Properties of pulsar subpulse drifting in different profile components

Abstract

We investigate the emission properties of the pulsars that exhibit different drifting subpulses in different pulse profile components based on the model for obliquely rotating pulsar magnetospheres of multiple emission states. Emission from subpulses is assumed coming from m discrete areas located on carousels around the magnetic axis and flowing at rates determined by the $\varvec{E}\times \varvec{B}$ drift. An emission state is defined by this drift, and a change in the emission state corresponds to a change in the drift velocity. The model is applied to eight pulsars, all displaying drifting subpulses of particular properties along two different drift-bands at different parts of the pulse profile at 21 cm observing wavelength. Simulation is performed to identify the emission states for the different drifting subpulses and the associated values of m in the emission region. The results show that drifting subpulses in different profile components correspond to particular emission states and m, and that different emission states and m can coexist in an emission region. We determine the circulation time for plasma around the magnetosphere and show that it is related to the obliquity angle. By comparing with three other pulsars that display drifting subpulses of a single drift-band, we find that drifting subpulses demonstrate similar properties when considering the different drift-bands separately. We speculate, based on our results, that the coexistence of different drifting subpulses within a pulse profile may require additional mechanism, and the phenomenon is more common in pulsars with small obliquity angle.

Appendices

Appendix A. The electric and magnetic fields

The electric field in the model for pulsar magnetosphere of multiple emission states is given by Equation (9) in Section 2 and reproduced below

$$\begin{aligned} \varvec{E} = (1 - y\, \varvec{bb}) \cdot \varvec{E}_{\mathrm{ind}} + (1 - y)\, \varvec{E}_{\mathrm{pot}}. \end{aligned}$$

(A1)

Here, the inductive electric field due to the obliquely rotating time-dependent magnetic dipole, $\varvec{p} =\varvec{p}(t-r/c)$, in vacuo is signified by

$$\begin{aligned} \varvec{E}_{\mathrm{ind}} (t,\varvec{x})=\frac{\mu _0}{4\pi } \bigg (\frac{\varvec{x}\times \dot{\varvec{p}}}{r^3} \bigg ) \end{aligned}$$

(A2)

and the vector potential due to the rotating dipole is (Melrose & Yuen 2014)

$$\begin{aligned} \varvec{A} = -\frac{\mu _0}{4\pi } \bigg (\frac{\varvec{x}\times \varvec{p}}{r^3} \bigg ), \end{aligned}$$

(A3)

where r and $\varvec{x}$ are the radial distance from the stellar center and position vector, respectively. Since $\varvec{p}$ is a function of $\alpha$ in the observer’s frame (see Appendix B), $\varvec{E}_{\mathrm{pot}}$ is also a function of $\alpha$. Both $\varvec{x}$ and $\varvec{p}$ can be determined uniquely at given coordinates for known $\zeta$ and $\alpha$. The unit vector of the dipolar magnetic field, $\varvec{B}_{\mathrm{dip}}$, is represented by the symbol $\varvec{b}$, with the magnetic field given by,

$$\begin{aligned} \varvec{B} = \frac{\mu _0}{4\pi } \Bigg [ \frac{3\varvec{xx}\cdot \varvec{p} - r^2\varvec{p}}{r^5} + \frac{3\varvec{xx}\cdot \dot{\varvec{p}} - r^2 \dot{\varvec{p}}}{r^4c} + \frac{\varvec{x} \times ( \varvec{x} \times \ddot{\varvec{p}} )}{r^3 c^2} \Bigg ], \end{aligned}$$

(A4)

where $1/r^3$ term corresponds to the dipolar term and the radiative terms are denoted by the terms $\propto 1/r^2$ and $\propto 1/r$. We note that all terms in Equation (A4) are functions of $\alpha$ (e.g., see the paper by Cheng et al. (2000)).

In spherical coordinates, the expression for $\varvec{E}_{\mathrm{ind}}$ has the form given by

$$\begin{aligned} \left( {\begin{array}{*{20}c} {E_{{{\text{ind}},r}} } \\ {E_{{{\text{ind}},\theta }} } \\ {E_{{{\text{ind}},\phi }} } \\ \end{array} } \right) = \frac{{\mu _{0} m\omega \sin \alpha }}{{4\pi r^{2} }}\left( {\begin{array}{*{20}c} 0 \\ { - \cos (\phi - \psi )} \\ {\cos \theta \sin (\phi - \psi )} \\ \end{array} } \right), \end{aligned}$$

(A5)

with the initial conditions chosen for the magnetic axis being in the plane $\phi =0$. Since $\varvec{E}_{\mathrm{ind}}$ is proportional to $\sin \alpha$, it vanishes for $\alpha =0$. Therefore, $\varvec{E}_{\mathrm{ind}}$ and $\varvec{E}_{\mathrm{pot}}$ are both functions of $\alpha$, and hence $\varvec{E}$ is also a function of $\alpha$.

Appendix B. Transformation matrices

The magnetic and rotation axes of a pulsar may be organized in Cartesian coordinates in such a way that $\hat{\mathbf{p}}={\hat{\mathbf{z}}}_b$ and ${{\hat{\omega }}_\star }={\hat{\mathbf{z}}}$. Here, the unit vectors relative to the magnetic and observer’s frames are given by ${\hat{\mathbf{x}}}_b,{\hat{\mathbf{y}}}_b ,{\hat{\mathbf{z}}}_b$ and ${\hat{\mathbf{x}}},{\hat{\mathbf{y}}},{\hat{\mathbf{z}}}$, respectively. The matrix for transformation between the unit vectors are given by

$$\begin{aligned} {\mathbf{R}} = \left( {\begin{array}{*{20}c} {\cos \alpha \cos \psi } & {\cos \alpha \sin \psi } & { - \sin \alpha } \\ { - \sin \psi } & {\cos \psi } & 0 \\ {\sin \alpha \cos \psi } & {\sin \alpha \sin \psi } & {\cos \alpha } \\ \end{array} } \right), \end{aligned}$$

(B1)

such that

$$\begin{aligned} \left( {\begin{array}{*{20}c} {{\hat{\mathbf{x}}}_{b} } \\ {{\hat{\mathbf{y}}}_{b} } \\ {{\hat{\mathbf{z}}}_{b} } \\ \end{array} } \right) = {\mathbf{R}}\left( {\begin{array}{*{20}c} {{\hat{\mathbf{x}}}} \\ {{\hat{\mathbf{y}}}} \\ {{\hat{\mathbf{z}}}} \\ \end{array} } \right)\,{\text{and}}\,\left( {\begin{array}{*{20}c} {{\hat{\mathbf{x}}}} \\ {{\hat{\mathbf{y}}}} \\ {{\hat{\mathbf{z}}}} \\ \end{array} } \right) = {\mathbf{R}}^{{\text{T}}} \left( {\begin{array}{*{20}c} {{\hat{\mathbf{x}}}_{b} } \\ {{\hat{\mathbf{y}}}_{b} } \\ {{\hat{\mathbf{z}}}_{b} } \\ \end{array} } \right), \end{aligned}$$

(B2)

where ${{\mathsf {\mathbf{{R}}}}}^\mathrm{T}$ is the transpose of ${{\mathsf {\mathbf{{R}}}}}$. The corresponding unit vectors for radial, polar and azimuthal in spherical coordinates are represented by ${\hat{\mathbf{r}}}, {{\hat{\theta }}}_b,{{\hat{\phi }}}_b$ and ${\hat{\mathbf{r}}}, {{\hat{\theta }}},{{\hat{\phi }}}$, with

$$\begin{aligned} \left( {\begin{array}{*{20}c} {{\hat{\mathbf{r}}}} \\ {\hat{\theta }} \\ {\hat{\phi }} \\ \end{array} } \right) = {\mathbf{P}}\left( {\begin{array}{*{20}c} {{\hat{\mathbf{x}}}} \\ {{\hat{\mathbf{y}}}} \\ {{\hat{\mathbf{z}}}} \\ \end{array} } \right). \end{aligned}$$

(B3)

Here,

$$\begin{aligned} {\mathbf{P}} = \left( {\begin{array}{*{20}c} {\sin \theta \cos \phi } & {\sin \theta \sin \phi } & {\cos \theta } \\ {\cos \theta \cos \phi } & {\cos \theta \sin \phi } & { - \sin \theta } \\ { - \sin \phi } & {\cos \phi } & 0 \\ \end{array} } \right) . \end{aligned}$$

(B4)

Then, the transformation of the rotating magnetic dipole, $\varvec{p}$, from the magnetic frame in Cartesian coordinates to the observer’s frame in spherical coordinates follows by using Equations (B2) and (B3).