2.1 Vlasov form of the response tensor

The Vlasov method gives the response tensor ${\rm K}_{ij} (\omega,\boldsymbol {k})$ as

(2.1)\begin{align} {\rm K}_{ij}(\omega,\boldsymbol{k}) & = \delta_{ij} + \sum \frac{q^2}{\varepsilon_0 \omega^2} \int {\rm d}^3\boldsymbol{p} \left[ b_i b_j \frac{v_\parallel}{v_\perp} \left(v_\perp \frac{\partial}{\partial p_\parallel} - v_\parallel \frac{\partial}{\partial p_\perp}\right)\right.\nonumber\\ & \quad \left.+ \sum_{s=-\infty}^\infty \frac{V_i(\boldsymbol{k},\boldsymbol{p};s)V_j^*(\boldsymbol{k},\boldsymbol{p};s)}{\omega-s\varOmega-k_\parallel v_\parallel} \left(\frac{\omega- k_\parallel v_\parallel}{v_\perp} \frac{\partial}{\partial p_\perp} + k_\parallel \frac{\partial}{\partial p_\parallel}\right) \right] f(\boldsymbol{p}), \end{align}

where $\delta _{ij}$ is the Kronecker delta; the unlabelled sum is over all (unlabelled) species of particles with charge $q$, mass $m$, relativistic gyrofrequency $\varOmega = \varOmega _e / \gamma$, distribution function $f(\boldsymbol {p})$, velocity $\boldsymbol {v}$ and 3-momentum $\boldsymbol {p} = \gamma m \boldsymbol {v}$; the parallel and perpendicular components, with respect to unit vector $\boldsymbol {b}$ along the magnetic field, of vector quantities are denoted respectively by subscripts $\parallel$ and $\perp$; and

(2.2)\begin{equation} \boldsymbol{V}(\boldsymbol{k},\boldsymbol{p};s) = \left( v_\perp \frac{s}{k_\perp R} {\rm J}_s(k_\perp R), -{\rm i} \epsilon v_\perp {\rm J}'_s(k_\perp R), v_\parallel {\rm J}_s(k_\perp R) \right), \end{equation}

where ${\rm J}_s (k_\perp R)$ is a Bessel function of the first kind with argument $k_\perp R$, where $R=p_\perp /|q|B$ is the radius of gyration, and ${\rm J}'(x) = {\rm d} {\rm J}(x) / {{\rm d}x}$. Here, it is assumed that there are only two species, electrons and positrons, labelled $\epsilon =\mp$, respectively.

2.1.1 Anti-Hermitian part of ${\rm K}_{ij}(\omega,\boldsymbol {k})$

The anti-Hermitian part of (2.1) is given by

(2.3)\begin{align} {\rm K}^A_{ij}(\omega,\boldsymbol{k}) & =- \sum{{\rm i}{\rm \pi} q^2\over\varepsilon_0\omega^2}\int {\rm d}^3\boldsymbol{p} \, \sum_{s=-\infty}^\infty V_i(\boldsymbol{k},\boldsymbol{p};s)V_j^*(\boldsymbol{k},\boldsymbol{p};s) \delta(\omega - s\varOmega_e / \gamma -k_\parallel v_\parallel)\nonumber\\ & \quad \times \left( \frac{s\varOmega_e}{ \gamma v_\perp} \frac{\partial}{\partial p_\perp} +k_\parallel \frac{\partial}{\partial p_\parallel} \right)f(\boldsymbol{p}), \end{align}

and describes dissipation due to gyroresonant interactions satisfying

(2.4)\begin{equation} \omega-s\varOmega_e/\gamma-k_\parallel v_\parallel=0, \end{equation}

where the gyroresonant number, $s$, is an integer. For a 1-D distribution, only $s=0,\pm 1$ contribute.

2.1.2 One-dimensional assumption

In the 1-D case, the perpendicular momentum, $p_\perp$, of all particles is identically zero, and the 3-D distribution function $f(\boldsymbol {p})$ may be replaced by the 1-D distribution function $g(u)$, where $u=p_\parallel /mc=\gamma \beta$ is the 4-speed (in units of $c$). Due to $v_\perp =0$, the argument of the Bessel functions in (2.2) is zero, and one has

(2.5a–c)\begin{equation} \boldsymbol{V}(\boldsymbol{k},\boldsymbol{p};0)=v_\parallel \boldsymbol{b}, \quad\boldsymbol{V}(\boldsymbol{k},\boldsymbol{p};\pm1)={\tfrac{1}{2}} v_\perp{\boldsymbol{\zeta}}(\pm1), \quad{\boldsymbol{\zeta}}(s)=(1,-{\rm i}s\epsilon,0), \end{equation}

with $\boldsymbol {V}$ being zero for all $|s|>1$ and where we choose $\boldsymbol {b} = (0, 0, 1)$ so that the magnetic field is along the 3-axis. Note that although the terms $\boldsymbol {V}(\boldsymbol {k},\boldsymbol {p};\pm 1)$ are zero for $v_\perp =0$ in the 1-D case, these terms need to be retained in the Vlasov form (2.1), because the $p_\perp$-derivative in (2.3) acts on $\delta (\,p_\perp )$ and hence one needs to partially integrate this term to evaluate it. One sets $v_\perp =0$ only after this partial integration. One has, for $s=\pm 1$,

(2.6)\begin{align} & \int{\rm d}^3\boldsymbol{p}\,V_i(\boldsymbol{k},\boldsymbol{p};s)V_j^*(\boldsymbol{k},\boldsymbol{p};s) \delta\left(\omega - s\varOmega_e/\gamma - k_\parallel v_\parallel\right) \frac{s\varOmega_e}{\gamma v_\perp}\frac{\partial}{\partial p_\perp} \left[\frac{\delta(\,p_\perp)}{2{\rm \pi} p_\perp}g(u)\right]\nonumber\\ & \quad =-\frac{sy}{2m}\,\zeta_i(s)\zeta_j^*(s)\int {\rm d} u\,\delta\left(u + s y - \gamma z\right)g(u), \end{align}

with $z=\omega /k_\parallel c$ and $y=\varOmega _e/k_\parallel c$.

2.2 Forward-scattering form for the dielectric tensor

An alternative general form for the dielectric tensor ${\rm K}_{ij}(\omega,\boldsymbol {k})$ is obtained using the forward-scattering method (Melrose Reference Melrose1987). This form is given by

(2.7)\begin{align} {\rm K}_{ij}(\omega,\boldsymbol{k}) & = \delta_{ij} - \sum\sum_{s=-\infty}^\infty{q^2\over\varepsilon_0m\omega^2} \int {\rm d}^3\boldsymbol{p}\, \frac{f(\boldsymbol{p})}{\gamma}\left\{\vphantom{\frac{{\rm J}_s(k_\perp R)}{\omega_s}}{\rm J}_s^2(k_\perp R) \tau_{ij}(\omega_s)\right.\nonumber\\ & \quad + \frac{{\rm J}_s(k_\perp R)}{\omega_s}\left[\tau_{im}(\omega_s) k_m V_j^*(\boldsymbol{k},\boldsymbol{p};s) +V_i(\boldsymbol{k},\boldsymbol{p};s) k_l\tau_{lj}(\omega_s) \right]\nonumber\\ & \quad \left.+ \frac{1}{\omega_s^2}\left[k_l k_m \tau_{lm}(\omega_s) - \frac{\omega^2}{c^2} \right] V_i(\boldsymbol{k},\boldsymbol{p};s) V_j^*(\boldsymbol{k},\boldsymbol{p};s) \right\}, \end{align}

where repeated subscripts $l$ and $m$ imply sums from 1 to 3 and with $\omega _s = \omega -s\varOmega _e/\gamma -k_\parallel v_\parallel$,

(2.8)\begin{equation} \tau_{ij}(\omega_s) = \begin{pmatrix} \dfrac{\omega_s^2}{\omega_s^2-\varOmega^2} & \dfrac{{\rm i}\epsilon\omega_s\varOmega}{\omega_s^2-\varOmega^2} & 0 \\ \dfrac{-{\rm i}\epsilon\omega_s\varOmega}{\omega_s^2-\varOmega^2} & \dfrac{\omega_s^2}{\omega_s^2-\varOmega^2} & 0 \\ 0 & 0 & 1 \end{pmatrix}. \end{equation}

As in (2.1), the sum in (2.7) is over all unlabelled species, with only electrons and positrons, $\epsilon =\mp$, relevant here. The form (2.7) may also be obtained from (2.1) by a tedious calculation involving partially integrating and using recursion relations and sum rules for the Bessel functions.

2.3 Averages over a 1-D distribution

Consider a pulsar plasma, composed of electrons, $\epsilon =-$, and positrons, $\epsilon =+$. The average of an arbitrary function $Q$ over the 1-D distribution function, ${g}_{\epsilon }(u)$, for electrons or positrons is given by

(2.9)\begin{equation} {n}^{\epsilon} {\left\langle{Q}\right\rangle}_{\epsilon} = \int {\rm d}u\,Q\,{g}_{\epsilon}(u) = \int {\rm d}\beta\, \gamma^3 Q\, {g}_{\epsilon}(u), \end{equation}

which defines the number density, ${n}^{\epsilon }$, for $Q=1$.Footnote ¹ The average of $Q$ over the combined distribution $g(u) = {g}_{+}(u) + {g}_-(u)$ is given by $n \left \langle {Q}\right \rangle = {n}^{+} {\left \langle {Q}\right \rangle }^{+} + {n}^- {\left \langle {Q}\right \rangle }^-$ with combined number density $n = {n}^{+} + {n}^-$. When electron and positron number densities are equal we have ${n}^{+} = {n}^- = n / 2$. The charge and current densities are then given by

(2.10a,b)\begin{equation} \eta = \sum_{\epsilon} \epsilon e {n}^{\epsilon},\quad \boldsymbol{J} = c \sum_{\epsilon} \epsilon e {n}^{\epsilon} {\left\langle{\beta}\right\rangle}^{\epsilon}. \end{equation}

Averages over primed quantities in the primed frame can be somewhat counter-intuitive. For example, setting $X=1$, the average $\left \langle {1/\gamma '}\right \rangle ^{{\epsilon }{\prime }} = \gamma _{s}^{-1} \left \langle {1/\gamma }\right \rangle ^{\epsilon }$ is much narrower than the average ${\left \langle {1/\gamma }\right \rangle }^{\epsilon }$ in the unprimed frame, and, setting $X=\gamma '^2$, the average of $\gamma '$ in the primed frame is much larger than the average of $\gamma$ in the unprimed frame, specifically, $\left \langle {\gamma '}\right \rangle ^{{\epsilon }{\prime }} = \gamma _{s} {\left \langle {\gamma (1+\beta ^2\beta _{s}^2}\right \rangle }^{\epsilon } \approx 2\gamma _{s} {\left \langle {\gamma }\right \rangle }^{\epsilon }$ for $\gamma,\gamma _{s}\gg 1$.

2.4 Response tensor for a 1-D distribution

In the 1-D case, the only contributions to the sum over $s$ in (2.7) are for $s=0,\pm 1$, corresponding to resonant denominators $1/\omega _0^2$ and $1/(\omega _0^2-\varOmega ^2)$, with $\omega _0=\omega -k_\parallel v_\parallel =\omega (z-\beta )/z$ and $\varOmega =\varOmega _e/\gamma =\omega y/z\gamma$. Assuming, without loss of generality, the magnetic field along the 3-axis and the wave vector in the 1–3 plane, one has

(2.11)\begin{equation} \boldsymbol{k}=(k_\perp,0,k_\parallel )=\frac{\omega}{zc}(\tan\theta,0,1). \end{equation}

With this notation, the dielectric tensor (2.7) for a pulsar plasma may be written in the form

(2.12)\begin{equation} {\rm K}_{ij}(\omega,\boldsymbol{k}) =\delta_{ij}+\frac{\varPi_{ij}(\omega,\boldsymbol{k})}{\varepsilon_0\omega^2}, \quad\text{with}\ \varPi_{ij}(\omega,\boldsymbol{k}) =-\sum_{\epsilon}\frac{e^2{n}^{\epsilon}}{m} {\left\langle{\frac{A_{ij}(\omega,\boldsymbol{k};\beta)}{\gamma}}\right\rangle}^{\epsilon}, \end{equation}

with $A_{ij}(\omega,\boldsymbol {k};\beta )\to A_{ij}$ given by

(2.13)\begin{equation} \left.\begin{gathered} A_{11}= A_{22}= \frac{\omega_0^2}{\omega_0^2-\varOmega^2},\quad A_{33}= \frac{\omega^2}{\gamma^2\omega_0^2} + \frac{\omega^2}{\omega_0^2-\varOmega^2}\left(\frac{\beta\tan\theta}{z}\right)^2,\\ A_{12}=-A_{21} = {\rm i}\epsilon \frac{\omega_0\varOmega}{\omega_0^2-\varOmega^2},\quad A_{13}= A_{31} = \frac{\omega_0\omega}{\omega_0^2-\varOmega^2}\left(\frac{\beta\tan\theta}{z}\right),\\ A_{23}=-A_{32}=-{\rm i}\epsilon \frac{\omega\varOmega}{\omega_0^2-\varOmega^2} \left(\frac{\beta\tan\theta}{z}\right). \end{gathered}\right\} \end{equation}

Writing the dielectric tensor in the form (2.12) facilitates Lorentz transforming of it, due to the tensor $\varPi _{ij}$ being the space components of a 4-tensor, as discussed in § 4.2.

2.5 Response tensor for a cold pair plasma

If the spread in energy in the pair plasma is neglected, the resulting model corresponds to a cold streaming pair plasma. In the rest frame $\mathcal {K}$, the response tensor for such a model follows by making the replacement $\textrm {d}u {g}_{\epsilon }(u) \to \textrm {d}u {n}^{\epsilon } \delta (u)$ when evaluating the average in (2.12). Assuming that electron and positron distributions have a common rest frame, $\mathcal {K}$, we obtain

(2.14)\begin{equation} \varPi_{ij}(\omega,\boldsymbol{k}) =-\sum_{\epsilon}\frac{e^2{n}^{\epsilon}}{m} \left. \tau_{ij}\left(\omega_s\right)\right|_{\beta = 0 }. \end{equation}

3.1 Interpretation of the $\pm$-solutions

In interpreting the $\pm$-solutions $\beta =\beta _\pm$ and $\gamma =\gamma _\pm$, first consider the symmetry properties between upgoing and downgoing waves. Let upgoing (downgoing) waves and particles be identified as $k_\parallel >0$ ($k_\parallel <0$) and $\beta >0$ ($\beta <0$), respectively. The transformation $z,y\to -z,-y$ interchanges the role of upgoing and downgoing waves and particles. The solutions $\beta _\pm$ reverse sign under $z,y\to -z,-y$, whereas $\gamma _\pm$ and the resonance conditions $z-\beta _\pm =sy/\gamma _\pm$ are unchanged. In the following discussion $z,y>0$ is assumed with solutions for $z,y<0$ following from these symmetry properties.

Figure 1 shows contour plots of $\beta _+$ (a) and $\beta _-$ (b) as a function of $y$ and $z$ for $\lvert \beta _{\pm }\rvert = 1$ (vertical dotted), 0 (dash-dotted) and $0.19, 0.38, 0.57, 0.76, 0.95$ (decreasing dash length). The negative contours are in thin blue and positive contours are in thick green. The thin grey curves indicate the line $1 + y^2 - z^2 = 0$.

Figure 1.

Contour plots of $\beta _+$ (a) and $\beta _-$ (b) as a function of $y$ and $z$ for $\lvert \beta _{\pm }\rvert = 1$ (vertical dotted), 0 (dash-dotted) and $0.19, 0.38, 0.57, 0.76, 0.95$ (decreasing dash length). The negative contours are in thin blue and positive contours are in thick green. The thin grey curves indicate the line $1 + y^2 - z^2 = 0$. The shaded regions correspond to the anomalous Doppler effect.

The $\pm$-solutions in (3.3) are real only for $z^2 \leq 1+y^2$. Equations (3.4a,b) imply that the anomalous Doppler resonance, $s=-1$, may be satisfied only for the $\beta = \beta _+$ solution over $0 \leq z < 1$, which is shown as a shaded region in figure 1; and the Doppler resonance, $s = +1$, may be satisfied for the $\beta = \beta _+$ solution over $1 < z < \sqrt {1+y^2}$, and for the $\beta = \beta _-$ solution over $0 \leq z < \sqrt {1+y^2}$. For the $+$-solution the transition between solutions for $s=+1$ and $s=-1$ occurs for $\gamma _+=\infty$, $\beta _+=+1$ (vertical dotted green line). It follows that, for the normal Doppler effect, $s=+1$, there is a solution $\beta =\beta _-$ for subluminal waves, $z<1$, and two solutions, $\beta =\beta _+, \beta =\beta _-$ for superluminal waves, $1 \leq z < \sqrt {1+y^2}$. The anomalous Doppler effect, $s=-1$, requires subluminal waves, $z<1$, and then only for $\beta =\beta _+$.

3.2 Plots of $\beta _\pm$ as functions of $z$ for fixed $x=\omega /\varOmega _e$

The resonant solutions, $\beta _\pm$, as functions of $z$ for fixed $x=\omega /\varOmega _e$ are of specific interest in the low-frequency limit, $x\ll 1$, and around the cyclotron resonance, where $x$ is of order unity. The dependence on $x$ may be shown by writing (3.3) and (3.4a,b) in the form

(3.6a,b)\begin{align} \beta_\pm= \frac{zx^2\pm \lvert z \rvert \left[x^2(1-z^2)+z^2\right]^{1/2}}{x^2+z^2},\quad \gamma_\pm= s \operatorname{sgn}{(z)} \frac{z \lvert z \rvert \pm \left[x^2(1-z^2)+z^2\right]^{1/2}}{x(z^2 - 1)}. \end{align}

Figure 2 shows contour plots of $\beta _+$ (a) and $\beta _-$ (b) as a function of $x$ and $z$ for $\lvert \beta _{\pm }\rvert = 1$ (vertical dotted), 0 (dash-dotted) and $0.17, 0.33, 0.5, 0.66, 0.83, 0.99$ (decreasing dash length). The negative contours are in thin blue and positive contours are in thick green. The thin grey curves indicate the line $x^2 (1 - z^2) + z^2 = 0$. The shaded region corresponds to the anomalous Doppler effect.

Figure 2.

Contour plots of $\beta _+$ (a) and $\beta _-$ (b) as a function of $x$ and $z$ for $\lvert \beta _{\pm }\rvert = 1$ (vertical dotted), 0 (dash-dotted) and $0.17, 0.33, 0.5, 0.66, 0.83, 0.99$ (decreasing dash length). The negative contours are in thin blue and positive contours are in thick green. The thin grey curves indicate the line $x^2 (1 - z^2) + z^2 = 0$. The shaded region corresponds to the anomalous Doppler effect.

Figure 3 shows plots of $\beta _{\pm }$ (a) and corresponding $\gamma _{\pm }$ (b) as a function of $z$ for $x = 0.1$ (solid) and $0.4, 1.2, 5$ (decreasing dash length). Contours of $\beta _+, \gamma _+$ are in thick green and those of $\beta _-, \gamma _-$ are in thin blue. The curves $\beta _+$ and $\beta _-$ extend to $z=\infty$ for $x<1$, and they form single closed curves for $x>1$ meeting at $\lvert z\rvert = x / (x^2 - 1)^{1/2}$. For $x\gg 1$, illustrated by $x=5$, the closed curve approaches the line $\beta = z$ for $z\leq 1$, which corresponds to the Cerenkov resonance. The shaded regions correspond to the anomalous Doppler effect, and are bounded on the right by $\beta _\pm = z$ and $\gamma _\pm = (1 - z^2)^{-1/2}$, which are the limits of $\beta _\pm$ and $\gamma _\pm$ as $x \to \infty$. In the low-frequency limit, $x\ll 1$, the solutions may be approximated by

(3.7)\begin{equation} \beta_\pm= \begin{cases} \dfrac{\pm \lvert z \rvert}{\left(x^2 + z^2\right)^{1/2}}, & \text{if}\ \lvert z \rvert \ll 1, \\[10pt] \pm \left(2 + z\right) \lvert z \rvert, & \text{if}\ \lvert z + 1 \rvert \ll 1, \\ \pm \left(2 - z\right) \lvert z \rvert, & \text{if}\ \lvert z - 1 \rvert \ll 1, \end{cases} \end{equation}

with corresponding Lorentz factors

(3.8)\begin{equation} \gamma_\pm= \begin{cases} - s \dfrac{z \lvert z \rvert \pm \left(x^2 + z^2\right)^{1/2}}{x}, & \text{if}\ \lvert z \rvert \ll 1, \\[10pt] s \dfrac{\left(z - 1\right) \left(\lvert z \rvert \mp 1\right)}{4x \left(z + 1\right)}, & \text{if}\ \lvert z + 1 \rvert \ll 1, \\[10pt] s \dfrac{\left(z + 1\right) \left(\lvert z \rvert \pm 1\right)}{4x \left(z - 1\right)}, & \text{if}\ \lvert z - 1 \rvert \ll 1, \end{cases} \end{equation}

and there is no restriction on $z$. For $x>1$ real solutions exist only for $z^2\leq x^2/(x^2-1)$, with the curves for the $\pm$-solutions joining smoothly at $z^2=x^2/(x^2-1)$.

Figure 3.

Contour plots of $\beta _{\pm }$ (a) and corresponding $\gamma _{\pm }$ (b) as a function of $z$ for $x = 0.1$ (solid) and $0.4, 1.2, 5$ (decreasing dash length). Contours of $\beta _+, \gamma _+$ are in thick green and those of $\beta _-, \gamma _-$ are in thin blue. The curves $\beta _+$ and $\beta _-$ extend to $z=\infty$ for $x<1$, and they form single closed curves for $x>1$ meeting at $z^2 = x^2 / (x^2 - 1)$. For $x\gg 1$, illustrated by $x=5$, the closed curve approaches the line $\beta = z$ (thin grey) for $z\leq 1$, which corresponds to the Cerenkov resonance. The shaded regions correspond to the anomalous Doppler effect.

3.3 Three RPDFs

We may write $A_{ij}$, using (2.13) and (3.3), as

(3.9)\begin{equation} \left.\begin{gathered} A_{11} = A_{22} = \frac{(z-\beta)^2}{(1+y^2)(\beta-\beta_+)(\beta-\beta_-)},\\ A_{33} = \frac{z^2}{\gamma^2(z-\beta)^2} + \frac{\beta^2\tan^2\theta}{(1+y^2)(\beta-\beta_+)(\beta-\beta_-)},\\ A_{13} =A_{31} = \frac{(z-\beta)\beta\tan\theta}{(1+y^2)(\beta-\beta_+)(\beta-\beta_-)},\\ A_{12} =-A_{21} = {\rm i}\epsilon \frac{y(z-\beta)}{\gamma(1+y^2)(\beta-\beta_+)(\beta-\beta_-)},\\ A_{23} =-A_{32} =- {\rm i}\epsilon \frac{y\beta\tan\theta}{\gamma(1+y^2)(\beta-\beta_+)(\beta-\beta_-)}. \end{gathered}\right\}\end{equation}

Writing

(3.10)\begin{align} \frac{1}{(\beta-\beta_+)(\beta-\beta_-)} \begin{bmatrix} 1 \\ \beta\\ \beta^2\\ z-\beta\\ (z-\beta)^2\\ (z-\beta)\beta \end{bmatrix} = \begin{bmatrix} 0\\ 0\\ 1\\ 0\\ 1\\ -1 \end{bmatrix} + \frac{1}{\beta_+- \beta_-} \sum_{\alpha={\pm}}\frac{\alpha}{\beta-\beta_\alpha} \begin{bmatrix} 1 \\ \beta_\alpha\\ \beta_\alpha^2\\ z-\beta_\alpha\\ (z-\beta_\alpha)^2\\ (z-\beta_\alpha)\beta_\alpha \end{bmatrix}, \end{align}

allows us to write the averages in (2.12) in terms of the three RPDFs

(3.11a–c)\begin{align} {W}_{\epsilon}(z)={\left\langle{\frac{1}{\gamma^3(\beta-z)^2}}\right\rangle}_{\epsilon}, \quad {R}_{\epsilon}(\beta_\alpha)= {\left\langle{\frac{1}{\gamma(\beta-\beta_\alpha)}}\right\rangle}_{\epsilon}, \quad {S}_{\epsilon}(\beta_\alpha)={\left\langle{\frac{1}{\gamma^2(\beta-\beta_\alpha)}}\right\rangle}_{\epsilon}, \end{align}

with the averages to be understood as defined by (2.9). The RPDF ${W}_{\epsilon }(z)$ arises from the Cerenkov resonance, and RPDFs ${R}_{\epsilon }(\beta _\alpha )$ and ${S}_{\epsilon }(\beta _\alpha )$ arise from the cyclotron resonances, $\beta =\beta _\alpha$. An alternative form for ${W}_{\epsilon }(z)$ is

(3.12)\begin{equation} {W}_{\epsilon}(z)=\frac{1}{{n}^{\epsilon}}\int{\rm d}\beta\,\frac{1}{\beta-z-{\rm i}0}\frac{{\rm d} {g}_{\epsilon}(u)}{{\rm d}\beta}, \end{equation}

which may be written as (Rafat et al. Reference Rafat, Melrose and Mastrano2019a)

(3.13)\begin{equation} {W}_{\epsilon}(z) = \begin{cases} {\left\langle{\dfrac{1}{\gamma^3(\beta-z)^2}}\right\rangle}_{\epsilon}, & \quad \text{for } \lvert z \rvert > 1, \\[11pt] \dfrac{1}{{n}^{\epsilon}}\left[{\rm i}{\rm \pi} \left.\dfrac{{\rm d} {g}_{\epsilon}(u)}{{\rm d} \beta} \right\vert_{\beta = z} + \wp \displaystyle\int {\rm d}\beta\, \dfrac{1}{\beta - z} \dfrac{{\rm d} {g}_{\epsilon}(u)}{{\rm d} \beta}\right], & \quad \text{for } \lvert z \rvert \leq 1, \end{cases} \end{equation}

where $\wp$ indicates a Cauchy principal value integral. The imaginary parts of these RPDFs follow using the Landau prescription, which gives the term $-\textrm {i}0$ in (3.12); the singularity contributes a semi-residue, found by replacing the resonant denominator by $+\textrm {i}{\rm \pi} \delta (\beta -z)$ in the numerator. The resonant parts of ${R}_{\epsilon }(\beta _\alpha )$ and ${S}_{\epsilon }(\beta _\alpha )$ follow by replacing $1/(\beta -\beta _\alpha )$ by $1/(\beta -\beta _\alpha -\textrm {i}0)$

(3.14a,b)\begin{align} {R}_{\epsilon}(\beta_\alpha)=\frac{1}{{n}^{\epsilon}}\int{\rm d} \beta\,\frac{\gamma^2}{\beta-\beta_\alpha-{\rm i}0} {g}_{\epsilon}(u), \quad {S}_{\epsilon}(\beta_\alpha)=\frac{1}{{n}^{\epsilon}}\int{\rm d} \beta\,\frac{\gamma}{\beta-\beta_\alpha-{\rm i}0} {g}_{\epsilon}(u), \end{align}

with the semi-residues giving the imaginary parts. We note that $\lvert \beta _\alpha \rvert < 1$ which implies that

(3.15)\begin{gather} {R}_{\epsilon}(\beta_\alpha) = \frac{1}{{n}^{\epsilon}}\left[{\rm i} {\rm \pi}\left.\gamma^2 {g}_{\epsilon}(u)\right\vert_{\beta = \beta_\alpha} + \wp \int {\rm d} \beta\, \frac{\gamma^2}{\beta - \beta_\alpha} {g}_{\epsilon}(u)\right], \end{gather}

(3.16)\begin{gather}{S}_{\epsilon}(\beta_\alpha) = \frac{1}{{n}^{\epsilon}}\left[{\rm i} {\rm \pi}\left.\gamma {g}_{\epsilon}(u)\right\vert_{\beta = \beta_\alpha} + \wp \int {\rm d} \beta\, \frac{\gamma}{\beta - \beta_\alpha} {g}_{\epsilon}(u)\right]. \end{gather}

Figure 4 shows plots of real (top row) and imaginary (bottom row) of RPDFs ${W}_{\epsilon } (z)$ (a), ${R}_{\epsilon } (\beta _\alpha )$ (b) and ${S}_{\epsilon } (\beta _\alpha )$ (c) for $\rho = 1$ (solid), 5 (long dashed) and 25 (short dashed). The magnitudes of real and imaginary components of the RPDFs have been scaled to unity while preserving their signs.

Figure 4.

Plots of real (thick) and negative of the imaginary (thin) of RPDFs $z^2 {W}_{\epsilon } (z)$ (a), $- \beta _\alpha {R}_{\epsilon } (\beta _\alpha )$ (b) and $- \beta _\alpha {S}_{\epsilon } (\beta _\alpha )$ (c) for $\rho ^\epsilon = 3.16$ (blue, dotted), 1 (orange, short dashed), 0.316 (green, long dashed) and 0.1 (red, solid). The real parts of the RPDFs are scaled to unity and the imaginary parts are scaled relative to the real parts.

Figure 5 shows colour plots of $\textrm {Re}\, {R}_{\epsilon } (\beta _\alpha )$ (first column), $\textrm {Re}\, {S}_{\epsilon } (\beta _\alpha )$ (second column), $\textrm {Im}\, {R}_{\epsilon } (\beta _\alpha )$ (third column) and $\textrm {Im}\, {S}_{\epsilon } (\beta _\alpha )$ (fourth column) for $\alpha = +$ (first and second rows) and $\alpha = -$ (third and fourth rows) with $\beta _\alpha = \beta _\alpha (z, y)$ (first and third rows) and $\beta _\alpha = \beta _\alpha (z, x)$ (second and fourth rows). We use $\rho = 1$ and contours of $\beta _\alpha$ from figures 1 and 2 are superimposed (transparent white). The magnitudes of real and imaginary components of the RPDFs have been scaled to unity while preserving their signs. We note that, when $\beta _\alpha = \beta _\alpha (z,y)$, a single point on the $(\beta _\alpha, {R}_{\epsilon }(\beta _\alpha ))$ 2-D curve maps to a 3-D path in the $(z, y, {R}_{\epsilon }(\beta _\alpha (z,y)))$ space. Projecting this path onto the $(z, y)$ plane and indicating the corresponding value of ${R}_{\epsilon }(\beta _\alpha (z,y))$ by a colour map results in mapping of a single point from $(\beta _\alpha, {R}_{\epsilon }(\beta _\alpha ))$ to a curve in the $(z, y)$ plane. These curves correspond to a single value of $\beta _\alpha (z, y)$ and hence the value of $(\beta _\alpha, {R}_{\epsilon }(\beta _\alpha ))$ is constant along these curves. The inclusion of contours of constant $\beta _\alpha$ from figures 1 and 2 are meant to serve as aids. The 2-D curve $(\beta _\alpha, {R}_{\epsilon }(\beta _\alpha ))$ is thus mapped to the 3-D surface $(z, y, {R}_{\epsilon }(\beta _\alpha (z,y)))$ (or to its projection onto a $(z, y)$ 2-D plane). The same comments apply to ${S}_{\epsilon }(\beta _\alpha )$ and also when $\beta _\alpha = \beta _\alpha (z, x)$ is considered.

Figure 5.

Colour plots of $\textrm {Re}\, {R}_{\epsilon } (\beta _\alpha )$ (first column), $\textrm {Re}\, {S}_{\epsilon } (\beta _\alpha )$ (second column), $\textrm {Im}\, {R}_{\epsilon } (\beta _\alpha )$ (third column) and $\textrm {Im}\, {S}_{\epsilon } (\beta _\alpha )$ (fourth column) for $\alpha = +$ (first and second rows) and $\alpha = -$ (third and fourth rows) with $\beta _\alpha = \beta _\alpha (z, y)$ (first and third rows) and $\beta _\alpha = \beta _\alpha (z, x)$ (second and fourth rows). We use $\rho = 1$ and contours of $\beta _\alpha$ from figures 1 and 2 are superimposed (transparent white). The magnitudes of real and imaginary components of the RPDFs have been scaled to unity while preserving their signs.

3.4 The components of $\varPi _{ij}(\omega,\boldsymbol {k})$ for a stationary Jüttner distribution

The components of the polarization tensor in the rest frame, $\mathcal {K}$, of a pulsar plasma are (Melrose et al. Reference Melrose, Gedalin, Kennett and Fletcher1999; Kennett, Melrose & Luo Reference Kennett, Melrose and Luo2000)

(3.17)\begin{equation} \left.\begin{gathered} {\varPi}{_{11}} ={\varPi}{_{22}} =-\sum_{\epsilon}\frac{e^2{n}^{\epsilon}}{m}\frac{1}{1+y^2} \left[{\left\langle{\frac{1}{\gamma}}\right\rangle}_{\epsilon}+\sum_{\alpha={\pm}}\frac{\alpha(z-\beta_\alpha)^2{R}_{\epsilon}(\beta_\alpha)}{\beta_+-\beta_-}\right],\\ {\varPi}{_{33}}=-\sum_{\epsilon}\frac{e^2{n}^{\epsilon}}{m}\left\{z^2{W}_{\epsilon}(z)+\frac{\tan^2\theta}{1+y^2}\left[{\left\langle{\frac{1}{\gamma}}\right\rangle}_{\epsilon}+\sum_{\alpha={\pm}}\frac{\alpha\beta_\alpha^2{R}_{\epsilon}(\beta_\alpha)}{\beta_+-\beta_-}\right] \right\},\\ {\varPi}{_{13}} ={\varPi}{_{31}} =-\sum_{\epsilon}\frac{e^2{n}^{\epsilon}}{m}\frac{\tan\theta}{1+y^2}\left[-{\left\langle{\frac{1}{\gamma}}\right\rangle}_{\epsilon} +\sum_{\alpha={\pm}}\frac{\alpha(z-\beta_\alpha)\beta_\alpha {R}_{\epsilon}(\beta_\alpha)}{\beta_+-\beta_-} \right],\\ {\varPi}{_{12}} =-{\varPi}{_{21}} =-{\rm i}\sum_{\epsilon}\frac{\epsilon e^2{n}^{\epsilon}}{m}\frac{y}{1+y^2}\sum_{\alpha={\pm}}\frac{\alpha(z-\beta_\alpha){S}_{\epsilon}(\beta_\alpha)}{\beta_+-\beta_-}, \\ {\varPi}{_{23}} =-{\varPi}{_{32}} ={\rm i}\sum_{\epsilon}\frac{\epsilon e^2{n}^{\epsilon}}{m}\frac{y\tan\theta}{1+y^2}\sum_{\alpha={\pm}}\frac{\alpha \beta_\alpha {S}_{\epsilon}(\beta_\alpha)}{\beta_+-\beta_-}. \end{gathered}\right\}\end{equation}

The terms in (2.12) involving ${R}_{\epsilon }(\beta _\alpha )$ describe the contribution of the cyclotron resonances to non-gyrotropic dispersion and the terms involving ${S}_{\epsilon }(\beta _\alpha )$ describe the contribution of the cyclotron resonances to gyrotropic dispersion.

4.1 Lorentz transformation to the streaming frame

Consider the Lorentz transformation between the rest frame $\mathcal {K}$ of the plasma and the pulsar frame $\mathcal {K}'$. We use 4-tensor notation with Greek indices $\mu$ running over $(0,1,2,3)$ where $\mu =0$ denotes the time component and $\mu =i$ denotes the $i$th spatial component. An event is described by the (contravariant) 4-vector (in natural units with $c=1$) $x^\mu =[t,\boldsymbol {x}]$ in $\mathcal {K}$ and $x^{\mu '}=[t',\boldsymbol {x}']$ in $\mathcal {K}'$. The wave 4-vector is $k^\mu =[\omega,\boldsymbol {k}]$, with $\boldsymbol {k} = (k_\perp, 0, k_\parallel )$, in $\mathcal {K}$ and $k^{\mu '}=[\omega ',\boldsymbol {k}']$ in $\mathcal {K}'$. The covariant components are $x_\mu =[t,-\boldsymbol {x}]$ and $k_\mu =[\omega,-\boldsymbol {k}]$.

The Lorentz transformation matrices between the frames are

(4.1a,b)\begin{align} L^{\mu'}{}_\mu(-\beta_{s})= \left(\begin{array}{cccc} \gamma_{s} & 0 & 0 & \gamma_{s} \beta_{s} \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ \gamma_{s} \beta_{s} & 0 & 0 & \gamma_{s} \end{array} \right),\quad L^\mu{}_{\mu'}(-\beta_{s})= \left(\begin{array}{cccc}\gamma_{s} & 0 & 0 & -\gamma_{s} \beta_{s} \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ -\gamma_{s} \beta_{s} & 0 & 0 & \gamma_{s} \end{array} \right), \end{align}

with $\gamma _{s}=(1-\beta _{s}^2)^{-1/2}$. The non-zero components of $L^{\mu '}{}_\mu =L^{\mu '}{}_\mu (-\beta _{s})$ and $L^\mu {}_{\mu '}=L^\mu {}_{\mu '}(-\beta _{s})$ are

(4.2)\begin{equation} \left.\begin{gathered} L^{0'}{}_{0}=L^{3'}{}_{3}=L^{0}{}_{0'}=L^{3}{}_{3'}=\gamma_{s},\quad L^{1'}{}_{1}=L^{2'}{}_{2}=L^{1}{}_{1'}=L^{2}{}_{2'}=1,\\ L^{0'}{}_{3}=L^{3'}{}_{0}=\gamma_{s}\beta_{s}, \quad L^{0}{}_{3'}=L^{3}{}_{0'}=-\gamma_{s}\beta_{s}. \end{gathered}\right\}\end{equation}

The frequency $\omega$ and the components $k_\parallel$ and $k_\perp$, parallel and perpendicular, respectively, to the magnetic field transform to

(4.3a–c)\begin{equation} \omega'=\gamma_{s}(\omega+k_\parallel c\beta_{s}), \quad k'_\parallel c=\gamma_{s}(k_\parallel c+\omega\beta_{s}), \quad k'_\perp=k_\perp. \end{equation}

In terms of the variables $z=\omega /k_\parallel c$ and $\theta =\arctan (k_\perp /k_\parallel )$ in the unprimed frame, $\mathcal {K}$, and $z'=\omega '/k'_\parallel c$ and $\theta '=\arctan (k'_\perp /k'_\parallel )$ in the primed frame, $\mathcal {K}'$, (4.3a–c) and the inverse transforms imply

(4.4a–d)\begin{align} z'=\frac{z+\beta_{s}}{1+\beta_{s} z}, \quad z=\frac{z'-\beta_{s}}{1-\beta_{s} z'}, \quad \tan\theta'=\frac{\tan\theta}{\gamma_{s}(1+\beta_{s} z)}, \quad \tan\theta=\frac{\tan\theta'}{\gamma_{s}(1-\beta_{s} z')}. \end{align}

4.2 Lorentz transformation of the response tensor

Several steps are involved in Lorentz transforming the dielectric tensor (e.g. Melrose Reference Melrose1973), as discussed by Rafat et al. (Reference Rafat, Melrose and Mastrano2019b) in the case where the cyclotron resonances are neglected.

The first step is to write the dielectric tensor in the form (2.12) and to note that the 3-tensor $\varPi _{ij}$ may be interpreted as the space components of the 4-tensor $\varPi ^{\mu }{}_{\nu }(k)$ that relates the 4-current, $\textrm {J}^\mu (k)$, to 4-potential, $A^\nu (k)$; specifically, the space components of $\textrm {J}^\mu (k)=\varPi ^\mu {}_\nu (k)A^\nu (k)$ imply the relation between the 3-current and the vector potential (in the temporal gauge), $\textrm {J}^i(k)=\varPi ^i{}_j(k)A^j(k)$, where the argument $k$ denotes the components of $k^\mu$. The 3-tensor $\varPi _{ij}$ is term by term equal to the mixed components of the 4-tensor $\varPi ^{\mu }{}_{\nu }$ with $\mu =i$, $\nu =j$.Footnote ²

The next step is to construct the full 4-tensor $\varPi ^{\mu }{}_{\nu }$ from the space components $\varPi ^{i}{}_{j}$ using the charge-continuity and gauge-invariance relations, $k_\mu \varPi ^\mu {}_\nu =0$ and $k^\nu \varPi ^\mu {}_\nu =0$, respectively. The 4-tensor components $\varPi ^0{}_0$, $\varPi ^i{}_0$, $\varPi ^0{}_j$ are given in terms of the mixed tensor components by

(4.5a,b)\begin{equation} \varPi^\mu{}_0=-\frac{\tan\theta}{z}\varPi^\mu{}_1-\frac{1}{z}\varPi^\mu{}_3, \quad \varPi^0{}_\nu=\frac{\tan\theta}{z}\varPi^1{}_\nu+\frac{1}{z}\varPi^3{}_\nu. \end{equation}

The third step is to apply the Lorentz transformation to $\varPi ^{\mu }{}_{\nu }(k)$ in $\mathcal {K}$, to find $\varPi ^{\mu '}{}_{\nu '}(k')=L^{\mu '}{}_{\mu }\varPi ^{\mu }{}_{\nu }(L^{-1}[k])L^{\nu }{}_{\nu '}$ in $\mathcal {K}'$, where $k'=L^{-1}[k]$ denotes the components $\omega ',\boldsymbol {k}'$ expressed in terms of $\omega,\boldsymbol {k}$. The transformed components $\varPi ^{i'}{}_{j'}$ are then identified (term by term) as the components of the (polarization) 3-tensor $\varPi '_{ij}$ in $\mathcal {K}'$. Finally, the dielectric tensor in $\mathcal {K}'$ is identified as

(4.6)\begin{equation} {\rm K}'_{ij}(\omega',\boldsymbol{k}')=\delta_{ij}+\frac{\varPi'_{ij}(\omega',\boldsymbol{k}')}{\varepsilon_0\omega'^2}, \end{equation}

which is interpreted as the dielectric tensor for the streaming distribution in the primed frame.

4.3 Transformed polarization tensor

The Lorentz transformation applied to the polarization 3-tensor $\varPi _{ij}$ gives the transformed 3-tensor $\varPi '_{ij}$

(4.7)\begin{align} \left.\begin{gathered} \varPi'_{11} = \varPi_{11},\quad \varPi'_{12} = \varPi_{12},\quad \varPi'_{21} = \varPi_{21},\quad \varPi'_{22} = \varPi_{22},\\ \varPi'_{13}= \gamma_s \left[\frac{\beta_s \tan\theta}{z} \varPi_{11} + \frac{z + \beta_s}{z} \varPi_{13}\right],\quad \varPi'_{31} = \gamma_s \left[\frac{\beta_s \tan\theta}{z} \varPi_{11} + \frac{z + \beta_s}{z} \varPi_{31}\right],\\ \varPi'_{23}= \gamma_s \left[\frac{\beta_s \tan\theta}{z} \varPi_{21} + \frac{z + \beta_s}{z} \varPi_{23}\right],\quad \varPi'_{32} = \gamma_s \left[\frac{\beta_s \tan\theta}{z} \varPi_{12} + \frac{z + \beta_s}{z} \varPi_{32}\right],\\ \varPi'_{33}= \gamma_s^2 \left[\left(\frac{\beta_s \tan\theta}{z}\right)^2 \varPi_{11} + \left(\frac{z + \beta_s}{z}\right) \left(\frac{\beta_s \tan\theta}{z}\right) \left(\varPi_{13} + \varPi_{31}\right) + \left(\frac{z + \beta_s}{z}\right)^2 \varPi_{33}\right]. \end{gathered}\right\}\end{align}

With $\varPi _{ij}$ given by (3.17), the transformed tensor (4.7) becomes

(4.8)\begin{align} \left.\begin{gathered} {\varPi'}_{11} ={\varPi'}_{22} =-\sum_{\epsilon}\frac{e^2{n}^{\epsilon}}{m}\frac{1}{1+y^2}\left[{\left\langle{\frac{1}{\gamma}}\right\rangle}_{\epsilon}+\sum_{\alpha={\pm}}\frac{\alpha(z-\beta_\alpha)^2{R}_{\epsilon}(\beta_\alpha)}{\beta_+-\beta_-}\right],\\ {\varPi'}_{33}=-\gamma_{s}^2\sum_{\epsilon}\frac{e^2{n}^{\epsilon}}{m}\left\{\vphantom{\left[{\left\langle{\frac{1}{\gamma}}\right\rangle}_{\epsilon}+\sum_{\alpha={\pm}}\frac{\alpha(\beta_{s}+\beta_\alpha)^2{R}_{\epsilon}(\beta_\alpha)}{\beta_+-\beta_-}\right]} (z+\beta_{s})^2{W}_{\epsilon}(z) +\frac{\tan^2\theta}{1+y^2}\left[{\left\langle{\frac{1}{\gamma}}\right\rangle}_{\epsilon}+\sum_{\alpha={\pm}}\frac{\alpha(\beta_{s}+\beta_\alpha)^2{R}_{\epsilon}(\beta_\alpha)}{\beta_+-\beta_-}\right] \right\},\\ {\varPi'}_{13} ={\varPi'}_{31}=-\gamma_{s}\sum_{\epsilon}\frac{e^2{n}^{\epsilon}}{m}\frac{\tan\theta}{1+y^2}\left[- {\left\langle{\frac{1}{\gamma}}\right\rangle}_{\epsilon} +\sum_{\alpha={\pm}}\frac{\alpha(z-\beta_\alpha)(\beta_{s}+\beta_\alpha){R}_{\epsilon}(\beta_\alpha)}{\beta_+-\beta_-} \right],\\ {\varPi'}_{12} =-{\varPi'}_{21} =-{\rm i}\sum_{\epsilon}\frac{\epsilon e^2{n}^{\epsilon}}{m}\frac{y}{1+y^2}\sum_{\alpha={\pm}}\frac{\alpha(z-\beta_\alpha){S}_{\epsilon}(\beta_\alpha)}{\beta_+-\beta_-},\\ {\varPi'}_{23} =-{\varPi'}_{32} ={\rm i}\gamma_{s}\sum_{\epsilon}\frac{\epsilon e^2{n}^{\epsilon}}{m}\frac{y\tan\theta}{1+y^2} \sum_{\alpha={\pm}}\frac{\alpha(\beta_{s}+\beta_\alpha){S}_{\epsilon}(\beta_\alpha)}{\beta_+-\beta_-}. \end{gathered}\right\}\end{align}

The result (4.8) is the polarization tensor in the frame $\mathcal {K}'$, expressed in terms of the plasma parameters in the (unprimed) rest frame $\mathcal {K}$ of the streaming distribution. An alternative form for this tensor is obtained by re-expressing (4.8) in terms of plasma parameters and RPDFs in the primed frame.

4.4 Lorentz-transformed RPDFs

In order to rewrite (4.8) in terms of variables in $\mathcal {K}'$, one need to relate the three RPDFs ${W}_{\epsilon }(z)$, ${R}_{\epsilon }(\beta _\alpha )$, ${S}_{\epsilon }(\beta _\alpha )$ as defined in $\mathcal {K}$ to the corresponding RPDFs in $\mathcal {K}'$. The latter are defined as

(4.9a–c)\begin{align} {W}^{\prime}_{\epsilon}(z') = {\left\langle{\frac{1}{\gamma'^3(\beta'-z')^2}}\right\rangle}^{\prime}_{\epsilon},\quad {R}^{\prime}_{\epsilon}(\beta_\alpha') = {\left\langle{\frac{1}{\gamma'(\beta'-\beta_\alpha')}}\right\rangle}^{\prime}_{\epsilon},\quad {S}^{\prime}_{\epsilon}(\beta_\alpha') = {\left\langle{\frac{1}{\gamma'^2(\beta'-\beta_\alpha')}}\right\rangle}^{\prime}_{\epsilon}. \end{align}

The averages of any quantity $X$ in the two frames are defined by (2.9), which implies

(4.10)\begin{equation} {\left\langle{\frac{X}{\gamma'}}\right\rangle}^{\prime}_{\epsilon} = \frac{1}{\gamma_{s}} {\left\langle{\frac{X}{\gamma}}\right\rangle}_{\epsilon}. \end{equation}

Using (4.10) and the relations (4.4a–d), one finds

(4.11)\begin{equation} \left.\begin{gathered} {W}^{\prime}_{\epsilon}(z') = \gamma_{s} (1 + \beta_{s} z)^2 {W}_{\epsilon}(z),\\ {R}^{\prime}_{\epsilon}(\beta'_\alpha) = \gamma_{s} (1 + \beta_{s} \beta_\alpha) \left[(1 + \beta_{s} \beta_\alpha) {R}_{\epsilon}(\beta_\alpha) + \beta_{s} {\left\langle{\frac{1}{\gamma}}\right\rangle}_{\epsilon}\right],\\ {S}^{\prime}_{\epsilon}(\beta'_\alpha) = (1 + \beta_{s} \beta_\alpha) {S}_{\epsilon}(\beta_\alpha), \end{gathered}\right\} \end{equation}

and the inverse transforms

(4.12)\begin{equation} \left.\begin{gathered} {W}_{\epsilon}(z) = \gamma_{s}^3(1 - \beta_{s} z')^2{W}^{\prime}_{\epsilon}(z'),\\ {R}_{\epsilon}(\beta_\alpha) = \gamma_{s}^3(1 - \beta_{s} \beta'_\alpha)\left[(1 - \beta_{s} \beta'_\alpha){R}^{\prime}_{\epsilon}(\beta'_\alpha) - \beta_{s} {\left\langle{\frac{1}{\gamma'}}\right\rangle}^{\prime}_{\epsilon}\right],\\ {S}_{\epsilon}(\beta_\alpha) = \gamma_{s}^2(1 - \beta_{s} \beta'_\alpha){S}^{\prime}_{\epsilon}(\beta'_\alpha). \end{gathered}\right\} \end{equation}

The identity (4.10) implies $\gamma _{s} {\left \langle {1/\gamma '}\right \rangle }^{\prime }_{\epsilon } ={\left \langle {1/\gamma }\right \rangle }_{\epsilon }$.

4.5 The polarization tensor in the primed frame

The response tensor $\varPi '_{ij}(\omega ',\boldsymbol {k}')$ in $\mathcal {K}'$ is only partly determined by the equalities (3.17); one also needs to express the unprimed parameters in terms of the primed parameters. In addition to the identities (4.3a–c), (4.4a–d), (5.1a–d) and (4.12) the following identities are useful in this context:

(4.13)\begin{align} \left.\begin{gathered} \gamma_{s}(1-z'\beta_{s}) = \frac{1}{\gamma_{s}(1+z\beta_{s})}, \quad y = \frac{y'}{\gamma_{s}(1-z'\beta_{s})}, \quad 1+y'^2-z'^2 = \frac{1+y^2-z^2}{\gamma_{s}^2 (1+z\beta_{s})^2},\\ \beta'_\pm= \frac{z'\pm y'[1+y'^2-z'^2]^{1/2}}{1+y'^2}=\frac{\beta_\pm+\beta_{s}}{1+\beta_\pm\beta_{s}}, \quad z-\beta_\pm= \frac{z'-\beta'_\pm}{\gamma_{s}^2(1-z'\beta_{s})(1-\beta'_\pm\beta_{s})},\\ \beta_+-\beta_-= \frac{\beta'_+-\beta'_-}{\gamma_{s}^2(1+\beta_+\beta_{s})(1+\beta_-\beta_{s})}, \quad (1+y^2)(\beta_+-\beta_-) = \frac{(1+y'^2)(\beta'_+-\beta'_-)}{\gamma_{s}^2(1-z'\beta_{s})^2},\\ \beta_+\beta_-= \frac{z^2-y^2}{1+y^2}, \quad \frac{1}{(1-\beta'_+\beta_{s})(1-\beta'_-\beta_{s})} = \frac{\gamma_{s}^2(1-z'\beta_{s})^2+y'^2}{\gamma_{s}^2(1+y'^2)}. \end{gathered}\right\} \end{align}

Using these expressions, a lengthy calculation leads to the following expression for $\varPi '_{ij}(\omega ',\boldsymbol {k}')$:

(4.14)\begin{equation} \left.\begin{gathered} \varPi'_{11}=-\sum_{\epsilon}\frac{e^2{n}^{\epsilon \prime}}{\gamma_{s}m(1+y'^2)}\left[ {\left\langle{\frac{1}{\gamma}}\right\rangle}_{\epsilon} +\sum_{\alpha={\pm}}\frac{\alpha(z'-\beta'_\alpha)^2{R}^{\prime}_{\epsilon}(\beta'_\alpha)}{\beta'_+-\beta'_-} \right],\\ \varPi'_{12} =-{\rm i}\sum_{\epsilon}\frac{\epsilon e^2{n}^{\epsilon \prime}}{\gamma_{s}m}\frac{y'}{1+y'^2}\sum_{\alpha={\pm}}\frac{\alpha(z'-\beta'_\alpha){S}^{\prime}_{\epsilon}(\beta'_\alpha)}{\beta'_+-\beta'_-}, \\ \varPi'_{23} ={\rm i}\sum_{\epsilon}\frac{\epsilon e^2{n}^{\epsilon \prime}}{\gamma_{s}m}\frac{y'\tan\theta'}{1+y'^2}\sum_{\alpha={\pm}}\frac{\alpha \beta'_\alpha {S}^{\prime}_{\epsilon}(\beta'_\alpha)}{\beta'_+-\beta'_-},\\ \varPi'_{13}=-\sum_{\epsilon}\frac{e^2{n}^{\epsilon \prime}\tan\theta'}{\gamma_{s}m(1+y'^2)} \left[-{\left\langle{\frac{1}{\gamma}}\right\rangle}_{\epsilon} +\sum_{\alpha={\pm}}\frac{\alpha(z'-\beta'_\alpha)\beta'_\alpha {R}^{\prime}_{\epsilon}(\beta'_\alpha)}{\beta'_+-\beta'_-} \right],\\ \varPi'_{33}=\sum_{\epsilon}\frac{e^2{n}^{\epsilon \prime}}{\gamma_{s}m}\left\{z'^2{W}^{\prime}_{\epsilon}(z') +\frac{\tan^2\theta'}{1+y'^2}\left[ {\left\langle{\frac{1}{\gamma}}\right\rangle}_{\epsilon} +\sum_{\alpha={\pm}}\frac{\alpha\beta'^2_\alpha {R}^{\prime}_{\epsilon}(\beta'_\alpha)}{\beta'_+-\beta'_-}\right]\right\}, \end{gathered}\right\} \end{equation}

where the substitution ${\left \langle {1/\gamma }\right \rangle }_{\epsilon }=\gamma _{s}{\left \langle {1/\gamma '}\right \rangle }^{\prime }_{\epsilon }$ is not made explicitly.

5.1 Lorentz transforming the distribution function

Any distribution function, ${f}_{\epsilon }(\boldsymbol {p})$ is a Lorentz invariant. In the 1-D case, the distribution function in $\mathcal {K}$ may be written as ${g}_{\epsilon }(u)$, with $u^\mu =p^\mu /m=(\gamma,u\boldsymbol {b})$, where $\boldsymbol {b}$ is the unit vector along the magnetic field and $u=\gamma \beta$ is the 4-speed. The Lorentz transformation implies

(5.1a–d)\begin{align} \gamma'=\gamma\gamma_{s}(1+\beta\beta_{s}), \quad\beta'=\frac{\beta+\beta_{s}}{1+\beta\beta_{s}};\quad \gamma=\gamma'\gamma_{s}(1-\beta'\beta_{s}), \quad \beta=\frac{\beta'-\beta_{s}}{1-\beta'\beta_{s}}. \end{align}

The distribution function in $\mathcal {K}'$ may be written as ${g}^{ \prime }_{\epsilon }(u')$, with $u'=\gamma '\beta '$. The normalizations in the two frames are

(5.2a,b)\begin{equation} \int_{-\infty}^\infty {\rm d}u\,{g}_{\epsilon}(u)={n}^{\epsilon}, \quad \int_{-\infty}^\infty {\rm d}u'\,{g}^{ \prime}_{\epsilon}(u')={n}^{\epsilon \prime}. \end{equation}

A 1-D Jüttner distribution is ${g}_{\epsilon }(u)={n}^{\epsilon } \exp (-\rho \gamma )/2\textrm {K}_1(\rho )$ where $\rho =mc^2/T$ is the inverse temperature in units of the electron rest energy. Transforming to $\mathcal {K}'$ gives

(5.3a,b)\begin{equation} {g}^{ \prime}_{\epsilon}(u') = \frac{{n}^{\epsilon \prime}}{\gamma_{s}} \frac{\exp({-\rho\gamma_{s}\gamma'(1-\beta_{s}\beta')})}{2{\rm K}_1(\rho)}, \quad {n}^{\epsilon \prime}={\gamma_{s}}{n}^{\epsilon}. \end{equation}

This result follows, for ${g}_{\epsilon }(-u)={g}_{\epsilon }(u)$, from $\textrm {d}u'=\textrm {d}(\beta '\gamma ')=\gamma '^3\,\textrm {d}\beta '$, $\textrm {d}u=\gamma ^3\,\textrm {d}\beta$ and $\textrm {d}\beta '/\textrm {d}\beta =\gamma ^2/\gamma '^2$ implying $\textrm {d}u'/\textrm {d}u=\gamma '/\gamma$, with $\gamma '$ and $\beta '$ given in terms of $\gamma$ and $\beta$ by (5.1a–d). The distribution function (5.3a,b) may be interpreted as a streaming Jüttner distribution function in the primed frame.

5.1.1 Response tensor for streaming Jüttner distribution

The response tensor for the streaming Jüttner distribution function (5.3a,b), evaluated in the frame $\mathcal {K}'$, is

(5.4)\begin{equation} \varPi'_{ij}(\omega,\boldsymbol{k})=-\sum_{\epsilon}\frac{e^2}{m}\int_{-\infty}^\infty {\rm d}u'\,{g}^{ \prime}_{\epsilon}(u') \frac{A'_{ij}(\omega',\boldsymbol{k}';\beta')}{\gamma'}, \end{equation}

where $A'_{ij}/\gamma '$ is related to $A_{ij}/\gamma$ by the Lorentz transformation that relates $\varPi '_{ij}$ to $\varPi _{ij}$, cf. (4.7).

The integral in (5.4) may be reduced to the same form as the integral in the non-streaming case by changing the variable of integration from $u$ to $u'$, with $\textrm {d}u'/\textrm {d}u=\gamma '/\gamma$ and replacing $\textrm {d}u'\,{g}^{ \prime }_{\epsilon }(u')\gamma '$ by $\textrm {d}u{g}_{\epsilon }(u)\gamma$, where ${g}_{\epsilon }(u)$ is the non-streaming distribution function. The resulting expression for the response tensor reproduces the form (4.14).

The primed frame, $\mathcal {K}'$, may be interpreted as the pulsar frame, provided that the electrons and positrons stream at the same speed (and that the rotation of the pulsar plasma is neglected). The transformed polarization tensor, in either form (4.8) or (4.14), may then be interpreted as the polarization tensor in the pulsar frame, expressed in terms of the unprimed and primed variables, respectively. The form (4.14) in terms of the primed variables might appear to be the more convenient because all quantities are defined in the frame of relevance to the observer. However, we find the form (4.8) to be more convenient because the RPDFs are usually defined in the rest frame of the distribution of particles. Alternatively one may use a mixed notation: starting from the form (4.14) one may use the relations (4.11) to rewrite the RPDFs in the primed frame in terms of the RPDFs in the unprimed frame, with the unprimed variables, including the arguments of the RPDFs, rewritten in terms of the primed variables using (5.1a–d).

6.1 Cold-plasma limit

The cold-plasma limits of the RPDFs follow from the definitions (3.11a–c), with $\textrm {d}u\, {g}_{\epsilon }(u) \to \textrm {d}u\, {n}^{\epsilon }\, \delta (u)$. The limiting forms are

(6.1a–c)\begin{equation} {W}_{\epsilon}({z}^{\epsilon}) \to \frac{1}{{z}^{\epsilon 2}}, \quad {R}_{\epsilon}({\beta}^{\epsilon}_{\alpha}) \to -\frac{1}{{\beta}^{\epsilon}_{\alpha}}, \quad {S}_{\epsilon}({\beta}^{\epsilon}_{\alpha}) \to -\frac{1}{{\beta}^{\epsilon}_{\alpha}}. \end{equation}

For a cold streaming plasma, these are replaced by

(6.2a–c)\begin{align} {W}^{\prime}_{\epsilon}({z}^{\epsilon \prime}) \to \frac{1}{{\gamma}^{\epsilon 3}_{s} ({z}^{\epsilon \prime} - {\beta}^{\epsilon}_{s})^2}, \quad {R}^{\prime}_{\epsilon}({\beta}^{\epsilon \prime}_{\alpha}) \to - \frac{1}{{\gamma}^{\epsilon}_{s} ({\beta}^{\epsilon \prime}_{\alpha} - {\beta}^{\epsilon}_{s})}, \quad {S}^{\prime}_{\epsilon}({\beta}^{\epsilon \prime}_{\alpha}) \to - \frac{1}{{\gamma}^{\epsilon 2}_{s} ({\beta}^{\epsilon \prime}_{\alpha} - {\beta}^{\epsilon}_{s})}, \end{align}

using the limit $\textrm {d}u'\, {g}^{ \prime }_{\epsilon }(u') \to \textrm {d}u'\, {n}^{\epsilon \prime }\, \delta (u' - {u}^{\epsilon \prime })$.

The cold-plasma limit of the polarization tensor (2.12) for a non-streaming Jüttner distribution reduces to

(6.3)\begin{equation} \varPi_{ij}(\omega,\boldsymbol{k}) \to - \sum_{\epsilon} \frac{e^2{n}^{\epsilon}}{m}A_{ij}(\omega,\boldsymbol{k};0), \end{equation}

with $A_{ij}(\omega,\boldsymbol {k};\beta )$ given by (2.13). The result (6.3) is reproduced by setting $g(u) \to {n}^{\epsilon } \delta (u)$ in (2.12).

The cold-plasma limit for a streaming Jüttner distribution reduces to

(6.4)\begin{equation} \varPi'_{ij}(\omega,\boldsymbol{k}) \to \sum_{\epsilon} \frac{e^2{n}^{\epsilon}}{\gamma_{s}m}A_{ij}(\omega',\boldsymbol{k}'; \beta_{s}), \end{equation}

with $\omega '=\gamma _{s}(\omega -k_\parallel \beta _{s}c)$, $k_\perp ^{\prime }=k_\perp$, $k_\parallel '=\gamma _{s}(k_\parallel -\omega \beta _{s}/c)$. The result (6.4) is implied directly by setting $g(u) \to {n}^{\epsilon } \delta (u-u_{s})$ in (2.12).

6.2 The highly relativistic limit $\rho \ll 1$

In the highly relativistic limit, $\rho \ll 1$ the RPDFs by be approximated by exponential-integral functions. One finds (Luo & Melrose Reference Luo and Melrose2004a; Melrose & Luo Reference Melrose and Luo2004)

(6.5)\begin{equation} \left.\begin{gathered} W(z,\rho)=-\rho\gamma_\phi^2\left\{1-{\tfrac{1}{2}}\rho\gamma_\phi z^2 \left[{\rm e}^{-\rho\gamma_\phi}{Ei}(\rho\gamma_\phi)-{\rm e}^{\rho\gamma_\phi}{Ei}(-\rho\gamma_\phi)\right]\right\},\\ R(z,\rho)=-{\tfrac{1}{2}}\rho\gamma_\phi^2z \left[{\rm e}^{-\rho\gamma_\phi}{Ei}(\rho\gamma_\phi)+{\rm e}^{\rho\gamma_\phi}{Ei}(-\rho\gamma_\phi) \right],\\ S(z,\rho)=-{\tfrac{1}{2}}\rho\gamma_\phi z\left[{\rm e}^{-\rho\gamma_\phi}{Ei} (\rho\gamma_\phi)-{\rm e}^{\rho\gamma_\phi}{Ei}(-\rho\gamma_\phi) \right]. \end{gathered}\right\}\end{equation}

For $\rho \gamma _\phi =\rho (1-z^2)^{-1/2}\gg 1$ one has

(6.6)\begin{gather} W(z,\rho) \sim-\rho \gamma_\phi^2 \left\{1 - z^2 \sum_{k = 0}^{\infty} \frac{(2k)!}{(\rho \gamma_\phi)^{2k}}\right\}\approx-\rho \gamma_\phi^2 \left\{1 - z^2 \left[1 + \frac{2}{(\rho\gamma_\phi)^2}\right]\right\}, \end{gather}

(6.7)\begin{gather}R(z,\rho) \sim-\frac{z}{\rho} \sum_{k = 0}^{\infty} \frac{(2k+1)!}{(\rho \gamma_\phi)^{2k}} \approx-\frac{z}{\rho}\left[1+\frac{6}{(\rho\gamma_\phi)^2}\right], \end{gather}

(6.8)\begin{gather}S(z,\rho) \sim- z \sum_{k = 0}^{\infty} \frac{(2k)!}{(\rho \gamma_\phi)^{2k}} \approx-z\left[1 + \frac{2}{(\rho\gamma_\phi)^2}\right]. \end{gather}

6.3 Approximations to the arguments of RPDFs

The arguments of the RPDFs $R$ and $S$ in (4.8) may be approximated in two ways: by making the WAA and by assuming highly relativistic streaming, $\gamma _{s}\gg 1$. The first of these approximations follows from $z'\approx 1/\cos \theta '$ and $y'\approx 1/x'\cos \theta '$, giving

(6.9a,b)\begin{align} \beta'_\alpha \approx \frac{x^{\prime 2} \cos\theta' + \alpha \left[1 - x^{\prime 2} \sin^2\theta'\right]^{1/2}}{1 + x^{\prime 2} \cos^2\theta'},\quad \gamma'_\alpha \approx s \frac{1 + \alpha \lvert \cos\theta' \rvert \left[1 - x^{\prime 2} \sin^2\theta'\right]^{1/2}}{x' \sin^2\theta'}, \end{align}

where $x'=\omega '/\varOmega _e$ is the ratio of the wave frequency to the cyclotron frequency in $\mathcal {K}'$. The second approximation follows by assuming $\beta _{s}\approx 1-1/2\gamma _{s}^2$, giving

(6.10a,b)\begin{equation} \frac{\beta'_\alpha-\beta_{s}}{1-\beta'_\alpha\beta_{s}} \approx-1+\frac{1}{2\varGamma_\alpha^2}, \quad \varGamma_\alpha^2=2\gamma_{s}^2\frac{1-\beta'_\alpha}{1+\beta'_\alpha}, \end{equation}

where $\gamma _{s}^2\gg (\gamma '_\alpha )^2$ is assumed. These approximations may be applied to the exponential-integral approximation (6.8) to the RPDFs with $z\to (\beta '_\alpha -\beta _{s})/(1-\beta '_\alpha \beta _{s})$ and $\gamma _\phi \to \varGamma _\alpha$.