak
scattering. As noted by 71, this approach does
not account for the relativistic quantum effects of strong magnetic
fields ($B>10^{12}\,{\rm G}$). When the photon energy exceeds $mc^{2}$
in the particle rest frame, the strong magnetic field significantly
lowers the Compton scattering cross section below and at the resonance.
71 developed expressions for the scattering of
ultrarelativistic electrons with $\gamma\gg1$ moving parallel to
the magnetic field. Because of the large Lorentz Factor of particle
$\gamma$, the photon incident angle $\psi$ gets Lorentz concentrated
to $\psi^{\prime}\approx\psi/2\gamma\approx0^{\circ}$ in the PRF.
The differential cross section in the rest frame of the particle can
be written as

\begin{equation}
\frac{{\rm d}\sigma{}_{\|,\perp}^{\prime}}{{\rm d}\cos\psi{}_{s}^{\prime}}=\frac{3\sigma_{_{{\rm T}}}}{16\pi}\frac{\epsilon{}_{s}^{\prime2}e^{-\epsilon{}_{s}^{\prime2}\sin^{2}\left(\psi_{s}^{\prime}/2\epsilon_{_{B}}\right)}}{\epsilon^{\prime}\left(2+\epsilon^{\prime}-\epsilon{}_{s}^{\prime}\right)\left[\epsilon{}_{s}^{\prime}+\epsilon^{\prime}\epsilon{}_{s}^{\prime}\left(1-\cos\psi{}_{s}^{\prime}\right)-\epsilon{}_{s}^{\prime2}\sin^{2}\psi{}_{s}^{\prime}\right]}\frac{1}{l!}\left(\frac{\epsilon{}_{s}^{\prime2}\sin^{2}\psi{}_{s}^{\prime}}{2\epsilon_{_{B}}}\right)G_{\|,\perp},\label{eq:cascade.cross_gonthier2}
\end{equation}

\begin{equation}
\end{equation}

\begin{equation}
\begin{split}\hat{G}_{\mathrm{no-flip}}^{\|}= & \int_{0}^{2\pi}\left|G_{\mathrm{no-flip}}^{\|,\|}\right|^{2}{\rm d}\phi^{\prime}=\int_{0}^{2\pi}\left|G_{\mathrm{no-flip}}^{\perp,\|}\right|^{2}{\rm d}\phi^{\prime}=\\
= & 2\pi\left\{ \left[\left(B_{1}+B_{3}+B_{7}\right)\cos\psi{}_{s}^{\prime}-\left(B_{2}+B_{6}\right)\sin\psi{}_{s}^{\prime}\right]^{2}+\left(B_{4}\cos\psi{}_{s}^{\prime}-B_{5}\sin\psi{}_{s}^{\prime}\right)^{2}\right\} ,\\
\hat{G}_{\mathrm{no-flip}}^{\perp}= & \int_{0}^{2\pi}\left|G_{\mathrm{no-flip}}^{\|,\perp}\right|^{2}{\rm d}\phi^{\prime}=\int_{0}^{2\pi}\left|G_{\mathrm{no-flip}}^{\perp,\perp}\right|^{2}{\rm d}\phi^{\prime}=\\
= & 2\pi\left[\left(B_{1}-B_{3}-B_{7}\right)^{2}+B_{4}^{2}\right],\\
\hat{G}_{\mathrm{flip}}^{\|}= & \int_{0}^{2\pi}\left|G_{\mathrm{flip}}^{\|,\|}\right|^{2}{\rm d}\phi^{\prime}=\int_{0}^{2\pi}\left|G_{\mathrm{flip}}^{\perp,\|}\right|^{2}{\rm d}\phi^{\prime}=\\
= & 2\pi\left\{ \left[\left(C_{1}+C_{3}+C_{7}\right)\cos\psi{}_{s}^{\prime}-\left(C_{2}+C_{6}\right)\sin\psi{}_{s}^{\prime}\right]^{2}+\left(C_{4}\cos\psi{}_{s}^{\prime}-C_{5}\sin\psi{}_{s}^{\prime}\right)^{2}\right\} ,\\
\hat{G}_{\mathrm{flip}}^{\perp}= & \int_{0}^{2\pi}\left|G_{\mathrm{flip}}^{\|,\perp}\right|^{2}{\rm d}\phi^{\prime}=\int_{0}^{2\pi}\left|G_{\mathrm{flip}}^{\perp,\perp}\right|^{2}{\rm d}\phi^{\prime}=\\
= & 2\pi\left[\left(C_{1}-C_{3}-C_{7}\right)^{2}+C_{4}^{2}\right].
\end{split}
\end{equation}

The imaginary terms and the $\phi^{\prime}$ dependence are isolated
in the polarisation components and in the phase exponentials, leading
to elementary integrations over the azimuthal angle, $\phi^{\prime}$

The differential cross section depends on the final Landau state $l$,
thus a sum must be calculated over all the contributing Landau states.
The energy of the scattered photon is given by [71]

\begin{equation}
\epsilon{}_{s}^{\prime}=\frac{2\left(\epsilon^{\prime}-l\epsilon_{_{B}}\right)}{1+\epsilon^{\prime}\left(1-\cos\psi{}_{s}^{\prime}\right)+\left\{ \left[1+\epsilon^{\prime}\left(1-\cos\psi{}_{s}^{\prime}\right)\right]^{2}-2\left(\epsilon^{\prime}-l\epsilon_{_{B}}\right)\sin^{2}\psi{}_{s}^{\prime}\right\} ^{\frac{1}{2}}},
\end{equation}

where $l$ is the final Landau level of the scattered particle. Each
final state has an energy threshold of $l\epsilon_{_{B}}$, thus the
maximum contributing Landau state $l_{{\rm max}}$ can be expressed
as: $\epsilon^{\prime}/\epsilon_{_{B}}-1<l_{{\rm max}}<\epsilon^{\prime}/\epsilon_{_{B}}$.
To obtain the energy-dependent cross section, the Romberg's method
can be used to numerically integrate the differential cross section
over $\psi{}_{s}^{\prime}$. For this particular case (scattering
of relativistic particles) there is only one resonance appearing at
the fundamental cyclotron frequency $\epsilon_{_{B}}=\beta_{q}=eB/\left(mc\right)$.

The values of $B$ and $C$ can be expressed as:

\begin{equation}
\begin{split}B_{1}= & \frac{2\epsilon^{\prime}-\epsilon^{\prime}\epsilon{}_{s}^{\prime}\left(1-\cos\psi{}_{s}^{\prime}\right)}{2\left(\epsilon^{\prime}-\epsilon_{_{B}}\right)},\\
B_{2}= & -\frac{\left(\epsilon^{\prime}-\epsilon{}_{s}^{\prime}\cos\psi{}_{s}^{\prime}\right)\left(2l\epsilon_{_{B}}-\epsilon{}_{s}^{\prime2}\sin^{2}\psi{}_{s}^{\prime}\right)+2l\epsilon_{_{B}}\epsilon^{\prime}}{2\epsilon{}_{s}^{\prime}\sin\psi{}_{s}^{\prime}\left(\epsilon^{\prime}-\epsilon_{_{B}}\right)},\\
B_{3}= & \frac{l\epsilon_{_{B}}\left(2l\epsilon_{_{B}}-2\epsilon_{_{B}}-\epsilon{}_{s}^{\prime2}\sin^{2}\psi{}_{s}^{\prime}\right)}{\epsilon{}_{s}^{\prime2}\sin^{2}\left[\psi{}_{s}^{\prime}\left(\epsilon^{\prime}-\epsilon_{_{B}}\right)\right]},\\
B_{4}= & -\frac{2\epsilon{}_{s}^{\prime}+\epsilon^{\prime}\epsilon{}_{s}^{\prime}\left(1-\cos\psi{}_{s}^{\prime}\right)-\epsilon{}_{s}^{\prime2}\sin^{2}\psi{}_{s}^{\prime}}{2\left[\epsilon^{\prime}\epsilon{}_{s}^{\prime}\left(1-\cos\psi{}_{s}^{\prime}\right)-\epsilon^{\prime}-\epsilon_{_{B}}\right]},\\
B_{5}= & -\frac{\left(\epsilon^{\prime}-\epsilon{}_{s}^{\prime}\cos\psi{}_{s}^{\prime}\right)\epsilon{}_{s}^{\prime}\sin\psi{}_{s}^{\prime}}{2\left[\epsilon^{\prime}\epsilon{}_{s}^{\prime}\left(1-\cos\psi{}_{s}^{\prime}\right)-\epsilon^{\prime}-\epsilon_{_{B}}\right]},\\
B_{6}= & \frac{l\epsilon_{_{B}}\cos\psi{}_{s}^{\prime}}{\sin\psi{}_{s}^{\prime}\left[\epsilon^{\prime}\epsilon{}_{s}^{\prime}\left(1-\cos\psi_{s}^{\prime}\right)-\epsilon^{\prime}+\epsilon_{_{B}}\right]},\\
B_{7}= & \frac{2l\left(l-1\right)\epsilon_{_{B}}^{2}}{\epsilon{}_{s}^{\prime2}\sin^{2}\psi{}_{s}^{\prime}\left[\epsilon^{\prime}\epsilon{}_{s}^{\prime}\left(1-\cos\psi{}_{s}^{\prime}\right)-\epsilon^{\prime}+\epsilon_{_{B}}\right]},\\
C_{1}= & \sqrt{2l\epsilon_{_{B}}}\frac{\epsilon^{\prime}}{2\left(\epsilon^{\prime}-\epsilon_{_{B}}\right)},\\
C_{2}= & -\sqrt{2l\epsilon_{_{B}}}\frac{2\epsilon^{\prime}+2\epsilon^{\prime}{}^{2}-\epsilon^{\prime}\epsilon{}_{s}^{\prime}\left(1-\cos\psi{}_{s}^{\prime}\right)-2l\epsilon_{_{B}}+\epsilon{}_{s}^{\prime2}\sin^{2}\psi{}_{s}^{\prime}}{2\epsilon^{\prime}{}_{s}\sin\psi{}_{s}^{\prime}\left(\epsilon^{\prime}-\epsilon_{_{B}}\right)},\\
C_{3}= & \sqrt{2l\epsilon_{_{B}}}\frac{\left(\epsilon^{\prime}-\epsilon_{s}^{\prime}\cos\psi{}_{s}^{\prime}\right)\left(2l\epsilon_{_{B}}-2\epsilon_{_{B}}-\epsilon{}_{s}^{\prime2}\sin^{2}\psi{}_{s}^{\prime}\right)}{2\epsilon{}_{s}^{\prime2}\sin^{2}\psi{}_{s}^{\prime}\left(\epsilon^{\prime}-\epsilon_{_{B}}\right)},\\
C_{4}= & -\sqrt{2l\epsilon_{_{B}}}\frac{\epsilon{}_{s}^{\prime}\cos\psi{}_{s}^{\prime}}{2\left[\epsilon{}_{s}^{\prime}\epsilon^{\prime}\left(1-\cos\psi{}_{s}^{\prime}\right)-\epsilon^{\prime}-\epsilon_{_{B}}\right]},\\
C_{5}= & \sqrt{2l\epsilon_{_{B}}}\frac{\epsilon{}_{s}^{\prime}\sin\psi{}_{s}^{\prime}}{2\left[\epsilon{}_{s}^{\prime}\epsilon^{\prime}\left(1-\cos\psi{}_{s}^{\prime}\right)-\epsilon^{\prime}-\epsilon_{_{B}}\right]},\\
C_{6}= & -\sqrt{2l\epsilon_{_{B}}}\frac{2\epsilon{}_{s}^{\prime}+\epsilon^{\prime}\epsilon{}_{s}^{\prime}\left(1-\cos\psi{}_{s}^{\prime}\right)-\epsilon{}_{s}^{\prime2}\sin^{2}\psi{}_{s}^{\prime}}{2\epsilon{}_{s}^{\prime}\sin\psi{}_{s}^{\prime}\left[\epsilon{}_{s}^{\prime}\epsilon^{\prime}\left(1-\cos\psi{}_{s}^{\prime}\right)-\epsilon^{\prime}+\epsilon_{_{B}}\right]},\\
C_{7}= & \sqrt{2l\epsilon_{_{B}}}\frac{\left(l-1\right)\epsilon_{_{B}}\left(\epsilon^{\prime}-\epsilon{}_{s}^{\prime}\cos\psi{}_{s}^{\prime}\right)}{\epsilon{}_{s}^{\prime2}\sin^{2}\psi{}_{s}^{\prime}\left[\epsilon{}_{s}^{\prime}\epsilon^{\prime}\left(1-\cos\psi{}_{s}^{\prime}\right)-\epsilon^{\prime}+\epsilon_{_{B}}\right]}.
\end{split}
\end{equation}

Although the expressions presented above describe the exact cross
section for ICS in strong magnetic fields, due to their complexity
their usage in cascade simulation is limited.

§.§.§ Approximate cross section (final states l=0)

An approximation to the exact $l=0$ differential cross section can
be given by assuming that the scattering is significantly below the
resonance, where $\epsilon^{\prime}<\epsilon_{_{B}}$ and also $\epsilon^{\prime}<1$.
71 showed that by keeping only terms to first
order in $\epsilon^{\prime}$ and $\epsilon{}_{s}^{\prime}$ in the
region of validity, it agrees very well with the exact $l=0$ cross
section. The approximation overestimates the exact $l=0$ cross section
above the region of validity $\epsilon^{\prime}>\epsilon_{_{B}}$.
However, the approximation is close to the total cross section for
both energy regions ($\epsilon^{\prime}<\epsilon_{_{B}}$ and $\epsilon^{\prime}>\epsilon_{_{B}}$
), even for high magnetic field strengths (see Figure <ref>).

According to 71, the polarisation-dependent and
averaged, approximate cross section can be calculated as:

\begin{equation}
\sigma{}^{\prime\|\rightarrow\|}=\sigma{}^{\prime\perp\rightarrow\|}=\frac{3\sigma_{_{{\rm T}}}}{16}\left[g\left(\epsilon^{\prime}\right)-h\left(\epsilon^{\prime}\right)\right]\left[\frac{1}{\left(\epsilon^{\prime}-\epsilon_{_{B}}\right)^{2}}+\frac{1}{\left(\epsilon^{\prime}+\epsilon_{_{B}}\right)^{2}}\right],
\end{equation}

\begin{equation}
\sigma{}^{\prime\|\rightarrow\perp}=\sigma{}^{\prime\perp\rightarrow\perp}=\frac{3\sigma_{_{{\rm T}}}}{16}\left[f\left(\epsilon^{\prime}\right)-2\epsilon^{\prime}h\left(\epsilon^{\prime}\right)\right]\left[\frac{1}{\left(\epsilon^{\prime}-\epsilon_{_{B}}\right)^{2}}+\frac{1}{\left(\epsilon^{\prime}+\epsilon_{_{B}}\right)^{2}}\right],
\end{equation}

\begin{equation}
\sigma{}_{{\rm avg}}^{\prime}=\frac{3\sigma_{_{{\rm T}}}}{16}\left[g\left(\epsilon^{\prime}\right)+f\left(\epsilon^{\prime}\right)-\left(1+2\epsilon^{\prime}\right)h\left(\epsilon^{\prime}\right)\right]\left[\frac{1}{\left(\epsilon^{\prime}-\epsilon_{_{B}}\right)^{2}}+\frac{1}{\left(\epsilon^{\prime}+\epsilon_{_{B}}\right)^{2}}\right],\label{cascade.cross_gonthier}
\end{equation}

\begin{equation}
\begin{array}{c}
\end{array}
\end{equation}

\begin{equation}
\end{equation}

\begin{equation}
h\left(\epsilon^{\prime}\right)=\left\{ \begin{array}{ll}
\frac{\epsilon{}^{\prime2}}{\sqrt{\epsilon^{\prime}\left(2-\epsilon^{\prime}\right)}}\arctan\left[\frac{\sqrt{\epsilon^{\prime}\left(2-\epsilon^{\prime}\right)}}{1+\epsilon^{\prime}}\right] & \mbox{ for \ensuremath{\epsilon^{\prime}<2}},\\
\frac{\epsilon^{\prime}{}^{2}}{2\sqrt{\epsilon^{\prime}\left(\epsilon^{\prime}-2\right)}}\ln\left[\frac{\left(1+\epsilon^{\prime}+\sqrt{\epsilon^{\prime}(\epsilon^{\prime}-2)}\right)^{2}}{1+4\epsilon^{\prime}}\right] & \mbox{ for \ensuremath{\epsilon^{\prime}>2}}.
\end{array}\right.
\end{equation}

Figure <ref> presents the total approximate
cross section of Compton scattering, the exact QED cross section (summed
over all contributing final electron/positron Landau states) and the
exact cross section for final Landau state $l=0$ as a function of
energy of the incident photon in PRF (in units of cyclotron energy,
$\epsilon^{\prime}/\epsilon_{_{B}}$). As mentioned above, the approximation
is valid in the region below the resonance, $\epsilon^{\prime}<\epsilon_{_{B}}$.
Although the approximation overestimates the cross section for $l=0$
final Landau state in the regime of high energetic photons ($\epsilon^{\prime}>\epsilon_{_{B}}$),
it can be used in this regime as the approximation of the total cross
section. In our simulation we use this approach to calculate the total
ICS cross section in both regimes, $\epsilon^{\prime}<\epsilon_{_{B}}$
and $\epsilon^{\prime}>\epsilon_{_{B}}$. Calculation of the cross
section for the resonance frequency ($\epsilon^{\prime}=\epsilon_{{\rm _{B}}}$)
is presented in the next section.

[Total cross section of ICS as a function of an incident photon energy] Total cross section of Compton scattering (in Thomson units) as
a function of an incident photon energy in PRF (in units of the cyclotron
energy) calculated for a magnetic field strength $B_{14}=3.5$. The
exact QED scattering cross section, summed over all contributing final
electron/positron Landau states, is indicated as the red dotted curve.
The cross section for final Landau states $l=0$ is plotted as a blue
dashed line.

§.§ Resonant Compton Scattering

This section describes an approach used to calculate the RICS cross
section for ultrastrong magnetic fields ($B>10^{12}\,{\rm G}$). For
weaker fields the calculations are much simpler and resonance is already
included in Equation <ref>.

The trend as $\beta_{q}$ increases is for the magnitude of the cross
section to drop at all energies. For weaker magnetic fields ($\beta_{q}<1$)
the width of the resonance increases with increasing $\beta_{q}$,
but for $\beta_{q}\ge1$ this width actually declines. Since the resonance
is formally divergent, the common practice (see 184, 112, 46, 51, 80, 9, 81, 12, 13)
is to truncate it at $\epsilon^{\prime}=\epsilon_{_{B}}$ by introducing
a finite width $\Gamma$. The procedure is to replace the resonant
$\left(\epsilon^{\prime}-\epsilon_{_{B}}\right)^{2}$ denominator
(see Equations <ref> and <ref>)
by $\left[\left(\epsilon^{\prime}-\epsilon_{_{B}}\right)^{2}+\Gamma^{2}/4\right]$.
In the $\beta_{q}\ll1$ regime, the cyclotron decay width assumes
the well-known result $\Gamma\approx4\alpha_{f}\epsilon_{_{B}}^{2}/3$
in dimensionless units. When $\beta_{q}\gg1$, quantum and recoil
effects generate $\Gamma\approx\alpha_{f}\epsilon_{_{B}}\left(1-1/\tilde{e}\right)$
where $\tilde{e}$ is Euler's number (e.g. see 14).
These widths lead to areas under the resonance being independent of
$\epsilon_{_{B}}$ in the magnetic Thomson regime of $\beta_{q}\ll1$
and scaling as $\epsilon_{_{B}}^{1/2}$ when $\beta_{q}\gg1$. These
results can be deduced using the $l=0$ approximation derived in Equation
<ref>. By using this approach the averaged,
approximate cross section can be written as

\begin{equation}
\sigma{}_{{\rm avg}}^{\prime}=\frac{3\sigma_{_{{\rm T}}}}{16}\left[g\left(\epsilon^{\prime}\right)+f\left(\epsilon^{\prime}\right)-\left(1+2\epsilon^{\prime}\right)h\left(\epsilon^{\prime}\right)\right]\left[\frac{1}{\left(\epsilon^{\prime}-\epsilon_{_{B}}\right)^{2}+\Gamma{}^{2}/4}+\frac{1}{\left(\epsilon^{\prime}+\epsilon_{_{B}}\right)^{2}}\right].\label{eq:cascade.sigma_gont}
\end{equation}

The common practice to calculate a resonant cross section in an ultrastrong
magnetic fields is to use the Dirac delta function as follows (e.g.

\begin{equation}
\sigma_{{\rm res}}^{\prime}\simeq2\pi^{2}\frac{e^{2}\hbar}{mc}\delta\left(\epsilon{}_{s}^{\prime}-\epsilon_{_{B}}\right)\label{eq:cascade.res_cross_a}
\end{equation}

This simplified approach, however, does not include scatterings of
photons whose energy in a particle rest frame is not equal but very
close to the resonance frequency. The relativistic quantum effects
of strong magnetic fields that are included in the approximate solution
increase the cross section, and thus the efficiency of the ICS process
in previous estimates could be underestimated.

According to 124 in ultrastrong magnetic fields the
ICS polarisation fraction is about $50\%$ (approximately $50\%$
of the photons are slightly above resonance and $50\%$ are slightly
below). Therefore, the polarisation of ICS photons is randomly assigned
in the ratio of one $\perp$ (perpendicular to the field) to every
$\parallel$ photon.

§.§ Particle mean free path

For the ICS process the calculation of the particle mean free path
$l_{{\rm ICS}}$ is not as simple as that of the CR process. Although
we can define $l_{{\rm ICS}}$ in the same way as we defined $l_{{\rm CR}}$,
it is difficult to estimate a characteristic frequency of emitted
photons. We have to take into account photons of various frequencies
with various incident angles. An estimation of the mean free path
of a positron (or electron) to produce a photon is [184]

\begin{equation}
l_{{\rm ICS}}\approx\left[\int_{\mu_{0}}^{\mu_{1}}\int_{0}^{\infty}\left(1-\beta\mu\right)\sigma^{\prime}\left(\epsilon,\mu\right)n_{{\rm ph}}\left(\epsilon\right){\rm d}\epsilon{\rm d}\mu\right]^{-1},
\end{equation}

where (as before) $\beta=v/c$ is the velocity in terms of speed of
light, $n_{{\rm ph}}$ represents the photon number density distribution
of semi-isotropic blackbody radiation (see Equation <ref>).
Here $\sigma^{\prime}$ is the average cross section of scattering
in the particle rest frame (see Equation <ref>).
We should expect two modes of the ICS process, i.e. Resonant ICS and
Thermal-peak ICS.

§.§.§ Resonant ICS

The RICS takes place if the photon frequency in the particle rest
frame is equal to the cyclotron electron frequency. Using Equation
<ref> we can write that the incident photon
energy is $\epsilon=\epsilon_{_{B}}/\left[\gamma\left(1-\beta\mu\right)\right]$.
For altitudes of the same order as the polar cap size we use $\mu_{0}=1$,
$\mu_{1}=0$ as incident angle limits for outflowing particles, and
$\mu_{0}=0$, $\mu_{1}=-1$ as incident angle limits for backflowing
particles. Thus, for outflowing particles the electron/positron mean
free path above a polar cap for the RICS process is

\begin{equation}
l_{{\rm RICS}}\approx\left[\int_{0}^{1}\int_{\epsilon_{{\rm _{res}}}^{^{{\rm min}}}}^{\epsilon_{{\rm _{res}}}^{{\rm ^{max}}}}\left(1-\beta\mu\right)\sigma^{\prime}\left(\epsilon,\mu\right)n_{{\rm ph}}\left(\epsilon\right){\rm d}\epsilon{\rm d}\mu\right]^{-1},\label{eq:cascade.ics_free_path}
\end{equation}

where limits of integration, $\epsilon_{{\rm _{res}}}^{^{{\rm min}}}$
and $\epsilon_{_{{\rm res}}}^{{\rm ^{max}}}$, are chosen to cover
the resonance. In our simulation we use such limits to include the
region where the integrated function decreases up to about two orders
of magnitude from its maximum:

\begin{equation}
\epsilon_{_{{\rm res}}}^{^{{\rm min/max}}}=\frac{\epsilon_{_{B}}\pm\frac{3}{2}\sqrt{11}\Gamma}{\gamma\left(1-\beta\mu\right)}.
\end{equation}

Here $\Gamma$ is the finite width introduced in Section <ref>
to describe the decay of an excited intermediate particle state.

Figure <ref> presents the dependence of
the integrand from Equation <ref> on the
incident photon energy for a given incident angle. The maximum of
the integrand shows a significant decline for stronger magnetic fields.
This is due to both the drop of the cross section at all energies
with an increasing magnetic field (see Section <ref>)
and due to the fact that for this specific incident angle resonance
is in a different range of photon energy. In stronger magnetic fields
resonance occurs not only for higher energetic photons but also the
width of the resonance is wider (see the right panel of Figure <ref>).

[Dependence of the integrand from Equation <ref>
on the energy of the incident photon]Dependence of the integrand from Equation <ref>
on the energy of the incident photon. Both panels were calculated
for surface temperature $T=3\times10^{6}\,{\rm K}$, cosine of the
incident angle $\mu=0.1$ and Lorentz factor of particle $\gamma=10^{3}$.
The left panel corresponds to resonance in magnetic field $B=10^{14}\,{\rm G}$,
while the right panel was obtained using $B=3\times10^{14}\,{\rm G}$.

Note that both plots do not include the dependence of the photon density
on distance from the stellar surface. Depending on whether the radiation
originates from the whole stellar surface or from the polar cap only,
the dependence of the photon number density on the height above the
surface can differ significantly (see Section <ref>).

§.§.§ Thermal-peak ICS

TICS includes all scattering processes of photons with frequencies
around the maximum of the thermal spectrum. In our simulation we adopt
$\epsilon_{{\rm _{th}}}^{{\rm ^{min}}}\approx0.05\epsilon_{{\rm _{th}}}$,
and $\epsilon_{{\rm _{th}}}^{^{{\rm max}}}\approx2\epsilon_{{\rm _{th}}}$
where $\epsilon_{{\rm _{th}}}=2.82kT/\left(mc^{2}\right)$ is the
energy, in units of $mc^{2}$, at which blackbody radiation with temperature
$T$ has the largest photon number density. The electron/positron
mean free path for the TICS process can be calculated as

\begin{equation}
l_{{\rm TICS}}\approx\left[\int_{\mu_{0}}^{\mu_{1}}\int_{\epsilon_{_{{\rm th}}}^{{\rm ^{min}}}}^{\epsilon_{{\rm _{th}}}^{{\rm ^{max}}}}\left(1-\beta\mu\right)\sigma^{\prime}\left(\epsilon,\mu\right)n_{{\rm ph}}\left(\epsilon\right){\rm d}\epsilon{\rm d}\mu\right]^{-1}.\label{eq:cascade.t_ics}
\end{equation}

Figure <ref> presents the dependence of the
integrand from Equation <ref> on photon energy for
two different incident angles of background photons. As the number
density depends exponentially on the photon energy, TICS is important
only for small incident angles ($\mu\approx1$). Note that for some
specific combination of magnetic field strength, the Lorentz factor
of the primary particle and the incident angle of background photons
the resonance is in region of thermal peak. In such a case the resonant
component is dominating (a much higher cross section) and the particle
mean free path should be calculated using the approach described in
the previous section.

[The integrand from Equation <ref> vs. photon number
density]Comparison of the integrand from Equation <ref>
with photon number density. The bottom panels present the dependence
of the photon number density on photon energy in OF. The red dashed
lines correspond to limits used to calculate the particle mean free
path for TICS. The top panels present the dependence of the integrand
on photon energy in PRF. Both panels were obtained using surface temperature
$T=3\times10^{6}\,{\rm K}$, Lorentz factor of the particle $\gamma=10^{2}$
and magnetic field strength $B=10^{12}\,{\rm G}$. The cosine of the
incident angle, $\mu=0.975$ and $\mu=0.96$, was used for the left
and right panel, respectively.

§.§.§ Calculation results

For ultrastrong magnetic fields quite a wide range of the particle
Lorentz factor falls into the peak of background photons (see Figure
<ref>). In such a case RICS is enhanced by the
fact that it involves photons with very high density. Furthermore,
the RICS process for such particles is indistinguishable from the
TICS (see Figure <ref>). For particles with
Lorentz Factor $\gamma\gtrsim10^{5}$, the dominant process of radiation
is CR. The exact value of this limit depends on conditions such as:
density of background photons, incident angles between particles and
photons, and curvature of magnetic field lines ($1/\Re$).

(show_le_gamma - Graphs, $B_{14}=2.$)

[Dependence of a particle mean free path on its Lorentz factor]Dependence of a particle mean free path on its Lorentz factor for
three different processes: CR, RICS and TICS. The calculations were
performed for magnetic field strength $B_{14}=2$, radius of curvature
of magnetic field lines $\Re_{6}=1$ (for the CR process) and hot
spot temperature $T_{6}=3$ (for RICS and TICS). Both RICS and TICS
were calculated for a full range of incident angles ($\mu_{0}=0$,
$\mu_{1}=1$). Note that for a Lorentz factor in the range of $\gamma\approx2\times10^{3}-10^{5}$
the particle mean free paths of RICS and TICS are equal as the resonance
falls into the peak of the background photons.

Figure <ref> presents the dependence of
a particle mean free path on the magnetic field strength and the particle
Lorentz factor for RICS. The minimum of the mean free path for relatively
weak magnetic fields ($B_{14}=0.5$) is for particles with Lorentz
factor $\gamma\approx2\times10^{3}$, while for relatively stronger
magnetic fields ($B_{14}=3.5$) the RICS is most efficient for particles
with energy an order of magnitude larger ($\gamma\approx2\times10^{4}$).
This is a natural consequence of the fact that resonance takes place
when the photon energy in PRF is equal to the electron cyclotron energy,
which in stronger fields is higher. As can be seen from the Figure,
the particle mean free paths for RICS in stronger magnetic fields
increase. This is due to the decreasing resonant cross section with
increasing magnetic field strength (see Figure <ref>).
Note, however, that this behaviour does not include the fact that
photon density in regions with weaker magnetic fields is considerably
smaller. In fact, the results of the cascade simulation presented
in Chapter <ref> show that RICS is efficient only in
the immediate vicinity of a neutron star since photon density at higher
altitudes drops rapidly.

(show_le3d_gonthier, $T_{6}=2$)

[Dependence of a particle mean free path on magnetic field strength
and the Lorentz factor of a particle [RICS] ]Dependence of a particle mean free path on magnetic field strength
($B_{14}$) and the Lorentz factor of a particle ($\gamma$) for the
RICS process. The particle mean free path was calculated for semi-isotropic
blackbody radiation ($\mu_{0}=0$, $\mu_{1}=1$) with temperature

§.§ Background photons

§.§.§ Photon density

One of the main parameters affecting ICS above the stellar surface
is photon density. The initial photon density (at altitude $z=0$)
highly depends on the temperature of the radiating surface. As shown
in Chapter <ref> (e.g. see Table <ref>),
the entire surface has the lowest temperature ($T_{6}\lesssim0.8$),
thus the initial photon density is up to about two orders of magnitude
lower than warm spot radiation ($T_{6}\lesssim3$) and up to about
three orders magnitude lower than hot spot radiation ($T_{6}\lesssim5$).
However, the density of the photons strongly depends on the distance
from the source of radiation (especially for the hot spot). Therefore,
we used the simplified method presented in Figure <ref>
to calculate photon density at a given point $L=\left(r,\,\theta,\,\phi\right)$.
Then the relative density of photons originating from the entire surface
can be calculated as

\begin{equation}
\frac{n_{{\rm st}}\left(\epsilon,\, T_{{\rm st}},\, L\right)}{n_{0}\left(\epsilon,\, T_{{\rm st}}\right)}=\sin^{2}\left(\frac{\Delta\theta_{{\rm st}}}{2}\right)=\left(\frac{R}{r}\right)^{2},\label{eq:cascade.n_ph_dist}
\end{equation}

where $n_{{\rm st},0}\left(\epsilon,\, T_{{\rm st}}\right)$ is the
density of photons with energy $\epsilon$ at the stellar surface
with temperature $T_{{\rm st}}$, and $\Delta\theta_{{\rm st}}$ is
the angular diameter of the star at a distance from the star centre

Likewise, we can write a formula for the relative density of photons
originating from a spot (warm or hot) as

\begin{equation}
\frac{n_{{\rm sp}}\left(\epsilon,\, T_{{\rm sp}},\, L\right)}{n_{{\rm sp},0}\left(\epsilon,\, T_{{\rm sp}}\right)}=\sin^{2}\left(\frac{\Delta\theta}{2}\right),
\end{equation}

where $n_{{\rm sp},0}\left(\epsilon,\, T_{{\rm sp}}\right)$ is the
density of photons with energy $\epsilon$ at the spot surface (either
hot or warm) with temperature $T_{{\rm sp}}$. The angular diameter
of the spot can be calculated as

\begin{equation}
\Delta\theta=\arccos\left(\frac{r_{1}^{2}+r_{2}^{2}-4R_{{\rm sp}}}{2r_{1}r_{2}}\right),
\end{equation}

here $R_{{\rm sp}}$ is the spot radius and $r_{1}$, $r_{2}$ are
the distances to the outer edges of the spot (see Figure <ref>).

[Simplified method used for calculating the background photon density]Simplified method used for calculation of a photon density originating
from an entire stellar surface (blue lines) and from a hot/warm spot
(red lines). Here $R_{{\rm sp}}$ is a spot radius (either hot or
warm). Let us note that the simplified method is valid for the entire
surface component regardless of the $\phi$ component of location
$L$, while for the spot component it can be used only for small values
of $\phi$. In a more general case the spot should be projected on
the surface perpendicular to the radius vector ${\bf r}$ and passing
through point $L$.

Figure <ref> presents the dependence
of the relative photon density ($n\left(z\right)/n_{0}$) on the distance
from the stellar surface. Due to the small size of a polar cap (hot
$R_{{\rm hs}}=50\,{\rm m}$) the density of the photons drops rapidly
and already at a distance of about $z=150\,{\rm m}$ it is one order
of magnitude lower than at the polar cap surface. On the other hand,
for a larger size of the warm spot ($R_{{\rm hs}}=1\,{\rm km}$) the
photon density is reduced by an order of magnitude at a distance of
about $z=3\,{\rm {\rm km}}$. From Equation <ref>
it can easily be seen that the photon density of radiation from the
entire stellar surface decreases by an order of magnitude at a distance
of about $z\approx3R\approx30\,{\rm km}$.

[Dependence of the relative photon density on the distance from the
stellar surface]Dependence of the relative photon density on the distance from the
stellar surface for three different thermal components (the entire
stellar surface, the warm spot and the hot spot). The following parameters
were used for the calculations: star radius $R=10\,{\rm km}$, warm
spot radius $R_{{\rm ws}}=1\,{\rm km}$ and hot spot radius $R_{{\rm hs}}=50\,{\rm m}$.

The very small size of the polar cap also has an additional implication
to the background photons' density. Namely, the density of the background
photons just above the polar cap highly depends not only on the distance
from the surface, but also on the position relative to the cap centre.

Figure <ref> presents the dependence
of the relative photon density originating from a polar cap (the hot
spot) on the distance from the stellar surface for three different
starting points on the polar cap. The distance was calculated for
points which follow the magnetic field structure of PSR B0656+14.
Note that for the extreme magnetic line (which starts at the cap edge)
already at a distance of about $z_{2}\approx5\,{\rm m}$ the photon
density decreases twice, while for the central ($\theta_{0}$) and
middle line ($\theta_{1}$) the distances are respectively $z_{0}\approx45\,{\rm m}$
and $z_{1}\approx30\,{\rm m}$. This result is important as the background
photon density directly translates to the particle mean free path
in ICS (see Section <ref>). This means
that for ICS-dominated gaps the sparks' height will vary depending
on their location. The breakdown of the gap (spark) in the central
region of a polar cap is easier to develop as the particle mean free
path is lower, and eventually it will result in lower heights of the
central sparks. This will influence the properties of plasma produced
in the central region of open magnetic field lines, and depending
on the conditions may result in the formation of plasma either suitable
to produce radio emission (core emission) or unsuitable to produce
radio emission (conal emission but with the line of sight crossing
the centre of the beam).

To find the dominant component of thermal radiation at a given altitude
we need to take into account the initial flux of radiation and how
it changes with the distance. Below we present the calculations of
a radiation flux (Figure <ref>)
for PSR B0656+14. The parameters of an entire surface and warm spot
components are in agreement with the observations (see Table <ref>),
while the hot spot component was calculated using parameters derived
from the modelling of a non-dipolar structure of the magnetic field
(see Chapter <ref>).

(show_hotspot, 430)

[Dependence of the relative photon density on the distance from the
stellar surface for a hot spot component [PSR B0656+14]]Dependence of the relative photon density on the distance from the
stellar surface for a hot spot component of PSR B0656+14. The relative
photon density was calculated for three different starting positions:
$\theta_{0}$ (central), $\theta_{1}$ (at the half distance to the
edge), and $\theta_{2}$ (the cap edge). The altitude ($z$) was calculated
for points which follow the magnetic field structure of
PSR B0656+14.

show(430 r_surf=2e6, r_ws=1.8e5, r_hs=5e3, t_surf=0.7e6,t_ws=1.2e6,
t_hs=2.9e6), plot_b0656

[Dependence of the radiation flux on the distance from the stellar
surface [PSR B0656+14]]Dependence of the radiation flux for three different components (the
entire stellar surface, the warm spot and the hot spot) on the distance
from the stellar surface for PSR B0656+14. The following parameters
were used for the calculations: entire stellar surface radiation,

$T_{{\rm st}}=0.7\,{\rm MK}$, $R_{{\rm st}}=20\,{\rm km}$; warm
spot, $T_{{\rm ws}}=1.2\,{\rm MK}$, $R_{{\rm ws}}=1.8\,{\rm km}$;
and hot spot, $T_{{\rm hs}}=2.9\,{\rm MK}$, $R_{{\rm hs}}=50\,{\rm m}$.

Already at a distance of $240\,{\rm m}$ the flux of the warm spot
radiation becomes higher than the flux of the hot spot radiation.
Furthermore, already at a height of $750\,{\rm m}$ flux the radiation
originating from the polar cap (hot spot) becomes lower than the flux
of radiation from the entire stellar surface. With an increasing distance
the flux of the warm spot decreases faster than the flux of the entire
surface radiation and at a distance of $6.3\,{\rm km}$ the thermal
radiation from the entire stellar surface becomes the dominant component
of the background photons.

The results may suggest that up to a height of about $240\,{\rm m}$
(for PSR B0656+14) the hot spot radiation should be the main source
of the background photons involved in ICS. However, the actual height
is smaller as the results do not include the efficiency of ICS, which
also depends on the incident angle between the photons and the particles
(see the next Section).

§.§.§ Photon incident angles

Another parameter that significantly affects the ICS is the incident
angle between the background photons and the relativistic particles.
Especially for Resonant Inverse Compton Scattering is the incident
angle of great importance. Figure <ref> presents
the dependence of a particle mean free path for ICS on a maximum value
of the incident angle $\psi_{{\rm crit}}$. If incident angles are
low, the resonance is outside of the photon spectrum and results in
very high values of particle mean free paths. The lower the energy
of the particle (lower Lorentz factor), the incident angles should
be larger to ensure that the resonance falls into an energy range
with high photon density.

[Dependence of the particle mean free path on the maximum value of
the incident angle] Dependence of the particle mean free path on the maximum value of
the incident angle $\psi_{{\rm crit}}$. The particle mean free path
$l_{{\rm p}}$ was calculated for magnetic field strength $B=10^{14}\,{\rm G}$
assuming background blackbody radiation with a temperature $T=3\,{\rm MK}$.
Two different particle Lorentz factors were used for the calculations:
$\gamma=10^{3}$ (dashed lines) and $\gamma=10^{4}$ (solid lines).
The red lines correspond to Resonant Inverse Compton Scattering, while
the blue lines correspond to Thermal-peak Inverse Compton Scattering.

TICS for a given magnetic field strength and the Lorentz factor of
particles is not significant (high particle mean free paths) unless
the angles of the incident photons are high enough. Note the characteristic
drop of the particle mean free path for TICS at $\psi_{{\rm crit}}\approx20^{\circ}$
(for $\gamma=10^{4}$) and $\psi_{{\rm crit}}\approx75^{\circ}$ (for
$\gamma=10^{3}$). For such high incident angles the resonance takes
place at the thermal peak of the background photons. Therefore, TICS
and RICS are indistinguishable, which results in an almost equal particle
mean free path (see the text above Figure <ref>
for more details).

Due to the very small size of the polar cap the influence of the hot
spot component will by lower not only because of the change of photon
density, but also because of the rapid change of the incident angle
between the photons and particles. Figure <ref>
presents the dependence of the maximum incident angle on the altitude
above the stellar surface for three thermal components (the entire
surface, the warm spot and the hot spot). As follows from the Figure,
already at an altitude of $z\approx90\,{\rm m}$ does the maximum
value of the incident angle between the photons from the hot spot
and the particles drop to $\psi_{{\rm crit}}=30^{\circ}$, which significantly
lowers the efficiency of ICS for this source of background photons
(see Figure <ref>). Since the size of the
warm spot component is larger, the warm spot radiation will be significant
for up to higher altitudes, but already at a distance of $z\approx1.5\,{\rm {\rm km}}$
the maximum value of the incident angle also drops to $\psi_{{\rm crit}}=30^{\circ}$.

show_three, plot_three_psi

[Dependence of the maximum incident angle on the distance from the
stellar surface] Dependence of the maximum incident angle on the altitude above the
stellar surface for three thermal components (the entire surface,
the warm spot and the hot spot radiation).

Note that in the Figure we have calculated the maximum value of the
intersection angle at altitudes which correspond to radial progression
from the stellar surface. In fact, the actual maximum value of the
incident angle also depends on the structure of the magnetic field.
Figure <ref> presents the actual
maximum value of the incident angle of photons originating from the
hot spot for three different magnetic field lines calculated for PSR
B0656+14. The actual values of the maximum incident angle just above
the surface exceed $90^{\circ}$, but its rapid decline (especially
for extreme lines) causes the radiation of the hot spot component
to become insignificant for ICS at relatively low altitudes $z\approx20\,{\rm m}$.

show_hotspot, plot_hotspot_psi

[Dependence of the maximum incident angle on the distance from the
stellar surface [PSR B0656+14]]Dependence of the maximum incident angle on the altitude above the
stellar surface for the hot spot component of PSR B0656+14. The maximum
incident angle was calculated for three different starting positions:
$\theta_{0}$ (central), $\theta_{1}$ (at the half distance to the
edge), and $\theta_{2}$ (the cap edge).

Both the decrease of photon density and the decrease of the maximum
inclination angle cause the parameters of plasma produced by RICS
to highly depend on the properties (size and temperature) of the background
photons source. The hot spot component will be the dominant source
of background photons for ICS in the gap region ($z\lesssim20\,{\rm m}$),
while the radiation of the warm spot and the entire surface will be
the main source of the background photons for ICS at higher altitudes.

CHAPTER: PHYSICS OF PULSAR RADIATION

§ INNER ACCELERATION REGION

§.§ Gamma-ray emission

In our model most of the $\gamma$-photons are produced in the Inner
Acceleration Region or in close vicinity of a neutron star. Due to
an ultrastrong surface magnetic field, the most energetic $\gamma$-photons
are produced by Inverse Compton Scattering in the PSG-on mode. If
a pulsar is in the PSG-off mode, Curvature Radiation produces fewer
energetic photons than ICS in the PSG-on mode. Photons produced in
IAR (both the ICS and CR) are absorbed by strong magnetic fields creating
positron-electron plasma in the gap region, thereby enhancing a cascade,
or just above the gap enhancing a secondary plasma population. The
absorption of $\gamma$-photons in close vicinity of NS makes it impossible
to directly observe the radiation produced in IAR. However, a characteristic
of this emission defines the parameters of the gap (e.g. multiplicity
in the gap region, gap height, etc.), and thus the parameters of secondary

§.§.§ PSG-off mode

In general, the existence of high potential in IAR (e.g. wide sparks
or $\eta\approx1$) results in solutions for which CR is responsible
for the emission of $\gamma$-photons. The energy of such radiation
depends on the Lorentz factor of primary particles and curvature of
the magnetic field lines. Figures <ref>
and <ref> present the histogram of
photons produced in IAR by CR for PSR B0628-28 and Geminga, respectively.
The curvature in IAR of Geminga is lower ($\Re_{6}\approx2.1$, see
Section <ref>), thus the primary particle should be
accelerated to higher energies in order to produce the required number
of photons in the gap region. Eventually the higher Lorentz factor
of primary particles will result in the emission of $\gamma$-photons
with energy up to $10\,{\rm GeV}$ for Geminga. On the other hand,
the curvature magnetic lines for PSR B0628-28 ($\Re_{6}=0.6$, see
Section <ref>) is higher, which reduces the photon
mean free path and it is possible to produce the required number of
photons in the gap region $N_{{\rm ph}}^{{\rm CR}}$ for lower the
Lorentz factor of primary particles.

In CR-dominated gaps we can distinguish three types of photons: (I)
radiation with energy below $1\,{\rm MeV}$ which is unaffected by
the magnetic field (except the splitting) and can be detected by a
distant observer, (II) soft $\gamma$-ray photons which create pairs
above ZPF, (III) and high energetic $\gamma$-photons responsible
for pair production below ZPF. In an ultrastrong magnetic field the
photons from the third group will produce particles just after reaching
the first threshold. Due to the fact that most CR photons are $\parallel$-polarised,
photon splitting is insignificant in cascade pair production in the
PSG-off mode.

t3 for 404 and 373

[Distribution of photons produced in IAR in the PSG-off mode [PSR
B0628-28]]Distribution of photons produced in IAR by a single particle for
PSR B0628-28. In the calculations we used parameters of the gap in
the PSG-off mode as presented in Table <ref>. We
also assumed a linear change in the acceleration electric field (see
Equation <ref>).

[Distribution of photons produced in IAR in the PSG-off mode [PSR
J0633-1746]]Distribution of photons produced in IAR by a single particle for
PSR J0633-1746. In the calculations we used parameters of the gap
in the PSG-off mode as presented in Table <ref>.

§.§.§ PSG-on mode

When the acceleration potential is low enough (narrow sparks with
$\eta<1$) to satisfy the condition for effective ICS ($l_{{\rm ICS}}\lesssim l_{{\rm acc}}$),
the gap will operate in the PSG-on mode. The energy of ICS radiation
in the gap region (RICS) depends on the Lorentz factor of primary
particles and the strength of magnetic field. In an ultrastrong magnetic
field of IAR implied by the PSG model, the primary particle loses
most of its energy during the scattering of background photons. Such
extremely energetic photons produce pairs on the zero-th Landau level
($\parallel$-polarised photons) or split to less energetic photons
before reaching the first threshold (see Section <ref>).
After the photons split the resulting photons are still very energetic
and create an electron-positron pair enhancing the avalanche production
of particles. In contrast to the PSG-off, most of the electron-positron
pairs in the PSG-on mode are created well below ZPF. Furthermore,
there is no additional radiation at lower energies ($\epsilon<1\,{\rm MeV}$)
which could be detected by a distant observer. Figures <ref>
and <ref> present the distribution
of photons produced by the first population of newly created particles
for PSR B0950+08 and PSR B1929+10, respectively. In both cases the
energy of the $\gamma$-photons ranges from $1\,{\rm GeV}$ to $\approx20\,{\rm GeV}$.
The narrow predicted spark half-width of PSR B0950+08 results in a
lower potential in IAR, thus increasing the efficiency of ICS (more
photons produced by the first population of particles). The particle
mean free path for ICS is smaller for backstreaming particles (see
Section <ref> for more details), thus most
photons in the PSG-on mode are produced in the direction towards the
stellar surface. Note that not all photons will produce electron-positron
pairs since some
$\gamma$-photons are produced so close to the stellar surface that
they reach its surface before they manage to reach the first threshold
for pair production.

t + plot_ics for 322 and 355

[Distribution of photons produced in IAR in the PSG-on mode [PSR
B0950+08]]Distribution of photons produced in IAR by the first population of
newly created particles for PSR B0950+08. In the calculations we used
the parameters of the gap in the PSG-on mode as presented in Table

[Distribution of photons produced in IAR in the PSG-on mode [PSR
B1929+10]]Distribution of photons produced in IAR by the first population of
newly created particles for PSR B1929+10. In the calculations we used
the parameters of the gap in the PSG-on mode as presented in Table

§.§ X-ray and less energetic emission

An negligible fraction of energy radiated by a primary particle in
the PSG-off mode falls in the X-ray band. What is more, in the PSG-on
mode all photons produced by ICS have energy which exceeds an electron's
rest energy by many orders of magnitude. Thus, IAR may be responsible
only for generating the thermal component of the X-ray spectrum in
the process of heating the stellar surface.

§.§.§ Thermal emission

As shown in Section <ref>, thermal emission is
a common feature of neutron stars. Due to the large uncertainties
in X-ray observations, it is not possible to distinguish all three
thermal components (entire surface radiation, warm spot component
and hot spot radiation) for one specific pulsar. Furthermore, only
for a few pulsars (e.g. Geminga, PSR B0656+14) was it possible to
distinguish two thermal components alongside the nonthermal one. In
this thesis we focus on an analysis of pulsars with a visible hot
spot component ($b>1$), since only for these pulsars is it possible
to estimate the size of the actual polar cap. Most of these pulsars
are old neutron stars and only for one of them (Geminga) was the whole
surface radiation found in the X-ray spectrum. Figure <ref>
presents the observed X-ray components of the Geminga pulsar: the
whole surface radiation, the polar cap (hot spot) and the nonthermal
component. The maximum of energy for the whole surface radiation is
in extreme ultraviolet and in soft X-rays for the hot spot component.
Taking into account the very small area of the polar cap, radiation
from the hot spot is unlikely to be observed in wavelengths off the

~/Programs/studies/phd/spectrum/spectrum.py (uncomment

[Observed flux of radiation [PSR J0633+1746]]Observed flux of radiation for PSR J0633+1746. In the figure we present
three components of radiation: the nonthermal one (green line), the
entire surface radiation (blue line), and the hot spot component (red
line). The dashed lines correspond to uncertainties in observations
(see Table <ref>).

Figure <ref> presents the X-ray spectrum
of PSR B1133+16. The small number of counts detected resulted in the
fact that only separate fits for the BB and PL components were performed.
Both the BB and PL fits describe the observed spectrum with similar
accuracy. In the Figure we present additional thermal components (the
entire surface radiation and the warm spot) which have not been determined
by the observations. The Figure shows that the overlapping thermal
components can mimic the power-law dependence of the spectrum at frequencies
below $2\,{\rm keV}$.

~/Programs/studies/phd/spectrum/spectrum.py (uncomment

[X-ray spectrum [PSR B1133+16]]X-ray spectrum of PSR B1133+16. In addition to the observed thermal
radiation (red solid line), two other thermal components are presented:
the warm spot radiation (green dashed line) and the entire surface
radiation (blue dotted line).

Although this specific combination of thermal components for PSR B1133+16
would result in a photon index greater than the observed one $\Gamma=2.51$,
the spectral fits for all pulsars should be extended to include more
BB components in order to examine the effect of thermal components
overlapping at lower frequencies. The results of our calculations
suggest that the nonthermal X-ray radiation should dominate the spectrum
at higher frequencies $\approx3-10\,{\rm keV}$, but the power-law-like
behaviour at lower frequencies could be the result of the overlapping
of thermal components anticipated in the PSG scenario (see Section

§.§.§ Nonthermal emission

The polarisation of ICS radiation in an ultrastrong magnetic field
is $50\%$ (one $\parallel$ to every $\perp$-polarised photon).
Synchrotron Radiation of secondary particles created by $\perp$-polarised
photons would generate hard X-ray photons, however, as was mentioned
in Section <ref>, these photons will split
before they reach the first threshold to produce pairs. Therefore,
regardless of whether the gap is dominated by CR or by ICS, Synchrotron
Radiation in IAR is not significant.

§.§.§ Warm spot component

Apart from the obvious X-ray component corresponding to the whole
surface radiation, the PSG model can explain both the hot and warm
spot radiation. The hot spot radiation is a natural consequence of
heating the actual polar cap region by the backstreaming particles
(see Section <ref>). As was mentioned in Section <ref>,
the warm spot component can have two different sources: (I) the drastic
difference of the crustal transport process due to the non-dipolar
structure of the surface magnetic field (for young and middle-aged
pulsars), (II) and a mechanism of heating the surface adjacent to
the polar cap. In this section we present the second mechanism, i.e.
heating of the surface adjacent to the polar cap, which can be applied
to both young and old pulsars.

Figure <ref> presents the mechanism
of heating the area adjacent to the polar cap for PSR B0950+08. When
the gap operates in the PSG-off mode the primary plasma (see Section
<ref>) will lose a significant part
of its energy via CR as the particles propagate through the region
of high curvature. For this particular magnetic line's configuration
the region of high CR extends up to an altitude about $4\,{\rm km}$
above the stellar surface. The most energetic CR photons emitted in
this region have a relatively short mean free path and they produce
electron-positron pairs in the region of open magnetic field lines.
However, both the less energetic CR photons and $\gamma$-photons
produced by SR have a large enough photon mean free path to produce
pairs in the region of the closed magnetic field lines. All newly
created pairs move along the closed magnetic field lines and heat
the surface beyond the polar cap on the opposite side of the star.

[The warm spot component [PSR B0950+08]]Global structure of magnetic field lines for PSR B0950+08. The structure
was obtained using two crust-anchored anomalies (see Section <ref>).
Green lines correspond to the outer open magnetic field lines, while
the red lines correspond to the closed magnetic field lines at which
secondary pairs are produced. Blue, yellow and red dots represent
the locations of secondary pair production for the outer left, the
middle and the outer right open field lines, respectively.

The fraction of energy transferred to the region of the closed field
lines highly depends on the region of open magnetic field lines considered
in CR/SR emission. In the Figure we use three different colours (blue,
yellow and red) to show the positions of pair creation for three characteristic
open magnetic field lines (the outer left, the middle and the outer
right). The simulation results in the following fractions of energy
transferred to the region of the closed field lines are: $0.02\%$,
$0.1\%$, $6\%$ for all three lines, respectively. For this specific
magnetic field configuration the transferred energy fraction increases
as we move towards the region with the highest curvature. We can roughly
estimate that for the proposed magnetic field configuration of PSR
B0950+08, about $1\%$ of the outflowing energy is responsible for
heating of the surface beyond the polar cap on the opposite side of
the star. Note that due to strong anisotropy of the outflowing and
backflowing stream of particles (see Section <ref>),
this fraction could be enough to obtain the warm spot component with
a luminosity equal or in some cases even higher than the luminosity
of the hot spot component.

[The warm spot component [PSR B0943+10]]Global structure of magnetic field lines for PSR B0943+10. The structure
was obtained using two crust-anchored anomalies (see Section <ref>).
Green lines correspond to the outer open magnetic field lines, while
the red lines correspond to the closed magnetic field lines at which
secondary pairs are produced. Blue, yellow and red dots represent
the locations of secondary pair creation for the outer left, the middle
and the outer right open field lines, respectively.

The fraction of energy transferred to the region of closed field lines
highly depends on the magnetic field configuration. A more complicated
structure of the magnetic field lines proposed for PSR B0943+10 (see
Figure <ref>) results in a much wider
area of closed field lines at which pairs are created, and hence higher
fractions of energy transferred to the region of closed field lines.
For this specific structure of the magnetic field these fractions
are: $7\%$, $1\%$, $2\%$ for three characteristic lines, respectively.
We can roughly estimate that about $3-5\%$ of the outflowing energy
is responsible for the heating. Note that the magnetic field structure
of PSR B0950+08 results in the heating of only one side beyond the
polar cap, while in the case of PSR B0943+10 the whole surface around
the polar cap is heated. The actual size of the warm spot also depends
on the magnetic field configuration in the heating zone, and can either
be decreased or increased.

§.§ Primary plasma

As we mentioned in Section <ref>, PSG-off and
PSG-on modes differ essentially by the Lorentz factor of primary particles
produced in the gap region. Furthermore, different scenarios of the
gap breakdown (due to surface overheating or due to production of
dense enough plasma) cause the evolution of primary particles in the
two modes to completely different.

We assume that in the PSG-off mode the gap breakdown is due to surface
overheating; hence the plasma cloud moving away from the stellar surface
is a mixture of ions and electron-positron plasma. In this scenario
the ions are the main source of charge density required to screen
the gap (see Equation <ref>). As the
plasma cloud moves away from the stellar surface both the spark height
and the spark width increase, which results in an increase of the
acceleration potential drop. When the particles gain the Lorentz factors
$\gamma\gtrsim10^{5}$, CR begins to produce $\gamma$-photons. In
the PSG-off mode most of the
$\gamma$-photons are created near ZPF (see Figure <ref>).
All particles created by $\gamma$-photons above the ZPF do not contribute
to the heating of the surface. Furthermore, the acceleration in the
upper parts of the gap is relatively weak, and electrons produced
in this region will also escape from the gap, thus not contributing
to the surface heating. Depending on the details of the cascade formation,
the process described above may result in the creation of strong streaming
anisotropies, where the flux of the backstreaming particles is considerably
smaller than the flux of the outstreaming particles. Note that the
density of the backstreaming particles required to overheat the surface
is significantly lower than the co-rotational density $n_{{\rm CR}}\ll n_{{\rm GJ}}$
(see Table <ref>).

In the PSG-on mode the quasi-equilibrium of the flux of backstreaming
particles and the flux of the polar cap radiation can cause the gap
to break only due to the production of dense enough plasma. Thus,
the surplus of positrons is the main source of the charge in the plasma
cloud moving away from the stellar surface. The ICS process responsible
for the cascade production of particles is effective only in the bottom
part of the gap. Hence, the backstreaming electrons will hit the surface
with a Lorentz factor $\gamma_{{\rm c}}$ well below the $\gamma_{{\rm max}}$.
As there is no strong pair production near (or above) ZPF, the backstreaming/outstreaming
anisotropy arises only due to the difference of the Lorentz factor
of electrons hitting the stellar surface and the Lorentz factor of
positrons accelerated in the gap $\gamma_{{\rm max}}/\gamma_{{\rm c}}\approx10$.
The actual density of newly created plasma to completely screen the
gap can be calculated only in the full cascade simulation. However,
as shown by 172, this density should significantly
exceed the co-rotational Goldreich-Julian density $n_{{\rm ICS}}\gg n_{{\rm GJ}}$.
We describe the difference between the co-rotational density and the
actual density of primary plasma required to completely screen the
ICS-dominated gap by factor  $N_{{\rm ICS}}=n_{{\rm ICS}}/n_{{\rm GJ}}\gg1$.

§ INNER MAGNETOSPHERE OF A PULSAR

§.§ Gamma-ray emission

In general there are three processes which can produce $\gamma$-ray
emission in the inner magnetosphere of a pulsar ($R_{{\rm pc}}\ll z\ll R_{{\rm LC}}$):
CR, ICS and SR. Which of them produces the majority of $\gamma$-photons
depends on the parameters of the primary particles, and thus mainly
depends on the mode in which the gap operates. Additionally, the efficiency
of the ICS process strongly depends on the source of the background

§.§.§ Curvature Radiation of primary particles

When the gap operates in the PSG-off mode, high-energetic particles
are produced
$\gamma_{{\rm c}}\gtrsim10^{6}$. As they pass the region with high
curvature ($\Re_{6}\approx1$) they radiate a significant part of
their energy through CR (see Section <ref>).

Figure <ref> presents the distribution of CR
photons produced by a single primary particle moving along the open
magnetic field line of PSR B1133+16 (see Section <ref>
for the details of the magnetic field configuration). The initial
Lorentz factor of the particle $\gamma_{{\rm max}}=1.7\times10^{6}$
was set according to the value presented in Table <ref>.
As the particle advanced through the region with high curvature, it
lost about $46\%$ of its initial energy, which was mainly converted
to high-energetic $\gamma$-photons with an energy up to about $2\,{\rm GeV}$.
The $\gamma$-photons are produced in a region of a strong magnetic
field, thus after passing a relatively short distance the most energetic
photons are absorbed by the magnetic field and electron-positron pairs
emerge. The red colour in the Figure corresponds to the final spectrum
(after photon splitting, pair production and SR) produced by a single
primary particle in the PSG-off mode. Most of the energy radiated
by the primary particle was converted into the secondary plasma (see
Section <ref>) and only about $5\%$
of the particle's initial energy ended in the form of radiation with
a cut-off at about $30\,{\rm MeV}$.

(read_data_final, plot_spectrum_final, 341_cr_1e04_10m_new)

[Final photon distribution produced by a single primary particle [PSR
B1133+16]]Final photon distribution produced by a single primary particle for
PSR B1133+16. The blue line corresponds to the initial CR distribution,
while the red line presents the final distribution with the inclusion
of photon splitting, pair production and SR.

To increase the amount of photons reaching the observer, the emission
zone, i.e. the region with the highest curvature, should by located
in the area with a weaker magnetic field. Such a configuration allows
a photon to travel a longer distance before it is absorbed by the
magnetic field. As a result the electron-positron pairs are created
at higher Landau levels, which enhances SR. Figure <ref>
presents the distribution of CR photons for PSR B0950+08. The calculations
were performed for the initial Lorentz factor of the particle $\gamma_{{\rm max}}=2.0\times10^{6}$
(see Table <ref>).

(read_data_final, plot_spectrum_final, 355_cr_1e04_10m_new)

[Final photon distribution produced by a single primary particle [PSR
B0950+08]]Final photon distribution produced by a single primary particle for
PSR B0950+08. The blue line corresponds to the initial CR distribution,
while the red line presents the final distribution with the inclusion
of photon splitting, pair production and SR.

Due to CR the primary particle lost about $40\%$ of its initial energy.
In this case about a half of the energy radiated by the primary particle
was converted into the secondary plasma and the same amount of energy
(about $20\%$ of the particle's initial energy) ended in the form
of radiation. For both PSR B1133+16 and PSR B0950+08, the maximum
of the curvature is of the same order. However, the maximum of curvature
PSR B1133+16 is located at an altitude of about $800\,{\rm m}$, while
for PSR B0950+08 it is located at an altitude of about $1.75\,{\rm km}$
(compare Figures <ref> and <ref>).

§.§.§ Inverse Compton Scattering of primary particles

In the PSG-on mode the maximum Lorentz factor of primary particles
is in the range of $10^{4}-10^{5}$ (see Table <ref>).
As it follows from Figures <ref> and <ref>,
the ICS process is most effective for particles with a Lorentz factor
in the range of $10^{3}-10^{4}$. Particles with high energies ($\gamma\gtrsim10^{5}$)
will upscatter thermal photons only just above the stellar surface,
where the density of the background photons is very high (see Section
<ref>). Thus, if there is no additional source
of background photons, the most energetic particles ($\gamma\gtrsim10^{5}$)
will escape from the inner magnetosphere without losing their energy
by ICS. However, the plasma cloud produced by the ICS-dominated gap
has a density exceeding the co-rotational Goldreich-Julian density
even by a few orders of magnitude (see Section <ref>).
Such a high charge density reduces the acceleration [172]
and, consequently, the bulk of particles will escape from the IAR
with lower Lorentz factors. It is not possible to estimate the actual
Lorentz factor of particles in the plasma cloud at the moment of gap
breakdown without performing a full cascade simulation. Thus, in this
thesis we assume that at the moment of gap breakdown most of the particles
will have an energy that is about the characteristic value at which
the acceleration is stopped by ICS in the bottom parts of the IAR
$\gamma_{{\rm c}}$. To increase readability for cascade simulations
with very low surface temperature, in all the Figures of the ICS distribution
we present $\gamma$-photons produced by $50$ primary particles with
Lorentz factors in the range of $0.5\gamma_{{\rm c}}-2\gamma_{{\rm c}}$.

In Figure <ref> we present the distribution
of ICS photons produced by the upscattering of surface thermal radiation
with temperature $T_{{\rm s}}=0.3\,{\rm MK}$ for PSR B0834+06. Even
for such a low surface temperature the whole surface radiation is
the dominant source of background photons for ICS up to altitudes
of about one stellar radii. During the scattering the primary particles
lose about $30\%$ of their initial energy while producing $\gamma$-photons
with energy up to $1\,{\rm GeV}$. Since the $\gamma$-photons are
very energetic and are produced in a region with a strong magnetic
field, they will be absorbed by the magnetic field, thus giving rise
to the secondary plasma population (see Section <ref>).
All pairs in the inner magnetosphere of a pulsar are created in the
nonzero Landau level, thus the pair production process is also accompanied
by strong SR (see the next section).

(read_data_ics, plot_hist_ics, 384_ics_1e03_10m_ics2_t03_fifty)

[Distribution of ICS photons for PSR B0834+06 [$T_{{\rm s}}=0.3\,{\rm MK}$]]Distribution of ICS photons produced by an upscattering of surface
($T_{{\rm s}}=0.3\,{\rm MK}$) for PSR B0834+06. The plot includes
all $\gamma$-photons upscatterd by $50$ primary particles with Lorentz
factors in the range of $2.5\times10^{3}-10^{4}$.

A natural way of increasing the number of $\gamma$-photons produced
by ICS in the inner magnetosphere is to increase the number of background
photons. Figure <ref> presents the distribution
of ICS photons produced by an upscattering of the surface thermal
radiation with temperature $T_{{\rm s}}=0.4\,{\rm MK}$ for PSR B0834+06.
During the ICS process the primary particles lose about $65\%$ of
their initial energy. For higher surface temperatures the ICS produces
$\gamma$-photons up to higher altitudes (about two stellar radii),
thus photons with lower energy emerge $\epsilon_{{\rm min}}\approx3\,{\rm MeV}$.
These less energetic photons will reach the observer, but their total
energy is significantly lower than the total energy of the secondary
plasma created by more energetic $\gamma$-photons.

(read_data_ics, plot_hist_ics, 384_ics_1e03_10m_ics2_t04_fifty)

[Distribution of ICS photons for PSR B0834+06 [$T_{{\rm s}}=0.4\,{\rm MK}$]]Distribution of ICS photons produced by an upscattering of the surface
radiation ($T_{{\rm s}}=0.4\,{\rm MK}$) for PSR B0834+06. The plot
includes all $\gamma$-photons upscatterd by $50$ primary particles
with Lorentz factors in the range of $2.5\times10^{3}-10^{4}$.

Note that although for PSR B0834+06 the X-ray spectral fit was performed
with only one BB component, the surface temperatures used in the calculations
($0.3\,{\rm MK}$ and $0.4\,{\rm MK}$) are in good agreement with
the predicted surface temperature of an old neutron star.

Another source of background photons which could be relevant for ICS
in the inner magnetosphere is the warm spot component. As shown in
Section <ref>, if the antipodal spot
operates in the PSG-off mode and if the magnetic field structure is
suitable then the warm spot is formed in the region adjacent to the
polar cap. With a temperature lower than the hot spot but a much larger
area, the warm spot is the main source of the background photons at
altitudes up to about half a stellar radius.

In Figure <ref> we present the distribution
of ICS photons produced by an upscattering of warm spot radiation
with temperature $T_{{\rm s}}=1.0\,{\rm MK}$ and radius $R_{{\rm ws}}=1\,{\rm km}$
for PSR B0834+06. When the warm spot is the main source of background
photons, the ICS process starts at lower altitudes. As a consequence,
the scattering produces photons with higher energy and the primary
particles lose up to $90\%$ of their initial energy. All these high
energetic $\gamma$-photons are absorbed by the magnetic field producing
electron-positron pairs. Note that for this specific pulsar the existence
of such a strong warm spot component is unlikely, but as mentioned
in Section <ref> the X-ray spectral
fits should be extended to include more thermal components to put
better constraints on the X-ray emission of pulsars.

(read_data_ics, plot_hist_ics, 384_ics_1e03_10m_ics1_t10_fifty_warm)

[Distribution of ICS photons for PSR B0834+06 [$T_{{\rm s}}=1.0\,{\rm MK}$,
$R_{{\rm ws}}=1\,{\rm km}$]]Distribution of ICS photons produced by an upscattering of warm spot
radiation ($T_{{\rm s}}=1.0\,{\rm MK}$, $R_{{\rm ws}}=1\,{\rm km}$)
for PSR B0834+06. The plot includes all $\gamma$-photons upscatterd
by $50$ primary particles with Lorentz factors in the range of $2.5\times10^{3}-10^{4}$.

§.§.§ Synchrotron Radiation

In both PSG-off and PSG-on modes SR plays a significant role in the
generation of soft $\gamma$-ray photons. Figure <ref>
presents the places at which SR-photons are generated (left panel)
and the SR spectrum (right panel) in the PSG-off mode for Geminga.
The most energetic photons are generated close to the stellar surface
($z\approx500\,{\rm m}$), while the less energetic ones are produced
at altitudes $z>2\,{\rm km}$, where the magnetic field is weaker.
SR in the PSG-off mode produces photons with energy in the range of
from $30\,{\rm keV}$ to $1\,{\rm GeV}$. Again, the high energetic
$\gamma$-photons produce electron-positron pairs in a strong magnetic
field, thus its observation is not possible.

~/Programs/studies/phd/lines/lines.py t0633_cr

[Synchrotron Radiation in the PSG-off mode [PSR J0633+1746]]Synchrotron Radiation in the PSG-off mode for PSR J0633+1746. The
left panel presents the places at which SR-photons are generated,
while the right panel presents the SR-photons distribution. Plots
were obtained in a cascade simulation calculated for a single primary
particle moving along the extreme left open magnetic field line.

In Figure <ref> we present the places
of SR-photon generation (left panel) and the energy distribution of
photons (right panel) in the PSG-on mode for Geminga.

~/Programs/studies/phd/lines/lines.py t0633_ics

[Synchrotron Radiation in the PSG-on mode [PSR J0633+1746]]Synchrotron Radiation in the PSG-on mode for PSR J0633+1746. The
left panel presents places at which SR-photons are generated, while
the right panel presents the SR-photons distribution. Plots were obtained
in a cascade simulation calculated for a single primary particle moving
along the extreme left open magnetic field line. The ICS process was
calculated using the whole surface radiation with temperature $T_{{\rm s}}=0.5\,{\rm MK}$
(see Table <ref>).

The production of SR-photons in the PSG-off mode starts at altitudes
about $z\approx1\,{\rm km}$ and ends at altitudes $z\approx4.5\,{\rm km}$.
Thus, the energy range of SR-photons is narrower, with the minimum
and maximum photon energy $\epsilon_{{\rm min}}\approx40\,{\rm keV}$
and $\epsilon_{{\rm max}}\approx50\,{\rm MeV}$, respectively. Note
the significant difference in the number of photons produced by SR
in the PSG-off and PSG-on modes. The difference is a direct consequence
of low secondary plasma multiplicity in the PSG-on mode (see Section

§.§ X-ray emission

The main source of X-ray photons produced in the inner magnetosphere
is SR. As mentioned in Section <ref>, to
increase the amount of photons reaching the observer the emission
zone should by located in the area with a weaker magnetic field. In
this section we focus on the results of PSR B0943+10 and PSR 1929+10
for which the proposed configuration of a magnetic field satisfies
this requirement (see Sections <ref> and <ref>).

In the PSG-off mode most of the X-ray photons are produced by the
SR of newly created electron-positron pairs. Figure <ref>
presents the final photon distribution produced by a single primary
particle of PSR B1929+10 in the PSG-off mode. For a single primary
particle we can estimate that only about $0.7\%$ of the total photon
energy is in the range of $1-10\,{\rm keV}$. The bulk of the energy
is carried away by newly created particles ($73\%$) and high energetic
photons ($27\%$).

t1929_cr_photons (read_data_final, plot_spectrum_final, 322_cr_1e04_10m_new)

[Final photon distribution produced by a single primary particle [PSR
1929+10]]Final photon distribution produced by a single primary particle for
PSR 1929+10 in the PSG-off mode. The blue line corresponds to the
initial CR photons distribution, while the red line presents the final
distribution with the inclusion of photon splitting, pair production
and SR.

In Figure <ref> we present the locations
and the photon distribution of SR for PSR 1929+10 in the PSG-off mode.
All SR-photons produced closer to the stellar surface will contribute
to $\gamma$-ray emission, while the SR-photons produced at higher
altitudes will produce photons in the X-ray band. As it results from
Figure <ref>, the curvature at an altitude
of $z\approx3.5\,{\rm km}$ is only $50\%$ higher than at $z\approx2\,{\rm km}$.
Furthermore, before the particle reaches the region with a relatively
low magnetic field ($z\approx3.5\,{\rm km}$), it radiates a significant
part of its energy at lower heights.

~/Programs/studies/phd/lines/lines.py t1929

[Synchrotron Radiation in the PSG-off mode [PSR B1929+10]]Synchrotron Radiation in the PSG-off mode for PSR B1929+10. The left
panel presents the places at which SR-photons are generated, while
the right panel presents the SR-photons distribution. Plots were obtained
in a cascade simulation calculated for a single primary particle moving
along the extreme left open magnetic field line.

To increase radiation in the $1-10\,{\rm keV}$ energy band we should
apply the magnetic field structure with considerably higher curvature
at altitudes where X-ray photons are generated. Although the curvature
will not directly affect the SR, it will enhance CR, and thus it will
increase the number of pairs produced in the region of a relatively
weak magnetic field. In Figure <ref> we present
the final photon distribution produced by a single primary particle
for PSR B0943+10 calculated using the magnetic field configuration
as presented in Section <ref>.

t0943_cr_photons (read_data_final, plot_spectrum_final, 911_cr_1e04_10m_leftline)

[Final photon distribution produced by a single primary particle [PSR
B0943+10]]Final photon distribution produced by a single primary particle for
PSR B0943+10 in the PSG-off mode. The blue line corresponds to the
initial CR photons distribution, while the red line presents the final
distribution with the inclusion of photon splitting, pair production
and SR.

For this magnetic field structure about $3\%$ of the total photon
energy is in the range of $1-10\,{\rm keV}$. The newly created particles
carry away about $63\%$ of the energy radiated by the primary particle,
while about $37\%$ of the energy remains in the form of photons.

The structure of the magnetic field of PSR B0943+10 allows enhanced
pair production in a region of a weaker magnetic field (see Figure
<ref>). The SR that accompanies pair
production at higher altitudes ($z>3\,{\rm km}$) essentially increases
the amount of energy radiated in the $1-10\,{\rm keV}$ energy band.
Note, however, that the fraction of energy radiated in this band is
still relatively low ($3\%$), and in order to be a substantial part
of the observed X-ray spectrum the strong anisotropy of backstreaming
and outstreaming plasma is required (see Section <ref>).

~/Programs/studies/phd/lines/lines.py t0943_cr

[Synchrotron Radiation in the PSG-off mode [PSR B0943+10]]Synchrotron Radiation in the PSG-off mode for PSR B0943+10. The left
panel presents the places at which SR-photons are generated, while
the right panel presents the SR-photons distribution. Plots were obtained
in a cascade simulation calculated for a single primary particle moving
along the extreme left open magnetic field line.

In the PSG-on mode even for a complicated structure of the magnetic
field most of the outflowing energy is converted to secondary plasma.
Figure <ref> presents the ICS-photons distribution
produced in the PSG-on mode for PSR B0628-28. The bulk of energy is
radiated in the form of high energetic $\gamma$-photons which are
responsible for pair production, and thus the formation of secondary
plasma. Taking into account not so high backstreaming/outstreaming
anisotropy in the PSG-on mode, the ICS process is not relevant for
the production of X-ray photons.

[Distribution of ICS photons for PSR B0628-28 [$T_{{\rm s}}=0.5\,{\rm MK}$]]Distribution of ICS photons produced by an upscattering of the whole
surface radiation ($T_{{\rm s}}=0.5\,{\rm MK}$) for PSR B0628-28.
The plot includes all photons upscatterd by $50$ primary particles
with Lorentz factors in the range of $3\times10^{3}-1.2\times10^{4}$.

The SR which accompanies the pair creation process in the PSG-on mode
mostly produces soft $\gamma$-photons (see the right panel of Figure
<ref>). Although the secondary pairs
are produced at similar altitudes in both modes, (compare the left
panels of Figures <ref> and <ref>),
the higher Lorentz factor of secondary plasma produced in the PSG-on
mode results in higher energy of the SR-photons. The results suggest
that when the gap operates in the PSG-on mode we should expect lower
efficiencies of nonthermal X-ray emission than in the PSG-off mode.
Note, however, that the final efficiency of X-ray radiation in the
PSG-off mode highly depends on the backstreaming/outstreaming anisotropy
and the structure of magnetic field lines.

~/Programs/studies/phd/lines/lines.py t0628_ics_xrays

[Synchrotron Radiation in the PSG-on mode [PSR B0628-28]]Synchrotron Radiation in the PSG-on mode for PSR B0628-28. The left
panel presents places at which SR-photons are generated, while the
right panel presents the SR-photons distribution. Plots were obtained
in a cascade simulation calculated for $50$ primary particles with
Lorentz factors in the range of $3\times10^{3}-1.2\times10^{4}$ moving
along the extreme left open magnetic field line.

§.§ Secondary plasma

The multiplicity of secondary particles in the PSG-off mode is much
higher than in the PSG-on mode. However, the primary plasma produced
in the IAR of CR-dominated gaps has a density considerably lower than
the Goldreich-Julian co-rotational density (see Equation <ref>).
Figure <ref> presents the energy histogram
of secondary plasma for Geminga (left panel) and PSR B1133+16 (right
panel). Despite major differences in the magnetic field structure
and conditions in the IAR for both pulsars, the secondary plasma distribution
shows many similarities. The only significant difference is the maximum
Lorentz factor of secondary plasma, which for Geminga is about $\gamma_{{\rm sec}}^{{\rm max}}\approx10^{4}$,
while for PSR B1133+16 is is a few times smaller $\gamma_{{\rm sec}}^{{\rm max}}\approx3\times10^{3}$.

By using the overheating parameters presented in Table <ref>
we can roughly estimate that the final multiplicity of particles in
the plasma cloud in the PSG-off mode ranges from $M=\kappa\cdot M_{{\rm sec}}\approx2$
(for Geminga) to $M=\kappa\cdot M_{{\rm sec}}\approx100$ (for PSR
B1133+16). Note, however, that these values do not take into account
the anticipated anisotropy of backstreaming and outstreaming particles.
The existence of such an anisotropy could further increase the final
multiplicity of particles in the plasma cloud leaving the inner magnetosphere.
Despite the fact that without a full cascade simulation in the IAR
we cannot unambiguously determine the final multiplicity in the plasma
cloud, it can be clearly seen that depending on the details of the
gap operating in the PSG-off mode, the produced plasma may be suitable
(e.g. PSR B1133+16) or unsuitable (e.g. Geminga) to generate radio
emission (see Section <ref>). The main
factor determining the parameters of the CR-dominated gap, and thus
determining whether it is possible to effectively produce radio emission,
is the radius of curvature of the magnetic field lines (see Section

t0633_cr_pairs, t1133_cr_pairs (change to right)

[Secondary plasma in the PSG-off mode [PSR J0633+1746, PSR B1133+16]]Energy histogram of secondary plasma in the PSG-off mode. The left
panel was obtained in a cascade simulation calculated for a single
primary particle moving along the extreme left open magnetic field
line of PSR J0633+1746, while the right panel corresponds to a cascade
simulation for PSR B1133+16.

In ICS-dominated gaps, on the other hand, the density of primary plasma
produced in IAR exceeds the co-rotational density. Thus, the development
of dense enough plasma for radio emission is much easier in the PSG-on
mode. In Figure <ref> we present
the locations of pair production and energy distribution of secondary
plasma in the PSG-on mode for PSR B0628-28. The final multiplicity
of particles in the plasma cloud in the PSG-on mode can be calculated
as $M=N_{{\rm ICS}}\times M_{{\rm sec}}$. As mentioned in Section
<ref>, the exact value of $N_{{\rm ICS}}$
can be found only by performing the full cascade simulation in IAR
but, as shown by 172, we should expect a full screening
of the acceleration region when $N_{{\rm ICS}}$ reaches a value as
high as $20-100$. Thus we can roughly estimate that for the whole
surface radiation with temperature $T_{{\rm s}}=0.3\,{\rm MK}$, the
final multiplicity of secondary plasma in the PSG-on mode for PSR
B0628-28 is of the order of $M\approx100$.

~/Programs/studies/phd/lines/lines.py t0628_ics_a50

[Secondary plasma in the PSG-on mode [PSR B0628-28, $T_{{\rm s}}=0.3\,{\rm MK}$]]Secondary plasma produced in the PSG-on mode for PSR B0628-28. The
left panel presents places at which pairs are produced, while the
right panel presents the histogram of particle energy. Plots were
obtained in a cascade simulation calculated for $50$ primary particles
with Lorentz factors in the range of $3\times10^{3}-1.2\times10^{4}$
moving along the extreme left open magnetic field line. The ICS process
was calculated using the whole surface radiation with temperature
$T_{{\rm s}}=0.3\,{\rm MK}$.

In the PSG-on mode the main factor which determines the final multiplicity
of secondary plasma is the source of the background photons. As shown
in Section <ref>, the polar cap radiation
(the hot spot component) has a negligible impact on the ICS process
above the IAR. Figure <ref> presents
the location of pair production and energy distribution of secondary
plasma for PSR B0628-28 calculated assuming the whole surface radiation
with temperature $T_{{\rm s}}=0.5\,{\rm MK}$. The increase in the
number of background photons results in enhancement of the ICS process,
and thus an increase of the secondary multiplicity $M_{{\rm sec}}\approx60$.
For such conditions the final multiplicity of secondary plasma in
the PSG-on mode is of the order of $M\approx10^{3}-10^{5}$.

~/Programs/studies/phd/lines/lines.py t0628_ics_b

[Secondary plasma in the PSG-on mode [PSR B0628-28, $T_{{\rm s}}=0.5\,{\rm MK}$]]Secondary plasma produced in the PSG-on mode for PSR B0628-28. The
left panel presents the places at which pairs are produced, while
the right panel presents the histogram of particle energy. Plots were
obtained in a cascade simulation calculated for $50$ primary particles
with Lorentz factors in the range of $3\times10^{3}-1.2\times10^{4}$
moving along the extreme left open magnetic field line. The ICS process
was calculated using the whole surface radiation with temperature
$T_{{\rm s}}=0.5\,{\rm MK}$.

CHAPTER: CONCLUSIONS

The hot spot component identified in X-ray observations implies the
non-dipolar structure of surface magnetic field. We used the Partially
Screened Gap model to explain both the
X-ray radiation of radio pulsars and production of secondary plasma
suitable for generation of radio emission.

§ A SPECIAL CASE (PSR B0943+10)

Our model predicts two additional sources of X-ray emission: (I) the
warm spot component and (II) enhanced SR radiation in the PSG-off
mode. The warm spot component is associated with particles originating
from the antipodal polar cap, while the high luminosity of X-ray photons
produced in the PSG-off mode is a result of strong anisotropy of backstreaming
and outstreaming particles.

Very recent results presented by 87 show the anti-correlation
of radio and X-ray emission of PSR B0943+10. The authors suggest an
unpulsed, non-thermal component in radio-bright mode and a $100\%$-pulsed
thermal component along with a nonthermal component in a radio-quiet
mode. In our model it is not possible to produce an unpulsed, nonthermal
X-ray component without the accompanying blackbody radiation of the
polar cap. Although it is possible to produce nonthermal X-ray radiation
which obscures the thermal component (strong SR in the PSG-off mode
with a high predominance of outstreaming particles), the resulting
radiation should be pulsed. We believe that the X-ray radiation of
PSR B0943+10 in the radio bright mode was misinterpreted as the nonthermal
one. As shown in Figure <ref> (panel d), for
a derived geometry of PSR B0943+10 the polar cap produces unpulsed,
thermal radiation. Furthermore, as reported by the authors, in the
radio-bright mode both the absorbed blackbody (BB) and the absorbed
power-law (PL) models fit the spectrum equally well (see Table S4
in 87). We believe that the observed radiation
modes of PSR B0943+10 correspond to a mode switch between the PSG-on
(radio-bright) and the PSG-off mode (radio-quiet). When pulsar is
in the PSG-on we observe both the radio emission and thermal radiation
which originates from the polar cap. In the PSG-off mode the secondary
plasma is not suitable to produce so strong radio emission as in the
PSG-on mode, but the polar cap radiation is accompanied by pulsed,
nonthermal emission produced by SR (see Sec. <ref>).

§ GAMMA-RAY PULSARS

As was shown in Sections  <ref> and <ref>,
$\gamma$-rays produced in IAR and the inner magnetosphere cannot
reach the observer due to efficient pair production in those regions.
Current models of $\gamma$-ray emission propose that the emission
comes from outer magnetospheric gaps. The non-dipolar structure of
a magnetic field has two key implications on $\gamma$-ray emission
models: (I) the formation of slot gaps is not possible as pairs are
produced along all open magnetic field lines, (II) the high density
of electron-positron plasma ($n_{p}\gg n_{{\rm GJ}}$) produced in
the inner magnetosphere prevents the outer gap formation. The high-density
plasma which crosses the null line will screen the outer magnetospheric
region due to plasma separation (acceleration of electrons and deceleration
of positrons). Thus, the formation of outer gaps is possible only
in special cases when the pulsar operates in the PSG-off mode and
produces secondary plasma with low density $n_{p}\approx n_{{\rm GJ}}$.

As recently reported by 4: “It is possible for
relativistic populations of electrons and positrons in the current
sheet of a pulsar’s wind right outside the light cylinder to emit
synchrotron radiation that peaks in the ${\rm sub-GeV}$ to ${\rm GeV}$
regime, with $\gamma$-ray efficiencies similar to those observed
for the Fermi/LAT pulsars.” We believe that the observed high-energetic
$\gamma$-rays are produced in the not yet well explored region right
outside the light cylinder.

§ RADIO EMISSION

Pulsed radio emission remains one of the most intriguing puzzles of
astrophysics. It is remarkable that despite the large ranges in $P$,
$B_{{\rm d}}$, the variations in the pulse profile between different
classes of neutron stars (young, old, millisecond, magnetars) are
similar to those within classes [125]. The radio emission
of most pulsars can be characterised by: a relatively narrow frequency
range, $\sim100\,{\rm MHz}$ to $\sim10\,{\rm GHz}$, and a high degree
of polarisation with a characteristic sweep of the position angle.
The extremely high brightness temperature of pulsar radio emission
(typically $T_{b}>10^{25}\,{\rm K}$) implies that a coherent emission
mechanism is involved. Many radio emission mechanisms have been proposed,
but no consensus on a specific emission mechanism has emerged. The
radio observations alone cannot identify the emission mechanism and,
hence, a model of the magnetosphere is needed to put constraints on
the radio emission model. An acceptable emission mechanism must involve
some form of instability to produce coherent radiation. The main difficulty
in finding a specific emission mechanism is that many of the predicted
features are common all proposed models. Furthermore, the polarisation
can also be regarded as generic rather than associated with a specific
emission mechanism [126].

The X-ray observations have allowed us to put constraints on the polar
cap region of pulsars. The non-dipolar structure of the surface magnetic
field causes plasma to form under similar conditions regardless of
the global configuration of the magnetic field. We have showed that
depending on the details of IAR, the resulting plasma either meets
the requirements for efficient radio emission (suitable multiplicity
and energy distribution of secondary plasma) or is not suitable to
produce efficient radio emission (e.g. Geminga). Furthermore, the
proposed drift model allows to find a connection between radio and
X-ray emission processes (see Section <ref>).

§ THE MIXED MODE

Although in the thesis we consider the PSG-on and PSG-off mode separately,
in a real case both of these modes can coexist either on two separate
polar caps or on the same polar cap occupying its different parts.
In the latter case the change of modes is associated with varying
degrees of intensity of the two modes. Furthermore, if specific conditions
are met, the ICS process can be a main source of $\gamma$-photons
in the lower parts of the gap, while the CR process can produce $\gamma$-photons
in the upper parts of the acceleration region. In such a case distinguishing
between the two modes is even more difficult.

§ SUMMARY

The main propositions associated with this thesis are as follows:

* The size of the hot spots implies that the magnetic field configuration
just above the stellar surface differs significantly from a purely
dipole one.

* The analysis of X-ray observations shows that the temperature of the
actual polar cap is equal to the so-called critical value, i.e. the
temperature at which the outflow of thermal ions from the surface
screens the gap completely.

* The non-dipolar structure of a surface magnetic field and the high
multiplicity of particles produced in IAR prevents the formation of
slot and outer gaps.

* The PSG model predicts the existence of two scenarios of gap breakdown:
the PSG-off mode for CR-dominated gaps and the PSG-on mode for ICS-dominated

* The two different scenarios of gap breakdown can in a natural way
explain the mode-changing phenomenon when both modes produce plasma
suitable to generate radio emission, and pulse nulling when the radio
emission is not generated in one of the modes.

* The mode changes of the IAR may explain the anti-correlation of radio
and X-ray emission in very recent observations of PSR B0943+10 [87].

* The regular drift of subpulses can be expected only when the gap operates
in the PSG-on mode. The proposed model of drift allows to connect
the drift information obtained by radio observations with the X-ray
data of rotation-powered pulsars.

CHAPTER: ACKNOWLEDGEMENTS

I would like to express my deep gratitude to Professor Giorgi Melikidze,
my research supervisor, for his patient guidance, enthusiastic encouragement
and useful critique of this research work. I would also like to thank
Professor Janusz Gil for his advice and support which allowed me to
complete this thesis. This research project would not have been possible
without the support of many people. I would like to thank all my colleagues
at the Institute of Astronomy who taught me a lot and never refused
to help: Professor Ulrich Geppert, Professor Dorota Gondek-Rosińska,
Professor Jarosław Kijak, Dr. Krzysztof Krzeszowki, Dr. Wojciech Lewandowski,
Professor Andrzej Maciejewski, Dr. Krzysztof Maciesiak, Dr. Olaf Maron,
Dr. Roberto Mignani, Dr. Marek Sendyk, and Dr. Agnieszka Słowikowska.
And a special thanks to Mrs Emilia Gil for her assistance in all the
administrative issues.

I would also like to extend my thanks to friends and family for their
support, sacrifice, patience and wisdom. My special thanks are extended
to my parents for their support and encouragement throughout my studies.

Thank you.

Image by Karolina Rożko

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