2.1 Analytic model for the visible point

We determine the visible point in terms of θ = θ_V(α, ζ; ψ) and ϕ = ϕ_V(α, ζ; ψ), or θ _b = θ _bV(α, ζ; ψ) and ϕ _b = ϕ _bV(α, ζ; ψ) by requiring that the line of sight be tangent to a field line,

(1) \begin{equation} ({\hat{\bf b}}\cdot {\hat{\bf n}})^2=1, \qquad {\hat{\phi }}_b\cdot {\hat{\bf n}}=0, \end{equation}

where $\hat{\bf b}$ is the unit vector along the magnetic field and $\hat{\bf n} = \sin \zeta \,{\hat{\bf x}}+\cos \zeta \,{\hat{\bf z}}$ is the unit vector along the line of sight.

Solving Equation (1) simultaneously in the magnetic frame gives (Gangadhara, Reference Gangadhara2004, Reference Gangadhara2005)

(2) \begin{eqnarray} \cos 2\theta _{b{\rm V}} = \frac{1}{3} \left(\cos \theta _b \sqrt{8+\cos ^2\theta _b} - \sin ^2\theta _b\right), & \nonumber \\ \tan \phi _{b{\rm V}} = \frac{\sin \zeta \sin \psi }{\sin \alpha \cos \zeta -\cos \alpha \sin \zeta \cos \psi }. \end{eqnarray}

The angles θ, ϕ and θ _b , ϕ _b are related by

(3) \begin{equation} \cos \theta _b \eqcellsep = \eqcellsep \cos \alpha \cos \theta +\sin \alpha \sin \theta \cos (\phi -\psi ), \\ \end{equation}

(4) \begin{equation} \tan \phi _b \eqcellsep = \eqcellsep \frac{\sin \theta \sin (\phi -\psi )}{\cos \alpha \sin \theta \cos (\phi -\psi )-\sin \alpha \cos \theta }, \end{equation}

(5) \begin{equation} \cos \theta \eqcellsep = \eqcellsep \cos \alpha \cos \theta _b-\sin \alpha \sin \theta _b\cos \theta_b, \\ \end{equation}

(6) \begin{equation} \tan (\phi -\psi ) \eqcellsep = \eqcellsep \frac{\sin \theta _b\sin \phi _b}{\cos \alpha \sin \theta _b\cos \phi _b + \sin \alpha \cos \theta _b}. \end{equation}

2.2 Examples of the trajectory

We use Mathematica ^® to plot the visible point for ψ ≔ [ − π, π] for chosen values of α ≔ [0, π/2] and ζ ≔ [0, π/2]. A point defined by (θ, ϕ) is plotted on the unit sphere corresponding to Cartesian coordinates {sin θcos ϕ, sin θsin ϕ, cos θ}.

As ψ varies from − 180° to 180° the visible point traces a continuous trajectory on the surface of the sphere. Figure 1 shows the visible point for α = 45° and ζ = 60°, with the brown dot at ψ = 0 and the trajectory shown by the dark closed curve (it is not a circle in general). A further example of the visible point and the trajectory are shown for α = 80°, ζ = 30° in Figure 2. A special case is for an orthogonal rotator, α = 90°, observed along the rotation axis, ζ = 0°; the trajectory is then circularly symmetric about the rotation axis, with the line of sight located inside the emission circle for α > ζ.

Figure 1.

This figure shows the viewing geometry of emission in three dimensions for α = 45°, ζ = 60°. A unit sphere, which represents a pulsar magnetosphere, is plotted in Cartesian coordinates. The rotation axis ω is along the z-axis, the red arrow represents the magnetic moment (m), and the line of sight (LOS) is represented by the blue arrow. The point in brown is the visible point as seen by the observer, in this case, it is at ψ = 0 with θ_V ≈ 55°. The visible point moves as the pulsar rotates from − π to π tracing out the dark curve.

Figure 2.

As for Figure 1, but for α = 80°, ζ = 30°.

3.1 Definition of the velocity

The visible point moves at an angular velocity ω_V with components

(7) \begin{equation} \omega _{{\rm V}\theta } = \omega _\star \frac{\partial \theta (\alpha ,\psi )}{\partial \psi }, \quad \omega _{{\rm V}\phi } = \omega _\star \frac{\partial \phi (\alpha ,\psi )}{\partial \psi }, \end{equation}

where ω_⋆ = dψ/dt is the angular speed of the star. The angular speed of the visible point is ω_V = |ω_V|. The motion of the visible point is periodic, with the same period as the star, but the motion can be far from uniform. The motion of the visible point corresponds to sub-rotation around ψ = 0, and super-rotation around ψ = π, with an average ⟨ω_V(ψ)⟩ = ω_⋆.

It is instructive to consider the change in velocity as α is decreased; in the aligned case α = 0 it can be shown that the visible point is stationary. The size of the trajectory decreases, with decreasing α, and the asymmetry in the speed of the visible point increases, becoming slower near ψ = 0 and faster near ψ = π. In the limit α → 0, the visible point is nearly stationary for nearly all ψ, with the exception being very rapid motion near ψ = π around a tiny trajectory.

3.2 Numerical calculation of ω_V

Figure 3 shows the variations of ω_V in units of ω_⋆ as a function of rotational phase for various combinations of α and ζ. For α = 90° and ζ = 0, the trajectory is a circle centred at the rotation axis, giving ω_V = ω_⋆. For other values of ζ and α, the maximum and minimum values of ω_V/ω_⋆ occur at ψ = 180° and 0, respectively. The extrema increase in magnitude with increasing α or ζ, maximizing at α = ζ = 90°. All curves intersect at ψ ~ ±90° with ω_V( ± 90°)/ω_⋆ = 1.

Figure 3.

The ratio of the angular frequency of the visible point, ω_V, to the spin frequency of the pulsar, ω_⋆, plotted against the rotational phase for α = 90°, ζ = 0 (blue), α = 10°, ζ = 5° (solid), α = 30°, ζ = 10° (dashed), and α = 45°, ζ = 15° (dot-dashed). The periodic motion over one pulsar period results in ⟨ω_V(ψ)⟩ = ω_⋆.

4.1 Last closed field line

When only the dipolar term is retained, the last closed field line is determined by the condition that the field line be tangent to the light cylinder, at rsin θ = r_L . For given field line, r = r ₀sin ²θ _b , ϕ _b = ϕ₀, this requires

(8) \begin{equation} \left.\frac{\partial (\sin ^2\theta _b\sin \theta )}{\partial \theta _b}\right|_{\phi _b}=0. \end{equation}

The derivative ∂θ/∂θ _b may be determined using (5), and the resulting equation solved for θ _b = θ _bL (ϕ _b ). The value of θ → θ _L (ϕ _b ) along the last closed field line follows from θ _b → θ _bL (ϕ _b ) in (5). The shape of the boundary of the open-field region is independent of r, and is given by plotting the function θ _b = θ _bL (ϕ _b ). The field line constant, r ₀ → r _L0(ϕ _b ), for the last closed field line is then determined by

(9) \begin{equation} r_{L0}(\phi _b)=\frac{r_L}{\sin ^2\theta _{bL}(\phi _b)\sin \theta _L(\phi _b)}. \end{equation}

The radial position along the last closed field line at ϕ _b is r(θ _b , ϕ _b ) = r _L0(ϕ _b )sin ²θ _b .

4.2 Minimum visible height

The minimum visible height, r _min, is determined by assuming that the trajectory is tangent to the locus of last closed field lines. This corresponds to θ _bV = θ _bL (ϕ _b ) at ϕ _b = ϕ _bV, and reproduces the result given by Gangadhara (Reference Gangadhara2004):

(10) \begin{equation} r_{\rm min} = \frac{r_L \sin ^2 \theta _{b{\rm V}}}{\sin ^2 \theta _{bL}(\phi _{b{\rm V}})\, \sin \theta _L(\phi _{b{\rm V}})}. \end{equation}

One may regard r _min as a function of α and ζ. Figure 4 shows a three-dimensional plot for r _min as functions of these variables.

Figure 4.

A three-dimensional surface plot of r _min as functions of ζ and α. The r _min increases as |β| increases and r _min → 0 for β → 0.

4.3 Probability of seeing a pulsar

The existence of a minimum height, below which any emission is not visible (because it is not along the observer’s line of sight) has a statistical implication. Consider a populations of neutron stars with α and ζ randomly distributed. This implies that the impact parameter, β = ζ − α, is also randomly distributed. Emission from low heights, r ≪ r_L , is restricted to a small cone, with half angle θ _c , equal to (r/r_L )^1/2 in the aligned case, and is visible only for |β| < θ _c . The probability of seeing emission from a particular neutron star is given by dividing the solid angle 2πθ² _c (for two poles) by 4π. This probability is r/2r_L for r/r_L ≪ 1. Based on the geometry alone, most visible emission from pulsars must come from relatively large heights. For example, if one assumes that at least 10% of all radio pulsars are visible, this implies r/r_L ≳ 0.2. This analytic estimate is supported by the numerical results in Figure 4, which show that near the minimum, r _min/r_L increases approximately proportional to β², implying that emission from low heights is visible only for small β.

4.4 Duty cycle

In the tangent model, the duty cycle of a pulsar is identified as the range of ψ between the two intersection points where the trajectory cuts the boundary of the open-field region. Figure 5 shows the viewing geometry in the magnetic frame. Along this portion of the trajectory, ω_V varies symmetrically about ψ = 0 (which defines the x_m -axis); ω_V is minimum at ψ = 0, and increases towards either of the two intersections. The visible point enters the open region, at A, with a certain angular speed, its speed gradually reduces, reaching its minimum at ψ = 0, and then increases again until it leaves the open-field region at H with the same angular speed as at A. The portion of the trajectory that is within the open-field region, and hence the duty cycle, increases with increasing r > r _min.

Figure 5.

The viewing geometry in the magnetic frame showing the observable visible point traces out a path through the open-field region. Four open regions are plotted for r = 1.2 r _min (green), 0.1r_L (gray), 0.2r_L (brown) and 0.6r_L (blue) for α = 45° and ζ = 60°. The trajectory of the visible point (black) intersects the green, gray, brown and blue curves at (D, E), (C, F), (B, G) and (A, H), respectively, between which an observer sees radiation. The red curve represents the path that traces out by the visible point in the conventional model, in which the line of sight is assumed to go through an origin.

5.1 Evolution of the PA

Pulsar radio emission has a linearly polarised component, described by its PA, which is assumed to be determined by the projection (perpendicular to the line of sight) of the magnetic field line at the visible point. The evolution of the PA with ψ in the RVM is assumed to give a characteristic S-shaped swing as the open-field region sweeps across the line of sight to the centre of the star (Radhakrishnan & Cooke, Reference Radhakrishnan and Cooke1969). The actual shape of the PA curve depends on α and β. Figure 6 shows the sweep in PA as predicted by the tangent model for α = 45° and three different values of ζ. The PA curve changes from I to III as β changes from negative to positive, with II corresponding to β = 0. For β < 0, the PA swing is an S-shaped curve, similar to a sine curve for β large and negative, with the slope of the S steepening as |β| decreases towards zero. For β ⩾ 0, the PA variation is monotonic, exhibiting a jump by 180°, which occurs at ψ = 0 for β = 0 (II) and at a non-zero ψ for β > 0 (III).

Figure 6.

The changes in polarisation position angle for α = 45° and ζ = 40° (solid), ζ = 45° (dashed) and ζ = 50° (dot-dashed) plotted against rotational phase. Integrated profiles, which are centred at ψ = 0 in the model, of different widths capture different information of the PA curve.

The observed PA curve depends on the pulse width. For a narrow pulse width the PA change can resemble an unbroken S-shaped curve, as illustrated by curve I in Figure 6, examples being PSR B0136+57, B0628-28 and Vela pulsar (Lyne & Manchester, Reference Lyne and Manchester1988; Becker et al., Reference Becker, Jessner, Kramer, Testa and Howaldt2005), or an abrupt jump, as illustrated by curve II in Figure 6, an example being PSR B0355+54 at 1.4 GHz (Gould & Lyne, Reference Gould and Lyne1998). For a larger pulse width the PA change illustrated by curve III in Figure 6 is similar to that observed from PSR J0738-4042 and PSR J1243-6423 (Karastergiou & Johnston, Reference Karastergiou and Johnston2006). The interpretation of PA curves suggests that effects other than geometry alone can be significant: ‘reversed’ PA curves indicate that pulsars rotate in the direction opposite to that we assume, and other unusual PA curves may be an indication of non-dipolar field structure, small scale distortions in the open-field region, or interstellar scattering effects (Karastergiou, Reference Karastergiou2009).

5.2 Comparison of RVM and tangent models

In the RVM the PA swing is calculated based on the path of the line of sight through the centre of the star as the open-field region sweeps across it (Radhakrishnan & Cooke, Reference Radhakrishnan and Cooke1969; Lyne & Manchester, Reference Lyne and Manchester1988). An example of the path implied by the RVM is the red curve in Figure 5, which is clearly different from the trajectory in the tangent model, shown by the black curve. Further examples showing the difference between the two models are illustrated by the red and black curves, for two different choices of parameters, in Figure 7. Such examples allow one to quantify the error introduced by use of the RVM.

Figure 7.

The trajectory of the visible point as predicted in the RVM (red) and the tangent model (black) for α = 45°, ζ = 60° (solid) and α = 80°, ζ = 30° (dashed) in the magnetic frame (see Figures 1 and 2). The leftmost point of intersection between a trajectory and the x_m axis represents ψ = 0.

As remarked in Section 1, the angular error introduced is $\arcsin (d/r)$ , where d is the perpendicular distance between the lines of sight in the RVM and the tangent model. This error is maximum, as a function of r, near the minimum value of r, at r = r _min, where it is d = r _minsin (θ _b − θ _bV). For given ζ and α, d has a minimum at ψ = 0 and a maximum at ψ = 180°, and it increases with increasing ζ and α.

5.3 Minimum visible height

Another way of comparing the RVM and the tangent model is in terms of the minimum visible height. The criterion for emission to be visible is that the line of sight, which is independent of r in both models, intersects the open-field region, which broadens with increasing r. In both models there is a minimum visible height, at which the line of sight intersects the boundary of the open-field region. Above this height, there is a range of ψ for which emission is visible. In Figure 8 we fix this height at a specific value, r = 0.2r_L , so that this corresponds to the emission height at the two extrema in ψ that define the edges of the pulse window. For ψ between these extrema emission is visible from a range of heights r_V < 0.2r_L with this range increasing to a maximum at ψ = 0, midway between the edges. Figure 8 compares the pulse window and the range of visible heights within the window for this particular example. The most obvious feature is that the RVM underestimates the size of the pulse window, compared with the tangent model.

Figure 8.

Variations in the emission height, r_V , along the open field lines where the trajectory cuts through the open region in the RVM (red) and the tangent model (black) for α = 45° and ζ = 30°. The geometry identifies the centre of the pulse at ψ = 0 where r_V = r _min, and r_V = 0.2r_L at the two boundaries representing the leading and trailing edges of the pulse profile.

5.4 Swing of PA

How significant is the error introduced by using the RVM to determine the swing in PA? To answer this question we calculate the swing in PA for both models with given values of α and ζ. The results are shown in Figures 9–12. The error introduced is small near ψ = 0, and indeed so small that we need to displace the curves, by 5°, in order to distinguish between the two models. One can conclude that the RVM is an excellent approximation for the purposes of calculating the PA swing, provided that the pulsar has a narrow pulse window.

Figure 9.

Differences in the trajectory through the open-field region in the magnetic frame (left) and the observed PA curve (right) between the RVM (red) and the tangent model (black), for α = 45° and ζ = 30°, and emission height at 0.2r_L . The red curve in the PA plot is shifted by + 5° for clarity. An S-shaped is predicted for the PA swing in the tangent model.

Figure 10.

Same as in Figure 9 but for ζ = 50°.

Figure 11.

Same as in Figure 9 but for α = 80° and ζ = 65°. Both with β = −15°.

Figure 12.

Same as in Figure 10 but for α = 80° and ζ = 85°. Both with β = 5°.

The difference in the PA curves between the two models increases as the pulse window increases. The range of ψ predicted by the RVM is narrower than that predicted by the tangent model, and this difference can be substantial for a broad pulse. The largest difference between the two models occurs for large β.

In summary, the error introduced by assuming that the line of sight passes through the origin include an underestimation of the height of emission, and hence either to an overestimation of α, for β > 0, or to an underestimation of α, for β < 0.

5.5 Visibility of the opposite pole

The integrated pulse profiles of some pulsars show an interpulse (IP), which is separated from the main pulse by approximately half the rotation period. Similar to main pulses, interpulses may also exhibit such phenomena as mode-changing, pulse-to-pulse intensity modulation and subpulse drifting (Weltevrede, Stappers, & Edwards, Reference Weltevrede, Stappers and Edwards2007). Assuming that the emission mechanism for the near and far magnetic poles is the same and both are actively radiating, the conditions that determine the visibility for the near magnetic pole also apply to visible emission from the far (opposite) pole. The analytic solution in Section 2 involves solving a quartic equation, and there are four solutions. Two of the solutions correspond to emission in the observer’s hemisphere, and the condition for an interpulse is that both be visible. (The other two solutions correspond to emission in the opposite hemisphere to the observer and are of no interest.)

The visibility conditions for an interpulse are that a portion of the trajectory of the visible point near the far pole is within the open field region, and that this occurs above the minimum visible height, r _{min, IP}. Figures 13–15 apply in the special case ψ = 0, when the magnetic axis, the rotation axis and the line of sight are in the same plane. These figures show the variations in r _min and r _{min, IP}, where IP refers to interpulse, as predicted in both the RVM and the tangent model as function of α for three values of ζ. Two emission heights are selected, at r = 0.1r_L and r = 0.2r_L , with the range of ± x_m from which emission is visible restricted to between the curves, which are symmetric about the horizontal axis.

Figure 13.

The minimum visible height for ψ = 0 is shown by the value of x_m for the main pulse (black) and interpulse (blue) in the tangent model (solid) and the RVM (dashed) as a function of 0 ⩽ α ⩽ 90° and ζ = 10°. The boundaries for the open regions are shown for r = 0.1r_L (dot-dashed gray) and 0.2r_L (solid gray) where each intersects the x_m axis at two points (x_m > 0 and x_m < 0, see Figure 9). The two magnetic poles, which are separated by θ = ψ = π, are plotted at the same origin for clarity and both are assumed with similar constrains on the emission height, and hence for β = 0 (intersection of the black curve with the x_m -axis), r _min = 0 for near pole but r _{min, IP} is large. Visible emission requires the visible point to be between the gray lines. Intersection of a curve with the horizontal axis occurs when ζ = α = 10°, but only for the main pulse.

Figure 14.

Same as in Figure 13 but for ζ = 50°; only the near pole is visible.

Figure 15.

Same as in Figure 13 but for ζ = 90°; both poles are visible for large α. The RVM (dashed curves) underestimates the range of α for which both poles are visible.

Three features are apparent from Figures 13–15. First, the far pole is visible only for large α and ζ, and large heights. Second, from Figure 15 one infers that the RVM underestimates the range of 90° − α over which the far pole is visible. Third, from Figure 13 it is evident that the RVM incorrectly predicts a range of small α and ζ where the far pole is visible. These differences illustrate the errors that can be introduced by using the RVM away from ψ ≈ 0, where the error is small.