Re: Counter Strike Source Patch 1.6 1.7 1.8.epub
Casandro Diveley
Plasma-filled pulsar magnetospheres contain thin current sheets wherein the charged particles are accelerated by magnetic reconnections to travel at ultra-relativistic speeds. On the other hand, the plasma frequency of the more regular force-free regions of the magnetosphere rests almost precisely on the upper limit of radio frequencies, with the cyclotron frequency being far higher due to the strong magnetic field. This combination produces a peculiar situation, whereby radio-frequency waves can travel at subluminal speeds without becoming evanescent. The conditions are thus conducive to Čerenkov radiation originating from current sheets, which could plausibly serve as a coherent radio emission mechanism. In this paper we aim to provide a portrait of the relevant processes involved, and show that this mechanism can possibly account for some of the most salient features of the observed radio signals.
Pulsars have been observed to radiate into a wide range of electromagnetic wavebands, going from sub-megahertz up to as high as gamma-ray frequencies. Traditionally, much of the discussion of the plausible emission mechanisms have concentrated on the so-called gaps wherein the electric field along the magnetic field does not vanish. For example, higher frequency incoherent signals are proposed to originate in outer (Cheng et al. 1986), slot (Arons & Scharlemann 1979), polar (Harding et al. 1978) or inner annular (Qiao et al. 2004) gaps. Enabled by sophisticated magnetohydrodynamic or particle-in-cell simulations, some of the more recent studies accounting for a plasma-filled force-free magnetosphere have turned attentions to thin current sheets (CSs; see Fig. 1 for a visual depiction) as alternative or additional sites of particle acceleration (Chen & Beloborodov 2014; Philippov et al. 2015; Melrose & Yuen 2016), and proposed for example, magnetic reconnections in them (Lyubarskii 1996; Uzdensky & Spitkovsky 2014) as the source of high frequency emissions. In this paper, we continue on along this path of exploring the roles played by the CSs, and we examine whether they could also be responsible for the less well understood (see e.g. Melrose & Yuen 2016 for a critical review of the existing models) coherent radio emissions.
An essential characteristic of the CSs (distinguishing them from gaps for example) is that they tend to be very thin, because the zero resistivity force-free regions have a tendency to squeeze them, driving the volume current density and thus charged particle speeds to large values. Outside of the CSs, the plasma frequency ωp for a neutron star magnetosphere is approximately in the tens to hundreds of gigahertz range (see Table 1 for parameter values that go into this estimate), at the top end of radio frequencies. On the other hand, the cyclotron frequency ωc is eight orders of magnitudes larger, instead of being much smaller as in the case with more familiar terrestrial plasma conditions, a fact that invalidates the intuition of the plasma frequency being a cutoff below which the waves become evanescent. Alternatively stated, the radio frequencies occupy a sweet spot of the pulsar magnetospheric environment, with the charged particles having sufficient time to respond to a passing radio wave (just like a simple pendulum will not move much under a driving force oscillating at frequencies higher than its natural frequency) and alter its dispersion relation, yet without being able to efficiently drain energy from it as their motions are severely constrained by the strong magnetic field. Consequently, at radio frequencies, the waves excited by those charged particles in a CS may well have phase speeds smaller than the speeds of the sourcing particles themselves, fulfilling the conditions for Čerenkov radiation. Therefore, the detailed radio emission mechanism we propose is Čerenkov,1 which can robustly produce direct (able to escape the magnetosphere without requiring additional mode conversions) and broad-band (radio emissions expand over ten octaves (Lorimer & Kramer 2012), without resonance peaks) non-thermal coherent radiation. Furthermore, the Čerenkov radiation can be toggled between on and off states in a sharp binary fashion, which is convenient for explaining pulsar nulling. Specifically, if the plasma becomes depleted and the magnetosphere enters into a more strongly charge-separated quasi-stable state, the refractive index would drop towards unity, shutting down Čerenkov radiation abruptly once the wave speeds exceed those of the sourcing particles. Such radio silence would occur concurrently to a starvation of the pulsar wind, explaining why pulsar nulling should be accompanied by a decline in the spindown rate, as is observed by for example, Kramer et al. (2006), even though radio emission only constitutes a very small fraction of the total energy flux.
We present a more quantitative discussion of the CS Čerenkov radiation (CSCR) model in Sect. 2 and have examined its compatibility with salient features of the observed radio signals in Sect. 3, before concluding in Sect. 4. Concretely, we have substituted numerical values into derivations when assessing the relative importance of various terms, and also when comparing with observations. We collect in Table 1 the typical parameter values (in SI units, which are adopted by much of the plasma literature) applicable to normal pulsars. All the parameters and formulae below are presented in the pulsar frame.
Current sheets (CSs) for an aligned rotator, whose cross-sectional shape matches that computed by Gralla et al. (2016). The yellow translucent surface represents the light cylinder, beyond which any co-rotating charged particles will have to travel superluminally and thus cannot exist. The CS outside of it resides on the equatorial plane, and those inside form a separatrix enclosure for the closed field line region. The concentrated currents flowing towards the neutron star at the centre of the figure along these sheets close the magnetosphere circuit (serve as the return current to those flowing out along open field lines), and provide the necessary boundary conditions that partition the open and closed field lines (allow the required discontinuities across the sheets, since open and closed lines have quite different toroidal components, see Eq. (1)).
The Čerenkov radiation is predicated on the charged particles in the CSs moving at very high speeds, which is indeed demonstrated by sophisticated particle-in-cell simulations. For example, Fig. 9 in Cerutti et al. (2015) shows the Lorentz factors achieved for electrons and positrons in the CSs (fed by energies released during magnetic reconnections). The electrons reach average Lorentz factors of (speed of ) in the separatrix CSs (while the positrons reach similar speeds in the equatorial CS outside of the light cylinder), with their distribution (Fig. 10 of Cerutti et al. 2015) exhibiting a tail going up to as high as 3000 or .
Intuitively, such extreme values are not surprising, as ultra-relativistic speeds are consistent with, as well as contribute positively to the maintenance of quasi-steady (stable in the direction along the magnetic field) thin CSs. The most obvious fact regarding consistency emerges when we recall that the purpose for the existence of the CSs is to divide two very different magnetic regimes. Specifically, we have that in the geometrized units (Zhang 2017)(1)
where (o) and (c) stand for the open and closed field line regions respectively, and μ is the magnetic dipole moment. The quantity is the normalized radius so on the stellar surface, and is the location angle against the magnetic axis (coinciding with the spin axis for our aligned rotator). Such an abrupt disparity means the derivative of the toroidal magnetic field across a very thin CS would be extremely large, requiring a large poloidal current density (thus high charged particle speeds, but also a high charge density when the speeds saturate at close to the speed of light) to sustain. This intuition can be (crudely) quantified by evoking the equilibrium Harris CS (Harris 1962), extended to the relativistic case by Hoh (1966) and extensively utilized in later pulsar CS studies such as Ptri (2013). The central prediction is that the magnetic field variation across a CS satisfies(2)
Regarding the maintenance of a quasi-steady-state, we note that the ideal infinite conductivity condition in the force-free regions of the magnetosphere would try to squeeze the CSs to become infinitesimally thin, forcing to diverge, leading to a large Reynolds number and instabilities, unless a counteracting dissipative process exists to introduce an effective resistivity that keeps vCS close to some finite equilibrium value. We posit that the Čerenkov radiation reaction constitute an ideal candidate mechanism to accomplish this task, which is then somewhat similar in intended functionality to dissipation through unbridled particle acceleration often assumed for the gaps (Melrose & Yuen 2016), but which turns out to be inefficient because violent instabilities still develop that force the gaps to become temporally highly variable (Levinson et al. 2005). In contrast, the Čerenkov radiation reaction is more genuinely dissipative (excess energy propagates away as waves; alternative kinetic mechanisms such as traditional dissipation by collisions tends to randomize but not remove the energy, thus instabilities may take on a different form but persist), and should result in the CSs being more stable.
In this section we examine wave propagation in the more regular parts of the magnetosphere, through which the waves sourced by the CSs must traverse. Many extensive studies on such waves exist in literature, assuming different magnetospheric conditions. For convenience though, we carried out an ab initio computation in the next subsection so we can extract intermediate results that are useful when examining whether the Čerenkov radiation conditions are satisfied. We note however, the fact that subluminal waves exist is not a new result, nor is it sensitive to the assumptions we adopt in this work.
b1e95dc632