Scintillation-Induced Circular Polarization in Pulsars and Quasars J.-P. Macquart1and D.B. Melrose2 Research Centre for Theoretical Astrophysics School of Physics, University of Sydney, NSW 2006, Australia. firstname.lastname@example.org,email@example.com ABSTRACT We present a physical interpretation for the generation of circular polarization resulting from the propagation of radiation through a magnetized plasma in terms of a rotation measure gradient, or ‘Faraday wedges’. Criteria for the observability of scintillation-induced circular polarization are identified. Application of the theory to the circular polarization in pulsars and compact extragalactic sources is discussed. Subject headings: Faraday rotation — polarization — turbulence — pulsars: general — galaxies: magnetic fields 1. Introduction The circular polarization (CP) of radio emission observed from pulsars (Manchester, Taylor & Huguenin 1975; Radhakrishnan & Rankin 1990; Han et al 1999) and from some quasars (Roberts et al 1975, Weiler & de Pater 1983; Saikia & Salter 1988) is not understood. In both cases, simple theory suggests that any polarization should be linear, determined by the direction of the projection of the magnetic field in the source region on the plane of the sky. In both cases, intrinsic CP is expected as a correction to first order in an expansion in the inverse of the Lorentz factor of the radiating particles, but it does not seem possible to account for the observations in terms of CP intrinsic to the emission process (Radhakrishnan & Rankin 1990; Radhakrishnan 1992; Saikia & Salter 1988). Another possibility is that the CP is imposed as a propagation effect due to the partial conversion of linear polarization into CP resulting from the ellipticity of the natural modes of the medium (Sazonov 1969; Pacholczyk 1973; Jones & O’Dell 1977a,b). However, this also fails to provide a satisfactory explanation for the properties of the observed CP. In this paper we describe a different propagation effect that can lead to CP for a source, independent of its degree of linear polarization. We refer to this as scintillation-induced CP. Although the observed CPs from pulsars and quasars are quite different, we suggest that both might be due to scintillation-induced CP, with the most obvious differences being due to pulsars scintillating in the diffractive regime and quasars scintillating in the refractive regime. In the formal theory of scattering in a turbulent, magnetized plasma (Melrose 1993a,b) there are differences in the scattering in the two oppositely circularly polarized wave modes of the
– 2 – medium due to their different refractive indices. Most of the terms contributing to the CP are too small to be of interest for scattering in the interstellar medium (ISM). However, we have recently shown (Macquart and Melrose 2000; hereinafter paper I ) that there is one much larger contribution to the CP that is a possible candidate for explaining the CP in pulsars and quasars. The CP identified in paper I is due to a nonzero variance in Stokes V , and the formal theory implies that the dominant contribution is of the form hV2i ∼ DV V(rref)hIi2, where DV V(r) is a phase structure function associated with the relative phase, φV, between the components in the opposite CPs. The distance rref= r2 of the Fresnel scale rF= (Lλ/2π)1/2, where L is the distance between the scattering screen and the observer’s plane and λ is the wavelength, and the diffractive scale rdiff, which is defined by writing the phase structure function in the forms D(r) = (r/rdiff)β−2for a power-law spectrum of turbulence. Our purposes in this paper are threefold: first, to provide a physical explanation for the mechanism that leads to this term, second, to use this interpretation to relax some of the restrictive assumptions made in paper I to obtain a more general semiquantitative expression for the predicted CP, and, third, to explore the suggested application to the observed CP in pulsars and extragalactic sources. F/rdiffis the refractive scale, which is defined in terms We start (section 2) by including birefringence in a simple model for strong scattering in which scattering is attributed to a large number of coherent patches of size ∼ rdiffwithin an envelope of size ∼ rrefon the scattering screen (e.g., Goodman & Narayan 1989, Narayan 1992, Gwinn et al 1998). When the birefringence is included, this model reproduces the result hV2i ∼ DV V(rref)hIi2 derived from the formal theory of scattering in a magnetized plasma in paper I. This simple model corresponds to strong diffractive scintillation, suggesting that our expression for hV2i applies only to a source that exhibits strong diffractive scintillations (which is the case for pulsars but not for quasars). Further physical interpretation of scintillation-induced CP is developed in section 3, where it is argued that it arises from a combination of two processes: a rippling of the wavefront, as in the conventional theory for scattering in a turbulent medium, combined with random refractions in the birefringent medium that cause a separation in the rays associated with the opposite CPs. This leads to the following interpretation: the ripples in the wavefront for the two CPs become spatially separated due to the random birefringent refractions, so that they do not overlap in the observer’s plane. This leads to alternate patches in which one CP and then the other dominates, leading to a nonzero hV2i. This interpretation is the basis for two important generalizations that we propose here. First, an essential requirement in this interpretation is that the images in the two CPs be displaced relative to each other by ∆x in the observer’s plane. Our interpretation of the formal theory is that this is due to random birefringent scatterings at the putative scattering screen. However, any mechanism that causes such a displacement of the rays corresponding to the two opposite CPs leads to a nonzero ∆x and hence to a nonzero hV2i. In particular, an alternative to random birefringent refractions is birefringent refraction at a single structure, which we refer to as a Faraday wedge, cf. section 4. Second, although the result derived from the formal theory applies for strong diffractive scintillations, our interpretation implies that a nonzero hV2i results for any source that exhibits scintillations. Consider a source with a scintillation index mIon a
– 3 – spatial scale rscintin the observer’s plane; it should exhibit fluctuations in the degree of CP, mV, that is of order ∆x/rscinttimes mI. In particular, a source that exhibits refractive scintillations should exhibit fluctuations in CP on a similar timescale with an amplitude smaller by the factor ∆x/rscint. For scintillation-induced CP to account for the observed CP in quasars and pulsars, two conditions need to be satisfied. First, the predicted degree of CP, mV, must be in the observed range, which appears to be ∼ 1 for a few pulsars and is∼> 10−3for relevant extragalactic sources. Second, the predicted timescale for the fluctuations in CP must be consistent with the observed variations in the CP. For pulsars this is the diffractive timescale, and for quasars it is the refractive timescale. These requirements are discussed for pulsars in section 5, and for extragalactic sources in section 6. 2. Strong Scattering in a Birefringent Medium In this section we show that the main result found in paper I is reproduced by a simple model for the scattering. This result is hV2i ∼ DV V(rref)hIi2, which was derived in Paper I for scattering at a single screen for a power-law spectrum of turbulence in a uniform magnetic field. The assumption that the fluctuations are only in the density implies that DV V(r) is proportional to the phase structure function, D(r), for the phase in the absence of birefringence. Assuming a power-law distribution for the turbulence, this implies (2-1) ?β−2 r ? DV V(r) = α2 (2-2) , rdiff with β = 11/3 for a Kolmogorov spectrum of turbulence, and with α = Y cosθ, Y = νB/ν, where νBis the electron cyclotron frequency and θ is the angle between the magnetic field and line of sight. The model for strong scattering that we introduce to interpret the result (2-1) involves regarding the amplitude of the wave at the observer’s screen as a sum of terms that each correspond to a point of stationary phase (e.g., Born & Wolf 1965, Goodman 1985, Gwinn et al 1998). The model is illustrated in Figure 1. There are two relevant contributions to the phase: a rippling on a scale rdiffdue to the postulated density fluctuations at the screen, and a geometric effect described by a curvature of the mean wavefront on a scale rF, with the points of stationary phase corresponding to points where the tangent to the wavefront is orthogonal to the line of sight, cf. Figure 1. Let there be N ∼ (rF/rdiff)4? 1 such points of stationary phase (e.g. Narayan 1992). The amplitude in this model is of the form N u = (2-3) Ajeiφj, X j=1
– 4 – where the Ajand φjare the amplitude and phase, respectively, associated with the jth coherent patch (the jth point of stationary phase). The φj are assumed to be a random set of phases uniformly distributed in the interval [0,2π]. For our purposes it suffices to assume that all N amplitudes are identical, writing Aj= A for all j. The mean is defined by averaging over the phases. The intensity is N * + hIi = huu∗i = A2ei(φj−φk) = NA2, (2-4) X j,k=1 and its variance is N * + hI2i = huuu∗u∗i = A4ei(φj+φk−φl−φm) = N(2N − 1)A4, (2-5) X j,k,l,m=1 For N ? 1 these imply hI2i = 2hIi2, which is the well-known result for diffractive scintillations. To include the polarization the wave amplitude, u, is separated into its right-hand, u+, and left-hand, u−, circularly polarized components, and the intensities in the two CPs are identified as Iσ= uσu∗ σwith σ = ±. The relevant Stokes parameters are given by I =1 2(I++ I−), V =1 2(I+− I−). (2-6) The difference in refractive index between wave in the two CPs leads to a phase difference between them, denoted σφV. Let σφV jbe the additional phase associated with the jth coherent patch. This assumption corresponds to N u = uσ= (2-7) Ajei(φj+σφV j), X X uσ, σ=± j=1 The mean CP is then N 1 2 * ei(φi+φV i−φj−φV j)− ei(φi−φV i−φj+φV j)i+ h hV i = A2 (2-8) X . i,j=1 We assume that the change in phase due to the birefringence at the screen is sufficiently small that the approximation ei(φj+σφV j)= (1 + iσφV j−1 applies. On substituting (2-9) into (2-8), an average over the random phases is nonzero only for i = j and the two terms in (2-8) cancel for i = j, giving hV i = 0. The variance in V is given by V j)eiφj (2-9) 2φ2 hV2i = hI2 ++ I2 −− 2I+I−i, (2-10)
– 5 – which becomes N * ?+ hV2i =A4 ?X ei(φi+φj−φk−φl) eiσ(φV i+φV j−φV k−φV l)− 2ei(φV i−φV j−φV k+φV l) , (2-11) X 4 σ=± i,j,k,l=1 On expanding as in (2-9), the only nonvanishing terms are for i = l and j = k. There are N2such terms, which give hV2i = N2A4DV V(rij), DV V(rij) = h(φV i− φV j)2i, (2-12) where DV V(rij) is the phase structure function for φV. The distance rijis the spatial separation of the ith and jth coherent patches, which is typically of order rref. With NA2= hIi according to (2-4), the result (2-12) with rij→ rrefreproduces the functional form of the relation (2-1). The numerical coefficient in the relation (2-1) can be evaluated for a power-law spectrum of the turbulence, as given by (2-2). A calculation given in Appendix A leads to a coefficient, g(β), of order unity. Thus (2-12) with (2-4) implies hV2i = g(β)DV V(rref)hIi2, (2-13) with DV V(r) given by (2-2). The expressions (2-12) or (2-13) are valid only for hV2i ? hIi2; this follows from the expansion made in (2-9) and a related expansion for DV V(r) ? 1 made in paper I. However, there is no reason in principle why this condition must be satisfied. If it is invalid, then the fluctuations in V can approach the maximum possible value, hV2i ∼ hIi2. This simple model for hV2i suggests the following interpretation for the result (2-1). The assumptions made in the calculation in paper I apply to a source that, in the absence of birefringence, exhibits strong diffractive scintillations. The inclusion of birefringence leads to random birefringent refractions that cause the two oppositely circularly polarized rays to emerge from the scattering screen propagating in slightly different directions. This angular separation between the rays implies that the peaks and troughs in the wavefronts in the opposite CPs are displaced from each other in the observer’s plane. This separation corresponds to alternating patches of opposite CPs, which is described by hV2i. 3. Random Birefringent Refractions In this section we discuss the interpretation of the function DV V(rref) in (2-1) in terms of random birefringent refractions. We then argue that this interpretation allows one to relax two restrictive assumptions made in paper I to obtain a more general, semiquantitative expression for the predicted scintillation-induced CP. Random refractions in an isotropic medium lead to an angular (‘scatter’) broadening of the source. In scattering theory this is described in terms of a bundle of rays diffusing in angle of
– 6 – propagation. For our purposes here a semiquantitative discussion suffices. In refractive scattering, the screen acts like a large lens, of size ∼ rref, that causes a phase change of order δφ ∼ [D(rref)]1/2. The phase change includes a tilt of the wavefront that corresponds to a change in the ray direction. The typical change in the ray angle is δψ ∼ (λ/2π)δφ/rref. When the birefringence is included, there is an additional phase change between the two wave modes, and this leads to a separation of the ray angles for the opposite CPs. The relative phase change between the wavefronts for the two CPs is of order δφV ∼ [DV V(rref)]1/2, and the angular separation between the rays in the two CPs is ∆ψV ∼ (λ/2π)δφV/rref. This angular separation times the distance, L, to the screen leads to a spatial displacement of the images in the two CPs of ∆x ∼ ∆ψVL, which implies DV V(rref) ∼(∆x)2 (3-1) . r2 diff The expression for hV2i obtained in paper I, and rederived in (2-13) above, then implies hV2i ∼(∆x)2 (3-2) hIi2. r2 diff We interpret the form (3-2) as follows. According to the model in section 2, the source is exhibiting diffractive scintillations that correspond to a rippling of the wavefront such that there are patches of constructive and destructive interference with a characteristic size rdiff. The birefringence causes a relative spatial displacement by ∆x of this pattern in the opposite CPs. For ∆x∼< rdiff, the partial separation of the images in the opposite CPs leads to an image with a degree of CP ∼ ∆x/rdiff. The foregoing discussion shows that two ingredients are essential for scintillation-induced CP: scintillations that produce a pattern of variations with a scale size rscintin the observer’s plane, and birefringent refraction that causes a displacement, ∆x, between the images in the opposite CPs. The resulting degree of CP is then given by ∆x rscint mI=[h(I2i − hIi2]1/2 mV =hV2i1/2 (3-3) mV ∼ mI, , . hIi hIi With the assumptions made in the models discussed above, the scintillations are diffractive, mI∼ 1, rscint= rdiff, and ∆x is due to random birefringent refractions. More generally, (3-2) should apply with mI, rscintcorresponding to the observed scintillations, and ∆x could be due to birefringent refraction either by a single structure (Faraday wedge) or by a spectrum of turbulence. For example, for refractive scintillations one has mV = (∆x/rref)mref, where mrefis the intensity modulation due to refractive scintillation. 4. The Faraday Wedge The random birefringent refractions that contribute to scintillation-induced CP through the factor DV V(rref) in (2-1) are due to gradients in rotation measure (RM) associated with
– 7 – the turbulence at the scattering screen. Scintillation-induced CP requires that the wavefront be rippled and that the ripples in the opposite CPs be displaced laterally along the wavefront, but it is not necessary that these two features be imposed at a single scattering screen. In this section we introduce the concept of a Faraday wedge in which the lateral displacement along the wavefront of the ripples in the opposite CPs is attributed to birefringent refraction at a single structure along the line of sight. We consider two idealized models for the wedge; in the first the wedge is modeled as a birefringent region with a density gradient and in the second the wedge is modeled as a prism with sharp edges. These two models lead to essentially the same semiquantitative result for the angular separation of the rays corresponding to the two CPs. 4.1. Refraction in a Birefringent Medium It is useful to view the effects of a Faraday wedge in terms of both geometric optics (rays) and physical optics (wavefronts). In terms of geometric optics, as illustrated in Figure 2, a Faraday wedge splits an incident ray into two oppositely circularly polarized emerging rays propagating in slightly different directions, separated by an angle ∆ζ say. After propagating the distance L from the wedge to the observer’s plane, the left- and right-hand images are separated by ∆x = ∆ζ L. For ∆ζ L∼> rscintan observer sees well-separated oppositely circularly polarized images of the source for a scintillating source. In terms of physical optics, as illustrated in Figure 3, the Faraday wedge causes an incident wavefront to split into two wavefronts whose normals are in slightly different directions, separated by ∆ζ. Each wavefront is rippled, and the effect of the Faraday wedge applies to the average (over the ripples) wavefronts. (One wavefront is also slightly delayed relative to the other, but this effect is neglected here.) At the observer’s plane, corresponding ripples on the two wavefronts are displaced along the wavefront by ∆x = ∆ζ L. The observer sees a net CP varying on the timescale associated with the scintillations as the whole pattern sweeps across the observer’s plane. The following estimate of the angle ∆ζ applies when there is a smooth gradient in RM. Consider a planar wavefront propagating parallel to the z-axis incident upon a slab of material of thickness ∆z, containing spatial variations in the plane transverse to the z-axis. As shown in Appendix A, using the paraxial approximation, the angular deviation of the ray in terms of the refractive index variations on the scattering screen is determined by d(nσκ) dz =∂nσ (4-1) ∂r, where σ = ± denotes the two wave modes, r = (x,y) is the plane orthogonal to the z-axis, κ = k/k is the direction of the wavevector and the arguments of the refractive index nσ(ω,κ,z,r) are suppressed. In the weak anisotropy limit one writes the refractive index in terms of polarization dependent and independent terms nσ= n + σ∆n/2 with ∆n = n+− n−. For a screen a distance L from the observer, the relative displacement of the centers of the images in the two modes on
– 8 – the observer’s plane is L∆ζ with ∆ζ = −∆n Z∆z Z∆z ∂ ∂r dz n +∂ dz ∆n. (4-2) ∂r n 0 0 The refractive indices depend on the magnetoionic parameters, X = (νp/ν)2, Y = νB/ν, where νp is the plasma frequency and νBis the electron cyclotron frequency. One has X ? 1, Y ? 1 here, with n = 1 −1 B. Then dominant contribution to ∆ζ is due to the gradient of where ∆RM is the contribution of the path length ∆z to the RM. We also introduce the RM phase, φRM= λ2RM, which is the relative phase difference between the two wavefronts. The displacement between the left- and right-circularly polarized images on the observer’s screen reduces to (further details are given in Appendix B) 2X, ∆n = XY cosθ, where θ is the angle between the wave-normal direction and R∆z dz XY cosθ = ∆RMλ3/2π, 0 ∆x(r) =Lλ (4-3) 2π∇⊥φRM, where ∇⊥denotes the gradient in the x-y plane. 4.2. Refraction at a Faraday prism An alternative model for a Faraday wedge is a prism. Let the prism have an apex angle of ψ and let its refractive index be n − n0greater than the refractive index (n0) in the surroundings. For simplicity, suppose that the prism is oriented nearly perpendicular to the line of sight, so that we are interested only rays at small angles relative to the direction perpendicular to the axis of the prism, cf. Figure 3. Then a ray incident at an angle θinemerges at an angle θoutgiven by θout= θin− 2(n − n0) tan(ψ/2). (4-4) The mean angular deviation, ∆θ = θout− θin, of a ray at the prism and the difference, ∆ζ, between the emerging ray in the two CPs are then given by ∆θ = 2(n − n0) tan(ψ/2), ∆ζ = 2∆n tan(ψ/2), (4-5) respectively. Let us compare the results (4-2) and (4-5). Retaining only the final term in (4-2), assuming the region to be uniform along the z direction and with a uniform gradient with a scale length L⊥in the perpendicular direction, (4-2) gives δ ζ = ∆n∆z/L⊥. Thus the two results coincide for 2tan(ψ/2) = ∆z/L⊥. More generally one can conclude that a Faraday wedge leads to a deviation of the rays with opposite CPs by an angle ∆ζ which is of order the difference, ∆n, in the refractive indices of the two modes within the wedge, times a geometric factor (∆z/L⊥or 2tan(ψ/2)) which is less than or of order unity.
– 9 – 4.3. Requirements on the Model A Faraday wedge leads to CP due to a displacement, ∆x, between ripples in the wavefronts corresponding to opposite CPs. Suppose that the ripples have a scale length rscintand that the modulation index, mI, of the intensity due to these ripples. Then one should observe fluctuations in CP with an rms degree of CP of (∆x/rscint 1 for ∆x ? rscint, for ∆x∼> rscint, Vrms I (4-6) ∼ mI on a characteristic timescale rscint/v, where v is the speed at which the pattern moves across the observer’s plane. For diffractive scattering one has rscint ∼ rdiff and mI ∼ 1, and for refractive scattering one has rscint∼ rrefand mI∼ (rdiff/rref)(β−4)/2, with (β − 4)/2 = −1/6 for a Kolmogorov spectrum β = 11/3. For a source of angular size θsto exhibit scintillations requires θ < rdiff/L and θ < rref/L for diffractive and refractive scattering, respectively. A semi-quantitative estimate for when these conditions are satisfied is deduced from equation (3-3) as follows. With φRM= RMλ2, we require ∆x∼> rdiff for diffractive scintillations and ∆x∼> rreffor refractive scintillations. Hence we require If the angular deviation, ∆ζ, between the rays in the two modes is dominated by a single structure (single Faraday wedge), which contributes ∆RM to the total RM, then one has ∇⊥φRM∼ ∆RMλ2/L⊥, where L⊥is a distance that characterizes the gradient in RM. The condition (4-7) includes the requirement ∆RMλ2/2πL⊥∼> θs. Assuming that the Faraday wedge has a density neand a magnetic induction B, ∆RM is proportional to neB times the line of sight distance, ∆z through the structure. Then the condition (4-7) reduces to RMλ3/2πL⊥∼> θs, and on inserting numerical values, this becomes rdiff L rref L∼> θs diffractive, ∼> θs ∇⊥RMλ3 2π (4-7) ∼> refractive. ??∆z ne ? ?B θs ? ? ? ? ∼> 109 (4-8) λ3 . 1G 1mas 1m−3 L⊥ Physically, the dependence on ∆z/L⊥is due to the requirement that refraction cause a large enough separation between the rays in the two CPs. The region acts like a prism with a small angle at its apex; the deviation of the ray is zero when this angle is zero (the prism reduces to a slab), and the deviation increases as this angle increases. The condition (4-8) is not easily satisfied: it requires an exceptionally dense, strongly magnetized region along the line of sight, and it is more easily satisfied at longer wavelengths.
– 10 – 5. Application to Pulsars The degree of CP observed in most pulsars is relatively small (e.g., Han et al 1999). However, the degree of CP usually quoted refers to the pulse-averaged quantity. Studies of single pulses are possible for a small subset of pulsars, and the CP in single pulses can be very much greater than the pulse-averaged CP (Manchester, Taylor & Huguenin 1975). Thus the pulse-averaged CP appears to be the mean value of a CP that can vary greatly on a timescale of order the pulse period. There is no satisfactory explanation for the pulse-averaged CP (e.g., Radhakrishnan & Rankin 1990; Kazbegi, Machabeli & Melikidze 1991; Radhakrishnan 1992). It is usually assumed this is imposed either by the emission process or by propagation in the pulsar magnetosphere. In this section we discuss the possibility that this rapidly varying component in the CP might be due to the Faraday wedge effect associated with diffractive scintillations induced by propagation through the ISM. 5.1. The Vela pulsar We apply the foregoing ideas to the Vela pulsar for which there is direct evidence for RM fluctuations. Three parameters are required to estimate the magnitude of the CP generated by the Faraday wedge: the magnitude of the RM gradient, the slope of the power spectrum of density inhomogeneities, β, and the refractive scale, rref. We list our best estimates of these parameters, and then use these to estimate the degree of CP expected according to the foregoing theory. Hamilton, Hall & Costa (1985) reported a linear change in RM across the Vela pulsar for the interval 1970-1985. The best fit to the RM gradient is 0.73 rad/m2/yr, which translates into a spatial RM gradient of 2.3 ×10−11v−1 to the line of sight in kms−1. We assume the Kolmogorov value of β = 11/3 for the power spectrum of density inhomogeneities for this pulsar. This is consistent with some measurements, although we note that the measurements of refractive flux variations are more consistent with β closer to 3.9 (Johnston, Nicastro & Koribalski 1998). The size of the scattering disk, as measured by Gwinn (1997), is ≈ 1.0 AU at around 2.3 GHz. (The actual measurement band was from 2.273 to 2.801 GHz). The refractive length, rref, scales as ν−[1+2/(β−2)]. km/sradm−3, where vkm/sis the speed of the wedge transverse In estimating the Faraday rotation phase change across the scattering disk, we choose an observing frequency of 600 MHz, where the effect of Faraday rotation is strong. Scaling the refractive length to this frequency, one has rref≈ 19AU. Combining this with our estimate of the RM gradient, the total RM change across the scattering disk, rref∇⊥φRM, is 15.9v−1 calculate the root-mean-square degree of CP we use this estimate of rrefin equation (3-3) for diffractive scintillation, giving phV2i(z)/hI2i(0) = 15%(4%), assuming that vkm/sis 100 (500). We conclude that measurement of scintillation-induced CP is feasible for the Vela pulsar, and that it is likely to be feasible for other pulsars at low frequency. Although not discussed in km/s. To
– 11 – detail here, there are several other pulsars known to exhibit variability in RM, including the Crab pulsar (Rankin et al. 1988) and PSR 1259-63 (Johnston et al 1986 ), which exhibits RM variations of ∼ 200 radm−2on timescales of 0.5 hr. An observational test of the scintillation-induced CP model requires statistics on the variance in the CP in individual pulses, and the way this variance changes with frequency of observation. 6. Application to Extragalactic Sources A small but significant degree of CP (∼< 0.1% to a few %) is observed in some compact extragalactic radio sources (e.g., Roberts et al 1975, Weiler & de Pater 1982, de Pater & Weiler 1982, Komesaroff et al 1984). The suggested interpretations include the intrinsic polarization associated with synchrotron radiation (Legg & Westfold 1968), and partial conversion of linear into circular polarization due to ellipticity of the natural wave modes of the cold background plasma (Pacholczyk 1973) or of the relativistic electron gas itself (e.g., Sazonov 1969, Jones & O’Dell 1977a,b). However none of these suggested interpretations has proved satisfactory in accounting for (a) the frequency dependence, (b) the temporal variations, and (c) the magnitude of the observed CP. Here we explore the possibility that the CP observed in compact extragalactic sources is scintillation-induced CP. We consider the requirements for scintillations to produce 0.1% CP due to scintillation either in our Galaxy or in the host object. 6.1. Extragalactic sources Scintillation-induced CP relies on birefringent refraction, which is determined by the gradient in RM. To estimate the degree of CP using equation (3-3) requires an estimate of the parameter ∆x, which is determined by (3-1) with (2-2). Our estimate for the CP then reduces to s hV2i(z) hI2i(0)∼ α ?β−2 ?rF rdiff (6-1) . where we ignore an observed slight excess in the relative fluctuations in RM in our Galaxy compared with the relative fluctuations in the electron density (Minter & Spangler 1996). We estimate the degree of CP at 5 GHz. For a screen at distance of L = 1kpc, the Fresnel scale is rF= pλL/2π = 5 × 108m. The length scale rdiffis estimated from the frequency, νt, at to ν2/(β−2). We assume a Kolmogorov spectrum of density inhomogeneities, β = 11/3. For an object located off the Galactic plane one typically has νt≈ 7GHz (Walker 1998), yielding rdiff≈ 3 × 108m at 5 GHz. Taking hBGcosθi = 3µG one has α ≈ 2.5 × 10−9which yields Vrms= 6 × 10−9hI2i1/2. Hence, we conclude that the contribution of Galactic RM fluctuations to the CP is negligible. which the scattering becomes strong, at rdiff= rF, and from the fact that rdiffscales proportional
– 12 – Next consider RM variations internal to an extragalactic source. The theory (developed in Paper I) needs to be modified slightly if the Faraday wedge is assumed to be located in the host object because the non-planarity of the wavefront cannot be ignored. The modifications for a spherical wavefront involve making the substitutions (Goodman & Narayan 1989) z1z2 z1+ z2, r → where z1 is the distance from the source to the scattering screen and z2 the distance from the screen to the observer. Using equations (4.9) in (5.21) in Paper I, this implies hV2i1/2≈ 2−1/2(z1∇⊥λ2RM/(2πrdiff/λ))(β−2)/2hI2i1/2. For the sake of discussion we choose the specific object 3C 345 for which there is an estimated dependence (Matveenko et al 1996) of RM = 3500(R/3.79h−1pc)−3rad/m2on radial distance R from the core, where R = 3.79h−1pc corresponds to an angular size of 1 mas. Consider the RM gradient required to produce 0.1% CP at 5 GHz. Since the scattering is likely to be concentrated around the AGN core we take an effective distance of z1= 100pc between the source and screen and rdiff= 107m. One then requires ∇⊥RM ≈ 3.6×10−11rad/m3. We conclude that variations in the large-scale RM are large enough to produce this degree of CP within the central 0.85h−3/4pc of the core. (6-2) z → z1 (6-3) z1+ z2r, The distance between the source and the screen is assumed to be z1= 100 pc. There is no direct evidence for this parameter and our choice is based on a plausibility argument in view of the known scattering parameters of Sgr A*. Sgr A* is an AGN-like object at our own Galactic center which appears to exhibit refractive scintillation (e.g., Zhao & Goss 1993), presumably due to turbulence in a medium ∼ 100 pc from the source (Backer 1978). Our plausibility argument is simply that compact extragalactic sources are likely to have some similarities to Sgr A*, and that it is plausible that they scintillate due to turbulence in the surrounding interstellar medium ∼ 100 pc from the source. The foregoing estimates apply when the source exhibits diffractive scintillation. CP can also result from refractive scintillation, for which the source size requirement is less stringent. However, the requirement on the separation of the two senses of CP is more stringent: the Faraday wedge must cause a separation that is a significant fraction of the scale on which refractive flux variations occur. Thus, one requires (cf. equation (3-3)) hV2i1/2≈ mrefrdiffλ2∇⊥RM, where mref≈ (rdiff/rF)2−β/2is the refractive modulation index. For example, the RM gradient in 3C 345 used above is sufficiently large that a centrally-located object small enough to exhibit refractive scintillation would produce > 0.1% CP at 5 GHz provided the RM gradient is greater than 3 × 10−8rad/m3. According to the RM model used above, such gradients are encountered within the central 0.15h−3/4pc of the core. Similar data available on large RM gradients on milliarcsecond scales near the cores of other extragalactic objects (Roberts et al 1990, Udomprasert et al 1997, Cawthorne et al. 1997, Taylor
– 13 – 1998) suggests that CP due to this effect is possible for a broad class of extragalactic sources. The timescale of variability in the CP depends on the relative speed, vs, of the screen and the source and on the relative speed, voof the observer and the screen. The speed at which the pattern moves across the observer’s plane is vs(z1+ z2)/z1, and the observer moves across this plane at vo. The pattern scale at the observer is rdiff(z1+ z2)/z1for diffractive scintillation and rref(z1+ z2)/z1for refractive scintillation. The timescale for variations in V is determined by the pattern scale of the oppositely circularly polarized regions divided by the relevant speed. For a point source, diffractive scintillation would be observed on a timescale rdiff/vs, and refractive scintillation would be observed on a timescale rref/vs, where it is assumed that the relevant speed is that of the screen relative to the source. This is the case for z1? z2, since the apparent speed of the scattering material across the line of sight dominates the motion of the observer across the pattern. Taking vs= 100km/s with the parameters assumed above, the timescale of diffractive variations is 102s and the refractive timescale is 3 × 104s. Based on these estimates, diffractive variations should exhibit reversals in sign of the CP on a timescale of a few minutes, which does not explain the observed variability over hours to days (Komesaroff et al 1984). Although the refractive timescale is closer to observed timescale of CP variations in extragalactic sources, evidence for reversals in the sense of the CP on this timescale is lacking. 7. Conclusions It was shown in Paper I that propagation of radio waves through a magnetized inhomogeneous plasma leads to fluctuations in circular polarization (CP), even if the scintillating source is itself unpolarized. Although the mean CP induced by propagation through the medium is zero, the variance, hV2i, is nonzero. The generation of this CP is attributed to a gradient in RM, called a Faraday wedge, leading to a lateral displacement at the observer’s plane of the wavefronts for opposite CPs. Inhomogeneities in the ISM introduce corrugations into the wavefront, and the displacement of these at the observer’s plane leads to alternate regions of opposite CP. As the ISM moves across the line of sight to a source, an observer samples these alternate patches of opposite CPs. The mean CP is zero, and the variance is nonzero. The effect is significant provided that the lateral displacement of the corrugations in the wavefronts for the opposite CPs is a significant fraction of (or larger than) the typical size of the corrugations. This scintillation-induced CP varies on the timescale associated with the scintillations. For a source undergoing diffractive scintillations, the CP varies on the diffractive timescale, and for a source undergoing refractive scintillations, the CP varies on the refractive timescale. Pulsars exhibit both diffractive and refractive scintillations, and one expects them to exhibit scintillation-induced CP on a timescale associated with the diffractive scintillation. The observed
– 14 – CP in pulsars can be several tens of percent, but it may be that some of this CP results from birefringence in the pulsar magnetosphere itself, which we do not consider here. Based on data on RM variations associated with the Vela pulsar we predict that scintillation-induced CP is likely to be observable. We suggest here that the pulse to pulse variation in CP observed in some pulsars is due to this effect. There is a strong case that the small (∼ 0.5−0.05%) degree of CP exhibited by some compact extragalactic sources is due to scintillation-induced CP. The high RMs (e.g., Udomprasert et al 1997) and proposed strong magnetic fields (Rees 1987) near the cores of AGN are sufficient to generate the observed degree of CP in association with refractive scintillation. A prediction is that scintillation-induced CP should reverse sign randomly, such that the average value is zero. There is some evidence for this for pulsars, for which the pulse-averaged CP is small compared with the relatively high CP in individual pulses. However, there is no strong evidence for reversals in the sense of CP for extragalactic sources. To determine whether or not this is compatible with the theory requires further consideration of the nature of RM fluctuations. One obvious point concerns the distinction between mean RMs and the variance of the RM. If the length scale over which the magnetic field changes orientation is short compared to the path length through the medium, the variance in the difference between the phases of the left- and right-hand polarized wavefronts may be very much larger than the mean value. This point is exemplified by studies of the ISM which seek to relate the mean electron density along a line of sight to its scattering properties. Based on the variation of hnei observed in the Galaxy, h(δne)2i is predicted to vary by 1.3 orders of magnitude, however the observed variation is considerably larger, at 4.2 orders of magnitude (Cordes, Weisberg & Borkiakoff 1985). For the same reason, the mean RM along a given line of sight may not reflect the degree of scintillation-induced CP expected. The generation of CP due to ultra fine scale structure in the magnetic field also needs to be addressed. If the observed density fluctuations in the ISM are the result of magnetohydrodynamic turbulence, one expects similarly fine scale structure in the magnetic field. This also is expected to boost the importance of scintillation-induced CP in the ISM. Further detailed modeling of this effect is warranted. We thank Mark Walker for suggesting the effect of RM gradients and Ron Ekers for helpful discussions relating to extragalactic sources. Appendix: Evaluation of g(β) A. We wish to evaluate the integral ?β−2 ?|xi− xj| rdiff Z hD(rij)i = d2xid2xjp(xi)p(xj)
– 15 – 1 ?β−2Z ? = dxidxjdyidyjp(xi)p(xj)p(yi)p(yj) rdiff (xi− xj)2+ (yi− yj)2i(β−2)/2. h (A1) × Using the expansion ∞ X i−1 Y xγ−iyi, (x + y)γ= (γ − j) (A2) i=0 j=0 equation (A1) becomes a series of products of 2-dimensional integrals: ∞ X m−1 k=0 1 ?β − 2 ?!Z ?β−2 dyidyjp(yi)p(yj)(yi− yj)2m. ? dxidxjp(xi)p(xj)(xi− xj)β−2−2m hD(rij)i = Y − k 2 rdiff m=0 Z (A3) × An observer sees speckles over the entire scattering disk of radius rref, and this provides a cutoff to the distribution p(xi). For the purposes of calculation, we approximate the outer boundary of the speckle pattern as a square of area rref. We then have ( 1/r2 0 |xi| < rref/2,|yi| < rref/2, otherwise. p(xi) = (A4) ref Making the substitutions ux= xi− xj, uy= yi− yj, vx= xi+ xjand vy= yi+ yj, we then have 1 4r4 ref rdiff m=0 k=0 ∞ X m−1 1 ?β − 2 ?!Z ?β−2 ? Aduxdvxuβ−2−2m hD(rij)i = Y − k x 2 Z (A5) Aduydvyu2m × y, where the integration area is taken into account as follows: Zrref−u Z0 Zrref + Zrref Z0 Z = dvy+ Aduxduy dux dux dvy u−rref dux 0 0 0 Z0 Z0 Zrref+u −rref−udvy+ (A6) dux dvy. −rref −rref 0 Using 4rβ−2m ref Z Aduxdvxuβ−2−2m = (A7) (β − 2m)(β − 2m − 1), x provided β − 2m 6= 1 and β − 2m 6= 0, and 4r2+2m ref Z = (A8) Aduydvyu2m (2 + 2m)(2m + 1), y
– 16 – we obtain ∞ X m−1 k=0 ?β − 2 1 ?!? ?β−2 ?rref rdiff ? hD(rij)i = 4 Y × − k 1 2 (2m + 2)(2m + 1) m=0 ? ? (A9) (β − 2m)(β − 2m − 1) ?β−2 ?rref rdiff g(β) (A10) ≡ . B. Derivation of ray-bending due to a Faraday wedge Here we derive the result of the ray-bending due to the Faraday wedge model discussed in §4. The relative displacement of the right- and left-polarized wavefronts is derived using Hamiltonian equations for a ray. These are dx dt=∂ωσ(k,x,t) dk dt= −∂ωσ(k,x,t) dω dt=∂ωσ(k,x,t) (B1) , , . ∂k ∂x ∂t In a stationary medium, assumed here, the third equation is not relevant. In the paraxial approximation one separates x into z, r, with the z axis along k = kκ. The relation k = nω/c is used after combining the first and second of equations (B1) to yield an expression for the angular deviation of the ray in terms of the refractive index variations on the scattering screen d(nσκ) dz =∂nσ (B2) ∂r, where the arguments of nσ(ω,κ,z,r) are suppressed. The change, δκσ, across a wedge of thickness ∆z is given by 1 Z∆z dz∂nσ δκσ= (B3) ∂r. nσ 0 In the weak anisotropy limit one writes the refractive index in terms of polarization dependent and independent terms nσ= n + σ∆n/2 with ∆n = n+− n−to simplify this to δκσ=1 n 0 1 −σ∆n ∂∆n ∂r Z∆z ?∂n ∂r+σ ?? ? (B4) dz , 2n 2 neglecting terms of order (∆n)2. Thus, if the screen is a distance L from the observer, the relative displacement of the centers of the images in the two modes on the observer’s plane is L∆ζ, as given by equation (4-2). REFERENCES Backer, D.C., 1978, ApJ, 222, L9
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– 19 – phase stationary phase points offset relative to image center Fig. 1.— An illustration of the scattered wavefront, showing the contribution of the geometric phase (dotted), and the combined contribution of the scattering medium and geometric phase (solid line). Several points of stationary phase are indicated on the diagram.
– 20 – RH LH plasma wedge Fig. 2.— A schematic of refraction of rays at a Faraday wedge: an incident ray splits into two circularly polarized rays propagating in slightly different directions.
– 21 – planar incident wavefront rotation measure gradient Fig. 3.— A schematic of the Faraday wedge from the viewpoint of physical optics. A RM gradient causes a difference in the ray paths of the left- and right-hand circularly polarized wavefronts. Upon arrival at the Earth, the scintillation pattern of one wavefront is slightly displaced with respect to the other, leading to variability in the CP.