A&A 445, 779-794 (2006)
DOI: 10.1051/0004-6361:20053040
P. K. Fung^{1} - D. Khechinashvili^{2,3} - J. Kuijpers^{2}
1 - Astronomical Institute,
Utrecht University, PO Box 80000, 3508 TA Utrecht, The
Netherlands
2 - Department of Astrophysics, IMAPP, Radboud
University Nijmegen, PO Box 9010, 6500 GL Nijmegen, The
Netherlands
3 - Centre for Plasma Astrophysics, K. U. Leuven,
Celestijnenlaan 200B, 3001 Leuven, Belgium; on leave for
Institute of Astronomy, University of Zielena Góra, Lubuska 2,
65-265, Zielena Góra, Poland / Astrophysical Observatory, Al.
Kazbegi ave. 2a, Tbilisi 0160, Georgia
Received 10 March 2005 / Accepted 12 July 2005
Abstract
The potential
role of a diocotron instability in causing drifting sub-pulses in
radio pulsar emission is investigated for aligned magnetic
rotators. It is assumed that the out-flowing plasma above a pulsar
polar cap consists of an initially axially symmetric, hollow beam
of relativistic electron positron pair plasma which carries an
electric charge as well as a current. The occurrence of
instability depends on shear in the angular velocity distribution
of the beam as a function of axial distance. Instability occurs
under typical pulsar conditions at mode numbers 40. It
destroys the symmetry of the equilibrium configuration and leads
to a carousel of density columns which rotates at fixed angular
pattern speed. The process is applied to two pulsars with observed
carousels of drifting sub-pulses, and the diocotron instability at
corresponding mode number and axial distance is used as a
diagnostic for the charge and current density of the polar flow.
Key words: stars: pulsars: general - plasmas - magnetohydrodynamics (MHD) - instabilities
Although radio pulsars have been observed for more than thirty-five years, our understanding of the emission process remains poor. The radiation is observed to have a high brightness temperature (Melrose 1995) and is believed to be emitted tangentially to open magnetic field lines above the neutron star's polar caps, somewhere between the stellar surface and the light cylinder. As the star rotates, the magnetic pole sweeps past our line of sight and we receive a radio pulse. The temporal structure of the observed pulse corresponds to the spatiotemporal emission along the strip cut out by our line of sight. The radio emission is probably generated by relativistic particles streaming out along the open field lines, but the precise physical conditions under which the radiation is produced, and the precise location of the emission region above the rotating neutron star, remain obscure. The most remarkable feature of the pulsar is the stability of the pulse profile when averaged over about a hundred subsequent pulses, whereas individual pulses are complex and highly variable. An individual pulse is built out of sub-pulses and sometimes micro-pulses of s duration. In a number of "ordinary'' (i.e., excluding millisecond) pulsars drifting sub-pulses have been observed which show a remarkable drift stability. It now appears that the occurrence of drifting sub-pulses is not a rare phenomenon but can be detected in half of the radio pulsars (Weltevrede et al. 2005; Ashworth 1981). Drifting sub-pulses are sub-pulses that drift in longitude from pulse to pulse at a nearly constant rate. For some pulsars, a "carousel'' model can be constructed, which consists of emission columns arranged along a circle centered at the stellar magnetic axis, which sub-rotate with respect to the star (Deshpande & Rankin 1999,2001). Understanding drifting sub-pulses is probably of critical importance for our understanding of the pulsar electrodynamics and pulsar emission mechanism.
To date, several models have been proposed to explain this phenomenon. In the model by Ruderman & Sutherland (1975, hereafter RS75), drifting sub-pulses are interpreted in terms of spark-associated plasma columns rotating around the magnetic axis due to the -drift, where is the background magnetic field and is the electric field in the frame corotating with the star. The assumption of the infinitely large binding energy of ions in the stellar surface, crucial for the spark model, has been severely criticized by several authors (see, e.g., Neuhauser et al. 1987,1986). Moreover, there is strong disagreement between the observed drift speed and the one predicted by the spark model (as shown by van Leeuwen et al. 2003, in the case of PSR B0809+74). In the model by Gil et al. (2003) the vacuum gap is partially screened due to thermionic emission of ions/electrons, and sparks still occur but at a smaller drift rate.
Another model (Clemens & Rosen 2004) ascribes the drifting sub-pulses to non-radial pulsations of the neutron star. A difficulty with this model is the exclusive selection of one particular high-order spherical eigenmode (with a spherical harmonic numbers in between and m=0) at the stellar surface. Also, there is no clear link of the internal oscillations of the star to the processes leading to generation of radio emission higher above the stellar surface. Alternatively, Kazbegi et al. (1991) explain the sub-pulse drift in terms of modulation of the emission region by large-scale "drift waves'', which alter the local curvature of the magnetic field lines and thus affect strongly the occurrence of the Cherenkov-drift resonance - a main mechanism of generation of pulsar radio emission, according to Kazbegi et al. (1991). A recent revised version of this model, by Gogoberidze et al. (2005), favours modulation either through modification of the distribution function of particles by the drift wave electric field, or through changes in the emission angle. The stability of a distinct drift pattern is achieved by the accumulation of the drift modes in a specific azimuthal eigenmode, resulting from their non-linear evolution. The model requires relatively weak magnetic field values, which prevail at distances corresponding to a significant fraction of the light cylinder radius while observations suggest that the emission processes take place at much closer distances to the pulsar surface, typically 50-100 stellar radii (see, e.g., Kijak & Gil 2003). Finally, a recent empirical model constructed by Wright (2003) attributes the formation of a drifting regular pattern of emission nodes to constructive interaction between electron and positron beams traveling, respectively, up and down in the magnetosphere, between inner and outer acceleration gaps. The model, though, does not elaborate on the detailed electrodynamics.
Here, the proposed explanation of drifting sub-pulses is the occurrence of a diocotron instability (also known as "rotational shear'' or "slipping stream'' instability) in the pair plasma on the open pulsar field lines. The diocotron instability is an instability of a sheared flow, much like the familiar Kelvin-Helmholtz instability in a quasi-neutral plasma, now, however, in a non-neutral plasma, which is why it is sometimes also called the "electrostatic'' Kelvin-Helmholtz instability (Hasegawa 1975). Specifically, the diocotron instability occurs in a charged plasma beam along a uniform background magnetic field and a vanishing background electric field component parallel to the background magnetic field (e.g., Davidson 1974, Sect. 2.10). The electric self-field of the charged beam would lead to the familiar drift around the cylinder axis which is now, however, modified by the effect of a magnetic self-field from the beam which, in general, carries an electric current as well as a charge. When the resulting differential rotation of the beam around the cylinder axis satisfies a certain condition, a non-axially symmetric perturbation in density, velocity and electric potential can grow over the shear layer. To linear order, the unstable surface modes are characterized by a radial eigenfunction, an azimuthal mode number l, an axial wave-number , and an angular frequency , whose imaginary part gives the growth rate and whose real part is related to the angular pattern speed through . It is this angular pattern speed which we identify with the rotation speed of the drifting sub-pulses when viewed from the lab frame.
In this paper we model the plasma outflow above the pair-creation front as an infinitely long, non-neutral annular beam composed of electrons and positrons propagating with relativistic speeds along the pulsar's magnetic field. Stability analysis reveals a large domain of realistic parameters, where the - linear stage of - diocotron instability may indeed occur under pulsar magnetospheric conditions. We assume that l and of the unstable mode can be linked to the observed number of sub-beams and pattern circulation time , where is the pulsar rotation frequency and is measured in units of pulsar periods P_{1}. Using these two assumptions, we then apply our model to the two pulsars with the best studied carousel patterns, viz. PSRs B0943+10 and B0826-34, and demonstrate that the the model is able to account in a natural way for the basic properties of the drift patterns observed in these pulsars. Furthermore, the model can explain drift direction reversals as in PSR B0826-34. For the relativistic beam to nearly corotate (only slightly sub-rotate) with the star, the model requires the charge difference in the beam to exceed the Goldreich-Julian charge density (Goldreich & Julian 1969; see also Eq. (1) below) many times, which is no problem for a typical pair plasma.
A similar instability, namely an electrostatic shear flow instability, has been studied before in the context of pulsar magnetospheres by Arons & Smith (1979). Although the authors state that their basic equations contain also the physics of other instabilities, including the diocotron instability, they do not treat the latter process, but rather concentrate on implications of the shear in parallel relativistic momentum. The modes excited due to such a shear are space charge waves, contrasting with the diocotron surface modes, treated in the present paper and leading to azimuthal fragmentation of the flow. It is also important to notice that Arons & Smith (1979) assume a small perturbation wavelength along the external magnetic field, whereas our approximation is just the opposite to that (i.e., we have ). In addition, Arons & Smith (1979) only found the electrostatic shear flow instability to appear at enormously large azimuthal wave-numbers (l > 10^{7}). We find the same to be true for the diocotron instability if one assumes that the flow is charge-separated, as in Arons & Smith (1979). However, here we will show that the diocotron instability can operate in a relativistic beam under pulsar conditions if the density distribution is not charge-separated, but instead is formed by a pair plasma.
In Sect. 2 we provide the physical background for the inner magnetosphere and describe a beam model with a minimum amount of detail but which does contain the essence of the various existing theories on pulsar electrodynamics. The diocotron instability of the beam is then investigated in a fluid approach and the analytical results are presented in Sect. 3 while the derivation is given in the Appendix. Numerical results on the instability, the pattern speed, and the growth rate for various choices in parameter space are presented in Sect. 4. The implications for pulsars and the observed drifting sub-pulses are given in Sect. 5. We end with a discussion and conclusions in Sect. 6. We use SI units throughout the paper.
In order to study the physics of the diocotron instability and construct a correct model of the non-neutral pair plasma beam, one should first understand the electrodynamics of the pulsar magnetosphere. Here, after brief review of existing magnetospheric theories, we discuss the nature of the "rotational'' drift in pulsars, in case of both aligned and inclined pulsars. As we will show by simple reasoning, the assumption of independence of the current and charge densities from each other is essential for "near-corotation'' of the open magnetic flux tube with the pulsar.
Most pulsar theories have in common the existence of a corotating closed
magnetosphere and two bundles of "open'' field lines defining the polar
caps on the stellar surface and passing through the light
cylinder. Everywhere in the corotating magnetosphere the plasma is
charge separated - it consists of one kind of charge only - and
the charge density is given by the Goldreich-Julian density
Figure 1: Sketch of various models for pair creation (hatched) on the open field lines in the pulsar magnetosphere with level curves of electrostatic potential and corresponding electric field : a) axially symmetric vacuum model à la Deutsch (1955); b) RS75, anti-aligned pulsar and infinite ion binding energy; c) Arons (1983); d) Muslimov & Tsygan (1992), general relativistic model. The magnetic moment is oriented along the vertical direction, and the rotation vector is . | |
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The field lines on the magnetic flux surface separating the incoming and the outgoing current are called the critical field lines by Goldreich & Julian (1969). The critical field lines are at the same electric potential as the interstellar medium. A simple estimate of their position on the polar cap can be obtained if one assumes that all over the polar cap the current density is just the Goldreich-Julian density at the stellar surface multiplied with - or + the speed of light depending on whether the current comes in or goes out. The feet of the critical field lines are then positioned at an angle as seen from the stellar center given by which differs only by a factor 0.96 from the feet of the field lines which pass through the cross-section of both the light cylinder and the null lines, and which are located at an angle at the stellar surface. It is interesting to note that the latter field lines define the outer emission cone in Wright (2003).
Figure 2: The current system I in the axially symmetric case; magnetic field , magnetic moment , rotation vector , null line n, light cylinder lc, critical field line c separating incoming from outgoing current. | |
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Figure 3: Level curves of the electrostatic potential on the open field lines in the oblique case; magnetic field , magnetic moment , rotation vector . The plasma circulation is on surfaces of constant . | |
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Figure 4: Axially symmetric case: Sketch of the level curves of the electrostatic potential in the frame corotating with the star on the left, and of the potential V in the lab frame on the right. In the middle are the level curves due to pure corotation with the star. Darker level of grey corresponds to a higher potential value. Hollow arrows mark the corresponding electric field E, with the arrow size being proportional to the field strength. The sum of the potentials on the left equals to the potential on the right. Magnetic field B (thin arrows), magnetic moment m and rotation vector are displayed. | |
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The field in the rupture zone near the foot-points is of
course not frozen into the plasma. Far above the acceleration
region, the situation becomes practically independent of altitude
and the electric field is entirely due to space charges inside the
beam and not to any surface-charges as is the case near the
stellar surface. In the RS75 model, such would apply already at a
distance of a few polar cap radii above the surface while in the
general relativistic model of Muslimov & Tsygan (1992) this would apply
above a few stellar radii. Since we are interested in the drift
motion of a relativistic plasma beam - which creates its own
transverse magnetic field
- the drift motion of
the plasma as obtained from the approximate force equilibrium
condition
is given by:
As to whether the magnetic field is strong enough to enforce corotation of any plasma with the stellar rotation, the answer depends on the global distribution of electric fields. If there is no potential drop between the stellar surface and a relativistic plasma beam higher up then in a Poynting-flux dominated situation, as is the case for the pulsar, the beam would corotate with the star. The required relatively strong electric field would simply be given by a strong super Goldreich-Julian density. But in a realistic pulsar, the relativistic nature of the beam is caused by a parallel voltage drop that allows for rupture of the magnetic field lines below the beam and may cause a strong sub-rotation of the beam with respect to the stellar rotation.
To study the diocotron instability, the relativistic macroscopic
fluid
description is used. Each species
is described by the coupled fluid-Maxwell's equations:
(14) | |||
(15) |
Figure 5: Top: model for the hollow beam. The density difference between positrons and electrons is in region II, and zero in regions I and III. At R > R_{3}, the plasma is assumed to be in solid-body rotation with the star. Bottom: the axial velocity of the beam particles across the beam is taken to be constant. | |
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The stability of the equilibrium under electrostatic perturbations
is investigated where each quantity is perturbed as follows:
(16) |
The surface waves are the solution of the linearized fluid-Maxwell equations. Before we analyze the equilibrium situation and the solution of the perturbed system, we first describe the geometry of the charged beam, because the assumptions we make enable us to simplify the set of equations.
We consider three possible geometries: a hollow beam, a hollow beam including a core component, and a hollow beam in which the current reverses sign.
As a first simple case, we model the beam in this paper as a
hollow beam of electrons and positrons with nearly equal axial
velocities (Fig. 5). The densities of electrons
and positrons are assumed to be constant in different regions of
the beam. Then, only the density difference
is relevant and is taken to be
different only in region II. We normalize
where
is not the GJ-density
Eq. (1) but an abbreviation involving the absolute value
of the local magnetic field strength B_{0}:
(17) |
For the motion of the two species, we take
independent of R. Additionally, particles are
only relativistic in the axial direction, so that
.
We express the axial current density
of the beam formally as
Two additional configurations are examined: the influence of a nonzero core and when part of the beam moves in the opposite direction (return current).
In Fig. 6 the model including a nonzero core is presented. The charge density difference in region I is . We expect that in this region no pair creation occurs because of the large radius of curvature of the magnetic field, and, therefore, that the beam is charge-separated. The axial velocity remains unchanged, i.e. both in region I and in region II, approximately the same axial velocity applies to all particles.
As for the return current, the axial velocity of the hollow beam
changes sign at the radial location
Figure 6: Similar to Fig. 5; now, a core component is included. The dimensionless charge difference in Region I is . | |
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Figure 7: Similar to Fig. 5, but now, part of the beam moves in the opposite direction (R_{p} is defined by Eq. (19)). | |
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If one neglects the first term one immediately derives the earlier
result
(cf. Eq. (7)). Actually,
the first term represents the inertial acceleration which can be
neglected even for large Lorentz factors because of the strength
of the background magnetic field. Defining
,
and
,
the angular
rotational velocity
for the hollow beam
(Fig. 5) reads:
For illustrative purposes, the rotational velocities for the cases
(24) to (26) with R_{1} = 0.3, R_{2} =
0.6 and R_{3} = 1 are shown in Fig. 8:
hollow beam with Q = 0 (solid), hollow beam with Q= 0.5 (long
dash), beam with core (
)
and Q = 0.5 (short dash) and beam including return current with
Q= 0.5 where R_{p} is determined by zero ring
current (Eq. (20), dotted) and another where p = 0.5(dash-dotted). Note that the presence of a return current in a
relativistic beam has the effect of pushing up the rotational
velocity toward corotation (compare dotted and long-dashed
solutions).
Figure 8: Examples of equilibrium rotational velocity distributions across the beam, where R_{1},R_{2} are the inner and outer boundary of the beam. See text for the cases and the beam parameters. | |
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Linearizing the fluid-Maxwell's equations is done by assuming that
the perturbations are electrostatic, i.e.
is small
compared to
in the momentum equation and Faraday's law.
With this, the perturbed electric field can be described as the
gradient of a scalar potential:
Now, since we are using step-functions for , Eq. (29) can be solved analytically. This applies even in the case of a "return current'', because in the derivation of Eq. (29) we have neglected the contributions of k_{z} and .
The general solution of
Eq. (29) for our model, is given by:
Figure 9: Equipotential for annular beam with Q = 0, R_{1} = 0.5, R_{2} = 0.6 and l = 5 at t = 0. Black areas denote negatively valued , and white the positive values. | |
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Obviously, extremes are formed at the two surfaces with five
azimuthal modes. Moreover, the surface waves are shifted in phase
with respect to each other. To demonstrated this, we use the
expression for the perturbed potential at the surfaces
Eq. (30), but now, C_{1}/C_{2} is given by:
(37) |
(38) |
Figure 10: Left: unstable diocotron modes in the parameter space of R_{1}/R_{3} (along the horizontal axis) and R_{2}/R_{3}(along the vertical axis). In levels of grey is plotted , which is the azimuthal wave number l for which the combination of R_{1}/R_{3} and R_{2}/R_{3} results in the largest growth rate (Eq. (35)). The parameter is found above each graph. Right: maximum growth rates (triangles) and pattern velocities (squares) as functions of . | |
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Figure 11: The same as in Fig. 10, but now including a charge-separated core in Region I (see Eq. (25) and Fig. 6). The corresponding value of is specified above each graph, and in all graphs. The growth rate and the pattern speed are displayed as squares and triangles, respectively. | |
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Figure 12: The same as in the left column of Fig. 10, but now including a return current (see Eq. (26) and Fig. 7). Above each graph are seen the corresponding parameters Q and p, the latter defining R_{p}, i.e. the radius where the axial velocity of the beam particles changes sign (see Eq. (19)). | |
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Figure 13: Maximum growth rate (squares) and pattern velocities (triangles) corresponding to the case in Fig. 12. | |
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Thus, in general, the two waves are shifted in phase, which is determined by the geometry. Due to the shift, the two surface waves are coupled to each other through their perturbed potential. To be more specific, under the condition determined by Eq. (34), perturbations at the outer surface can influence the perturbations at the inner surface, and vice versa, leading to exponential growth.
It is this system that probably will evolve into a non-linear stage where particles are trapped and vortices form.
For a hollow beam, in which Q ranges from 0 to 0.8 in steps of 0.2, is presented in Fig. 10. In the case of Q = 0 (i.e. either the beam is nonrelativistic or the beam carries no current), the beam displays instability for all modes. The general trend for all Q's is that thinner beams, i.e. , are unstable for higher l.
An increase in Q leads to a shift of the regions in parameter space and a decrease in their area, until, for small , instability eventually vanishes. In the case of Q = 0.9, almost any hollow beam is stable and therefore, the corresponding parameterspace is not presented. This is in agreement with the results of Arons & Smith (1979) for a charge-separated, relativistic, particle beam. In that case: fequals h, the first term for both G_{1} and G_{2} becomes negligible when we assume , and solutions for are very stable unless (see Eqs. (33) and (34)).
Figure 11 depicts the influence of adding a charge separated core component by increasing in steps from 0.1 to 0.9 (in region II: Q = 0.4). Since it is charge separated, we have . Clearly, when the shear is still present (see Eq. (25)) and we observe a diocotron instability.
In Figs. 12 and 13 is shown the effect of a return current on the occurence of instability. In the case where R_{p} is determined by the condition that the net ring current is zero, the rotational velocities at R_{1} and R_{2}are identical to those of Q=0 in the hollow beam case. Since the instability condition depends only on these velocities (Eqs. (31)-(34)) there is no need to locate the unstable domains in the parameter space again.
The data available from the literature is shown in Table 1.
Table 1: Observationally inferred pulsar data. Note that in Esamdin et al. (2005) no value for is quoted. Also, we infer P_{3} from their values of P_{2} and using the relation for the drift rate , which is a good approximation for this system, as it is nearly aligned. Positive and negative values are obtained due to the definition of drift rate.
For R_{1}/R_{3} = 0.65 and R_{2}/R_{3} = 0.92, we calculate in the parameterspace (l, Q) (see Fig. 14). Apparently, growth rate is not at optimum for l = 20. Since we do not have more information on Q, we conclude that for l = 20 to develop, we need . By combining this with measured , we find from Eq. (36) that .
Since the values for R_{1}/R_{3} and R_{2}/R_{3} are just estimations, it is important to see what the influence is if we change the beam configuration. The results are shown in Figs. 15. The values of Q-parameter, providing instability for l = 20, are listed in Table 2. If we apply the measured pattern velocity to all five cases shown in Table 2 and plug the corresponding values into Eq. (36), we obtain that for the instability to develop, the charge density difference should be .
Table 2: The results of Q and for possible beam configurations for PSR B0943+10 by assuming l = 20 and . The charge density difference is found to be . Initial guess for R_{1}/R_{3} and R_{2}/R_{3} is taken from Fig. 10 in DR01.
Figure 14: Growth rates as functions of Q, for different values of azimuthal mode number l (indicated in the plot), for a hollow beam with R_{1}/R_{3} = 0.65 and R_{2}/R_{3} = 0.92. | |
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Figure 15: Similar to Fig. 14, but now for different combinations of the inner and outer radii of the beam. Top: R_{1}/R_{3} = 0.6 ( left) and R_{1}/R_{3} = 0.8 ( right). Bottom: R_{2}/R_{3} = 0.86 ( left) and R_{2}/R_{3} = 0.93 ( right). | |
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Recently, two detailed observational studies have been made of this pulsar. Gupta et al. (2004) studied the main pulse (III) with the GMRT at 318 MHz. They claim that the apparent sign reversal is merely an effect of the geometry and aliasing. Our line-of-sight is very near the magnetic axis , and this axis in turn is close to the rotation axis, (probably ). In this way, the observations are still compatible with the Ruderman & Sutherland model. They concluded that the carousel has about 15 sparks (at 318 MHz), rotating with an angular frequency of . Another observation was made with Parkes at 1.374 GHz by E05. They studied the whole pulse profile and used a method called "phase tracking'' where (as the name suggests) the phase of the pulse throughout the whole profile is tracked, which resulted in 13 sub-beams. At this frequency, the interpulse is much stronger than the main pulse, therefore, they came to the conclusion that we observe two nested cones, (see Fig. 10 in their paper, note that both cones have 13 sub-beams). They argue that the drift reversal cannot be accounted for in terms of geometry and aliasing (which would be too much of a coincidence), but that it is a true reversal, implying that the beams are rotating and per rotation period in either direction, corresponding to pattern velocities of and 0.99, respectively. Contrary to the model of Rutherman and Sutherland, the diocotron instability can account for a reverse drift. Therefore, we will illustrate the drift reversal in this context.
Here, we assume that the two observed sub-beam rings are formed by an instability of two nested hollow beams. The treatment of the inner hollow beam is unaffected by the outer beam, since the electric field causing the drift is only determined by the enclosed charge. The outer beam, on the other hand, cannot be described correctly within the context of our model without further assumption, since two different hollow beams would lead to an eigenvalue equation for of 4th degree instead of the quadratic Eq. (31). For simplicity, we therefore assume that the outer beam can be treated independently. Although E05 suggested a polar cap geometry, R_{1}/R_{3} and R_{2}/R_{3} are not well enough determined. So, unlike in PSR B0943+10, we cannot first constrain Q by fixing the hollow beam geometry. Instead, we allow different Q and plot the instability region for l = 13 and Q = 0 - 0.7 (Fig. 16; white regions are stable). Inside the domain we separate - by using greyscales and black - the regions according to equal "angular'' velocities (black represents , and the other two have: , ). Note that instead of (R_{2}/R_{3},R_{1}/R_{3}), we now use the center of the beam, R_{b} = (1/2)(R_{1} + R_{2}) and the beam half-width, .
To have a reverse drift, our model implies
that the charge density differences are changing around a critical value. For
instance, if we take Q = 0.7, and for the beam radius of the inner cone
(a rough estimation from Fig. 10 in E05),
then for l = 13, the maximum of
occurs at
,
which gives us
.
For exact rotation with the star, we need
(which corresponds to
). Thus, for slightly
larger or smaller ,
the pattern speed will be either sub-
or super-rotational as compared to the stellar rotation. Also, note
that the beam thickness
increases with the beam radius
for the same (l,
), just like
E05 suggested for the two rings on the basis of their observations
(see their Fig. 10). If we choose, e.g.,
for
the outer cone,
then, for this beam to be unstable for l = 13 and to have the same pattern velocity as the inner beam, we
find a maximum of
,
now corresponding to
,
which is indeed slightly larger than the beam thickness of the
inner cone.
Figure 16: Domain of instability for l=13 as applied to PSR B0826-10. The three regions denote different intervals of (see legend). | |
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Despite the absence of observationally well-determined values of and , our results demonstrate the onset of instability for the required l-value and pattern speed over a broad range of thin beams.
We find that for charge separated flow, only the modes with very high azimuthal harmonic number, , are unstable. However, as we have demonstrated, when the flow consists of both positrons and electrons, the charge density and the current density become largely independent of each other, and the beam becomes unstable in the relativistic case already for moderate azimuthal numbers, .
We apply our model to two pulsars with sufficiently detailed carousel observations, namely PSRs B0943+10 and B0826-34, identifying the azimuthal number l of the unstable surface modes with the number of sub-beams in a particular pulsar profile. In both cases we have demonstrated that the hollow beam model can account for the observed sub-pulse drift. It is important that since the drift direction is determined by the non-dimensional charge density h (i.e., relative excess of the charge of one sign over another), together with the normalized current density Q, direction reversal of the drift is possible within our model. This is observed in PSR B0826-34. Such an application opens up the possibility to determine details of the polar cap plasma, such as charge and current densities of the particles, within the framework of the diocotron instability model.
However, an exact comparison between the model and the observations is far from being complete. The number of sub-beams observed in a particular pulsar depends sensitively on our line-of-sight. Furthermore, a detailed description of carousels underlying observed drifting sub-pulses is only known for very few pulsars. Empirical models of the emission geometry point to radiation coming from a number of nested cones plus a "core'' component. On the other hand, for more than one cone, the number of discontinuities to match becomes more than two. This results in a higher-order polynomial equation, instead of the quadratic Eq. (31), which is not accessible to analytical treatment but rather requires involvement of numerical methods. This applies, in particular, to our treatment of PSR B0826-10 where E05 suggest that the whole carousel drifts with one speed.
In both applications we find that a strong super-GJ charge density difference is needed for the relativistic beam to nearly corotate with the star. There are no difficulties for the net charge of a pair plasma beam to achieve such densities. This is in contrast with a charge-separated relativistic beam which has a low rotation rate ( and is small in Eq. (24)). This has been noticed before by Fawley et al. (1977), and results from the approximate balance of the repelling electric radial force by the pinching force of the self-magnetic field. Of course, when a relativistic beam is in approximate corotation with the star there is only a small parallel voltage drop deep below the resonance region, close to the stellar surface, and conversely when the beam is not rotating with the star there exists a large parallel voltage drop at low heights which allows for strong slippage of the magnetospheric field with respect to the star.
As mentioned above, we have limited ourselves to purely perpendicular wave modes, which is a textbook approach to diocotron instability of surface waves at the beam boundary (see, e.g., Davidson 1974). On the other hand, if one includes a finite wave number parallel to the vertical direction (as done by Chen & Uhm 1985, in the case of a relativistic pure electron beam), the analysis becomes more complicated, whereas for the application to drifting sub-pulses we expect only a gradual change in the results by extending the geometry of fluted columns to spiralling structures. To justify the use of invariance in the vertical direction in our treatment, we note that at the pulsar surface a feedback exists between the potential structure at and above the stellar surface, the charge density perturbations in the beam and the density perturbations. Even though the particles themselves stream out at approximately the speed of light, the perturbations in potential are left behind, which in turn determine perturbations in the energy of primary particles. The latter might affect the density (multiplicity) of out-flowing pair plasma, causing filamentation of the flow, and therefore modulating the radio emission pattern generated by this plasma.
In the above analysis we have simplified the problem by assuming that the pulsar is aligned, . Since it now appears that the majority of pulsars exhibit the drifting sub-pulse phenomenon, the diocotron instability should be examined in oblique rotators, where the equilibria become time-dependent. Future work on this instability should incorporate this, as well as the other points mentioned above.
Although we have investigated the linear stage of the diocotron instability only, practical applications would correspond to solutions in the non-linear regime. In published work on the non-linear stage of the diocotron instability, the surface waves interact with each other through the perturbed potential, and result in stable vortices (Rosenthal et al. 1987; Davidson et al. 1991; Driscoll & Fine 1990). The diocotron instability modulates and fragments a charged cylindrical beam in the azimuthal direction as a result of initial shear of the plasma angular velocity around the axis. We expect that drifting sub-pulses correspond to such vortices from the non-linear phase of the diocotron instability instead of the linear structures.
The radio emission from pulsars presents more puzzles than the drifting sub-pulse phenomenon, the major one being the mechanism of the radio emission itself. Here we note that Mahajan et al. (1997) have suggested that the diocotron instability can couple non-escaping longitudinal Langmuir waves to escaping electromagnetic waves.
Acknowledgements
We would like to thank Axel Jessner and Jérôme Pétri for stimulating discussions. D.Kh. thanks the Netherlands Research School for Astronomy (NOVA) for support, and the Department of Astrophysics at the Radboud University Nijmegen for hospitality during his stay there.This work has been funded under a joint research project between the Centre for Plasma Physics and Radiation Technology (CPS) and The Netherlands Research School for Astronomy (NOVA).
The coupled fluid-Maxwell's
equations (8)-(13) are linearized and
electrostatic perturbations are applied. Using the
assumption (27), the set of first order
equations becomes:
Equations (A.1)-(A.5) result in one equation for
:
(A.7) | |||
(A.8) | |||
(A.9) | |||
(A.10) | |||
(A.11) | |||
(A.12) |