Predicting Pulsar Scintillation from Refractive Plasma Sheets
MMNRAS , 1–14 (2018) Preprint May 16, 2018 Compiled using MNRAS L A TEX style file v3.0
Predicting Pulsar Scintillation from Refractive PlasmaSheets
Dana Simard, , , (cid:63) Ue-Li Pen , , , Canadian Institute for Theoretical Astrophysics, University of Toronto, 60 Saint George Street, Toronto, ON M5S 3H8, Canada Department of Astronomy and Astrophysics, University of Toronto, 50 Saint George Street, Toronto, ON M5S 3H4, Canada Dunlap Institute for Astronomy and Astrophysics, University of Toronto, 50 Saint George Street, Toronto, ON M5S 3H4, Canada Canadian Institute for Advanced Research, Program in Cosmology and Gravitation, Toronto, ON M5G 1Z8, Canada Perimeter Institute for Theoretical Physics, 31 Caroline Street North, Waterloo, ON N2L 2Y5, Canada
Accepted XXX. Received YYY; in original form ZZZ
ABSTRACT
The dynamic and secondary spectra of many pulsars show evidence for long-lived,aligned images of the pulsar that are stationary on a thin scattering sheet. One ex-planation for this phenomenon considers the effects of wave crests along sheets inthe ionized interstellar medium, such as those due to Alfv´en waves propagating alongcurrent sheets. If these sheets are closely aligned to our line-of-sight to the pulsar,high bending angles arise at the wave crests and a selection effect causes alignmentof images produced at different crests, similar to grazing reflection off of a lake. Us-ing geometric optics, we develop a simple parameterized model of these corrugatedsheets that can be constrained with a single observation and that makes observablepredictions for variations in the scintillation of the pulsar over time and frequency.This model reveals qualitative differences between lensing from overdense and under-dense corrugated sheets: Only if the sheet is overdense compared to the surroundinginterstellar medium can the lensed images be brighter than the line-of-sight image tothe pulsar, and the faint lensed images are closer to the pulsar at higher frequenciesif the sheet is underdense, but at lower frequencies if the sheet is overdense.
Key words: pulsars: general – ISM: general – ISM: structure
Observations of pulsar scintillation, the variation in intensityover frequency and time due to propagation effects inducedby the interstellar medium (ISM), have revealed signifi-cant structure in the secondary spectrum (the 2-dimensionalpower spectrum of the dynamic spectrum, the intensityof the pulsar over time and frequency). In particular, aparabolic distribution of power in the secondary spectrumhas been found to be common in pulsars imaged with suf-ficient dynamic range and resolution (Putney et al. 2005;Stinebring et al. 2001). In some pulsars, inverted arclets withapexes along the main parabolic arc are also present (Hillet al. 2005; Stinebring 2007); see Brisken et al. (2010, figure1) for a particularly striking example. These parabolic arcscan arise if the scattering is highly anisotropic and local-ized at a thin scattering screen along our line-of-sight, whileinverted arclets in the secondary spectrum can result fromindividually distinguishable images on the screen (Cordeset al. 2006; Walker et al. 2004). In this picture, each in- (cid:63)
E-mail: [email protected] verted arclet is due to the interference of one lensed imagewith the other images of the pulsar, while the main parabolais due to the interference of the bright, line-of-sight imageof the pulsar with the scattered images. The discrete imageshave been observed to persist for weeks (Hill et al. 2003,2005) and show minute changes in their locations with fre-quency (Brisken et al. 2010; Hill et al. 2005), suggestingsmall, long-lived substructures within the screen, which areinconsistent with the expected characteristics of isotropicturbulence in the ISM. Not only are the substructures, ob-served to be (cid:46) . AU (Brisken et al. 2010; Hill et al. 2005)in size, much smaller than those expected from interstellarturbulence, but such large free electron densities, n e ≈ cm − (Hill et al. 2005), are required to produce the observedscattering angles that these structures would be out of pres-sure equilibrium with the ISM and therefore rare, which isinconsistent with the prevalence of pulsar scintillation arcs.Pen & Levin (2014) suggest that scintillation is insteadcaused by corrugated sheets, such as current sheets alongwhich Alfv´en waves with amplitudes larger than the thick-ness (depth) of the sheet create many crests. If the sheetis closely aligned with our line-of-sight to the pulsar and © a r X i v : . [ a s t r o - ph . GA ] M a y D. Simard & U.-L. Pen corrugated in a perpendicular direction, a high gradient infree electron column density is achieved at each crest, result-ing in large refraction angles near the crest. If many wavecrests are distributed over the sheet, grazing refraction off ofthe sheet will result in a linear series of images, in analogyto grazing reflections off waves on a lake. (See Fig. 1 in Liuet al. (2016) for an example.) This is due to a selection effect- bending angles close to our line-of-sight can be achieved bysmaller, and more common, wave crests. (Note that in con-trast to surface waves on a lake, in this picture the entiredepth of the sheet is perturbed by the waves, so that it re-sembles a flag in the wind.) If refraction is occurring dueto a corrugated, closely-aligned thin sheet, the curvature ofthe corrugations relieves the requirement for very high elec-tron densities within the sheet; it is the combination of thecurvature and the difference in the electron density betweenthe sheet and the ambient ISM that leads to large refractionangles. This both alleviates the tension between the preva-lence of pulsar scintillation and the large overpressures ofthe inferred structures in the ISM and has the implicationthat both underdense and overdense lenses can produce theobserved refraction angles; we will thus consider both caseshere.A number of lensing models have already been consid-ered in the context of both pulsar scintillation and quasar ex-treme scattering events (ESEs) (Fiedler et al. 1987), lensingof quasars by the ISM. Clegg et al. (1998) model two ESEsusing a Gaussian-shaped lens with a free electron overden-sity. They find that the lens can produce the overall shapeof the light curves observed, but that the parameters of thelens must be fine-tuned at each frequency band. Pen & King(2012) consider the effect of a Gaussian-shaped underdenselens on a point source such as a pulsar or quasar, and findthat a double-peaked light curve, such as those characteristicof ESE’s, is produced. Bannister et al. (2016); Tuntsov et al.(2016) model the dynamic spectrum of an ESE in an attemptto determine the electron column density and shape of thelens. They consider two lens shapes, one which is isotropicand one which is anisotropic, but find that the data holdsno preference for one over the other. They find that the pa-rameters used to model the ESE at one observing band arenot suitable at another band. These results indicate thatsuccessfully modeling the lensing behaviour at a single fre-quency band and epoch is not enough to suggest consistencyof the model with observations; a successful model must alsopredict changes in the scattering with time and frequency.In this paper, we investigate the effects of a thin, corru-gated plasma sheet closely aligned to our line-of-sight, likethe current sheets corrugated by Alfv´en waves discussed byPen & Levin (2014), on emission from a pulsar. In Section 2we construct a model of this lens and examine it analytically,while in Section 3 we present some numerical examples ofthis model. In both Sections 2 and 3 we examine the mag-nifications and angular separation between the images whenmultiple images form at a single crest and we explore theevolution of the lensing with time and frequency. Observa-tions which can be compared to this model are discussedin Section 4 and extensions to this model are considered inSection 5. We finish with concluding remarks in Section 6.
In the picture of scattering from a corrugated sheet pre-sented by Pen & Levin (2014), each wave crest along the in-clined sheet produces an image of the pulsar and each crestcan be parameterized from properties of the wave and theobserving geometry, as shown in Fig. 1. The lensing in thispicture can be explored in different ways. One can considerhow properties of the sheet, including the thickness of thesheet and the distribution of waves, determine the statisticalproperties of the scattering, such as the angular distributionof the scattered radiation, which can be compared to the ob-served scattering tails or Very Long Baseline Interferometry(VLBI) correlated flux densities of pulsars. In this paper wetake a different approach, and focus on the case where in-dividual scattered images of the pulsar are distinguishable,for example as inverted arclets in the secondary spectrum.In this regime, the magnification and position of each imageis related to properties of the crest producing that image.Furthermore, by tracking the image locations and magnifi-cations through frequency and time, the dependence of thescattering on both the observing frequency and the proxim-ity of the pulsar to the crest can be compared to the model.With this in mind, we begin our investigation by consid-ering a single wave crest. We choose z to be the line-of-sightdirection, and x and y to be in the plane of the sky, with theorigin at the crest. We will consider a sheet of thickness T corrugated in the x direction, so that all refraction occurs inthe x direction. We will model the crest itself as a parabolain the x - z plane. In truth, we do not know the orientationof the sheet or the corrugations in the x - z plane, but sinceit is the projected curvature that determines the lensing be-haviour, differences in inclination can be accounted for bychanging the curvature of the parabola. We will focus on asingle wave crest, and write the equation for this crest as x = z R , (1)where R is the radius of curvature projected along the x direction at the apex of the crest.We will use the lensing geometry shown in Fig. 2. Inthis geometry, the lens equation is θ = β + s ˆ α , (2)where θ is the observed position of the source, β is the trueposition, ˆ α is the bending angle, s = − d lens / d psr , d psr isthe distance to the pulsar plane, and d lens is the distance tothe lens plane. We determine ˆ α by considering Φ , the phasechange imparted by the lens, Φ ( x ) = πλ ∫ dz ( n ( x ) − n ) , (3)where n is the index of refraction outside of the lens, n is theindex of refraction inside the lens, and λ is the wavelengthof observations. The bending angle is related to the gradientof the phase change by ˆ α ( x ) = − λ π ∇ x Φ ( x ) . (4)Assuming that the index of refraction is constant inside ofthe lens, this reduces to ˆ α ( x ) = − ∆ n ∇ x Z ( x ) , (5) MNRAS000
In the picture of scattering from a corrugated sheet pre-sented by Pen & Levin (2014), each wave crest along the in-clined sheet produces an image of the pulsar and each crestcan be parameterized from properties of the wave and theobserving geometry, as shown in Fig. 1. The lensing in thispicture can be explored in different ways. One can considerhow properties of the sheet, including the thickness of thesheet and the distribution of waves, determine the statisticalproperties of the scattering, such as the angular distributionof the scattered radiation, which can be compared to the ob-served scattering tails or Very Long Baseline Interferometry(VLBI) correlated flux densities of pulsars. In this paper wetake a different approach, and focus on the case where in-dividual scattered images of the pulsar are distinguishable,for example as inverted arclets in the secondary spectrum.In this regime, the magnification and position of each imageis related to properties of the crest producing that image.Furthermore, by tracking the image locations and magnifi-cations through frequency and time, the dependence of thescattering on both the observing frequency and the proxim-ity of the pulsar to the crest can be compared to the model.With this in mind, we begin our investigation by consid-ering a single wave crest. We choose z to be the line-of-sightdirection, and x and y to be in the plane of the sky, with theorigin at the crest. We will consider a sheet of thickness T corrugated in the x direction, so that all refraction occurs inthe x direction. We will model the crest itself as a parabolain the x - z plane. In truth, we do not know the orientationof the sheet or the corrugations in the x - z plane, but sinceit is the projected curvature that determines the lensing be-haviour, differences in inclination can be accounted for bychanging the curvature of the parabola. We will focus on asingle wave crest, and write the equation for this crest as x = z R , (1)where R is the radius of curvature projected along the x direction at the apex of the crest.We will use the lensing geometry shown in Fig. 2. Inthis geometry, the lens equation is θ = β + s ˆ α , (2)where θ is the observed position of the source, β is the trueposition, ˆ α is the bending angle, s = − d lens / d psr , d psr isthe distance to the pulsar plane, and d lens is the distance tothe lens plane. We determine ˆ α by considering Φ , the phasechange imparted by the lens, Φ ( x ) = πλ ∫ dz ( n ( x ) − n ) , (3)where n is the index of refraction outside of the lens, n is theindex of refraction inside the lens, and λ is the wavelengthof observations. The bending angle is related to the gradientof the phase change by ˆ α ( x ) = − λ π ∇ x Φ ( x ) . (4)Assuming that the index of refraction is constant inside ofthe lens, this reduces to ˆ α ( x ) = − ∆ n ∇ x Z ( x ) , (5) MNRAS000 , 1–14 (2018) redicting Pulsar Scintillation pulsarobserverlens i (a) xz i A A R (b) Figure 1.
The geometry of lensing from an inclined sheet. Figure 1a shows how multiple wave crests result in many images of the pulsar.The line of sight image is shown with the solid line, while the dotted lines indicate the lensed images. The grey solid line shows theorientation of the sheet, and i is the inclination angle between the sheet and the line of sight to the pulsar. The specific ray deflectionsshown are those that would result if the corrugated sheet were underdense. Figure 1b shows the relation between the radius of curvatureat the location where the column density gradient is maximized and the parameters of the wave perturbing the sheet. The thick curverepresents the sheet, perturbed by a sinusoidal wave with wavelength λ A and amplitude A , and inclined by an angle i relative to theline-of-sight to the source. The column density through the lens is maximized at the origin, and the dashed circle is the osculating circle,with radius R , at this point. Note that these figures are not to scale: We’ve drawn the lensing sheet to occupy the whole distance betweenthe observer and the pulsar, while in reality it only occupies a small percentage, we expect dozens of crests along the current sheet ratherthan the few that we’ve drawn, and the angle i is much less than 1. where ∆ n = n − n and Z is the extent of the lens in the z -direction. In the regime x (cid:29) T / , Z = T d l d x , where d l isthe length element of the lens, d l = d z + d x . The factor of2 comes from the two sides of the parabola. Using equation(1) for the shape of the lens, we find ∇ x Z = T d l d x = − T R x (cid:112) R / x + , (6)so that we can write ˆ α ( x ) = ∆ n T R x (cid:112) R / x + . (7)For a plasma, the index of refraction is given by n = (cid:115) − ω p ω , (8)where ω p = (cid:112) π c r e n e is the characteristic frequency of theplasma expressed in terms of the electron density, n e , andthe classical electron radius, r e = π(cid:15) e m e c where (cid:15) is the vacuum permittivity and m e is the mass of the electron. For n e = . cm − , a typical value for the ISM, this charac-teristic frequency is 1.5 kHz, much smaller than the GHzfrequencies of typical radio observations. Therefore, we ap-proximate the index of refraction as n (cid:39) − λ π n e r e . (9)We now write the bending angle in terms of the electrondensity, ˆ α ( x ) = − λ π ∆ n e r e T R x (cid:112) R / x + , (10)where ∆ n e is the difference between the free electron densi-ties inside and outside of the sheet. Since we measure angu-lar positions on the sky, rather than physical ones, we willuse the dimensionless variables θ = x / d lens , r = R / d lens and θ T = T / d lens , and write: ˆ α ( θ ) = − λ π ∆ n e r e θ T r θ (cid:112) r / θ + . (11) MNRAS , 1–14 (2018)
D. Simard & U.-L. Pen xz lens planeobserverpulsar d psr d lens - T T (a) x ∆ N e lens planeobserverpulsar d psr d lens - T T ˆ αβ θ (b) Figure 2.
Lensing geometry and definitions of variables that we will use in this paper. The grey curve in Fig. 2a represents a singlecrest the sheet, while the solid curve in Fig. 2b is a sketch of the column density profile of the lens. The angles θ and β are measuredclockwise from the line from the observer to the crest of the corrugation, so that β < as drawn. The angle ˆ α is measured clockwise fromthe line between the pulsar and the image of the pulsar on the lens plane. Note that in practice the angles are all much smaller than 1and the crest covers only a very small portion of the line-of-sight to the pulsar. The specific deflection angle and column density profileshown are those for an underdense corrugated sheet. We now write the lens equation for this system, θ = β − s λ π ∆ n e r e θ T r θ (cid:112) r / θ + . (12)We see that increasing the over or underdensity, radius ofcurvature, or thickness of the lens results in a larger deflec-tion.Under the approximation that θ (cid:28) r / , equation (12)simplifies to θ (cid:39) β − s r e λ √ π ∆ n e θ T √ r θ / (13) (cid:39) β + s √ ∆ n θ T √ r θ / , (14)where we have used equation (9) to write the lens equationin terms of ∆ n . We see that under this approximation thelens equation depends only on the frequency of observation,the distances to the pulsar and lens, and a single physicalparameter describing the lens itself, ∆ n θ T √ r ; in other words,the column density and radius of curvature of the lens aredegenerate. Throughout this paper, we will use this approx-imation to analyze the behaviour of the lens; however allnumerical results are calculated using the full form of thelens equation, equation (12).The solution to the lens equation gives the observedangular positions, θ , of the pulsar for a given true angularposition, β . Due to conservation of surface brightness, the magnification of the lensed image is µ = (cid:18) d β d θ (cid:19) − . (15)Using equation (13), this evaluates to µ (cid:39) (cid:32) + s √ ∆ n θ T √ r θ / (cid:33) − . (16)We will now consider two possibilities - either the sheetis underdense or it is overdense. We will approximate thesheet as a thin sheet to simplify the following analytic anal-ysis. Within this approximation, we will consider the lens tobe at θ > , with no lens at θ < . If the sheet is overdense,then the index of refraction in the sheet is less than thatin the ambient ISM ( ∆ n < ), and the bending angles pro-duced are negative (see equation (11)). If instead the sheetis underdense, ∆ n > and the bending angles produced arepositive.We can understand the general behaviour of the lens byconsidering the function β ( θ ) : β = θ − s ∆ n θ T (cid:112) r / θ / . (17)When θ is large, β ≈ θ , while when θ is small, β ∝ θ − / if thelens is overdense and β ∝ − θ − / if the lens is underdense.There is a lensed image at β = when θ ≡ (cid:16) s ∆ n θ T (cid:112) r / (cid:17) / , (18) MNRAS000
Lensing geometry and definitions of variables that we will use in this paper. The grey curve in Fig. 2a represents a singlecrest the sheet, while the solid curve in Fig. 2b is a sketch of the column density profile of the lens. The angles θ and β are measuredclockwise from the line from the observer to the crest of the corrugation, so that β < as drawn. The angle ˆ α is measured clockwise fromthe line between the pulsar and the image of the pulsar on the lens plane. Note that in practice the angles are all much smaller than 1and the crest covers only a very small portion of the line-of-sight to the pulsar. The specific deflection angle and column density profileshown are those for an underdense corrugated sheet. We now write the lens equation for this system, θ = β − s λ π ∆ n e r e θ T r θ (cid:112) r / θ + . (12)We see that increasing the over or underdensity, radius ofcurvature, or thickness of the lens results in a larger deflec-tion.Under the approximation that θ (cid:28) r / , equation (12)simplifies to θ (cid:39) β − s r e λ √ π ∆ n e θ T √ r θ / (13) (cid:39) β + s √ ∆ n θ T √ r θ / , (14)where we have used equation (9) to write the lens equationin terms of ∆ n . We see that under this approximation thelens equation depends only on the frequency of observation,the distances to the pulsar and lens, and a single physicalparameter describing the lens itself, ∆ n θ T √ r ; in other words,the column density and radius of curvature of the lens aredegenerate. Throughout this paper, we will use this approx-imation to analyze the behaviour of the lens; however allnumerical results are calculated using the full form of thelens equation, equation (12).The solution to the lens equation gives the observedangular positions, θ , of the pulsar for a given true angularposition, β . Due to conservation of surface brightness, the magnification of the lensed image is µ = (cid:18) d β d θ (cid:19) − . (15)Using equation (13), this evaluates to µ (cid:39) (cid:32) + s √ ∆ n θ T √ r θ / (cid:33) − . (16)We will now consider two possibilities - either the sheetis underdense or it is overdense. We will approximate thesheet as a thin sheet to simplify the following analytic anal-ysis. Within this approximation, we will consider the lens tobe at θ > , with no lens at θ < . If the sheet is overdense,then the index of refraction in the sheet is less than thatin the ambient ISM ( ∆ n < ), and the bending angles pro-duced are negative (see equation (11)). If instead the sheetis underdense, ∆ n > and the bending angles produced arepositive.We can understand the general behaviour of the lens byconsidering the function β ( θ ) : β = θ − s ∆ n θ T (cid:112) r / θ / . (17)When θ is large, β ≈ θ , while when θ is small, β ∝ θ − / if thelens is overdense and β ∝ − θ − / if the lens is underdense.There is a lensed image at β = when θ ≡ (cid:16) s ∆ n θ T (cid:112) r / (cid:17) / , (18) MNRAS000 , 1–14 (2018) redicting Pulsar Scintillation which has a single real solution if the sheet is underdensebut no solutions if the sheet is overdense. Finally, we seethat local extrema occur when d β d θ = , or θ ext ≡ (cid:16) − s ∆ n θ T (cid:112) r / (cid:17) / . (19)If the lens is underdense, there are therefore no local ex-trema, but if the lens is overdense there is one, a local min-imum.Combining this information, we see that if the lens isunderdense there is a single lensed image for all β , or all truepositions of the pulsar. In addition, the unlensed image of thepulsar is visible if β < . As a result, two images are visibleif β < , the unlensed image and a lensed image. If β > ,only one, lensed image is seen, but it tends towards the line-of-sight position for large β . If the lens is overdense, thenthere are two lensed images for β > β ( θ ext ) > , one at small θ and one at θ ≈ β . There is only the line-of-sight image at β < , and there are no images at all for for < β < β ( θ ext ) .From equation (16), differences in the brightnesses ofthe lensed images between the two lenses are apparent. Ifthe lens is underdense, ∆ n > , and µ < , so that lensedimages brighter than the unlensed image of the pulsar can-not be produced. When β = and θ = θ , we find that themagnification of the image is , regardless of the lens pa-rameters. If the lens is overdense, ∆ n < , so that µ > or µ < . In this case, either the lensed image is brighter thanthe unlensed image of the pulsar, or the image is inverted.In practice, as discussed later in Section 4, we do notmeasure the absolute magnifications and locations of thelensed images, but rather the magnification ratio and an-gular displacement between two images of the pulsar. Tocompare with observations, we will therefore investigate themagnification ratios and angular separations between im-ages in the cases where two images are formed at a singlecrest.We can derive a relation between the magnification ra-tio and angular separation by assuming that the brightestimage is unlensed, which for the underdense lens is true,and for the overdense lens is a good approximation when β (cid:29) (cid:0) s | ∆ n | θ T (cid:112) r / (cid:1) / . We will write the location and mag-nification of the brightest image as θ (cid:39) β and µ (cid:39) re-spectively. In this case, the angular separation is ∆ θ (cid:39) θ − β (cid:39) s √ ∆ n θ T √ r θ / (20) (cid:39) . s / ∆ n . − θ T − (cid:114) r (cid:18) θ (cid:19) − / , (21)where θ is the angular location of the fainter image. (SeeSection 3.1 for the origin of the fiducial values chosen for ∆ n , θ T , r and θ . For most pulsar scintillation systems, the screenis not associated with the pulsar and is midway betweenthe observer and the pulsar, so we choose a fiducial value s = / .) This allows us to write θ in terms of the angularseparation, ∆ θ : θ (cid:39) (cid:32) s √ ∆ n θ T √ r ∆ θ (cid:33) / . (22)In this regime, we expect the magnifications of the faintimages to be much less than one. We can therefore simplify equation (16): µ (cid:39) √
23 1 s θ / ∆ n θ T √ r . (23)We now combine equations (22) and (23) to write the mag-nification ratio in terms of the angular separation: (cid:12)(cid:12)(cid:12)(cid:12) µµ (cid:12)(cid:12)(cid:12)(cid:12) (cid:39) (cid:32) s √ | ∆ n | θ T √ r (cid:33) / | ∆ θ | − / (24) (cid:39) . (cid:32) s / | ∆ n | . − θ T − (cid:114) r (cid:33) / (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∆ θ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − / . (25)Equation (24) can be rewritten as (cid:12)(cid:12)(cid:12)(cid:12) µµ (cid:12)(cid:12)(cid:12)(cid:12) (cid:39) (cid:12)(cid:12)(cid:12)(cid:12) ∆ θ ref ∆ θ (cid:12)(cid:12)(cid:12)(cid:12) / , (26)where ∆ θ ref = (cid:18) s √ | ∆ n | θ T √ r (cid:19) / (27) (cid:39) . (cid:32) s / | ∆ n | . − θ T − (cid:114) r (cid:33) / . (28)Therefore, we expect (cid:12)(cid:12) µµ (cid:12)(cid:12) ∝ | ∆ θ | − / for large separations.This relation allows us to determine ∆ θ ref , and therefore | ∆ n | θ T √ r , from observations of the angular separations andrelative magnifications of the images from a single wavecrest. Note that if the sheet is underdense, ∆ θ ref is the angu-lar separation when the pulsar is directly behind the crest, ∆ θ ( β = ) = θ . (See equation (18).)We can also consider the frequency evolution of the lens-ing qualitatively. The bending angle ˆ α ∝ − ∆ n θ / when θ (cid:28) r / and the index of refraction scales with wavelength, ∆ n ∝ λ ,so that at higher frequencies, images near the crest of thewave must form at smaller θ . In the underdense case, theline-of-sight image of the pulsar is at β < when a secondlensed image is visible, so that as the lensed image moves to-wards the crest of the wave the angular separation betweenthe line-of-sight and lensed images decreases. In contrast, ifthe lens is overdense two images are visible when β > , thebright image of the pulsar at approximately the line-of-sightto the pulsar and a faint image close to the crest of the wave.In this case, as the wavelength increases the fainter imagemoves to smaller θ while the brighter image moves very lit-tle with wavelength, so that the angular separation betweenthe two images increases. Thus, we see that if the lens isunderdense, the angular separation between the two imagesdecreases with frequency, while if it is overdense the angularseparation increases with frequency. This behaviour allowsone to distinguish between the two cases, and determine thesign of the lensing parameter ∆ n θ T √ r . In order to construct some numerical examples of this model,we first put some values to parameters in the model in Sec-tion 3.1. We then consider the behaviour of a single wavecrest for lenses of varying strengths in Section 3.2. Finally,
MNRAS , 1–14 (2018)
D. Simard & U.-L. Pen we present an example of constraining the lens parameterand predicting the evolution of the lensed images with timeand frequency in Section 3.3.
For our numerical model, we will adopt parameters from oneof the most thoroughly studied examples of pulsar scintilla-tion arcs, PSR B0834+06. We will assume the distances tothe pulsar and lens are those measured by Liu et al. (2016)for PSR B0834+06 and one of its lensing screens, d psr = pc, d lens = pc. We will assume an observing frequencyof 314.5 MHz. For a typical lensed image, we will consideran angular displacement from the line-of-sight image of thepulsar of 10 mas and a magnification of 0.01, parameterssimilar to those measured by Brisken et al. (2010) for PSRB0834+06. Finally, in order to predict the temporal evolu-tion of this system, we will need to know the speed withwhich the pulsar moves behind the lens in the direction ofscattering. For this, we will also use the value measured byLiu et al. (2016), 172.4 km s − , or an angular relative veloc-ity of 1.12 mas/week.We will assume values for T , the thickness of the sheet,and R , the radius of curvature of the crest, and vary onlythe electron density, and thus the index of refraction, insidethe lens, but keep in mind that this model only constrainsthe combination of parameters ∆ n θ T √ r and not ∆ n itself.In physical parameters, for a sheet with an inclination angle i relative to the line-of-sight to the pulsar, the radius ofcurvature at the point where the tangent is in the z -directionis R = λ A π A cos ( i ) (cid:114) − λ A π A tan ( i ) (29) ≈ λ A π A , (30)where λ A and A are respectively the wavelength and ampli-tude of the Alfv´en wave. See Figure 1b for a sketch of thewave and the resulting radius of curvature. In order to cor-rugate the current sheet, the amplitude of the wave must bemuch greater than the thickness of the sheet, and we expecta very small inclination angle of the sheet with respect toour line-of-sight, i (cid:28) , in order to produce the linear seriesof images observed. We expect the projected wavelength, λ A sin ( i ) , to be similar to the separation between images,which in some cases is as small as 0.05 AU for the PSRB0834+06 system (Brisken et al. 2010).We choose an Alfv´en wavelength λ A = AU, an in-clination of the sheet with respect to our line-of-sight tothe pulsar i = − rad, a thickness of the sheet T = . AU or θ T = . mas, and an amplitude of the Alfv´en wave A = . AU. These parameters give a projected wavelength of1 AU, and a projected radius of curvature at the apex of theparabola of R = . kpc or r = . Assuming a volume fillingfactor of 10% (Draine 2011) for the warm ionized interstellarmedium, and an average electron density throughout the en-tire ISM of 0.03 cm − , we assume a typical electron densityof 0.3 cm − in the warm ionized interstellar medium. Whileour analysis does not depend on this value, it does informthe physically allowed electron density differences between the lens and the surrounding medium: We will consider elec-tron density differences between inside and outside the lensof ∆ n e = ± . , ± . cm − . Using equations (12) and (15), we can calculate the angularpositions and magnifications of images for lenses of varyingstrengths, shown in Fig. 3. When examining these figures,recall that the part of the lens we are considering is at θ ≥ θ T / , with maximum strength at θ = θ T / . From Fig. 3,we can confirm that multiple images are produced by theunderdense lens when the pulsar is not behind the crest, andmultiple images are produced by the overdense lens when thepulsar is behind the crest. We also note that magnificationslarger than 1 occur only in the overdense case. If the lens isoverdense, the fainter image is inverted, as indicated by thenegative magnification, and there is a range in β for whichwe see no images at all as the line-of-sight to the pulsar isobscured by the lens and the lens is bending light out of oursight.The numerical relation between the angular separationand flux ratio of two images formed at a single wave crestis shown in Fig. 4. We see that after each curve is scaled bythe reference separation, ∆ θ ref (equation (27)), the relationbetween magnification ratio and angular separation is inde-pendent of the lens parameters. By measuring the angularseparation and flux ratio for a pair of images, we can deter-mine the value of ∆ θ ref required to place the point along thecurves in Fig. 4, and therefore | ∆ n | r √ θ T , provided that thedistance to the pulsar is known. Once the free parameter of the lens, | ∆ n | θ T √ r , is con-strained from the magnification ratios and angular separa-tions, changes in the locations and magnifications of the im-ages with frequency can be predicted. If the angular velocityof the pulsar relative to the screen in the x -direction (parallelto the axis along which the images are formed; see Fig. 2), ( V psr − V lens ) (cid:107) d psr , is also known, we can predict changes over time.Since we do not know if the lens is overdense or underdenseat this stage, we consider both cases.As an example, we will consider two images with a mag-nification ratio of 0.01 and an angular separation of 10 masat a frequency of 314.5 MHz. Using these values, we calcu-late that the lens has, for the parameters given in Section3.1, | ∆ n e | = . cm − . From the lens equation and theangular separation, we now determine the true position ofthe pulsar relative to the crest, β , for the underdense andoverdense cases, -9.9 and 10.2 mas respectively. Let’s assumethat the angular separation is decreasing as the pulsar movestowards the crest. In the case of the overdense lens, after 4weeks, the magnification ratio of the 2 images has increasedto 0.028, and the angular separation has decreased by 4.6mas. The magnification ratio of the images will continue toincrease and the images will continue to get brighter untilthe two images have zero angular separation after 7.6 weeks,after which the lensed image will disappear. In the case ofthe underdense lens, after 4 weeks, the angular separation MNRAS000
For our numerical model, we will adopt parameters from oneof the most thoroughly studied examples of pulsar scintilla-tion arcs, PSR B0834+06. We will assume the distances tothe pulsar and lens are those measured by Liu et al. (2016)for PSR B0834+06 and one of its lensing screens, d psr = pc, d lens = pc. We will assume an observing frequencyof 314.5 MHz. For a typical lensed image, we will consideran angular displacement from the line-of-sight image of thepulsar of 10 mas and a magnification of 0.01, parameterssimilar to those measured by Brisken et al. (2010) for PSRB0834+06. Finally, in order to predict the temporal evolu-tion of this system, we will need to know the speed withwhich the pulsar moves behind the lens in the direction ofscattering. For this, we will also use the value measured byLiu et al. (2016), 172.4 km s − , or an angular relative veloc-ity of 1.12 mas/week.We will assume values for T , the thickness of the sheet,and R , the radius of curvature of the crest, and vary onlythe electron density, and thus the index of refraction, insidethe lens, but keep in mind that this model only constrainsthe combination of parameters ∆ n θ T √ r and not ∆ n itself.In physical parameters, for a sheet with an inclination angle i relative to the line-of-sight to the pulsar, the radius ofcurvature at the point where the tangent is in the z -directionis R = λ A π A cos ( i ) (cid:114) − λ A π A tan ( i ) (29) ≈ λ A π A , (30)where λ A and A are respectively the wavelength and ampli-tude of the Alfv´en wave. See Figure 1b for a sketch of thewave and the resulting radius of curvature. In order to cor-rugate the current sheet, the amplitude of the wave must bemuch greater than the thickness of the sheet, and we expecta very small inclination angle of the sheet with respect toour line-of-sight, i (cid:28) , in order to produce the linear seriesof images observed. We expect the projected wavelength, λ A sin ( i ) , to be similar to the separation between images,which in some cases is as small as 0.05 AU for the PSRB0834+06 system (Brisken et al. 2010).We choose an Alfv´en wavelength λ A = AU, an in-clination of the sheet with respect to our line-of-sight tothe pulsar i = − rad, a thickness of the sheet T = . AU or θ T = . mas, and an amplitude of the Alfv´en wave A = . AU. These parameters give a projected wavelength of1 AU, and a projected radius of curvature at the apex of theparabola of R = . kpc or r = . Assuming a volume fillingfactor of 10% (Draine 2011) for the warm ionized interstellarmedium, and an average electron density throughout the en-tire ISM of 0.03 cm − , we assume a typical electron densityof 0.3 cm − in the warm ionized interstellar medium. Whileour analysis does not depend on this value, it does informthe physically allowed electron density differences between the lens and the surrounding medium: We will consider elec-tron density differences between inside and outside the lensof ∆ n e = ± . , ± . cm − . Using equations (12) and (15), we can calculate the angularpositions and magnifications of images for lenses of varyingstrengths, shown in Fig. 3. When examining these figures,recall that the part of the lens we are considering is at θ ≥ θ T / , with maximum strength at θ = θ T / . From Fig. 3,we can confirm that multiple images are produced by theunderdense lens when the pulsar is not behind the crest, andmultiple images are produced by the overdense lens when thepulsar is behind the crest. We also note that magnificationslarger than 1 occur only in the overdense case. If the lens isoverdense, the fainter image is inverted, as indicated by thenegative magnification, and there is a range in β for whichwe see no images at all as the line-of-sight to the pulsar isobscured by the lens and the lens is bending light out of oursight.The numerical relation between the angular separationand flux ratio of two images formed at a single wave crestis shown in Fig. 4. We see that after each curve is scaled bythe reference separation, ∆ θ ref (equation (27)), the relationbetween magnification ratio and angular separation is inde-pendent of the lens parameters. By measuring the angularseparation and flux ratio for a pair of images, we can deter-mine the value of ∆ θ ref required to place the point along thecurves in Fig. 4, and therefore | ∆ n | r √ θ T , provided that thedistance to the pulsar is known. Once the free parameter of the lens, | ∆ n | θ T √ r , is con-strained from the magnification ratios and angular separa-tions, changes in the locations and magnifications of the im-ages with frequency can be predicted. If the angular velocityof the pulsar relative to the screen in the x -direction (parallelto the axis along which the images are formed; see Fig. 2), ( V psr − V lens ) (cid:107) d psr , is also known, we can predict changes over time.Since we do not know if the lens is overdense or underdenseat this stage, we consider both cases.As an example, we will consider two images with a mag-nification ratio of 0.01 and an angular separation of 10 masat a frequency of 314.5 MHz. Using these values, we calcu-late that the lens has, for the parameters given in Section3.1, | ∆ n e | = . cm − . From the lens equation and theangular separation, we now determine the true position ofthe pulsar relative to the crest, β , for the underdense andoverdense cases, -9.9 and 10.2 mas respectively. Let’s assumethat the angular separation is decreasing as the pulsar movestowards the crest. In the case of the overdense lens, after 4weeks, the magnification ratio of the 2 images has increasedto 0.028, and the angular separation has decreased by 4.6mas. The magnification ratio of the images will continue toincrease and the images will continue to get brighter untilthe two images have zero angular separation after 7.6 weeks,after which the lensed image will disappear. In the case ofthe underdense lens, after 4 weeks, the angular separation MNRAS000 , 1–14 (2018) redicting Pulsar Scintillation (cid:45) (cid:45)
10 10 20 Β (cid:64) mas (cid:68) (cid:45) (cid:45) Θ (cid:64) mas (cid:68) (cid:45) (cid:45) (cid:68) n e (cid:64) cm (cid:45) (cid:68) (a) Underdense lens (cid:45) (cid:45)
10 10 20 Β (cid:64) mas (cid:68) (cid:45) (cid:45) Θ (cid:64) mas (cid:68) (cid:68) n e (cid:64) cm (cid:45) (cid:68) (b) Overdense lens (cid:45) (cid:45)
10 10 20 Β (cid:64) mas (cid:68) (cid:45) (cid:45) Μ (cid:45) (cid:45) (cid:68) n e (cid:64) cm (cid:45) (cid:68) (c) Underdense lens (cid:45) (cid:45)
10 10 20 Β (cid:64) mas (cid:68) (cid:45) (cid:45) (cid:45) Μ (cid:68) n e (cid:64) cm (cid:45) (cid:68) (d) Overdense lens Figure 3.
The observed position of the pulsar, θ , and magnification, µ , as a function of the true position, β . The legend indicates thedifference between the free electron density inside and outside of the lens, ∆ n e when the parameters in Section 3.1 are assumed. Recallthat the lens we are considering is at θ > θ T / , or θ > . mas. In the underdense lens case, a lensed image is observed along withthe true, unlensed image of the pulsar before the pulsar passes behind the crest, as the lens is bending light from the pulsar into ourline-of-sight. The angular separation between these two images reaches a minimum, non-zero value just as the pulsar passes behind thecrest ( β = ), and the magnification of the lensed image at this point is , independent of the lens parameters. After the pulsar passesbehind the crest, the line-of-sight image to the pulsar is obscured by the lens, and a single, lensed image is observed. In the case ofan overdense lens, only the line-of-sight image is seen before the pulsar is passing behind the crest as the lens bends light out of ourline-of-sight. Just after the pulsar has passed behind the lens, no images are seen, as the line-of-sight image is obscured by the lens, andthe lens is still bending light out of our line-of-sight. There is a minimum β for which two images are produced, and at which the twoimages have zero angular separation and are both strongly magnified. The angular separation between the images grows as the pulsarmoves to larger β . The unlensed image is at θ = β and µ = for all lenses when β < − θ T / .MNRAS , 1–14 (2018) D. Simard & U.-L. Pen (cid:45) (cid:45) (cid:45) (cid:200) (cid:68)Θ (cid:144) (cid:68)Θ ref (cid:200) (cid:200) Μ (cid:144) Μ (cid:200) (cid:45) (cid:45) (cid:68) n e (cid:64) cm (cid:45) (cid:68) Figure 4.
The ratio of the magnification of the fainter image, µ , to the magnification of the brighter image, µ , as a function of theangular separation, ∆ θ , between two images formed at a single wave crest, scaled by the reference value, ∆ θ ref (equation (27)). The legendindicates the difference between the electron density inside and outside the lens, ∆ n e when the parameters in Section 3.1 are assumed.The reference separations are 0.58 and 3.68 mas for ∆ n e of ± . and ± . cm − respectively, and the solid black line shows the analyticalrelation at large separations, equation (26). After these shifts in angular separation, the curves for various values of the lens parameterlie directly on top of one another, although the curves for an overdense sheet extend to smaller angular separations due to the minimumangular separation in the underdense case. This allows one to measure the parameter describing the lens, ∆ n e θ T √ r , up to the sign of ∆ n e , from measurements of | µ / µ | and ∆ θ by determining the reference separation required to align the point with the curves. Furtherobservations, such as the change in angular separation with wavelength, can then be used to determine whether the lens is overdense orunderdense. between the two images has decreased by 4.4 mas, and themagnification of the lensed image is 0.026.Similarly, we can consider changes with frequency. Us-ing the same example, we examine how the angular separa-tion between the two images varies over a 32-MHz frequencyband from 310.5 MHz to 342.5 MHz and find that, for the un-derdense lens, the angular separation will decrease by 0.019mas moving from the lowest to highest frequencies in theband, while the magnification ratio decreases by 0.0012. Forthe overdense lens, the angular separation increases by 0.022mas from the lowest to highest frequencies in the band, while | µ / µ | decreases by 0.0013. We fit a power law to the rela-tion between angular separation and wavelength over thisband for both the underdense and overdense lenses. Thebest fit power law exponents are shown in Table 1. In theunderdense case, the power law exponent is positive, mean-ing that at higher frequencies the images are closer together,while in the overdense case it is negative, so that at higherfrequencies the images are further apart. This is also ap-parent from Figs. 3a and 3b: Since ∆ n ∝ − ∆ n e λ , observingat longer wavelengths has the same effect as increasing thestrength of the lens, and one can see from Figs. 3a and 3bthat when the lens is stronger (or at lower frequencies) theangular separation is larger if the corrugated sheet is un-derdense, but smaller if the corrugated sheet is overdense.Brisken et al. (2010) statistically combine arclets in the sec-ondary spectrum of PSR B0834+06 and measure positive Table 1.
Power law fits of the form ∆ θ = B λ γ to the change inseparation with wavelength for a lens that produces two imageswith an angular separation of 10 mas and a flux ratio of 0.01at 314.5 MHz. The fit is done over 310.5 MHz to 342.5 MHz.While the exponents here are small, exponents of the same orderof magnitude have been measured over the same bandwidth with10 to 20% uncertainties by Brisken et al. (2010). ∆ n e [cm − ] γ -0.0067 0.0190.0067 -0.023 power law exponents, . ± . and . ± . , overthis 32-MHz band for two of three identified arclet groups(for the third group, they find that the separation does notvary with wavelength within the measurement uncertain-ties), suggesting that the sheet is underdense and indicatingthat the small exponents predicted by this model are mea-surable. The main phenomenon this model hopes to reproduce is theexistence of parabolic arcs in the secondary spectra of pul-sars, which can arise from highly anisotropic scattering at a
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MNRAS000 , 1–14 (2018) redicting Pulsar Scintillation thin sheet along our line-of-sight to the pulsar. Using globalVLBI to obtain the secondary cross-spectrum between twostations, the angular locations of the images relative to theline-of-sight image of the pulsar can be measured (Briskenet al. 2010). Under the thin sheet approximation, the dis-tance and velocity of the sheet can be determined from therelationship between the measured angular separations oftwo images and the Doppler frequency and delay of thefeature in the secondary spectrum resulting from the inter-ference between these images. If inverted arclets, evidenceof distinguishable images on the screen, are present in thesecondary spectrum, the location of these in the secondaryspectrum combined with the distance and velocity of thesheet can be then used to obtain the positions of the im-ages to ≈ accuracy (Brisken et al. 2010), a techniqueknown as ‘back-mapping’. Using additional techniques, suchas holography, the precision of these measurements can beimproved even further (Pen et al. 2014). The relative fluxesof inverted arclets in the secondary spectrum encodes therelative fluxes of the images of the pulsar on the scatter-ing screen, providing us with the rest of the information re-quired to constrain the model. To test temporal evolution ofthe scattering against this model, we also need to know thewhether the angular separation between an image and theline-of-sight image to the pulsar is increasing or decreasing.If the Doppler frequency of the arclet corresponding to animage is negative, then the pulsar is moving towards the im-age and the angular separation is decreasing. If the Dopplerfrequency is positive, then the pulsar is moving away fromthe image, and the angular separation is increasing.This model may also be applied to pulse echoes, suchas those observed in the Crab pulsar (eg. Backer et al. 2000;Lyne et al. 2001), PSR B2217+47 (Michilli et al. 2018) andat least one other pulsar (Oslowski et al. , in prep.). In thesecases, radio interferometry can be used to measure the an-gular position of the echo relative to the main pulse, whilethe pulse profile itself can be used to determine the magnifi-cation of the echo. Combining the angular information withthe delay of the echo relative to the main pulse allows oneto determine the geometry and velocities (if one has multi-epoch observations of the system) of the scattering system.By correlating the unlensed pulse with the echo, the phaseimparted by the lens can be retrieved. If the image is in-verted, the waveform of the image will be distorted (Dai &Venumadhav 2017), allowing a direct test for the inversion ofthe image in this case. Some pulsars have giant pulses, short,intense bursts of radiation, that exhibit scattering tails frominterstellar scattering. Giant pulses allow the response of thelens to be determined from the scattering of a single pulse(Main et al. 2017), making these systems even more promis-ing for detecting image inversion. − θ T / < θ < θ T / So far, we have considered only the portion of the crest where θ > θ T / . However, we expect additional images from theregion − θ T / < θ < θ T / The maximum extent of the lensalong the line-of-sight occurs at θ = θ T / , so that the gradi-ent ∇ x Z has opposite sign for | θ | < θ T / and θ > θ T / , and therefore the bending angle, which is proportional to thisgradient, switches direction at θ = θ T / .We can treat the region − θ T / < θ < θ T / as a separatelens, and we can parameterize the lens as being bounded bythe line z = (cid:112) ( R + T )( x + T / ) . (31)The thickness of the lens is then Z = (cid:112) ( R + T )( x + T / ) , (32)and, once again assuming the index of refraction is constantwithin the lens, the bending angle induced by the lens is ˆ α ( x ) = − ∆ n ∇ x Z (33) = − ∆ n (cid:115) R + Tx + T / (34) ˆ α ( θ ) = − ∆ n (cid:115) r + θ T θ + θ T / , (35)so that the lens equation for this system is: θ = β − s ∆ n (cid:115) r + θ T θ + θ T / . (36)From equation (36), we see that the region − θ T / < θ < θ T / corresponds to β ≥ θ T + s | ∆ n | (cid:113) r + θ T θ T if the lens is underdense,and to β ≤ θ T − s | ∆ n | (cid:113) r + θ T θ T if the lens is overdense.Once again, we can determine the magnification of theimage produced from the derivative of the lens equation: µ = (cid:18) d β d θ (cid:19) − (37) = (cid:18) − s ∆ n (cid:112) ( r + θ T )/ ( θ + θ T / ) / (cid:19) − . (38)We see that if θ T (cid:38) ( s | ∆ n | (cid:112) ( r + θ T )/ ) / , a very highlymagnified image can occur in the region − θ T / < θ <θ T / and would need to be considered. However, if θ T (cid:28)( s | ∆ n | (cid:112) ( r + θ T )/ ) / , which holds for the parameters in Sec-tion 3.1, the magnification is instead maximized when θ = θ T / , and reaches a value | µ | max ≈ θ / T s | ∆ n | (cid:112) ( r + θ T )/ . (39)For the parameters in Section 3.1 and ∆ n e = . cm − , | µ | max = − , much fainter than the images producedat θ > θ T / . We see from equation (39) that if θ T (cid:28)( s | ∆ n | (cid:112) ( r + θ T )/ ) / , the images produced by the region − θ T / < θ < θ T / are very faint compared to the unlensedimage.The constraints on the angular positions of the pulsar atwhich this faint image appears result in a minimum angularseparation between the faint image and the bright image, orline-of-sight image, of the pulsar, | ∆ θ | min (cid:39) θ T / − β ( θ T / ) (40) = s | ∆ n | (cid:115) r + θ T θ T / . (41) MNRAS , 1–14 (2018) D. Simard & U.-L. Pen
This is exact when the sheet is overdense, and the faint im-age from − θ T / < θ < θ T / appears simultaneously withthe line-of-sight image to the pulsar. If the sheet is un-derdense, the faint image appears simultaneously with thelensed image at positive β , but as this faint image from − θ T / < θ < θ T / appears only at large β , where the brightlensed image from θ > θ T / appears at θ ≈ β , this approxi-mation is very good. For the parameters in Section 3.1 and ∆ n e = . cm − , | ∆ θ | max ≈ mas. In contrast, in thesecondary spectrum of PSR B0834+06, scattered flux ap-pears up to an angular separation of 28 mas. Since thisis less than the minimum angular separation between theline-of-sight image of the pulsar and the faint image from − θ T / < θ < θ T / , we see once again that we can ignore theimages from this region of the lens. At larger thicknesses,the lensed images become brighter and move to lower angu-lar separations, where they can complicate the analysis andshould be considered.Due to the change in lensing direction at θ = θ T / , wemay be able to place independent constraints on the thick-ness of the sheet. Consider the case where a faint imagefrom a region θ > θ T / is moving towards smaller θ as thepulsar moves away from the crest. Once the image reaches θ = θ T / , if the pulsar continues to move further from thecrest, the lens will no longer be able to bend the pulsar lightinto our line-of-sight, and the faint image and correspond-ing echo or arclet in the secondary spectrum will disappear.This sets a maximum angular separation between the twoimages when θ maxsep . = θ T / . If this maximum angular sep-aration could be observed, then from the angular separationand magnifications of the images we could determine theparameter of the lens, the true position, β , and the lensedposition, θ min = θ T / from ∆ θ ref ( e.g. Section 3.2). If we knowthe distance to the lens, we can also calculate the physicalthickness of the lens, T . For the parameters in Section 3and ∆ n e = . cm − , the maximum angular separation is3000 mas, at which the flux ratio between the two imagesis − − . Typical secondary spectra analyses are not sen-sitive to angular separations larger than approximately 30mas, so the maximum angular separation is not expectedto be observable as a sudden disappearance of the lensedimage. In previous sections, we have considered the effects of a lensprofile with a discontinuous derivative. This is due to ourassumption that the density within the sheet is constant,so that the density does not vary smoothly across the sheet.We can relax this assumption with a more general treatment,with which we can consider any density profile through thesheet, ∆ n e , sheet ( d ) , where d is the depth through the unbentsheet, normal to the plane of the sheet.We write the bending angle, α as: α = − s λ π r e ∇ x ∆ N e ( x ) , (42)where ∆ N e ( x ) = ∫ ∞−∞ dz ( n e ( x , z )− n e , ) and n e , is the free elec-tron density in the ambient ISM. We can approximate N e ( x ) as the convolution of the profile of the corrugated sheet and the electron density profile through the sheet, ∆ n e , sheet ( d ) : ∆ N e ( x ) = ∫ ∞−∞ d X ∆ n e , sheet ( X ) l d x (cid:12)(cid:12)(cid:12)(cid:12) x − X . (43)This is a good approximation when the tangent to the cor-rugated sheet is near parallel with our line of sight to thepulsar, such as near the crest of a wave on the sheet. In orderto compare between different density profiles, we normalizethe electron density profile through the sheet so that ∫ ∞−∞ d d ∆ n e , sheet ( d ) = T ∆ n e , (44)where T = . AU and ∆ n e = . cm − .We consider a Gaussian form of the electron densityprofile, ∆ n e , sheet ( d ) = ∆ n e T √ π s exp (cid:18) − d s (cid:19) , (45)where s = T √ ( ) and ∆ n e , sheet ( d ) has a FWHM of T , as anexample of a smoothly-varying density profile and contrastthis with a top-hat electron density profile, ∆ n e , sheet ( d ) = ∆ n e Θ ( T / − d ) Θ ( T / + d ) , (46)where Θ is the Heaviside step function. This top-hat electrondensity profile, allows us to extend the model discussed inprevious sections of this paper, where we investigated a top-hat electron density profile using analytic approximations,to the entire lens, including those regions where our previousapproximations did not hold.The lens profiles are shown in Fig. 5, and the solutionsto the lens equation, θ ( β ) and µ ( β ) , are shown in Fig. 6, wherewe see that the behaviour at | θ | (cid:29) θ T (0.08 mas for these ex-amples) does not depend on the specifics of the density pro-file through the lens. However, when | θ | (cid:46) θ T , the bendingangles in the case of the top-hat sheet profile diverge. This isin contrast to the Gaussian sheet, which adheres to the odd-image theorem and produces additional images compared tothe top-hat sheet. These images are normally very faint butare highly magnified when d β d θ = . These highly-magnifiedevents occur for large bending angles, when the pulsar is farfrom the crest of the wave, and for only a small range in β , and thus in time. Typical secondary spectrum analysesare sensitive to | ∆ θ | (cid:46) mas, and therefore do not need toaccount for these highly magnified images. In addition, dueto the chromatic nature of the lensing, these events occurat a specific β for only a small range in frequency, and thusare expected to be much less apparent when observing witha wide bandwidth.We can also look at how the number of images pro-duced by the lensing system varies with both the observingfrequency and the offset between the pulsar and the crestof the wave, shown in Fig. 7. When observing these images,keep in mind that during the transition between regions ofone and three images events of high magnification occur. Asexpected from our discussion in previous sections, we seethat additional high-magnification events occur in the over-dense case compared to the underdense case. In practice, we often observe many lensed images of a pulsar.In the picture presented in this work, each of these images
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MNRAS000 , 1–14 (2018) redicting Pulsar Scintillation x / T N e / n e T GaussianTop Hat
Figure 5.
The lens profiles as a function of x for a top hat (orange dashed line) and a Gaussian (blue solid line) electron density profilethrough the sheet. Note that the lens profile is continuously differentiable for the Gaussian electron density profile, but that the lensprofile from a top hat electron density profile, like that considered analytically in the sections above, has a discontinuous derivative at x = − T / and x = T / . As a result, the corrugated Gaussian sheet adheres to the odd-image theorem while the corrugated top hat sheetdoes not. ( m a s ) Gaussian Top Hat40 20 0 20 40 (mas) ( m a s ) (mas) -5e-4-2e-402e-45e-4 (a) Underdense sheet ( m a s ) Gaussian Top Hat40 20 0 20 40 (mas) ( m a s ) (mas) -5e-4-2e-402e-45e-4 (b) Overdense sheet Figure 6.
The solution to the lens equation for the lens profiles in Fig. 5. From the top two panels of each subfigure, we see that when θ (cid:29) θ T , the lens behaviour does not depend on the electron density profile through the sheet apart from the extra image formed near θ = when the sheet has a Gaussian profile. In the bottom two panels, we have adjusted the axis to focus on region | θ | < θ T and highlightthe differences that arise between the two electron density profiles. We have drawn the extra image produced when a smooth Gaussiandensity profile through the sheet is considered with a thin line to distinguish it from the other images.MNRAS , 1–14 (2018) D. Simard & U.-L. Pen
400 200 0 200 400 (mas) f o b s ( G H z ) Number of images (a) Underdense sheet
400 200 0 200 400 (mas) f o b s ( G H z ) Number of images (b) Overdense sheet
Figure 7.
The dependence of the number of images produced on observing wavelength, λ obs , and the location of the pulsar relative tothe crest of the wave, β , for a sheet with a Gaussian electron density profile. Both an underdense (Fig. 7a) and overdense (Fig. 7b) sheetare considered. We see that this model adheres to the odd-image theorem, as there are only regions of 1 or 3 images. is from a different wave crest on the sheet. Each wave crestmay have a different value of ∆ n θ T √ r , and so the analysisabove can be done for every wave crest, or equivalently everyinverted arclet in the secondary spectrum or every echo ofthe pulsar.While in some pulsars the individual images are distin-guishable in the secondary spectrum as individual invertedarclets, in many cases these inverted arclets are not resolved,adding ambiguity to modeling the lensing in terms of indi-vidual wave crests. Holographic techniques ( e.g. Pen et al.2014; Walker et al. 2008) can retrieve the electric field dueto the combination of images, which may assist in identi-fying individual images, and we can additionally considerstatistical phenomena that arise when we have lensing frommultiple crests, such as the the scattering tail of a singlepulse, the spread of pulse power over time due to the de-lays imparted by the various crests. This requires numericalsimulations that take into account the distribution of wavesalong the sheet, and is deferred to future work.
Although we have focused on pulsar scintillation, the pic-ture discussed within this paper can also lead to symmetriclensing events. Consider two consecutive crests which ex-tend from the plane of the sheet in opposite directions. Ifthe crests are very close together, as shown in Fig. 8, thena very high column density through the lens is achieved at x = , and the column density profile is symmetric in x . Fur-ther quantitative analysis, outside of the scope of this paper,is necessary to confirm the qualitative similarities betweenthis set-up and the observed characteristics of symmetriclensing events such as ESEs. In particular, the ability of thismodel to predict both the light curves and frequency evolu-tion of these events must be studied. While these events areexpected to be generic, as with any corrugated sheet there is an inclination angle for which the line of sight to the pul-sar is parallel to the tangent through the sheet at x = , anapproximation of the prevalence of this event requires as-sumptions of both the distribution of plasma sheets in theISM and the distribution of wavelengths and amplitudes ofwaves along those sheets, and is thus beyond the scope of thepaper. However, this feature of the corrugated sheet modelpresented here may also explain why both symmetric andand asymmetric features are seen in the echoes of the Crabpulsar (Lyne et al. 2001). We have investigated the effects of a corrugated plasma sheetas the source of pulsar scintillation arcs. Using geometric op-tics, we calculate the number of images and magnificationsof these images as a pulsar moves behind a wave crest on oneof these sheets. We find that in the limit θ (cid:28) r / the lens canbe described by a single parameter, ∆ n θ T √ r , which can beconstrained from observations of the pulsar. Once this pa-rameter is known, this model can be used to predict how themagnifications and locations of the images will change overtime and frequency, providing a concrete test of this model.In particular, we see that over a small band of 32 MHz weexpect changes in the locations of the images comparable tothose measured by Brisken et al. (2010). We also see that weexpect very different behaviour from an overdense sheet oran underdense sheet: In the overdense case, the angular sep-aration between two images increases at higher frequencies,while in the underdense case it decreases at higher frequen-cies. Brisken et al. (2010) find that the separation decreaseswith frequency for two of three identified groups of images inthe secondary spectrum of PSR B0834+06, suggesting thatthe lens is underdense. There are other qualitative differ-ences between the underdense and overdense lenses: magni-fications greater than 1 occur only when the sheet is over- MNRAS000
Although we have focused on pulsar scintillation, the pic-ture discussed within this paper can also lead to symmetriclensing events. Consider two consecutive crests which ex-tend from the plane of the sheet in opposite directions. Ifthe crests are very close together, as shown in Fig. 8, thena very high column density through the lens is achieved at x = , and the column density profile is symmetric in x . Fur-ther quantitative analysis, outside of the scope of this paper,is necessary to confirm the qualitative similarities betweenthis set-up and the observed characteristics of symmetriclensing events such as ESEs. In particular, the ability of thismodel to predict both the light curves and frequency evolu-tion of these events must be studied. While these events areexpected to be generic, as with any corrugated sheet there is an inclination angle for which the line of sight to the pul-sar is parallel to the tangent through the sheet at x = , anapproximation of the prevalence of this event requires as-sumptions of both the distribution of plasma sheets in theISM and the distribution of wavelengths and amplitudes ofwaves along those sheets, and is thus beyond the scope of thepaper. However, this feature of the corrugated sheet modelpresented here may also explain why both symmetric andand asymmetric features are seen in the echoes of the Crabpulsar (Lyne et al. 2001). We have investigated the effects of a corrugated plasma sheetas the source of pulsar scintillation arcs. Using geometric op-tics, we calculate the number of images and magnificationsof these images as a pulsar moves behind a wave crest on oneof these sheets. We find that in the limit θ (cid:28) r / the lens canbe described by a single parameter, ∆ n θ T √ r , which can beconstrained from observations of the pulsar. Once this pa-rameter is known, this model can be used to predict how themagnifications and locations of the images will change overtime and frequency, providing a concrete test of this model.In particular, we see that over a small band of 32 MHz weexpect changes in the locations of the images comparable tothose measured by Brisken et al. (2010). We also see that weexpect very different behaviour from an overdense sheet oran underdense sheet: In the overdense case, the angular sep-aration between two images increases at higher frequencies,while in the underdense case it decreases at higher frequen-cies. Brisken et al. (2010) find that the separation decreaseswith frequency for two of three identified groups of images inthe secondary spectrum of PSR B0834+06, suggesting thatthe lens is underdense. There are other qualitative differ-ences between the underdense and overdense lenses: magni-fications greater than 1 occur only when the sheet is over- MNRAS000 , 1–14 (2018) redicting Pulsar Scintillation pulsarobserverlens xz Figure 8.
A geometry of the corrugated sheet which can produce symmetric events. Due to the orientation of the sheet combined withthe locations of the two crests, the lens has a symmetric column density profile that results in large bending angles at x = . Note inthe picture we are considering, the sheet only occupies a small percentage of the distance between the source and the observer, and theangle between the sheet and the line-of-sight to the pulsar is much less than one. dense, both images of the pulsar disappear when the pulsaris just behind the crest of the wave in the overdense case,and only the overdense lens produces inverted images.Qualitatively, there are two major differences betweenthis model and previous models of ESEs and pulsar scintil-lation arcs. Other authors ( e.g. Bannister et al. 2016; Clegget al. 1998; Pen & King 2012; Tuntsov et al. 2016) have con-sidered smoothly varying electron column densities, whichadhere to the odd image theorem and produce one or threeimages. In this work, we see that when we consider an abruptchange in density at the sheet boundary, the column densityconsidered is not continuously differentiable, and the lensproduces one or two images. When we consider a smoothdensity profile within the sheet, our lens does adhere tothe odd-image theorem. However, due to the thinness of thesheet considered, the additional images formed are faint atangular separations of interest to pulsar scintillation studies,and can be safely ignored. In addition, we have consideredan asymmetric lens that may explain asymmetric dispersionand scattering events, such as the anomalous dispersion mea-sure variations seen in PSR J1713+0747 (Jones et al. 2017;Lentati et al. 2016) and asymmetric echo events observed inthe Crab pulsar (Lyne et al. 2001).In future work, we will apply this test to observationsof pulsar scintillation and of pulsar echoes. The simplesttest comes from observations of pulsars with well-definedinverted arclets in their secondary spectra, as these arcletscan be mapped to images on the sky through VLBI observa-tions (Brisken et al. 2010) and therefore to individual wavecrests in the current sheet. Ideal observations will be at lowfrequencies where scintillation effects are strongest and overa wide bandwidth in order to measure the changes in the sec-ondary spectrum over frequency. We can also compare thismodel to observed changes in the secondary spectrum overtime. For this, we desire many observations of the pulsar on week to month long timescales. Finally, we can simulate andtest features of this model that arise when we consider manyimages of a single pulsar being created by multiple crests,such as the scattering tail of a highly scattered pulsar. Theresults of these tests will assess the consistency of the cor-rugated, closely aligned, sheet model with observations ofpulsar scintillation.
ACKNOWLEDGEMENTS
We thank Marten van Kerkwijk and Robert Main for manyvaluable discussions from the early stages of this work. Wealso thank Dan Stinebring and Barney Rickett for usefulsuggestions, and Peter Martin for helpful discussions. Wethank the referee for comments that have much improved thepaper. DS acknowledges funding from NSERC. The DunlapInstitute for Astronomy and Astrophysics is funded throughan endowment established by the David Dunlap family andthe University of Toronto.
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