Photon Propagation Pair Production Or Photon Splitting

Published Date: 02 Nov 2017

Chapter 4

Cascade simulation

The cascade simulation can be divided into three parts:

• propagation and photon emission of the primary particle (positron or electron),

• photon propagation, pair production or photon splitting,

• propagation and photon emission of the secondary1 electrons and positrons.

Both photons and particles are tracked in the "co-rotating" frame of reference (the frame which rotates with the star), and any bending of the photon path due to rotation is ignored. This is expected to be valid since the simulation includes regions far inside the light cylinder (Medin & Lai, 2010). The approach presented below also ignores any effects of general relativity on the photon/particle trajectory.

Figure 4.1 presents a summary flowchart of the algorithm used to calculate the properties of sec- ondary plasma and the spectrum of radiation for a given structure of a neutron’s star magnetic field and gap parameters.

Figure 4.1: Flowchart of the algorithm used to calculate a cascade simulation.

4.1 Curvature Radiation

As we have shown in Chapter 2, an ultrastrong surface magnetic field (Bs > 1014 G) is accom- panied by high curvature (curvature radius ℜ6 ≈ 0.1 − 10). This suggests that one of the important processes of radiation which should be considered is Curvature Radiation (CR).

CR is quite similar to ordinary synchrotron radiation (radiation of ultrarelativistic particles in the magnetic field), the only difference being that the radius of circular motion (the gyroradius) is in fact the curvature radius of magnetic field lines. Due to beaming effects the radiation appears to be concentrated in a narrow set of directions about the velocity of the particle. The angular width of the cone of emission is of the order ∼ 1/γ, where γ is a Lorentz factor of an emitting particle (for more details see Rybicki & Lightman,

1979).

After the primary particle leaves the acceleration zone (the gap region) it moves out- ward along the field line in a stepwise fashion. The length of the step ∆s is chosen so as to achieve sufficient accuracy even for large curvature of the magnetic field line,

1 In this thesis the term "secondary" refers to any cascade particle except for the primary particles, e.g. the third generation of electrons and positrons are all "secondary" particles.

∆s ≈ 0.01ℜmin, where ℜmin is the minimum radius of curvature. The distribution of CR

photon energy can be written as (see Equation 14.93 in Jackson, 1998)

[eq.] (4.1)

where E = 4πe2γ4/3ℜ is the total energy radiated per revolution, ǫCR = 3γ3~c/(2ℜ) is the characteristic energy of curvature photons, and K5/3 is the n = 5/3 Bessel function of the second kind. The total energy radiated by an electron after it passes the length ∆s, E∆s , can be written as

[eq.] (4.2)

Thus, by using Equations 4.1 and 4.2 we can write the formula for the distribution on

CR photon energy after an electron passes length ∆s

[eq.] (4.3) It is convenient to divide the spectrum of photon energy into discrete bins. Then, the

number of photons in each energy bin can be calculated as

[eq.] (4.4)

where ǫi is the lowest energy for the i-th bin and ∆ǫ is the energy bin width. Our simulation uses 50 bins with an energy range of ǫ0 = 4 × 10−2 keV (soft X-ray) to ǫ49 = 4 × 105 MeV

(hard γ-rays).

The polarisation fraction of CR photons is between 50% and 100% polarised parallel to the magnetic field, depending on the photon frequency (see Jackson, 1998, Rybicki & Lightman, 1979). Therefore, the polarisation is randomly assigned in the ratio of one ⊥ (perpendicular to the field) to every seven k (parallel to the field) photons, corresponding to

75% parallel polarisation (Medin & Lai, 2010).

4.2 Photon propagation

To explain some of the properties of pulsars and their surroundings (e.g. nebulae ra- diation), large magnetospheric plasma densities exceeding the Goldreich-Julian density (see Equation 3.7) by many orders of magnitude are required. In order to simulate the process of generation of such dense plasma it is necessary to check the conditions of photon decay into electron-positron pairs.

A photon with energy Eγ > 2mc2 and propagating with a nonzero angle Ψ with respect

to an external magnetic field can be absorbed by the field and, as a result, an electron-positron pair is created. The concurrent process is photon splitting γ → γγ, which may occur even if the photon energy is below the pair creation threshold (Eγ < 2mc2).

In the cascade simulation the photon is emitted (or scattered in the case of ICS) from

point Pph in a direction tangent to the magnetic field line ∆sk. The direction vector is calculated as the value of the magnetic field at the point of photon creation (see Equations

2.1, 2.2 and 2.7) normalised so that its length is equal to the desired step ∆sk = B∆s/B. However, the direction of the magnetic field at the point of photon emission does not take into account the randomness of the emission direction due to the relativistic beaming effect. In Section 4.2.1 we describe a procedure to include the beaming effect in the emission process which alters ∆sk → ∆sph . Finally, we can write that at the point of curvature emission photons are created with energy ǫph, polarisation k or ⊥, weighting factor Nǫ (number of photons), and with both optical depths (for pair production τ and for photon splitting τsp) set to zero. The photon propagates in a straight line from the point of emission. In each step

the photon travels a short distance ∆sph through the magnetosphere. In the co-rotating frame of reference in every step we need to take into the account aberration due to pulsar rotation. In order to do so, in every step we alter the photon position according to the procedure described in Section 4.2.2. At the new photon position the change in the optical depth for pair production, ∆τ , and for photon splitting ∆τsp, are calculated as (Medin & Lai,

2010):

[eq.] (4.5)

k,⊥where Rk,⊥ and Rsp

[eq.] (4.6)

are the attenuation coefficients for k or ⊥ polarised photons for

pair production and photon splitting, receptively.

4.2.1 Relativistic beaming (emission direction)

Due to relativistic beaming the emission direction should be modified by an additional emission angle of order ∼ 1/γ. We use the following steps to include the beaming effect in our simulation (see Figure 4.2).

Figure 4.2: Relativistic beaming effect of photon emission (for both CR and ICS). In the simulation we include the beaming effect by performing three steps: (I) rotation of the xyz frame of reference in order to align the z-axis with ∆sk, (II) transformation of the step vector from a Cartesian to a spherical system of coordinates and alteration of the θ and φ components with random values 1/γ cos Λ and Π, respectively. (III) transformation of the step vector from a spherical to a Cartesian system of coordinates and rotation back to the original system of reference. Note that after these steps we get a new vector

∆sph inclined to the primary one, ∆sph , at an angle ranging from 0 to 1/γ.

(I) The first step is rotation of the xyz frame of reference in order to align the z-axis with

∆sk. In our calculations we used rotation by angle ςy around the y-axis, Ry (ςy ), and rotation by angle ςx around the x-axis, Rx (ςx). The final rotation matrix can be written as

[eq.] (4.7) The Euler angles for rotations can be calculated as

[eq.] (4.8) Note that in order to increase readability, the ∆ symbol and k index were discarded

(e.g. sx = ∆sk,x).

(II) The second step is the transformation of the step vector’s coordinates in the double

rotated frame of reference ∆s′′

= s′′ , s′′, s′′

to spherical system of coordinates and

ph x y z

and alteration of the θ and φ components as follows

[eq.] (4.9)

where Λ and Π are random angles between 0 and 2π. The inverse tangent denoted in the φ- coordinate must be suitably defined by taking into account the correct quadrant (see the "atan2" description in the footnote on page 22).

(III) The last step is the transformation of vector components to the Cartesian system of

coordinates, s′′

= s′′ sin (s′′) cos s′′

, s′′ sin (s′′) sin s′′

, s′′ cos (s′′) and rotation back

ph r θ

φ r θ

φ r θ

phto the original coordinate system ∆sph = (Ryx)−1 s′′ .

The rotation matrix of this transformation can be written as

[eq.] (4.10)

4.2.2 Aberration due to pulsar rotation

Note that in our frame of reference (co-rotating with a star) the path of the photon should be curved with the angular deviation from a straight line growing approximately as sphΩ/c = sph/RLC (see Harding et al. 1978). This curved path modifies the growth of the photon-magnetic field intersection angle and, therefore, the location of photon decay.

In our simulation we include the aberration effect by alteration of photon position Pph

in every step ∆sph (see Figure 4.3).

Figure 4.3: Aberration due to pulsar rotation. We use the following procedure to include the aber- ration effect: (I) rotation around the y-axis to align Ω with µ, (II) rotation by angle ω = 2π∆sbm / (cP ) around the z-axis (which reflects the pulsar rotation), (III) rotation back to the original frame of reference (in which µ is aligned with the z-axis).

We use the three-step procedure to alter the photon position.

ph(I) Rotation of the xyz frame of reference around the y-axis by angle α, P′

= Ry (α) Pph .

Note that here α refers to the inclination of the magnetic axis with respect to the rotation axis

and we assume that the pulsar’s angular velocity vector Ω lies in the xz-plane. The rotation matrix of this transformation can be written as

[eq.] (4.11) (II) After step (I) the z-axis is aligned to Ω, and in order to include the rotation of the pulsar we

need to again rotate the frame of reverence by angle ω = 2π∆sbm / (cP ) around the z-axis,

P′′ ′

ph = Rz (ω) Pph . We use the following rotation matrix

[eq.] (4.12)

ph

ph(III) The final step is a rotation back to the original frame of reference, P′′′

= (Ry (α))−1 P′′ ,

using the following rotation matrix

[eq.] (4.13)

4.2.3 Pair production attenuation coefficient

The attenuation coefficient for pair production can be written as

[eq.] (4.14) where R′ is the attenuation coefficient in the "perpendicular" frame (i.e. the frame where the photon propagates perpendicular to the local magnetic field), Ψ is the angle of inter-

section between the photon and the local magnetic field. Please note that we suppress the subscripts k and ⊥ , but the pair production attenuation coefficient has to be calculated

separately for both polarisations.

R′The total attenuation coefficient for pair production is given by R′ = Pjk R′j,k , where

j,k is the attenuation coefficient for the process in which the photon produces an electron in Landau level j and a positron in Landau level k, and the sum is taken over all possible

states for the electron-positron pair. Since pair production is symmetric with respect

jk

= R

jk

kjto the electron and positron, R′ ′

, for simplicity’s sake we will use R′

to represent the

combined probability of creating the pair in either the (jk) or (kj) state. For given Landau levels j and k, the threshold condition for pair production is (Medin & Lai, 2010)

n[eq.] (4.15)

γwhere E′

= Eγ sin Ψ is the photon energy in the perpendicular frame and E′

= mc2√1 + 2ǫB n

is the minimum energy of an electron/positron in Landau Level n. This condition can be

written in a dimensionless form as

[eq.] (4.16)

where Ç«B = ~eB/ (mc) is the cyclotron energy of a particle in magnetic field B in units of mc2.

The first nonzero attenuation coefficients for both polarisations (⊥ and k) are (Daugh- erty & Harding, 1983)

[eq.] (4.17) [eq.] (4.18) [eq.] (4.19) [eq.] (4.20)

[eq.] (4.21)

⊥,00where a0 is the Bohr radius (let us note that R′

= 0). In the above equations the atten-

uation coefficients are multiplied by a factor of two for all channels except 00 (see the text

above Equation 4.15).

The optical depth is defined as:

[eq.] (4.22) We can assume Ψ ≪ 1, because all high-energy photons (x > 1) will produce pairs

much earlier than Ψ reaches a value near unity. In this limit sin Ψ ≃ sph/ℜ, so the relation

between x and sph can be expressed by

Equation 4.22 can be rewritten as

[eq.] (4.23)

τ = τ1 + τk,2 + τ⊥,2 + ...;

[eq.] (4.24)

where s0 and s1 are distances which the photon should pass in order to have energy x0 and x1 , respectively (in the perpendicular frame of reference). Let us note that s0, s1 and s2 are of the same order, and if s < s0 the attenuation coefficient is zero.

The optical depth to reach the second threshold for pair production is

[eq.] (4.25)

where s0 is the distance travelled by the photon to reach the threshold x0 ≡ 1, and s1 is the distance travelled by the photon to reach the second threshold x1 ≡ 1 + √1 + 2ǫB /2.

Figure 4.4: Panel (a) presents the dependence of the optical depth for pair production on the magnetic field strength (βq = B/Bq ). Panel (b) presents the dependence of the optical depth on photon energy in the perpendicular frame of reference (x = ǫ sin Ψ/ 2mc2 ). On both panels the photon energy is ǫ = 500 MeV, while panel (b) was obtained for magnetic field strength βq = 1.

4.2.4 Photon mean free path

As was shown in the previous section (see Figure 4.4) for strong magnetic fields (e.g. βq & 0.2 for ǫ = 500 MeV), τ1, τk,2, and τ⊥,2 are much larger than one. Therefore, the pair production process takes place according to two scenarios (see also Medin & Lai, 2010). If βq & 0.2 photons produce pairs almost immediately upon reaching the first threshold, the created pairs will be in the low Landau levels (n . 2). If βq . 0.2, the photons will travel longer distances to be absorbed and the created pairs will be in the higher Landau levels.

Thus, for strong magnetic fields (βq & 0.2) the photon mean free path can be approximated

as

[eq.] (4.26)

while for relatively weak magnetic fields (βq . 0.2) we can use the asymptotic approxi- mation derived by Erber (1966):

[eq.] (4.27) [eq.] (4.28)

4.2.5 Photon-splitting attenuation coefficient

kBased on the kinetic selection rule (Adler, 1971, Usov, 2002, Baring & Harding, 2001) only the ⊥→kk process is allowed (Medin & Lai, 2010). Thus, for k-polarised photons the attenuation coefficient for photon splitting is zero (Rsp = 0). For ⊥-polarised photons

we use the formula adopted from the numerical calculation of Baring & Harding (1997)

to calculate the splitting attenuation coefficient in the perpendicular frame:

[eq.] (4.29) For x ≤ 1 this expression reproduces the results of Baring & Harding (1997) with a dis-

R′spcrepancy less than 10% at both βq ≤ 0.5 and βq ≫ 1 while underestimating the results at βq = 1 by less than 30%. As can be seen from Figure 4.5, the attenuation coefficient

⊥ drops rapidly with the magnetic field strength for βq < 1, thus photon splitting is unimportant for βq . 0.5 (e.g. Baring & Harding, 2001; Medin & Lai, 2010).

Figure 4.5: Dependence of the photon -splitting attenuation coefficient on the energy of the photon in the perpendicular frame (x = ǫ sin Ψ/ 2mc2 , vertical axis) and on the strength of the magnetic field (βq = B/Bq , horizontal axis).

4.2.6 Pair creation vs photon splitting

Even though the attenuation coefficient for photon splitting above the first threshold (x > x0) is much smaller than for pair production (see Figure 4.6), in ultrastrong magnetic fields (βq & 0.5) the ⊥-polarised photons split before reaching the first threshold (see Figure 4.7). On the other hand, the k-polarised photons produce pairs in the zeroth Landau level.

Figure 4.6: Attenuation coefficients of pair production and photon splitting in the perpendicular frame of reference. Panel (a) was obtained using photon energy Eγ = 103 MeV and magnetic field strength B = Bq = 4.414 × 1013 G (βq = 1). Panel (b) presents calculations for photon energy Eγ = 103 MeV and magnetic field strength B = 2.5 × 1014 G (βq = 5.7).

Figure 4.7: Optical depth for pair production and photon splitting for ⊥-polarised photons. Panel (a) presents results for Eγ = 103 MeV and B = Bq = 4.4 × 1013 G (βq = 1), while panel (b) was obtained using the same photon energy but a stronger magnetic field B = 2.5 × 1014 G (βq = 5.7). If βq = 1 the photon will create an electron-positron pair, while in an ultrastrong magnetic field (βq = 5.7) the photon will split before it reaches the first threshold, x = x0.

4.2.7 Secondary plasma

In the simulation whenever τ ≥ 1 and the threshold for pair production is reached (x = x0 for k-polarised photons and x = x1 for ⊥-polarised photons), the photon is turned into an electron-positron pair. Whereas if τsp ≥ 1 the photon is turned into two photons. To simplify the calculation we assumed that each photon takes half of the energy of the parent photon (see Baring & Harding, 1997). At the point of splitting a new photon is created with an energy 0.5ǫph and weighting factor 2∆Nǫ . The new photon is k-polarised and is assumed to travel in the same direction as the parent photon, ∆sph . In fact, the photon should be destroyed only with probability 1−e−τ , but in practice it has a negligible effect on the cascade (Medin & Lai, 2010).

For βq . 0.1 pairs are created in high Landau levels with energy half of the photon energy each (see Daugherty & Harding, 1983). We assume that the newly created particles travel in the same direction as the photon. When βq & 0.1 the electron and positron are created in low Landau levels (in the calculations we choose the maximum allowed values of j and k).

Figure 4.8 presents the spectrum of Curvature Radiation for a dipolar and non-dipolar struc- ture of magnetic field lines. Note the characteristic three peaks in the CR spectrum for the non-dipolar structure. Formation of the peaks is caused by the fact that the particle passes regions with three different types of curvature: (i) just above the stellar surface, z ≈ 1 km, where curvature is the highest; (ii) at altitudes where the influence of anomalies is compa- rable with the global dipole, z ≈ 2.5 km, also with strong curvature; (iii) and at altitudes where the influence of anomalies is negligible, z & 3.1R, with approximately dipolar cur- vature (see Figure 2.11). Hence, the spectrum is a sum of radiation generated in a highly non-dipolar magnetic field (high energetic and soft γ-rays) and with radiation at higher alti- tudes where the magnetic field is dipolar (X-rays). The primary particle loses about 63% and

1% of its initial energy in the non-dipolar and dipolar case, respectively. As can be seen from the Figure, to get high emission of CR photons and, thus, a significant density of secondary plasma, a non-dipolar structure of the magnetic field is required.

Figure 4.8: Spectrum of CR produced by a single primary particle for a dipolar (blue line) and non-dipolar (red line) structure of the magnetic field. The minimum radius of curvature in the dipolar case is

min

min

ℜ6 ≈ 50, while in the non-dipolar case ℜ6 ≈ 2. In both cases the radiation was calculated up to

a distance of D = 100R, and with an initial Lorentz factor of the primary particle γc = 3.5 × 106.

The high energetic photons produced in a strongly non-dipolar magnetic field will either split or create electron-positron pairs. Figure 4.9 presents the distribution of particle energy created by CR photons. Note that for βq . 0.1 the pairs are created in high Landau levels and in order to get the final distribution of secondary plasma energy we should consider the loss of particle energy due to Synchrotron Radiation (see 4.3).

Figure 4.9: Distribution of particle energy created by CR photons calculated for a non-dipolar structure of the magnetic field. For this specific magnetic field configuration and initial parameters (see the caption of Figure 4.8) the secondary plasma multiplicity is Msec ≈ 6 × 103. Note that this result does not include Synchrotron Radiation and the actual energies of the created pairs are lower as they lose their transverse momenta (see Section 4.3).

4.3 Synchrotron Radiation

When pairs (electrons and positrons) are created in high Landau Levels they radiate away their transverse momenta through Synchrotron Radiation (SR). The secondary electron (or positron) is created with energy γmc2 and pitch angle Ψ , which corresponds to a specific value of Landau Level n. For the calculations it is convenient to choose the frame in which the particle has no momentum along the magnetic field direction and only moves transverse to the field in a circular motion ("circular" frame). The energy of the created particles in the circular frame of reference, E⊥ = γ⊥mc2, is related to that in the co-rotating frame by (Medin & Lai, 2010)

[eq.] (4.30) The power of synchrotron emission, PSR , can be written as

[eq.] (4.31)

In the circular frame, E⊥ is radiated away through synchrotron emission after a particle travels a distance

[eq.] (4.32)

The particle (electron or positron) mean free path for SR is much shorter than for other relevant cascade processes (see Section 4.1 for Curvature Radiation, and Section 4.4 for ICS). In fact, it is so short that we assume that the particle loses all of its perpendicular momentum p⊥ due to SR before moving from its initial position (see Daugherty & Harding,

1982; Medin & Lai, 2010). The final energy of the particle once it reaches the ground

Landau level (n = 0, p⊥ = 0) is given by

[eq.] (4.33)

here β = v/c = p1 − 1/γ2 is the particle velocity in units of speed of light.

To simplify the simulation the approach proposed by Medin & Lai (2010) was used which assumes that in the circular frame synchrotron photons are emitted isotropically in the plane of motion such that there is no perpendicular velocity change of the particle (the Lorentz factors γ and γ⊥ decrease but γk is constant). Thus, the frame corresponding to the circular motion of the particle does not change during the synchrotron emission and Equation 4.33 remains valid until the particle reaches the ground state. In order to simulate the full SR process the following procedure was adopted: in the circular frame the particle Lorentz factor γ⊥ drops from its initial value to γ⊥ = 1 (i.e., n = 0) in a series of steps. In each step one synchrotron photon is emitted, with energy ǫ⊥ , which depends on the current value of γ⊥. After the photon is emitted the energy of the particle is reduced by ǫ⊥ ,

∆γ⊥ = ǫ⊥ /mc2. In the next step the particle with reduced energy emits a photon with

a new value of ǫ⊥ . This process continues until the particle is at n = 0 Landau level.

Depending on the Landau level n, the photon energy ǫ⊥ of the synchrotron radiation is

chosen in one of three ways.

(I) If the particle is created in a high Landau Level (n ≥ 3), the energy of the photon is chosen randomly but with a probability based on the asymptotic synchrotron spectrum (e.g. Sokolov & Ternov (1968), Harding & Preece (1987)):

[eq.] (4.34)

where

[eq.] (4.35)

is the characteristic energy of the synchrotron photons, f = 1 − ǫ⊥ / (γ⊥mc2) is the frac-

xtion of the electron’s energy remaining after photon emission, F (x) = x ´ ∞ K

5/3

(t) dt, and

G (x) = xK2/3 (x). The functions K5/3 and K2/3 correspond to modified Bessel func- tions of the second kind. The expression in Equation 4.34 differs from the classical

synchrotron spectrum (e.g. Rybicki & Lightman, 1979) in two ways: (i) by a factor of f = 1 − ǫ⊥ / (γ⊥mc2) which appears in several places in Equation 4.34 (when the pho- ton energy is equal to the electron energy, ǫ⊥ = γ⊥mc2, the asymptotic expression goes to zero), (ii) by a term containing the function G (x). While the term G (x) appears in the classical expressions for the radiation spectra of both k and ⊥-polarised photons, in the classical expression for the total radiation spectra these terms cancel out. However, when the quantum effect of the electron spin is considered there is asymmetry between the perpendicular and parallel polarisations such that term G (x) term (Medin & Lai,

2010).

(II) When n = 2, the energy of the photon is either that required to lower the particle energy to its first excited state (n = 1) or to the ground state (n = 0) with a probability that depends on the local magnetic field strength. To calculate the transition rates for n = 2 we use the simplified prescription based on the results of Herold et al. (1982) (see also Harding & Preece, 1987): if βq < 1, the energy of the photon is chosen to be that which is

required to lower the particle to the first excited state, ǫ⊥ = mc2 p1 + 4βq − p1 + 2βq .

If βq & 1 the energy of the photon is randomly chosen to be that which is required to lower the electron

(or positron) to either the ground state (ǫ⊥ = mc2 p1 + 4βq − 1 ), 50% of time, or the

first excited state, 50% of time.

(III) If n = 1, the energy of the photon is that required to lower the particle energy to its ground state, ǫ⊥ = mc2 p1 + 2βq − 1 . If the particle is not in the ground state

after emission of the synchrotron photon, γ⊥ is recalculated and a new photon energy is chosen.

The energy of the photon in the co-rotating frame can be written as

[eq.] (4.36) The emitted synchrotron photon is with the same weighting factor ∆Nǫ as the secondary

particle that emitted it. Because the photon is emitted in a random direction perpendicular

to the magnetic field in the circular frame, in the co-rotating frame the angle of emission can be calculated using Equations 4.30 and 4.33 as follows

[eq.] (4.37)

where Π is a random number from 0 to 2π. In our simulation we include this angle of emission by using the same approach as presented in Section 4.2.1, but as the maximum value we use Ψ instead of 1/γ.

For SR the polarisation fraction is in the range of 50% to 100% polarised perpendicular to the magnetic field (the exact opposite of the CR case). In our simulation we randomly as- sign the photon polarisation in the ratio of one k to every seven ⊥ photons (corresponding

to a 75% perpendicular polarisation).

Figure 4.10 presents the spectrum of SR produced by a single secondary particle. To show the nature of the spectrum, a relatively high pitch angle was used. Note that when a

particle is created at a distance where the magnetic field is relatively weak (e.g. βq = 10−5 for

γ = 102) then most of the energy is radiated in the range of 1 − 10 keV. Thus, we believe that

if a strong enough instability forms (that increases the particle’s pitch angle), the SR process could be responsible for the production of a non-thermal component of the X-ray spectrum.

Figure 4.10: Spectrum of SR produced by a single secondary particle with Lorentz factor γ =

102. We have assumed that the particle was created in a region where the magnetic field strength was B = 4.14 × 108 (βq = 10−5) and with a pitch angle Ψ = 7◦. For such a relatively high pitch angle the particle loses most of its energy ending with Lorentz factor γend ≈ 6.

Figure 4.11 presents the final spectrum produced by a single primary particle with an initial Lorentz factor of γc = 3.5×106 for a non-dipolar configuration of the surface magnetic field of PSR J0633+1746 (see Section 2.4.2). Due to CR the particle loses about 68% of its initial energy (∆ǫ = 2.2 × 106mc2), which is radiated mainly in close vicinity of a neutron star, where curvature of the magnetic field is the highest. As the γ-photons propagate they will split (only if the magnetic field is strong enough) and eventually most photons will be absorbed by the magnetic field – as a result electron-positron pairs emerge. These pairs radiate away their transverse momenta through SR, producing mainly X-ray photons (at larger distances) and only a few γ-photons (in a strong magnetic field just above the stellar surface). Note that at the end (after pair production) only 14% of the primary particle’s energy (∆ǫph = 4 × 105mc2) is converted into photons and the bulk of its energy, 54% (∆ǫpairs = 1.8 ×

106mc2), is allocated into secondary plasma. The multiplicity for this specific simulation is of the order Msec = 104. Note that we use Msecto describe the multiplicity of secondary plasma in contrast to M, which describes particle multiplicity in the IAR.

Figure 4.11: Final spectrum produced by a single primary particle. The red line corresponds to the initial CR spectrum for a non-dipolar structure of the magnetic field, while the blue line presents the final spectrum with the inclusion of photon splitting, pair production and SR.

Figure 4.12 presents the distribution of particle energy created by CR photons but with the inclusion of SR emission (red line). Note that synchrotron emission both lowers the particle energy (after SR maximum at γ ≈ 5 − 8, while without SR at γ ≈ 15 − 20) and increases the multiplicity of secondary plasma Msec ≈ 104.

Figure 4.12: Distribution of particle energy created by CR photons calculated for a non-dipolar structure of the magnetic field. For this specific magnetic field configuration and initial parameters (see the caption of Figure 4.8) the secondary plasma multiplicity is Msec ≈ 104. Note that this result does not include Synchrotron Radiation and the actual energies of the created pairs are lower as they lose their transverse momenta (see Section 4.3).

4.4 Inverse Compton Scattering

The Inverse Compton Scattering (hereafter ICS) process in the neutron star vicinity has been studied extensively by Xia (1982); Kardash¨ev et al. (1984); Xia et al. (1985); Daugherty & Harding (1989); Dermer (1989, 1990); Bednarek et al. (1992); Chang (1995); Sturner (1995); Zhang & Qiao (1996); Zhang et al. (1997); Zhang & Harding (2000); Hard- ing et al. (2002), etc. According to these studies, the ICS process may play a significant

role in the physics of a neutron star’s magnetosphere. Relativistic particles (positrons and electrons) can Compton-scatter thermal radiation from the neutron star surface. As a particle with a certain relativistic velocity scatters the thermal photons with a blackbody distribution, it will produce radiation in quite a wide energy range. However, we can dis- tinguish two characteristic frequencies of upscattered photons: one is the frequency due to resonant scattering, another is the frequency range contributed by the scattering of photons at the "thermal-peak" frequency. The Resonant Inverse Compton Scattering (RICS) corresponds to a scenario when the scattering cross section is largest. On the other hand, Thermal-peak Inverse Compton Scattering (TICS) corresponds to interactions with photons with the maximum number density. These two modes are very different when it comes to the nature of the process. The photons’ energy in RICS depends on the strength of the magnetic field, thus at low altitudes (where the field is very strong), it can power pair cascades, while TICS can be responsible for magnetospheric radiation at much higher altitudes. Note that for some specific combinations of magnetic strength and distribution of background photons, RICS and TICS are indistinguishable as the resonance frequency falls into the thermal peak range.

4.4.1 The cross section of ICS

Due to the rapid time scale for synchrotron emission (see section 4.3), a particle in an excited Landau level almost instantaneously de-excites to the ground level. The particle motion is therefore strongly confined to the magnetic field direction. In our calculations we consider the geometry illustrated in Figure 4.13. In the observer’s frame of reference (OF), a particle with Lorentz factor γ travelling along the magnetic field line scatters a photon. Let ψ = arccos µ be the angle between the magnetic field line (particle propagation) and the direction of photon propagation in OF and ψ′ = arccos µ′ in the particle rest frame (PRF). The energy of the photon in PRF is given by

[eq.] (4.38)

sAfter scattering, the photon energy is denoted by ǫ′

in PRF and Ç«s in OF. The angle

between the direction of propagation of the scattered photon and B (which describes the

s

sdirection of particle propagation) is denoted by ψs = arccos µs in OF and ψ′

= arccos µ′ ,

swhere µ′

= (µs − β) / (1 − βµs) in PRF (Dermer, 1990).

Figure 4.13: Reproduction of the Figure from Dermer (1990). Geometry of the ICS event in the observer’s frame (left) and the particle rest frame (right). A particle with Lorentz factor γ, beamed along the direction of the magnetic field, scatters a photon with energy ǫ directed at angle ψ with respect to the magnetic field line. After scattering, the energy and angle of the photon are denoted by ǫs and ψs, respectively. Quantities in the particle rest frame are denoted by a prime.

4.4.1.1 ICS cross section in the Thomson regime

Restriction to the Thomson regime requires that γǫ (1 − µ) ≪ 1. In the particle rest frame, the angle ψ′ = arcsin {γ−1 [sin ψ/ (1 − β cos ψ)]}, and when γ ≫ 1, |µ′| → 1. In the Thomson regime the only important Compton scattering process involves transitions between ground-state Landau levels. Daugherty & Harding (1989) and Dermer (1989) calculated the differential cross section (after summing over polarisation modes and inte-

s

sgrating over azimuth) for a photon scattered from ψ′ = 0◦ into angle ψ′

= arccos µ′ as

follows

[eq.] (4.39)

Bwhere Γ = 4αf ǫ2 /3 is the resonant width (Daugherty & Ventura, 1978; Xia et al., 1985), σT is the Thomson cross section, and αf = e2/~c is the fine-structure constant. In the Thomson limit ǫ′ ≪ 1, and thus the scattered photon energy in PRF can be approximated as

[eq.] (4.40)

sEquations 4.39 and 4.40 show that a differential magnetic Compton scattering cross sec- tion when γ ≫ 1 is similar in form to a nonmagnetic Thomson cross section. The important difference is that the magnitude of the cross section is enhanced when ǫ′ approaches ǫB and is depressed at energies ǫ′ < ǫB . The total cross section for magnetic Compton scat- tering, obtained by integrating Equation 4.39 over µ′ , was calculated by Dermer (1989);

Zhang et al. (1997) and is given by

[eq.] (4.41)

where σI C = σT , σT is the Thomson cross section, u = ǫ′/ǫB , a =

23 αf ǫB .

4.4.1.2 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 γǫ (1 − µ) & 1. 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 σIC = σT we can use the following relationship

[eq.] (4.42) In an extreme relativistic regime ǫ′ ≫ 1 the Klein-Nishina formula can be simplified to

[eq.] (4.43)

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

4.4.1.3 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 Gonthier et al. (2000), this approach does not account for the relativistic quantum effects of strong magnetic fields (B > 1012 G). When the photon energy exceeds mc2 in the particle rest frame, the strong magnetic field significantly lowers the Compton scattering cross section below and at the resonance. Gonthier et al. (2000) developed expressions for the scattering of ultrarelativistic electrons with γ ≫ 1 moving parallel to the magnetic field. Because of the large Lorentz Factor of particle γ , the photon incident angle ψ gets Lorentz concentrated to ψ′ ≈ ψ/2γ ≈ 0◦ in the PRF. The differential cross section in the rest frame of the particle can be written as

where

[eq.] (4.44) [eq.] (4.45)

and

[eq.] (4.46)

The φ′ dependence and the imaginary terms are isolated in the polarisation components and in the phase exponentials, thus leading to elementary integrations over the azimuthal angle, φ′ (Gonthier et al., 2000). The values of B and C can be expressed as:

[eq.] (4.47) Since the differential cross section depends on the final Landau state l, a sum must be

performed over all the contributing Landau states. The energy of the scattered photon is

given by

[eq.] (4.48)

where l is the final Landau level of the scattered particle. Each final state has an energy threshold of lǫB , thus the maximum contributing Landau state lmax can be expressed as:

ǫ′/ǫB − 1 < lmax < ǫ′/ǫB . To obtain the energy-dependent cross section, the differential

scross section can be numerically integrated over ψ′

using the Romberg integration method.

For this particular case (scattering of relativistic particles) there is only one resonance occurring at the fundamental cyclotron resonance ǫB = βq = eB/ (mc).

Although the expressions presented in this section describe the exact cross section

for ICS in strong magnetic fields, due to their complexity (and thus high computing requirements) their usage in cascade simulation is limited.

4.4.1.4 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 ǫ′ < ǫB and also ǫ′ < 1.

sGonthier et al. (2000) showed that by keeping only terms to first order in ǫ′ and ǫ′

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 ǫ′ > ǫB . However, the approximation is close to the total cross section for both energy regions (ǫ′ < ǫB and ǫ′ > ǫB ), even for high magnetic field strengths (see Figure 4.14).

According to Gonthier et al. (2000), the polarisation-dependent and averaged, approx- imate cross section can be calculated as:

[eq.] (4.49) [eq.] (4.50) [eq.] (4.51) [eq.] (4.52) [eq.] (4.53)

[eq.] (4.54)

Figure 4.14 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 the

incident photon energy in PRF (in units of cyclotron energy, ǫ′/ǫB ). As mentioned above, the approximation is valid in the region below the resonance, ǫ′ < ǫB . Although the

approximation overestimates the cross section for l = 0 final Landau state in the regime of high energetic photons (ǫ′ > ǫB ), 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, ǫ′ < ǫB and ǫ′ > ǫB . Calculation of the cross section for the resonance frequency (ǫ′ = ǫB ) is presented in the next section.

Figure 4.14: 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 B14 = 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.

4.4.2 Resonant Compton Scattering

This section describes an approach used to calculate the RICS cross section for ultrastrong mag- netic fields (B > 1012 G). For weaker fields the calculations are much simpler and reso- nance is already included in Equation 4.41.

The trend as βq increases is for the magnitude of the cross section to drop at all energies, while the width of the resonance increases (for βq ≥ 1 this width actually declines). Since the resonance is formally divergent, the common practice (see Xia et al., 1985; Latal, 1986; Daugherty & Harding, 1989; Dermer, 1990; Harding & Daugherty, 1991; Baring et al.,

2005; Harding & Lai, 2006; Baring & Harding, 2007, 2008) is to truncate it at ǫ′ = ǫB

2

−by introducing a finite width Γ. The procedure is to replace the resonant (ǫ′ ǫB )

2

denominator (see Equations 4.44 and 4.51) by (ǫ′ − ǫB )

+ Γ2/4 . In the βq ≪ 1 regime,

Bthe cyclotron decay width assumes the well-known result Γ ≈ 4αf ǫ2 /3 in dimensionless

βq ≫ 1, quantum and recoil effects generate Γ ≈ αf ǫB (1 − 1/e˜) where e˜

units. When

Bis Euler’s number (e.g. see Baring et al., 2011). These widths lead to areas under the resonance being independent of ǫB in the magnetic Thomson regime of βq ≪ 1 and scaling as ǫ1/2 when βq ≫ 1. These results can be deduced using the l = 0 approximation derived in Equation 4.51. By using this approach the averaged, approximate cross section can

be written as

[eq.] (4.55)

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. Medin & Lai, 2010)

[eq.] (4.56) 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.

Based on the assumption that approximately 50% of the photons are slightly below resonance and 50% are slightly above, in ultrastrong magnetic fields the ICS polarisation fraction is about 50% (Medin & Lai, 2010). Therefore, the polarisation is randomly assigned in the ratio of one ⊥ (perpendicular to the field) to every k photon.

4.4.3 Particle mean free path

For the ICS process the calculation of the particle mean free path lICS is not as simple as that of the CR process. Although we can define lICS in the same way as we defined lCR , 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 (Xia et al., 1985)

[eq.] (4.57)

where (as before) β = v/c is the velocity in terms of speed of light, nph represents the photon number density distribution of semi-isotropic blackbody radiation (see Equation

3.33). Here σ′ is the average cross section of scattering in the particle rest frame (see Equation 4.55). We should expect two modes of the ICS process, i.e. Resonant ICS and Thermal- peak ICS.

4.4.3.1 Resonant ICS

The RICS takes place if the photon frequency in the particle rest frame is equal to the cyclotron electron frequency. Using Equation 4.38 we can write that the incident photon energy is ǫ = ǫB / [γ (1 − βµ)]. For altitudes of the same order as the polar cap size we use

µ0 = 1, µ1 = 0 as incident angle limits for outflowing particles, and µ0 = 0, µ1 = −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

[eq.] (4.58)

min

max

where limits of integration, Ç«res and Ç«res , 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:

[eq.] (4.59) Here Γ is the finite width introduced in Section 4.4.2 to describe the decay of an excited

intermediate particle state.

Figure 4.15 presents the dependence of the integrand from Equation 4.58 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 4.4.2) 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 4.15). Note that both plots do not include the dependence of photon density on the 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 4.4.4).

Figure 4.15: Dependence of the integrand from Equation 4.58 on the energy of the incident photon. Both panels were calculated for surface temperature T = 3 × 106 K, cosine of the incident angle

µ = 0.1 and Lorentz factor of particle γ = 103. The left panel corresponds to resonance in magnetic field B = 1014 G, while the right panel was obtained using B = 3 × 1014 G.

4.4.3.2 Thermal-peak ICS

TICS includes all scattering processes of photons with frequencies around the maximum

min

max

of the thermal spectrum. In our simulation we adopt ǫth ≈ 0.05ǫth , and ǫth ≈ 2ǫth

2 2

where Ç«th = 2.82kT / (mc ) is the energy, in units of mc , 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

[eq.] (4.60) Figure 4.16 presents the dependence of the integrand from Equation 4.60 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 (µ ≈ 1). 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.

Figure 4.16: Comparison of the integrand from Equation 4.60 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 on the right and left were obtained using surface temperature T = 3×106 K, Lorentz factor of the particle γ = 102 and magnetic field strength B = 1012 G. The cosine of the incident angle, µ = 0.975 and µ = 0.96, was used for the left and right panel, respectively.

4.4.3.3 Calculation results

For ultrastrong magnetic fields quite a wide range of the particle Lorentz factor falls into the peak of background photons (see Figure 4.17). 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 4.16). For particles with Lorentz Factor γ & 105 , 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/ℜ).

Figure 4.17: 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 B14 = 2, radius of curvature of magnetic field lines ℜ6 = 1 (for the CR process) and hot spot temperature T6 = 3 (for RICS and TICS). Both RICS and TICS were calculated for a full range of incident angles (µ0 = 0, µ1 = 1). Note that for a Lorentz factor in the range of γ ≈ 2 × 103 − 105 the particle mean free paths of RICS and TICS are equal as the resonance falls into the peak of the background photons (see Figure 4.16).

Figure 4.18 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 (B14 = 0.5) is for particles with Lorentz factor γ ≈ 2 × 103, while for relatively stronger magnetic fields (B14 = 3.5) the RICS is most efficient for particles with energy an order of magnitude larger (γ ≈ 2 × 104). 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

4.15). 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 5 show that RICS is efficient only in the immediate vicinity of a neutron star since photon density at higher altitudes drops rapidly (see Section

4.4.4).

Figure 4.18: Dependence of a particle mean free path on magnetic field strength (B14) and the Lorentz factor of a particle (γ) for the RICS process. The particle mean free path was calculated for semi-isotropic blackbody radiation (µ0 = 0, µ1 = 1) with temperature T6 = 2.

4.4.4 Background photons

4.4.4.1 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 1 (e.g. see Table 1.4), the entire surface has the lowest temperature (T6 . 0.8), thus the initial photon density is up to about two orders of magnitude lower than warm spot radiation (T6 . 3) and up to about three orders magnitude lower than hot spot radiation (T6 . 5). 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 4.19 to calculate photon density at a given point P = (r, θ, φ). Then the relative density of photons originating from the entire surface can be calculated as

[eq.] (4.61)

where nst,0 (ǫ, Tst) is the density of photons with energy ǫ at the stellar surface with temperature Tst, and ∆θst is the angular diameter of the star at a distance from the star centre r.

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

[eq.] (4.62)

where nsp,0 (Ç«, Tsp) is the density of photons with energy Ç« at the spot surface (either hot or warm) with temperature Tsp. The angular diameter of the spot can be calculated as

[eq.] (4.63)

here Rsp is the spot radius and r1, r2 are the distances to the outer edges of the spot (see

Figure 4.19).

Figure 4.19: 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 Rsp 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 φ component of location P , while for the spot component it can be used only for small values of φ. In a more general case the spot should be projected on the surface perpendicular to the radius vector r and passing through point P .

Figure 4.20 presents the dependence of the relative photon density (n (z) /n0) on the dis- tance from the stellar surface. Due to the small size of a polar cap (hot spot, Rhs = 50 m) the density of the photons drops rapidly and already at a distance of about z = 150 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 (Rhs = 1 km) the photon density is reduced by an order of magnitude at a distance of about z = 3 km. From Equation 4.61 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 ≈ 3R ≈ 30 km.

Figure 4.20: 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 km, warm spot radius Rws = 1 km and hot spot radius Rhs = 50 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 4.21 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 z2 ≈ 5 m the photon density decreases

twice, while for the central (θ0) and middle line (θ1) the distances are respectively z0 ≈ 45 m and z1 ≈ 30 m. This result is important as the background photon density directly translates to the particle mean free path in ICS (see Section 4.4.3). 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).

Figure 4.21: 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: θ0 (central), θ1 (at the half distance to the edge), and θ2 (the cap edge). The altitude (z) was calculated for points which follow the magnetic field structure of PSR B0656+14.

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 4.22) for PSR B0656+14. The parameters of an entire surface and warm spot components are in agreement with the observations (see Table 1.4), while the hot spot component was calculated using parameters derived from the modelling of a non-dipolar structure of the magnetic field (see Section A.3). Already at a distance of 240 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 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 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 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).

Figure 4.22: 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, Tst =

0.7 MK, Rst = 20 km; warm spot, Tws = 1.2 MK, Rws = 1.8 km; and hot spot, Ths =

2.9 MK, Rhs = 50 m.

4.4.4.2 Photon incident angles

Another parameter that significantly affects the ICS is the incident angle between the back- ground photons and the relativistic particles. Especially for Resonant Inverse Compton Scat- tering is the incident angle of great importance. Figure 4.23 presents the dependence of a particle mean free path for ICS on a maximum value of the incident angle ψcrit. 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. 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 ψcrit ≈ 20◦ (for γ = 104) and ψcrit ≈ 75◦ (for γ = 103). 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 4.16 for more details).

Figure 4.23: Dependence of the particle mean free path on the maximum value of the incident angle ψcrit. The particle mean free path lp was calculated for magnetic field strength B = 1014 G assuming background blackbody radiation with a temperature T = 3 MK. Two different particle Lorentz factors were used for the calculations: γ = 103 (dashed lines) and γ = 104 (solid lines). The red lines correspond to Resonant Inverse Compton Scattering while the blue lines correspond to Thermal-peak Inverse Compton Scattering.

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 4.24 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 ≈ 90 m does the maximum value of the incident angle between the

photons from the hot spot and the particles drop to ψcrit = 30◦ , which significantly lowers the

efficiency of ICS for this source of background photons (see Figure 4.23). As the size of

the warm spot component is larger it will be significant for up to higher altitudes but already at a distance of z ≈ 1.5 km the maximum value of the incident angle also drops to ψcrit = 30◦.

Figure 4.24: 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 4.25 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