The microarcsecond structure of an AGN jet via interstellar scintillation
J.-P. Macquart, L. E. H. Godfrey, H. E. Bignall, J. A. Hodgson
aa r X i v : . [ a s t r o - ph . C O ] J a n Draft version March 4, 2018
Preprint typeset using L A TEX style emulateapj v. 5/2/11
THE MICROARCSECOND STRUCTURE OF AN AGN JET VIA INTERSTELLAR SCINTILLATION
J.-P. Macquart , L.E.H. Godfrey, H.E. Bignall, J.A. Hodgson ICRAR/Curtin University, Bentley, WA 6845, Australia
Draft version March 4, 2018
ABSTRACTWe describe a new tool for studying the structure and physical characteristics of ultracompactAGN jets and their surroundings with µ as precision. This tool is based on the frequency dependenceof the light curves observed for intra-day variable radio sources, where the variability is caused byinterstellar scintillation. We apply this method to PKS 1257–326 to resolve the core-shift as a functionof frequency on scales well below ∼ µ as. We find that the frequency dependence of the position ofthe scintillating component is r ∝ ν − . ± . (99% confidence interval) and the frequency dependenceof the size of the scintillating component is d ∝ ν − . ± . . Together, these results imply that thejet opening angle increases with distance along the jet: d ∝ r n d with n d > .
8. We show that theflaring of the jet, and flat frequency dependence of the core position is broadly consistent with a modelin which the jet is hydrostatically confined and traversing a steep pressure gradient in the confiningmedium with p ∝ r − n p and n p &
7. Such steep pressure gradients have previously been suggestedbased on VLBI studies of the frequency dependent core shifts in AGN.
Subject headings: galaxies: jets — techniques: high angular resolution — quasars: individual(PKS 1257 − INTRODUCTION
The brightest, most compact feature of an AGNjet, the “core”, is identified with the part of thejet at which the optical depth ( τ ν ) is of order unity(Blandford & K¨onigl 1979), and is often referred to asthe τ ν = 1 surface, or photosphere. Due to positionalvariation of the opacity in the jet and/or surround-ing medium, the position of the τ ν = 1 surface is fre-quency dependent, and therefore, so too is the absoluteposition of the core (eg. K¨onigl 1981; Lobanov 1998;Kovalev et al. 2008; Sokolovsky et al. 2011). The fre-quency dependent position of the core is referred to sim-ply as the core shift.The core shift effect provides an observational toolwith which to investigate the structure and physical con-ditions in parsec-scale AGN jets. Moreover, modellingthe effect may provide information about the confine-ment mechanism and pressure gradients in the externalmedium. The core shift effect is also relevant to the questfor high precision absolute astrometry for the Interna-tional Celestial Reference Frame, as it can introduce asignificant offset in positions determined using group de-lay measurements (Porcas 2009).The magnitude of the core shift between 2.3 and8.4 GHz is typically of order a few hundred µ asor less (O’Sullivan & Gabuzda 2009; Sokolovsky et al.2011; Pushkarev et al. 2012), and therefore detecting thiseffect requires very high accuracy registration of im-ages at two or more frequencies. Despite the techni-cal challenges, the core-shift effect obtained from VLBIimaging has been reported for an ever-increasing num-ber of radio galaxies (Marcaide & Shapiro 1984; Lobanov1998; Kovalev et al. 2008; O’Sullivan & Gabuzda 2009; ARC Centre of Excellence for All-Sky Astrophysics (CAAS-TRO) Now at Max-Planck-Institute f¨ur Radioastronomie, Auf demH¨ugel 69, 53121, Bonn, Germany
Sokolovsky et al. 2011; Pushkarev et al. 2012). More re-cently, Kudryavtseva et al. (2011) have employed an in-direct method to measure the core shift effect based onfrequency-dependent time lags of flares observed usingsingle-dish data spanning several years.The frequency dependence of core position is typicallyassumed to follow a power-law of the form r ∝ ν − /k r .In many sources for which core shifts can be measuredwith VLBI imaging, the absolute core position variesapproximately with the inverse of the frequency (i. e. k r = 1) (O’Sullivan & Gabuzda 2009; Sokolovsky et al.2011). This situation is consistent with the standardmodel of a conical jet in which the plasma is in a stateof equipartition between particle and magnetic energydensities (Blandford & K¨onigl 1979). However, valuesof k r much greater than unity are observed in somesources, which may be due to free-free absorption in theimmediate vicinity of the jet, or due to rapid changesin pressure in the external medium if hydrostatic con-finement is important (Lobanov 1998). Lobanov (1998)has shown that while k r ∼ k r increasestowards the jet base. Kudryavtseva et al. (2011) haveshown that the value of k r is time-dependent, and corre-lated with flux density. Finally, the pc-scale jet of M87is observed to deviate from a conical geometry near tothe core (Asada & Nakamura 2012). Further investiga-tion into the frequency dependence of core position istherefore warranted, and highly relevant to the study ofultracompact AGN jets.Here we present a potentially powerful new methodfor the study of ultracompact jets in AGN, which enablessimultaneous measurement of the core shift effect and jetgeometry to very high precision. This new technique usesauto- and cross-correlation analysis of multi-frequencylight curves of a rapidly scintillating AGN to measurethe frequency dependence of the position and size of thescintillating component.In section 2 we present the observations and data anal-ysis. In Section 3 we discuss the mathematical formula-tion of the auto- and cross-correlation analysis, and de-rive the frequency dependent source position and size forPKS 1257-326. In Section 4 we discuss the implicationsof our findings, and model the jet in terms of a hydrostat-ically confined jet traversing a steep pressure gradient.Finally, in Section 5 we present our conclusions. OBSERVATIONS AND DATA CALIBRATION
PKS 1257 −
326 was observed at the ATCA for tenhours on 15 January 2011, with two 2 GHz bands, cen-tred on frequencies of 5.5 and 9.0 GHz. The output dataincluded all four polarisation products and 2048 spectralchannels each 1 MHz wide in each of the two bands. Flag-ging and calibration of the data were performed using the
Miriad software package. The ATCA primary calibratorPKS 1934 −
638 was used to correct the overall flux den-sity scale and the spectral slope. In order to solve for thebandpass and to correct gain amplitudes as a function oftime and pointing for each antenna, the secondary cali-brator PKS 1255 − ◦ from PKS 1257 − − ∼
1% or better, based on the PKS 1255 − − −
638 iswell known, archival data on the secondary calibratorPKS 1255 −
316 shows it to be variable by up to ∼ −
326 is relatively smooth across the entirerange of frequencies, suggesting that the calibration isaccurate. Moreover any residual constant offsets in theflux density scale which may be present between differentfrequencies have no effect on the cross-correlation analy-sis presented in this paper.Figure 1 shows the large, rapid intra-hour variationsexhibited by PKS 1257 − DERIVATION OF JET STRUCTURE
Interpretation
The rapid fluctuations observed in the centimetrewavelength flux density of the intra-hour variable quasarPKS 1257 −
326 are due to interstellar scintillation (Big-nall et al. 2003, 2006). This is established from the mea-surement of a time delay in the arrival time of the vari-ations between telescopes separated by several thousandkilometers. The timescale of the variations also under-goes an annual modulation due to relative motion of theEarth about the Sun, which in turn moves relative to theinterstellar medium responsible for the variations.Inspection of Figure 1 reveals that there is a time offsetin the arrival time of the intensity variations between dif-ferent frequencies, with the variations at high frequencyleading those at lower frequencies. This behaviour is con-sistent with observations in previous epochs. Bignall etal. (2003) reported that variations at 4.8 and 8.6 GHz areclosely correlated, and there is a systematic time delaybetween the variations at these two frequencies. Themagnitude of the time delay was observed to follow anannual cycle which is not identical to the annual cycle invariability timescale but, it was argued, can be explainedon the basis of it, as we discuss below.PKS 1257-326 was monitored with the ATCA at 4.8and 8.6 GHz at 19 epochs between 2001 February and2002 April. Typical observations were over a 12 hourperiod, and 6 epochs covered 2 ×
12 hours in 48 hoursessions as part of a multi-source monitoring program.The minimum duration of each light curve is ∼
10 timesthe length of the characteristic timescale. In every oneof these epochs, the time delay between 4.8 and 8.6 GHzhas the same sign, with 8.6 GHz variations always lead-ing. Moreover, a clear annual cycle is observed in thetwo-frequency time delay, with the longest delays beingobserved from late July to mid-August. Such an annualcycle is expected for a core-shift which remains stableover the course of the year, and the observed annual cy-cle is well modelled by such a shift on a scale of order10 µ as (Bignall et al. 2003), although the precise magni-tude and direction of the core shift could not be uniquelydetermined. These data provide strong evidence that thecore shift effect dominates over any refractive effects or”jitter” in the ISS pattern.We argue that the temporal offset in the present dataarises as a direct consequence of an angular offset be-tween two compact components within the scintillatingsource. The effect may be understood as follows. Whenan angular separation, θ , is present between two com-ponents, this results in a spatial displacement of theirrespective scintillation patterns across the plane of anobserver by an amount D θ , whe re D is distance be-tween the observer and the scattering material (Little &Hewish 1966). Since the scintillation patterns are in mo-tion across this plane with some velocity v , the resultis a separation in the arrival time of the scintillationsassociated with each component. In the present case, adisplacement in the lightcurves between closely-spacedfrequencies arises because there is an angular offset inthe image centroids between the respective frequencies.For any pair of frequencies, the time delay is, in terms ofthe centroid offset θ (see Appendix A),∆ t = − D θ · v + ( R − D θ × ˆ S )( v × ˆ S ) v + ( R − v × ˆ S ) , (1) PKS 1257-326, ATCA, 2011 Jan 15
16 18 20 22 24Time (hr UT)0.140.160.180.200.220.24 F l u x den s i t y ( Jy ) Fig. 1.—
PKS 1257 −
326 flux density measurements as a function of time, plotted with 1-minute averaging, showing each 128 MHz bandin a different colour. Frequency increases from top to bottom. The heavy black points show the average of each 2 GHz wide-band dataset,corresponding to the centre frequencies of 5.5 and 9 GHz. where R is the anisotropy ratio of the scintillation patternand ˆ S = (cos β, sin β ) is the direction of its major axis,which we measure with respect to the RA axis. The scin-tillation parameters have been derived from annual cycleand two-station time-delay measurements, and are givenin Table 1. It is evident that this delay is modulated bothby the annual cycle in the magnitude of v and by changesin the angle of the velocity vector with respect to θ ; thislatter effect causes the annual cycle experienced by ∆ t to differ from the annual cycle in scintillation velocity.Phase gradients in the ISM may, in principle, also causetemporal offsets of lightcurves as a function of frequencyin a scintillating source. However, the offset observedhere is difficult to attribute to such an extrinsic causefor several reasons: (i) the sense and magnitude of thedelay is constant throughout the dataset; upon dividingthe dataset in two halves (in time) and deriving timeoffsets based on these two halves separately, we findthe same offsets to within the margin of error of theestimates. (ii) The delay is observed over a timescaleof 10 hours, whereas refractive phase gradients in theISM in the regime of weak scintillation for a Kolmogorovspectrum of phase inhomogeneities would occur on thetimescale associated with the scintillations, and the timeoffset should converge to zero as the average is performedover an increasing number of scintles. Any small jitter inthe offset between individual scintles appears to be dom-inated by the systematic offset. (iii) An annual cycle inthe time offset is reported by Bignall et al. (2003), in- dicating that the offset persists on a timescale of greaterthan a year. Time delay measurement
To determine the relative time delay between each pairof lightcurves, I ( t, ν ) and I ( t, ν ), we computed thecross-correlation function, C (∆ t ; ν , ν ) = h [ I ( t ′ , ν ) − ¯ I ( t, ν )][ I ( t ′ + ∆ t, ν ) − ¯ I ( t, ν )] i p var[ I ( t, ν )]var[ I ( t, ν )] . (2)A peak in the cross correlation at positive delay, ∆ t ,indicates that the fluctuations at frequency ν precedethose at ν . We fitted a gaussian of the form, C ( t ) = A exp (cid:20) − ( t − t ) B (cid:21) , (3)to the inner part of the cross-correlation function, C (∆ t ; ν , ν ) (equivalent to the auto-correlation func-tion), to obtain an estimate of the time delay betweeneach frequency-lightcurve pair and its associated error.An example cross-correlation function and its associatedfit is shown in Fig. 2. Typical errors in the estimated de-lay are 50 s. The derived delays as a function of ν and ν are shown in Fig. 3. The estimated 50 s uncertainties inthe delay between each frequency pair are derived fromleast-squares fitting of a gaussian to the peak of the de-lay, thus the errors are directly related to the width of TABLE 1The parameters of the scintillations in PKS 1257 − Parameter a Symbol B06 value W09 valuescreen distance D
10 pc 10 pcscintillation velocity, v , v α .
12 km s − . − on observation date v δ − .
05 km s − . − anisotropy ratio R
12 6anisotropy orientiation β − ◦ − ◦ a Parameters are derived on the basis of the annual cycle in the vari-ability timescale (Bignall et al. 2003) and measurements of the timedelay observed between two stations (Bignall et al. 2006, henceforthB06). An alternate fit to the scintillation data provided by Walker,de Bruyn & Bignall (2009) (henceforth W09) is also listed. The scin-tillation velocity quoted here is the addition of Earth’s relative to theSun velocity on the observation date and the velocity of the scatteringmedium relative to the Sun. the peak in the cross-correlation functions . These un-certainties are in turn derived from the least squares fitto the cross-correlation function; the errors in the indi-vidual points in the cross-correlation function are dom-inated, at low time lags, by Poisson errors associatedwith the number of independent measurements of thecross-correlation measureable from the lightcurve. Thiserror is indicated by the scatter between points in theCCF. Fitting to the CCF yields a typical formal errorof ≈
20 s. However the high degree of cross correlationbetween the lightcurves in our observations means thatthe error in the time delay estimate for any given fre-quency pair is not completely independent from the timedelay estimate of any other frequency pair. It is neces-sary to take the cross-correlation into account because itbecomes an important factor when assessing the signifi-cance of the fit to the frequency dependence of the timedelay using a least-squares approach (see below), whichis the primary reason to estimate the error in the timelag. Failure to take into account this interdependence intime lag estimates would lead to an overestimate of thesignificance of the fit to the frequency dependence of thetime shift. It was found empirically, from examinationof the reduced chi-squared in the fit procedure that thiscross-correlation is taken into account in the error anal-ysis with errors that are 2 . ν is fitted to a power law, ∆ θ = Aν − ζ ,where A and ζ are constants to be determined. Sincethe time delay is linearly proportional to | θ | , we fitted afunction of the form ∆ t ( ν , ν ) = K ( ν − ζ − ν − ζ ) to thedelays. The best-fitting parameters are ζ = 0 .
10 and K = (1 . ± . × s MHz ζ , with the 99% confidenceinterval of ζ extending over the range [ − . , . . This error is significantly smaller than the time delay errorsthat might be estimated on the basis of the data presented in Big-nall et al. (2003). As is evident from Fig. 2 in Bignall et al. (2003),the data in most of those epochs contain considerably few scintlesfrom which to estimate the time lag. Thus one would expect theerrors to be larger relative to the 2011 data. The dataset obtainedon 4 Jan 2001 contains a comparable number of scintles, but herethe temporal sampling of the lightcurve was too sparse to estimatea time delay. The 99% confidence limits are calculated, in the conventionalway, from the change in the sum of the squares of the residuals
The best-fitting exponent indicates a core-shift depen-dence on frequency that is much shallower than the value ζ = 1 typically found in other quasars on the basis ofVLBI measurements (e.g. O’Sullivan & Gabuzda 2009;Sokolovsky et al. 2011).We also measure the timescale of the scintillations,which is derived from the parameter B in eq. (3) (andwith ν = ν ). Figure 4 shows that the timescale is well-fit by a relation scaling as ν − . ± . ; a fit using abroken power law reveals the both the 4-5 and 8-9 GHztimescales follow the same scaling with frequency withinthe (small) margin of error.In the regime in which the angular size of the source, θ src , exceeds the angular size of the Fresnel scale at thedistance of the scattering screen, θ F = p c/ πνD , thescintillation timescale is linearly proportional to the sizeof the scintillating component. In the opposite regime, θ src < θ F , the scintillation timescale is determined by Dθ F /v = r F /v , which scales as ν − . .We wish to determine whether the timescaleof the scintillations measures θ F or θ src .The expected Fresnel crossing timescale is1 . × ( D/
10 pc) / ( ν/ − / ( v/
54 km s − ) − s.The observed scintillation timescale at 5.0 GHz is1 . × s, so if the source is unresolved there mustbe an error in the nominal scintillation parameters; ifthe scintillation velocity is held at its nominal valueone must have D >
19 pc, or if the screen distance isheld at its nominal value one must have v <
40 km s − .The latter option is viable for the scintillation velocityderived by Walker, de Bruyn & Bignall (2009), so weconclude that it is plausible that the source is unresolvedby the scintillations. However, the fact that the scalingof scintillation timescale is significantly different from ν − . suggests that the source is at least partiallyresolved in any case. The smooth trend evident in bothtimescale and scintillating amplitude with frequencysuggests that the source remains resolved over the entirefrequency range 5-10 GHz.Although tangential to the objectives of this paper, wecan, in addition, estimate the amplitude of the scintil- of the fit. Now, in practice, since the lightcurves are correlatedbetween frequencies, the time lag estimate between adjacent fre-quency pairs are strictly not independent, and this influences thechi-squared estimate. However, this effect is treated by accountingfor the cross-correlation in the estimates of the time delays for eachfrequency pair. C r o ss c o rr e l a ti on Fig. 2.—
The cross correlation between the lightcurves at 8040and 4540 MHz, and the associated best-fit gaussian. The peak ofthe gaussian represents the location of the time-delay. The positiveoffset of the peak indicates that the variations at 8040 MHz leadthose at lower frequency. lating component of the source. For an extended sourceof size θ src in the regime of weak scintillation caused bya Kolmogorov spectrum of turbulent fluctuations, theobserved rms, h δI i / can be expressed in the form(Narayan 1992), h δI i / h I i ≈ (cid:18) r diff r F (cid:19) / ((cid:16) θ F θ src (cid:17) / θ src > θ F θ src < θ F , (4)where h I i is the flux density of the scintillating compo-nent of the source and r diff ∝ ν / is the diffractive scale,which is determined by the properties of the interstellarturbulence (see Narayan 1992). In the regime of interme-diate scattering one has r diff ≈ r F , with equality at thetransition frequency, which likely occurs in the range 3-7 GHz based upon the modelling of Walker (1998). Onethen solves for h I i using the measurements of h δI i / .Assuming that the source size exceeds the Fresnel angu-lar scale, one determines θ src /θ F from the ratio t scint /t F .We performed a fit using the measured values of h δI i and t scint , and subject to the assumption that r diff = r F at 4 GHz, to derive a rough estimate of the flux densityof the scintillating component: S scint = 19 (cid:16) ν (cid:17) . mJy . (5)We caution that the spectral index of the component de-rived here is only approximate because the expression forthe modulation index in eq.(4) is only approximate at fre-quencies where r diff ≈ r F . A more sophisticated estimatewould employ a more complicated approximation to themodulation index near the transition frequency and takeinto account the anisotropy of the scintillations. Overall source scale
Although the frequency scaling of the core position andsize are the primary results of our analysis, it is possibleto roughly relate these measurements back to physicalscales within the source.The Schwarzschild radius of the black hole at the cen-tre of PKS 1257 −
326 is R S = 2 . × M m, whereM = 10 M ⊙ is the BH mass, which is estimated frommeasurements of the broad line emission width to be10 . M ⊙ (D’Elia, Padovani & Landt, 2003). At theredshift of PKS 1257 − z = 1 . µ as subtends ν GHz40006000800010000 ν GHz 0200400600Time offset s
Fig. 3.—
The time delay associated with each pair of lightcurvesmeasured at frequencies ν and ν . The red surface correspondsto the best-fitting model of the form ∆ t = A ( ν − ζ − ν − ζ ) (see textfor details). W i d t h s Fig. 4.—
The timescale of the scintillations as a function of fre-quency. The line represents the best-fitting model of the form τ = Kν − β (see text for details). Note that the intrinsic scat-ter between adjacent points is smaller than the error bars: this isbecause the lightcurves are highly correlated leading to a smallerscatter than is reflective of the errors. . × − pc, equivalent to 89 /M Schwarzschild radii .The time delay is related to angular structure in thesource using eq. (1) and the scintillation parameters inTable 1. This also depends on the angle, ξ , that θ makeswith the right ascension axis, which is unknown. For atime offset ∆ t , the expected amplitude of the angularoffset is given by | θ | = ∆ t (cid:26) (27 . ξ + 38 . ξ ) − µ as , B06 , (43 . ξ + 61 . ξ ) − µ as , W09 , (6)where the two solutions denote the scintillation parame-ters found by Bignall et al. (2006) (B06) and Walker etal. (2009) (W09). For instance, the angular separationimplied by the 520 s delay observed between the lowest(4540 MHz) and highest (9960 MHz) frequency bands ofour observations is 19 µ as (12 µ as) for the scintillationparameters of B06 (W09) if ξ = 0 (i.e. if the angular sep-aration is aligned parallel to the right ascension axis). We use H = 70 km s − Mpc − , Ω m = 0 .
27 and Ω Λ = 0 . This translates to a physical scale of 0.16 pc (0.10 pc) atthe source.However, these estimates are subject to uncertaintiesin both the screen distance and the orientation of theseparation ξ . For instance, doubling D above its nominalvalue of 10 pc would result in estimates of | θ | lower bya factor of two. They thus serve only as a guide to theorder of magnitude of scales which are probed by thesemeasurements.In the same vein, we also estimate the angular scale ofthe source from the variability timescale, using eq. (A10)in Appendix A to find the characteristic angular scale ofthe scintillation pattern, θ src ( ν ) = (cid:20) (cid:21) (cid:16) ν .
96 GHz (cid:17) − . (cid:18) D
10 pc (cid:19) − µ as (cid:20) B W (cid:21) . (7)The minor axis of the scintillation pattern has a size θ src / √ R while the major axis has a characteristic sizeof θ src √ R . DISCUSSION
Observational Constraints
The foregoing analysis indicates that the centroid ofthe scintillating component has a frequency dependenceof the form: r core ∝ ν − . ± . . In the terminology ofK¨onigl (1981) and Lobanov (1998), r core ∝ ν − /k r , wehave k r >
3. We proceed under the assumption that thesource is resolved, as suggested by the frequency depen-dence of the scintillation time-scale (Section 3.2). In thatcase, the jet diameter has a frequency dependence of theform: d ∝ ν − . ± . . Taken together, these results im-ply d ∝ r n d core with n d > .
8. The scenario implied by ouranalysis is illustrated in Figure 5. This is in contrast tothe frequency dependence of the core position in VLBIstudies: typically r core ∝ ν − , consistent with expecta-tions for a conical jet in which the particle and magneticenergy densities are in equipartition (Sokolovsky et al.2011). !" $% ’" τ ν !"! r core ∝ v − k r : k r > d ∝ r coren d : n d > Fig. 5.—
A cartoon illustrating the constraints implied by ouranalysis.
Interpretation in terms of hydrostatic jetconfinement in a steep pressure gradient
Lobanov (1998) suggested that the changes in jet opac-ity observed for a sample of sources appear consistentwith a self-absorbed jet propagating through a regionwith a steep pressure gradient of the form p ∝ r − .Here we consider such a model to explain the inferred frequency dependence of the core size and position inPKS 1257–326. Specifically, we consider whether sucha model can simultaneously account for the increase inopening angle along the jet ( d ∝ r n d core with n d > . r core ∝ ν − /k r , with k r > p ∝ r − n p , are (see Appendix B) n d = n p k r = ( n p / . α )(2 + m B ) − − (1 . α ) n δ . αα core = 52 − k r h n p (cid:16) m B (cid:17) + n δ i Here m B is a parameter determined by the magnetic fieldgeometry as discussed in Appendix B, and α is the op-tically thin spectral index which directly relates to theelectron energy distribution N ( γ ) ∝ γ − (2 α +1) , as op-posed to α core , which is the optically thick spectral indexof the self-absorbed core. We assume α = 0 . m B = 1(i. e. a predominantly toroidal magnetic field configu-ration) and take n p = 7 . n d = 1 .
8. In thatcase, the jet Lorentz factor goes as Γ / Γ = ( r/r ) . ,so that for reasonable values of the jet viewing angle(5 ◦ < φ < ◦ ) and initial Lorentz factor (2 . Γ . − . . n δ . .
5, which implies:2 . . k r . . , (8)0 . . α core . . . There is excellent agreement between the predictedand observed frequency dependence of core position andradial dependence of core size. The agreement be-tween the predicted and observed core spectral indexis poorer, but this is perhaps unsurprising since, as re-marked above, observational determination of the corespectral index is marred by uncertainties in the modelused to derive it. We therefore suggest that this simplemodel of hydrostatic confinement is broadly consistentwith the data, provided n p & .
2. The required pressuregradient in the external medium ( n p & .
2) appears verysteep, however, such pressure gradients have previouslybeen suggested in studies of core shifts in AGN (Lobanov1998).Finally, we note that the frequency dependence of coreposition may also be influenced by free-free absorptionin the immediate environment of the jet (e. g. Lobanov1998). However, we find no evidence for a significantrotation measure, and such an interpretation cannot ac-count for the flaring jet geometry, so that an additionalmechanism, such as entrainment, would also be requiredto explain these results. CONCLUSIONS
We have developed a new technique to probe the struc-ture of AGN jets on micro-arcsecond scales, by usinginterstellar scintillation to simultaneously determine theshift in the position of the AGN core as a function of fre-quency, and the frequency scaling of the core size, to highprecision. This approach is amenable to sources whichharbour compact ( . µ as) features.This method was applied to broadband observationsof PKS 1257–326 with the Australia Telescope CompactArray. The scaling of the core-shift is found to be re-markably shallow with frequency; the best fit to the posi-tion of the scintillating component in the source scales as r ∝ ν − . , with the 99% confidence interval of the indexextending over the range [ − . , . r ∝ ν − . The scaling of the jet size isalso determined, based on the scaling of the scintillationtimescale with frequency. This shows that the jet sizescales formally as d ∝ ν − . ± . . It is possible to de-termine the scaling of the core shift and jet diameter tohigh precision because they do not depend critically oncomplete knowledge of the properties of the scatteringmedium responsible for the scintillations. Determina-tion of the absolute physical scale of the core shift does,however, require knowledge of the scintillation param-eters, and we can only determine these quantities ap-proximately. The observed 520 s time offset between thescintillations at the the lowest (4540 MHz) and highest(9960 MHz) frequency bands of our observations impliesan angular separation of & µ as, for an assumed scat-tering screen distance of 10 pc. This translates to a phys-ical scale of ∼ .
10 pc at the source. This physical scaleprobed is an order of magnitude smaller than typical coreshifts obtained with VLBI measurements. We furthernote that this technique easily detects the core-shift be-tween frequency pairs separated by only ∼
300 MHz, thusproviding sub-microarcsecond resolution of the jet struc-ture.A major conclusion arising from our analysis is thatthat the often assumed frequency scaling of core posi-tion, r core ∝ ν − (e. g. Pushkarev et al. 2012), may notbe applicable to all sources, in line with similar findingsof Lobanov (1998) and Kudryavtseva et al. (2011). Fur-ther, our results hint at a physical difference between per-sistent intra-day variable (IDV) sources, and the broaderpopulation of AGN.To place these results in a physical context, we haveexplored a simple model based on a hydrostatically con-fined jet traversing a pressure gradient. The pressureprofile implied by this model is steep: p ∝ r − n p ; n p & r core ∝ ν − scaling observed in sources studied withVLBI may arise because different types of source lendthemselves to different types of analysis. Only the largestcore shifts can be detected with VLBI imaging. In ananalysis of 277 objects by Kovalev et al. (2008), only10% gave reliable core shift measurements. The sourcesused in the detailed, multi-frequency VLBI study of thecore shift effect by Sokolovsky et al. (2011) were selectedbased on their known large core shift, in addition to theirbright optically thin jet features that were required toenable accurate registration of the images.More specifically, we suggest that the peculiar natureof the core-shift frequency dependence in PKS 1257–326is related to a number of other remarkable source prop-erties. The implied brightness temperature of the scin-tillating component in PKS 1257-326 is high, > K(Bignall et al. 2006), and the source has exhibited suchbright emission at least as long as IDV has been observed,since 1995. The persistence of this IDV over more than15 years is relatively rare amongst IDV sources. Duringthe 4-epoch, one year duration MASIV survey, Lovell etal. (2008) found that sources which consistently exhibitedIDV over all four epochs accounted for only 20% of allIDV sources observed. Moreover, the stability of the an-nual cycle in the source variability timescale (Bignall etal. 2003, 2006) suggests that the size of the source is re-markably stable on a timescale of years. Physically, thisimplies that the ultracompact jet in this source is bothremarkably bright and stable. It is tempting to spec-ulate, therefore, that the hydrostatic confinement pro-vided by the strong pressure gradient suggested by oursimple model is responsible for the observed stability ofthe jet.The Australia Telescope is funded by the Common-wealth of Australia for operation as a National Facilitymanaged by CSIRO. The observations presented herewere made by JAH as part of the CSIRO Astronomy& Space Science (CASS) Vacation Scholarship Program.JAH thanks Dominic Schnitzeler for assistance with theobserving setup. Parts of this research were conductedby the Australian Research Council Centre of Excellencefor All-sky Astrophysics (CAASTRO), through projectnumber CE110001020.
Facilities:
ATCA ()
APPENDIX
DERIVING SOURCE STRUCTURE FROM THE TIME DELAY BETWEEN SCINTILLATION LIGHTCURVES
Here we relate the time delay observed between the scintillations at two frequencies to the angular displacement θ between the centroids of the source emission at the two frequencies. The time delay depends not only on the velocity,but also on the axial ratio, R , and orientation of the interstellar scintillation pattern.For intensity fluctuations ∆ I ( r , t ) measured at two locations r ′ and r ′ + r on the observer’s plane at time t , theintensity correlation function is defined as, ρ ( r , t ) = h ∆ I ( r ′ , t ′ )∆ I ( r ′ + r , t ′ + t ) ih ∆ I i . (A1)We follow the treatment of Coles & Kaufman (1978), in which the simplest approximation is that the contours of equalspatial correlation comprise a family of similar ellipses, and the model for the spatial correlation function takes theform ρ ( r ,
0) = f (cid:18) | C r | σ (cid:19) , (A2)where f is a monotonically decreasing function of | C r | . If the scintles are elliptical with axial ratio R and x axis isinclined at an angle β with respect to the major axis, then C takes the form C = (cid:20) cos β/ √ R sin β/ √ R −√ R sin β √ R cos β (cid:21) . (A3)The scintillation pattern is assumed frozen onto a screen that moves past the observer at velocity v , so there is adirection relation between the spatial and temporal dependence of the autcorrelation, ρ ( r , t ) = f (cid:18) | C ( r − v t ) | σ (cid:19) . (A4)Surfaces of constant correlation correspond to curves of constant | C ( r − v t ) | . The maximum of ρ ( r , t ) occurs at atime lag ∆ t given by ∆ t = a · r (A5)where a = ( C T C ) · v | C · v | . (A6)Equations (A2) − (A6) are derived in Coles & Kaufman (1978), and are shown here for completeness.Now suppose a source possesses similar structure at two frequencies ν and ν , but that they are displaced by anangle θ . Then the scintilation fluctuations at ν , ∆ I ( r , t ), are identical to those at ν , ∆ I ( r , t ), except that they aredisplaced by a linear scale D θ (see, e.g. Little & Hewish 1966): I ( r ) = I ( r − D θ ) , (A7)where D is the scattering screen distance. The cross-correlation between I and I takes the form h ∆ I ( r ′ , t ′ )∆ I ( r ′ + r , t ′ + t ) ih ∆ I ∆ I i = h ∆ I ( r ′ , t ′ )∆ I ( r ′ + r − D θ , t ′ + t ) ih ∆ I ∆ I i ≡ ρ I I ( r − D θ , t ) . (A8)Once again, we assume that ρ I I is a montonically decreasing function of r that takes the form given by equation(A2). Now, if the scintillations are measured at identical locations, we are interested in the maximum of ρ I I ( − D θ , t ).In this case the time delay measured between I and I is the same as that given by eq. (A5) above, with r replacedby − D θ , namely, ∆ t = − D a · θ = − D θ · v + ( R − D θ × ˆ S )( v × ˆ S ) v + ( R − v × ˆ S ) . (A9)where ˆ S = (cos β, sin β ) is the direction along which the scintles are oriented. The cross product of a vector with ˆ S isthe component of that vector that points orthogonal to the elongation axis. For instance, r × ˆ S is the component of r orthogonal to the long axis of the scintillation pattern.Finally, we note that the timescale of the scintillations can also be derived from the foregoing formalism, and is givenby, t scint = σ √ R q v + ( R − v × ˆ S ) , (A10)where σ , defined by eq. (A2), is the overall scale factor of the scintillation pattern. CORE SHIFT IN THE PRESENCE OF A PRESSURE GRADIENT
As discussed by Lobanov (1998), the frequency dependence of core position may be influenced by the pressuregradients in the external medium if hydrostatic confinement is important. Lobanov (1998) plotted the frequencydependence of core position as a function of the power-law index of the pressure profile, but neglected the effect of thechanging Doppler factor along the jet. Accordingly, we present a derivation of the properties of the core (diameter,distance along the jet, and spectral index) for a simple model of a hydrostatically confined jet in the presence of apower-law pressure profile, accounting for the effect of a radially dependent Doppler factor.Let L be the path length through the source, d the diameter of the jet perpendicular to the jet axis, D A theangular size distance, φ the jet viewing angle, and α ′ ν ′ the absorption coefficient in the source co-moving frame, atthe rest frequency ν ′ = zδ ν . Assuming the jet opening angle is small compared to the jet viewing angle (that is, weapproximate the jet as a cylinder), we can approximate the optical depth as, τ ν = Lα ν ≈ d sin φ α ν = d sin φ (1 + z ) δ α ′ ν ′ , (B1)since να ν is a relativistic invariant. For a power law electron distribution, the absorption coefficient is, α ′ ν ′ = C ( α ) k e B α +1 . (cid:18) zδ ν (cid:19) − ( α +2 . , where C ( α ) is a constant which depends only on the optically thin spectral index α . So now, τ ν = d sin φ C ( α ) k e B α +1 . (cid:18) zδ (cid:19) − ( α +1 . ν − ( α +2 . . (B2)Hence, at the τ ν = 1 surface, which we identify with the scintillating component, or core: (cid:18) νν (cid:19) ( α +2 . = d ( r ) d ( r ) k e ( r ) k e ( r ) (cid:20) δ ( r ) B ( r ) δ ( r ) B ( r ) (cid:21) . α , (B3)where r is the position of the τ ν = 1 surface, or core, at frequency ν .Hydrostatic confinement implies that the pressure inside the jet adjusts to the pressure of the surrounding medium.In that case, the diameter of the jet, d ( r ), is determined entirely by the pressure gradient of the external medium,and the lateral expansion of the jet dictates the run of magnetic field, particle density, and Lorentz factor along thejet. Consider a jet with ultra-relativistic equation of state, and relativistic particle energy distribution of the form N ( γ ) = K e γ − a between some minimum and maximum Lorentz factor, γ min and γ max , confined by an ambient mediumwith pressure p ext ∝ r − n p . In that case, conservation equations imply that the following relations hold: d ( r ) ∝ r n p / (B4)Γ( r ) ∝ r n p / B || ∝ r − n p / B ⊥ ∝ r − n p / n ∝ r − n p / γ min ∝ r − n p / k e ∝ r − np ( α +1 . Following K¨onigl (1981) and Lobanov (1998), let us define k r such that r core ∝ ν − /k r . (B5)and m B such that B ∝ r − mBnp . (B6)In that case, m B = 1 corresponds to a predominantly toroidal (perpendicular) magnetic field, while m B = 2 correspondsto a predominantly poloidal (parallel) magnetic field. Approximating the radial dependence in Doppler factor as apower law of the form δ ∝ r n δ , Equations B3 and B4 give k r = ( n p / . α )(2 + m B ) − − (1 . α ) n δ . α . (B7)Note that n δ may be positive or negative depending on the initial Lorentz factor, Γ( r ), and jet viewing angle φ , andtherefore the effect of the Doppler factor may be to steepen or flatten the relationship between r and ν .For a self-absorbed synchrotron source, the flux density at the peak frequency (which we associate with the fluxdensity at the observed frequency) is, S m ∝ ν / m θ d B − / δ / . (B8)This expression, when combined with the functions d(r), B(r), δ (r) and r( ν ), allows a prediction of the core spectralindex (see K¨onigl 1981, equation 12). Here we define the spectral index of the core, α core , such that S ν ∝ ν α core , and α core = 52 − k r h n p (cid:16) m B (cid:17) + n δ i . (B9)0(B9)0