THE ASTROPHYSICAL JOURNAL, 531 :902È916, 2000 March 102000. The American Astronomical Society. All rights reserved. Printed in U.S.A.(

SIZE OF THE VELA PULSARÏS EMISSION REGION AT 13 CENTIMETER WAVELENGTH

C. R. GWINN,1 M. C. BRITTON,1,2 J. E. REYNOLDS,3 D. L. JAUNCEY,3 E. A. KING,3 P. M. MCCULLOCH,4J. E. J. LOVELL4,5 C. S. FLANAGAN,6 AND R. A. PRESTON7

Received 1998 August 25 ; accepted 1999 November 4

ABSTRACTWe present measurements of the size of the Vela pulsar in three gates across the pulse, from obser-

vations of the distribution of intensity. We calculate the e†ects on this distribution of noise in theobserving system, and measure and remove it using observations of a strong continuum source. We alsocalculate and remove the expected e†ects of averaging in time and frequency. We Ðnd that e†ects ofvariations in pulsar Ñux density and instrumental gain, self-noise, and 1 bit digitization are undetectablysmall. E†ects of normalization of the correlation are detectable, but do not a†ect the Ðtted size. The sizeof the pulsar declines from 440^ 90 km (FWHM of best-Ðtting Gaussian distribution) to less than200 km across the pulse. We discuss implications of this size for theories of pulsar emission.Subject headings : ISM: general È pulsars : individual (Vela pulsar) È radio continuum: stars È

scattering

1. INTRODUCTION

1.1. Pulsar Radio EmissionPulsars emit meter- to centimeter-wavelength radiation

with brightness temperatures of T [ 1023 K, greater thanany other astrophysical sources. The ultimate energy sourcefor pulsar emission is rotation of the neutron star. Com-bined with this rotation, the neutron starÏs magnetic Ðeldgenerates forces sufficient to accelerate electrons from thestellar surface. These energetic electrons or the gamma-raysthey emit can pair-produce after a few centimeters. Theconsequent cascade of electron-positron pairs permeatesthe magnetic Ðeld of the pulsar out to the light cylinder. Theelectrons and positrons follow the lines of the strong mag-netic Ðeld closely. Particles on the ““ open ÏÏ Ðeld lines, whichpass through the light cylinder, gradually carry away partor all of the rotational kinetic energy of the pulsar as anelectron-positron wind, leading to the observed spin-downof the pulsar. The ““ polar cap ÏÏ is the region deÐned by theopen Ðeld lines at the surface of the neutron star, fromwhich the wind is drawn. This basic picture has long beenunderstood (Goldreich & Julian 1969 ; Sturrock 1971 ;Ruderman & Sutherland 1975 ; Arons 1983). Despite thisunderstanding, the physical mechanism for conversion of asmall fraction of the spin-down power to a beam of radioemission, which sweeps past the observer each rotationperiod to produce the observed pulses, remains uncertain.Observational tests of models for pulsar emission are diffi-cult because the emission is so compact. It cannot be resolv-ed with instruments of Earth-like or smaller dimensions.

1 Physics Department, University of California, Santa Barbara, Califor-nia, 93106.

2 Swinburne Centre for Astrophysics and Supercomputing, SwinburneUniversity of Technology, Hawthorne, Victoria 3122, Australia.

3 Australia Telescope National Facility, Epping, New South Wales,2121, Australia.

4 Physics Department, University of Tasmania, Hobart, 7001, Tasma-nia, Australia.

5 Institute of Space and Astronautical Science, 3-1-1 Yoshinodai,Sagamihara, Kanagawa 229, Japan.

6 Hartebeesthoek Radio Astronomy Observatory, Krugersdorp,Transvaal, South Africa.

7 Jet Propulsion Laboratory, California Institute of Technology,Pasadena, California, 91109.

Melrose (1996) and Asseo (1996) recently reviewed theo-ries of pulsar emission. In all models, collective behavior ofthe electron-positron plasma results in coherent emissionfrom groups of particles. Melrose divides theories into fourcategories, according to location of the emission region : (1)at a pair-production front at the polar cap ; (2) in theelectron-positron wind on open Ðeld lines above the polarcap ; (3) in the electron-positron wind far above the cap,near the light cylinder ; and (4) at, or even outside, the lightcylinder. Theories from all four categories appear viable.Indeed, more than one emission mechanism may act, toproduce the wide range of observed pulse morphologies(Rankin 1983).

Pulsars exhibit rich temporal structure, including compli-cated variations of structure of individual pulses about anaverage and large pulse-to-pulse variations in Ñux density ;these o†er clues to the emission process (Hankins 1972 ;Lyne & Graham-Smith 1998). Because the fundamentalmodes of lower-frequency electromagnetic waves arelinearly polarized, either across or along the pulsarÏs mag-netic Ðeld (Blandford & Scharlemann 1976 ; Melrose &Stoneham 1977 ; Arons & Barnard 1986 ; Lyutikov 1998a),the strong linear polarizations of pulsars yield the directionof the pulsarÏs magnetic Ðeld at the locus of emission(Radhakrishnan & Cooke 1969). Polarization observationsusually suggest that emission arises near the magnetic poleof the neutron star, which is o†set from the rotation pole.

Pulsars show stable, sometimes complicated long-termaverage proÐles, but with large pulse-to-pulse variations.The most successful classiÐcation of these proÐles is that ofRankin (Rankin 1983, 1990), who divides emission into““ core ÏÏ and ““ cone ÏÏ components, corresponding respec-tively to elliptical pencil beams and hollow cones directedfrom above the magnetic pole of the neutron star. A host ofassociated pulse properties support this classiÐcation(Rankin 1986 ; Lyne & Manchester 1988 ; Radhakrishnan &Rankin 1990). The existence of these di†erent componentsmay indicate the presence of more than one emissionprocess (Lyne & Manchester 1988 ; Weatherall & Eilek1997). A few core emitters show interpulses, suggesting thattheir magnetic poles lie near the rotation equator. Compari-son of pulse width with period for these pulsars, and exten-sion of this relation to all core emitters, closely tracks the

902

VELA PULSAR SIZE 903

relation expected if the size of the polar cap sets the angularwidth of the beam for core emission (Rankin 1990). Rankinconcludes that core emission arises very near the surface ofthe neutron star.

Interstellar scattering o†ers the possibility of measuringthe sizes of pulsars and the structure of the emission region,and its changes over the pulse, using e†ective instruments ofAU dimensions (Backer 1975 ; Cordes, Weisberg & Boria-ko† 1983 ; Wolszczan & Cordes 1987 ; Smirnova et al. 1996 ;Gwinn et al. 1997, 1998). In this paper, we describe mea-surement of the size of the Vela pulsar, from the distributionof Ñux density in interstellar scintillation, observed on ashort interferometer baseline.

1.2. Interstellar ScatteringDensity Ñuctuations in the interstellar plasma scatter

radio waves from astrophysical sources. These Ñuctuationsact as a corrupt lens, with typical aperture of about 1 AU.Like such a lens, scattering forms a di†raction pattern in theplane of the observer. For a spatially-incoherent source, theintensity of the pattern is the convolution of the intensity ofthe pattern for a point source, with an image of the source(Goodman 1968). The scattering system has resolutioncorresponding to the di†raction-limited resolution of the““ scattering disk,ÏÏ the region from which the observerreceives radiation.

The Ðnite size of the scattering disk sets a minimumspatial scale for the di†raction pattern in the plane of theobserver, at the di†raction limit. This scale is j/h, where j isthe observing wavelength and h is the angular standarddeviation of the scattering disk, as seen by the observer. Ifthe plasma Ñuctuations are assumed to remain ““ frozen ÏÏ inthe medium, while the motions of pulsar, observer, andmedium carry the line of sight through it at speed thisV

M,

spatial scale sets the timescale of scintillations, tISS \ j/hVM.

Di†erences in travel time from the di†erent parts of thescattering disk set the minimum frequency scale of thepattern, at *l\ (c/2n)(hD)~2[(1/D)] (1/R)]~1, where D isthe characteristic distance from observer to scatterer, and Ris the characteristic distance from scatterer to source.Observations with time resolution shorter than andtISSwith frequency resolution Ðner than *l are in the ““ specklelimit ÏÏ of interstellar scattering.

For an extended, spatially-incoherent source, overlap ofdi†erent parts of the image of the source in the observerplane, after the convolution, reduces the depth of modula-tion of scintillation : ““ stars twinkle, planets do not.ÏÏ Therelation of Cohen et al. (1967) and Salpeter (1967) quantiÐesthis fact as a relation between the size of the source and thedepth of modulation of scintillations.

1.3. T he Vela PulsarThe Vela pulsar is particularly well-suited for studies of

pulsar emission because it is strong, heavily scattered, andrelatively nearby. In a typical ““ scintle ÏÏ of observing time

and bandwidth *l, a typical terrestrial radio telescopetISScan attain a signal-to-noise ratio of a few, for observationsof the Vela pulsar at decimeter wavelengths. A short obser-vation can thus sample many scintles. Moreover, the scat-tering disk is large enough at these wavelengths that thescale of the di†raction pattern is on the order of the size ofthe Earth, so that di†erent radio telescopes on Earth cansample di†erent scintles. The nominal linear resolution ofthe scattering disk, acting as a lens, is about 1000 km at the

pulsar. For the Vela pulsar, the diameter of the light cylin-der is 8500 km. In contrast, the size of the polar cap is tensof meters for a dipole magnetic Ðeld and a D10 km radiusfor the neutron star. Thus, observations of the scintillationpattern can distinguish among di†erent theories of pulsaremission, as discussed in ° 1.1.

Among the properties that place the Vela pulsarÏs pulseinto the core class are its single component, varying little inwidth with frequency ; its short period and rapid spin-downrate ; its circular polarization ; and its weak, irregular pulse-to-pulse variations (Rankin 1983, 1990). However, Vela alsoshows characteristics of cone emission, including strong,well-ordered linear polarization and a double proÐle in X-and gamma rays (Strickman et al. 1996). Indeed, Radhak-rishnan et al. (1969) Ðrst observed in the Vela pulsar theS-shaped variation of the direction of linear polarizationcharacteristic of cone pulsars. Manchester (1995) has pro-posed that VelaÏs pulse (and those of some other youngpulsars) could represent one side of broad cone, whichwould help to explain the apparent inconsistency. On theother hand, Romani & Yadigaroglu (1995) propose that theradio pulse is indeed core emission, perhaps from very closeto the polar cap, and that the double-peaked X- andgamma-ray emission arises far out in the magnetosphere,near the light cylinder.

Krishnamohan & Downes (1983) performed an extensivestudy of 87,040 pulses from the Vela pulsar, and found thatthe pulse shape varied with peak intensity. Notably, strongpulses tend to arrive earlier. They also found that the rate ofchange of angle of linear polarization changes with pulsestrength and interpreted this as activity from emissionregions at di†erent altitudes and magnetic longitudes. Theyinferred variations in the location of the emission region byabout 400 km in altitude, and by about 4¡ in longitude.Emission earlier in the pulse was found to arise further fromthe star, and from a larger region.

2. DISTRIBUTION OF INTERFEROMETRIC CORRELATION

Interferometers present several advantages over singleantennas for observations of scintillating sources. Becausenoise does not correlate between antennas, no baselineresponse need be subtracted from observations. Moreover,interferometers are nearly immune to interference and emis-sion from extended or unrelated nearby sources. The sta-tistics of interferometer response to signals and noise is wellstudied (see, e.g., Thompson, Moran, & Swenson 1986). Fora short interferometer baseline, interferometers measure theintensity of the source, commonly expressed as Ñux density.Disadvantages of interferometers include the facts that theyrequire coordinated observations at multiple antennas andso often yield smaller e†ective apertures than the largestavailable single antenna, and they often a†ord narrowerbandwidths than some observing modes at single antennas.For the present work, careful treatment of statistics is essen-tial and interferometers are preferred. We focus on obser-vations of scintillating sources on short interferometricbaselines in this paper.

2.1. Distribution of Flux Density for a Small SourceFor a point source in strong scintillation, sampled in the

speckle limit, the distribution of Ñux density follows anexponential distribution (Scheuer 1968 ; Goodman 1985) :

P(S) \ 1/S0 exp M[S/S0N . (1)

904 GWINN ET AL. Vol. 531

This equation holds if the source is small, compared withthe resolution of the scattering disk treated as a lens, so that

where M \ D/R is the magniÐcation of thatkMhps> 1,

lens, and k \ 2n/j where j is the observing wavelength. Thesize of the source, parametrized as the standard deviation ofa circular Gaussian distribution of intensity, is Ifp

s. kMhp

sis Ðnite but small compared with 1, then the distribution ofÑux density is the weighted sum of three exponentials(Gwinn et al. 1998). If, for example, the scattering disk ismodeled as an elliptical Gaussian and the source is modeledas a circular Gaussian, then the exponential scales are

S0 , S1m\S0(kMhm ps)2 , and S1g\S0(kMhg p

s)2 , (2)

where and are the angular standard deviations of thehm hgscattering disk along the major and minor axes, and setsS0the scale of Ñux density. We assume that this(kMhps)> 1 ;

is equivalent to the assertion that the scattering disk,treated as a lens, does not resolve the source. The probabil-ity distribution for Ñux density is then

P(S)\ S0(S0[ S1m)(S0[ S1g)

expA[ S

S0

B

] S1m(S1m [ S0)(S1m [ S1g)

expA[ S

S1m

B

] S1g(S1g [ S1m)(S1g [ S0)

expA[ S

S1g

B. (3)

At large Ñux density, the Ðrst term dominates and the dis-tribution remains exponential, but with di†erent scale S0and normalization than for a point source. At small Ñuxdensity, the probability density falls to zero at zero Ñuxdensity, in a way that depends on source size The largerp

s.

the source, the more sharply the distribution of Ñux densityis peaked near the average Ñux density. Figure 1 showssample distributions. In this paper, we use this expression,with modiÐcations for the e†ects of noise, to Ðnd the size ofthe Vela pulsar in three gates across the pulse.

Source size reduces the intensity variation due to scintil-lation. The Cohen-Salpeter relation formalizes this fact as arelation between angular size and modulation index,m\ (SI2T [ SIT2)1@2/SIT. Here the angle brackets denotean ensemble average, approximated in practice by a timeaverage. For a circular Gaussian source viewed through aelliptical scattering disk, in strong scattering (Gwinn et al.1998),

m2\ [1] (2kMhg ps)2]~1@2[1] (2kMhg p

s)2]~1@2 . (4)

This expression is valid for arbitrarily large source size ps.

The modulation index calculated from equation (3) is con-sistent with this expression in the limit kMhg p

s> 1.

2.2. E†ects of Averaging in T ime and FrequencyBecause the scintillation pattern varies with time and fre-

quency, integration over either reduces the depth of modu-lation and narrows the distribution of amplitude. It thusa†ects the distribution in a way similar to Ðnite source size.Averaging leaves the mean Ñux density SST unchanged butreduces the mean square Ñux density SS2T, and so reducesthe modulation index. We can calculate the e†ects ofaveraging on the modulation index, and liken them to thee†ects of source size, to assess e†ects of averaging on esti-mates of source size.

FIG. 1.ÈProbability distribution P(S) for Ñux density S, for a pointsource (dotted line), and for sources with small size solidp

s(kMhm p

s\ 0.15 :

line) and with larger size long-dashed line). Vertical lines(kMhmps\ 0.40 :

near the top of the Ðgure indicate the scales and for the smallerS1m, S1m, S0source ; those near the bottom indicate these scales for the larger source.The mean Ñux density is the same for all three sources ; SST \ 1. In theseexamples, the elongation of the scattering disk is hm/hg\ 1.4.

For a source with pointlike or Gaussian intensity dis-tribution, seen through a Gaussian scattering disk, thecorrelation function of Ñux density with frequency follows aLorentzian distribution (see, e.g., Gwinn et al. 1998) :

SS(l)S(l] dl)T \ SS(l)2TC 11 ] (dl/*l)2] 1

D, (5)

where l and l] dl are the observing frequencies, and *l isthe decorrelation bandwidth of the scintillations. For asmall source with a Gaussian distribution of intensity, thecorrelation function has the same form, with larger decor-relation bandwidth. For non-Gaussian spectra of densityÑuctuations in the scattering material, the correlation func-tion shows the same behavior, with a di†erent functionalform (Codona et al. 1986). After averaging in frequency byconvolving with a ““ boxcar ÏÏ of bandwidth B, the meanintensity is unchanged, and the mean square intensitybecomes

SS1 (l)2T \T 1

B2P0

BdxP0

BdyS(l] x)S(l] y)

U

\ 2BP0

Bdu(B[ u)SS(l)S(l] u)T

\ SS(l)2TCtan~1 (B/*l)

B/*l] 1

2

[ ln (1] (B/*l)2)2(B/*l)2

D4 SS(l)2T f

f. (6)

No. 2, 2000 VELA PULSAR SIZE 905

Here the bar on denotes averaging. This expressionS1deÐnes the convenient factor f

f.

A similar approach yields e†ects of averaging in time. Thespatial scintillation pattern for a point source is the Fouriertransform of the screen phase (Goodman 1985 ; Cornwell,Anantharamaiah, & Narayan 1989 ; Gwinn et al. 1998).Commonly, it is assumed that scintillation in time rep-resents the motion of the scintillation pattern across theinstrument, while the screen and pattern remain unchanged.For an extended source, the pattern is the convolution ofthe point-source response with an image of the source. Ifboth source and scattering disk follow Gaussian distribu-tions of intensity, then the correlation function in time isGaussian :

SS(t)S(t ] dt)T \ SS(t)2T exp M[ ln (2)(dt/tISS)2N , (7)

where t and t ] dt are the times of observation and istISSthe timescale of interstellar scintillation. If the observeraverages over time interval q, the mean square Ñux densityafter that averaging is

SS1 2T \ SS2TC1a

Jn erf (a)] 1a2 e~a2 [ 1

a2D

4 SS2T ft, (8)

where a \ Jln 2q/t ISS.The e†ects of averaging can be expressed as a reductionin modulation index or an increase in inferred source size.The postaveraging modulation index is related to them6preaveraging value by

m6 2\ ff

ft(m2] 1)[ 1 . (9)

Using equation (4) we can infer a postaveraging size fromp6sand so relate to the true size using knowledge of them6 p6

sps,

averaging bandwidth and time. This is the approach takenbelow.

2.3. Noise and Self-Noise for Scintillating SourcesThe e†ects of noise can be important for studies of scintil-

lating pulsars because the scintles of lowest Ñux density areimportant in determining the size of the source. Similarly,noise a†ects the distribution of intensity most strongly atthe lowest intensities. Thus, noise is important in determin-ing source size.

2.3.1. Noise and Self-Noise for Interferometric Correlation

The interferometrist observes the signal from the source,along with noise, at 2 stations 1 and 2 and measures theircorrelation : TheC12 \C

x] iC

y\ (1/N

q) £

p/1Nq (V1p V2p* ).correlation is often called the interferometric visibilityC12and is often expressed in units of Ñux density, using themeasured or estimated gains of the antennas. For interfero-metric observations, is complex (Thompson et al. 1986).C12In this paper, we use boldface characters to denote complexquantities. Here are the electric Ðelds in the observedV1,2polarization, at the two antennas ; and is the number ofN

qindependent samples, the product of integration time andbandwidth. System noise may be in or out of phase with thesignal, and so contributes to both real and imaginary com-ponents. Signals from astrophysical sources are intrinsicallynoiselike, and their stochastic variations represent anothersource of noise, known as self-noise. Without loss of gener-ality, we take the phase of the signal to deÐne the real axis.

For a strongly-scintillating point source, the measuredcorrelation is drawn from the distribution :C12

P(C12) \P

du1

2np2 expG

[ 12

(Cx[ u)2 ]C

y2

p2H

]P

dSfAuS

; Nq

B 1S0

expG

[ SS0

H, (10)

where is the s-squared distribution withf (u/S ; Nq) N

qdegrees of freedom (see, e.g., Meyer 1975). Again, is theS0mean Ñux density of the source, averaged over many scin-tles. We measure the correlation in units of Ñux density,using the gains of the antennas. The system noise in units ofÑux density is where are thep2\ (T

S1/!1)(TS2/!2)/Nq, T

S1,2system temperatures at stations 1 and 2, and are the!1,2gains at stations 1 and 2. This equation assumes that thenoise from the source has Gaussian statistics. In practice theradiation from all known astrophysical sources does appearto have Gaussian statistics. In particular, radiation frompulsars follows Gaussian statistics closely, with time-varying amplitude (Rickett 1975). For some pulsars, theamplitude varies quite dramatically over short times withinthe pulse (Hankins 1972 ; Lyne & Graham-Smith 1998),although the Vela pulsar seems to be relatively stable in thisrespect.

When the number of independent samples is large, theNqs-squared distribution is well approximated by a Gaussian,

and the distribution takes the form

P(C12) \P0

`=dS

12np

xexp

G[ 1

2(C

x[ S)2px2

H

]1

2npy

expG

[ 12

Cy2

py2H 1

S0exp

G[ S

S0

H. (11)

The variances of the real and imaginary parts of the corre-lation are

px2\ 1

Nq

ATS1

!1] SBAT

S2!2

] SB

(12)

and

py2\ p

x2 . (13)

The Ñux density of the source, S, appears in the denomina-tor of the exponent in equation (11), through This reÑec-p

x.

ts self-noise.For two identical antennas, system temperature and gain

are equal, so that and and the corre-TS1 \T

S2 !1\!2,lation is drawn from the distributionC12\ Cx] iC

y,

P(C12) \P0

`=dS

Nq

2nS0(TS/!] S)

expG[ (C

y)2

2(TS/!] S)2/N

q

H

]1

(TS/!] S)

expG[ (C

x[ S)2

2(TS/!] S)2/N

q

H

] expG[ S

S0

H. (14)

Unfortunately, this integral is not easily simpliÐed. Figure 2shows plots of the distribution of the amplitude of the corre-lation, for a scintillating point source, for sampleP( o C12 o ),values of and When is large, self-noise is!S0/TS

Nq. N

qnegligible and the ratio is the average signal-S0!(Nq)1@2/T

Sto-noise ratio. When is small, self-noise becomes impor-Nqtant. As the Ðgure shows, self-noise increases the number of

906 GWINN ET AL. Vol. 531

FIG. 2.ÈUpper panel : Distribution of amplitude of correlation, o C12 ofor a scintillating point source, observed by two identical antennas, includ-ing e†ects of system noise and self-noise (see eq. [14]). The mean Ñuxdensity of the source is The signal-to-noise ratio (excluding self-S0\ 1.noise) is or 2, as indicated for the two families of curves.!S0N

q1@2/T

s\ 10

For each family, the number of samples is 3000 (solid line), 30 (dottedNqline), and 10 (dashed line). The system temperature is and the antennaT

stemperature is E†ects of self-noise are greatest when the signal is large!S0.compared with noise, and is small. L ower panel : The same curves,Nqplotted on a logarithmic scale.

points at low amplitude, and Ñattens the distribution athigh amplitude.

2.3.2. Distribution for a Point Source with Noise

If the number of independent samples is large, thenNqself-noise can be ignored. The distribution of amplitude,

then takes the often-useful formC12\ o C12 o ,

PS(C12, S0, p)\ P(u)]

dudC12

\ 1

Jn1S0

expA14

b2Bu

]P0

nd/ exp M[u2 sin2 /[ bu cos /N

]C1 ] erf

Au cos /[ 1

2bBD

]1

J2p, (15)

where erf ( ) is the error function (Meyer 1975). The noise, inÑux density units, is We have adoptedp 4p

x\ p

y\ (T

S/!).

the scaled parameters

u \ C12/J2p and b \ J2p/S0 . (16)

If the antennas are not identical, the distribution has thesame form in this limit, but the deÐnitions of the scaled

parameters are slightly di†erent. Because self-noise canoften be ignored, as Figure 2 suggests, this expression isoften appropriate. For scintles much stronger than noise

and average Ñux density not much smaller than(C12? p),noise (so that this distribution approaches theS0?p2/C12),purely exponential form

P(C12) BGexp

A p22S02BH 1

S0e~C12@S0 . (17)

The constant of the exponential is the same as that for anoise-free scintillating point source, but the normalizationis di†erent. In e†ect, noise shifts the distribution towardgreater amplitude.

2.3.3. Distribution for a Small Source with Noise

When the source has small but Ðnite size, and self-noisecan be ignored, the distribution of takes the form sug-C12gested by the combination of equations (3) and (15) :

P(C12) \S0

(S0[ S1m)(S0[ S1g)P

S(C12, S0, p)

] S1m(S1m [ S0)(S1m[ S1g)

PS(C12, S1m, p)

] S1g(S1g [ S1m)(S1g [ S0)

PS(C12, S1g, p) , (18)

Here equation (2) deÐnes and and equation (15)S0, S1m, S1g,deÐnes This relatively simple relationshipPS(C12, S

n, p).

holds even though the e†ects of noise on the amplitudedistribution cannot be described as a convolution. For scin-tles with amplitude much larger than noise, and averageÑux density not much less than noise, and(C12 ? p S0 ?

the limits of eq. [17]), only the Ðrst of the threep2/C12,terms is important, and this distribution also approachesthe purely-exponential form

P(C12) BGexp

A p22S02BHG

[1[ (kMhm pS)2]

] [1[ (kMhgpS)2]H~1 1

S0e~C12@S0 . (19)

In this expression, e†ects of source size appear only throughthe normalization. This fact demonstrates that source sizea†ects the form of the distribution only at small amplitude.

2.4. E†ects of Variations in Flux Density andInstrumental Sensitivity

If the gain of the observerÏs instrument varies with fre-quency or time, then the distribution of observed Ñuxdensity will be the superposition of several distributions ofthe form given by equation (1) (or eq. [3], if the source isextended). For example, consider a scintillating pointsource observed with a distribution of gains f (!), where ! isthe gain. We take S to be the true Ñux density of the source,and to be the antenna temperature, the observedT

A\ !S

quantity. The distribution of is then the convolution ofTAthe distribution of S with that for f :

P(TA) \P0

=d!f (!)

P0

=dSP(S)d(!S [ T

A)

\P0

=d!f (!)

1S0!

expA[ T

A!S0

B. (20)

No. 2, 2000 VELA PULSAR SIZE 907

Similarly, if the Ñux density of the source is not constant,then the distribution of measurements includes contribu-tions from exponential distributions with di†erent scales.The overall distribution of Ñux density will be the same (eq.21), although the e†ective gain variations will be stochasticrather than deterministic. In either case, the distributionfunction will be concave upward on a semilog scale, becausethe distributions with highest average Ñux density fall o†most slowly. Figure 3 shows examples for Ñat distributionsof gain about the mean. The e†ect is signiÐcant when thedistribution of gains includes very small or zero values.

2.5. Normalized Correlation and the Van V leck Relationfor Scintillating Sources

Practical interferometers measure approximations to thevisibility In particular, interferometers almost alwaysC12.measure the normalized correlation (Thompson et al. 1986,pp. 214, 248) :

o0\ gC12

J(TS1/!1] S)(T

S2/!2] S). (21)

Here g is a correlator-dependent constant and S is the Ñuxdensity of the source. Note that the denominator in equa-tion (21) does not include the number of samples TheN

q.

denominator is simply a normalization factor ; it does notarise from, or characterize, the noise in the measurement.

Most signals are digitized before correlation. For 1 bitsampling, as for the Haystack correlator used for the obser-vations described here, the measured correlation coefficient

FIG. 3.ÈExpected distributions of amplitude of correlation for a scintil-lating point source, including e†ects of variations in instrumental gain (or,equivalently, intrinsic variations in Ñux density). The solid curve shows theexponential distribution expected for a point source in strong scintillation,with mean and exponential scale equal to the mean amplitude of thesource, 1 in this example. Variations in gain change this distribution bysuperposing distributions with di†erent exponential scales. The model dis-tributions of gain are Ñat, extending from 0.8 to 1.2 times the mean (dottedcurve) and from 0 to 2 times the mean (dashed curve). Distributions withsigniÐcant contributions at zero or very low gain di†er the most from thecase of constant gain.

is related to by the Van Vleck relation :o2 o0

o2\ 2n

sin~1 o0 . (22)

Often correlators tabulate the correlation coefficient scaledby the linearized Van Vleck correction Because(n/2)o2.normalization and digitization become nonlinear at largeamplitude, and noise and source structure are important atsmall amplitude, their e†ects are relatively easily separable.

Because normalization and digitization are important atlarge amplitude, where the distribution of amplitude isexponential even in the presence of source structure andnoise, we investigate e†ects of normalization and digitiza-tion on such a distribution : ForP(S) \ (1/S0) exp M[S/S0N.two identical antennas observing an unresolved source,

and visibility is equal to the Ñuxo0\ C12/(TS/!] S),

density, Then the distribution of will beC12\ S. o0

P(o0) \TS

!S0 g1

(1[ o0/g)2 expG

[ TS

!S0

o0/g1 [ o0/g

H. (23)

Note that the characteristic scale of the exponential is thesignal-to-noise ratio in a single sample, It is inde-!S0/TS

.pendent of integration bandwidth or time. At small o0,can be approximated by an exponential distribution :P(o0)

P(o0) BTS/!

S0 gexp

G[AT

S/!

S0[ 2Bo0H

. (24)

The additive constant ““ 2 ÏÏ in the exponential arises fromthe factor of before the exponential in equa-(1 [ o0/g)~2tion (23).

We can also include the e†ects of 1 bit sampling, byincluding the Van Vleck relation in calculating this distribu-tion. For the scaled correlation coefficient,

PAn2

o2B

\ TS/!

S0

cos [(n/2)o2]/g(1[ sin [(n/2)o2]/g)2

] expG[ T

S/!

S0

sin [(n/2)o2]/g1 [ sin [(n/2)o2]/g

H. (25)

We show sample distributions of and inP(o0) P[(n/2)o2]Figure 4. As this Ðgure demonstrates, the distribution ofcorrelation departs from an exponential distribution atlarge amplitudes when the signal-to-noise ratio is large in asingle sample.

2.6. Note on Size Estimate From Modulation IndexThe modulation index, m\ (SS2T/SST2[ 1)1@2, para-

metrizes source size p, as equation (4) shows. One can esti-mate the modulation index directly using this relation, byreplacing the ensemble averages with Ðnite averages over aset of observations. Such a procedure replaces SST with

and SS2T with and yields an1/N £2/1N S2, 1/N £2/1N S22,estimate for the modulation index. Unfortunately, thesemüÐnite sums are weighted by Ñux density, so that they aremost sensitive to the scintles with the greatest Ñux density.For a nearly exponential distribution of Ñux density, asexpected for a small source in strong scattering, these strongscintillations are rare. Therefore, the sums are slow to con-verge. The measurement is heavily subject to the shot noiseof scintillation (J. M. Cordes 1998, private communication).Moreover, source size a†ects the weakest parts of the dis-tribution, to which these sums are least sensitive ; whereas

908 GWINN ET AL. Vol. 531

FIG. 4.ÈProbability distribution of normalized correlation ando0,scaled 2-bit correlation for a strongly-scintillating source with expo-n/2o2,nential distribution of Ñux density. The average Ñux density, and scale ofthe exponential distribution, is Antenna temperature is andS0. !S0,system temperature is so is signal-to-noise ratio for a singleT

S, !S0/TSsample. For each of three values of the probability distribution is!S0/Ts

,shown for the normalized correlation (solid curve : eq. [23]), the scaledo01-bit correlation (dashed curve : eq. [25] with g \ 1), and then/2o2limiting exponential form for small correlation (dotted curve : eq. [24]). Thecurves di†er only when the source is strong relative to system noise. Allforms assume that is large so that self-noise can be ignored.N

q

most of the systematic e†ects discussed above are greatestfor the strongest parts of the distribution.

To exclude the possibility that the estimate of size isbiased by e†ects other than source structure, such as thosediscussed in °° 2.4 through 2.2, a careful study of the fulldistribution is the most accurate approach. The modulationindex expresses the distribution as a single value. Weakestscintles are also the most common, so the important partsof the distribution can be measured with greater accuracythan can the modulation index. Although the shot noise ofindividual scintles can limit the accuracy of a measurementbased on a Ðt to the distribution, the accuracy is muchbetter than provided by estimates of modulation index.

2.7. Summary : Expected Distribution of Flux DensityA pointlike scintillating source produces an exponential

distribution of Ñux density, so that one expects an exponen-tial distribution of visibility amplitude for an interferometerif the baseline is short compared to the spatial scale of thedi†raction pattern. Averaging in frequency or in time canreduce the modulation due to scintillation. We show thatvariations in system gain or intrinsic Ñux density of thesource a†ect this distribution, particularly if the variationsare of order 100%. System noise and self-noise also changethe distribution. System noise a†ects the form of the dis-tribution function at small amplitudes, and by a change inthe normalization at large amplitudes. Self-noise a†ects thedistribution at all amplitudes when the number of samples

is small. Normalization of the correlation function, andNq

1 bit sampling, a†ect the correlation at large amplitudes,particularly when the source is strong relative to systemnoise.

For a source of small, but Ðnite size, the distribution ofÑux density is the weighted sum of three exponentials. By““ small ÏÏ we mean here that the source is small comparedwith the resolution of the scattering disk seen as a lens.Scales of two of these exponentials depend on the size of thesource, and are thus useful for size determination, thesubject of this paper. These two exponentials a†ect the dis-tribution at small Ñux density. Because e†ects of systemnoise and source size are both important at small corre-lation amplitude, both e†ects must be included in datareduction.

Considerations of Ðnite size, noise, self-noise, normal-ization, and quantization can be important for single-dishobservations of scintillating pulsars as well. The e†ects ofthese factors depend on the details of the detection andsampling schemes used at the antenna. Jenet & Anderson(1998) describe some of the important e†ects for digitizationand autocorrelation.

3. OBSERVATIONS AND DATA REDUCTION

3.1. Observations and CorrelationWe observed the Vela pulsar and comparison quasars on

1992 October 31 to November 1 using radio telescopes atTidbinbilla (70 m diameter), Parkes (64 m), and Hobart(25 m) in Australia ; Hartebeesthoek (25 m) in South Africa ;and the seven antennas of the Very Long Baseline Array ofthe US National Radio Astronomy Observatory8 thatcould usefully observe the Vela pulsar. Each antennaobserved right-circular polarized radiation in 14 ] 2 MHzbands between 2.273 and 2.801 GHz. These bands aresometimes referred to as ““ IF bands ÏÏ or ““ video converters ;ÏÏhere we call them frequency bands or simply bands. Thedata were digitized to 1 bit (that is, we measured only thesign of the electric Ðeld), and recorded with the Mark IIIrecording system.

We correlated the data with the Mark IIIB correlator atHaystack Observatory, with time resolution of 5 s and with160 time lags. Fourier transform of the correlation functionsto the frequency domain yields cross-power spectra with 80channels, with frequency resolution of 25 kHz. Because ofcorrelator limitations, we correlated only six of the 14recorded bands for most scans. Correlations are tabulatedas normalized fractional correlation with one-bit sampling,

as given by equations (21) and (22).o2,For each baseline, we correlated the data in three gatesacross the pulse. Gate 1 covers approximately the 13 milli-periods up to the peak of the pulse, Gate 2 covers the next13 milliperiods, and Gate 3 covers the next 25 milliperiods.The three gates have average intensity ratios of about0.7:1:0.5, as discussed further in ° 4.1 below. The processorset the gates so that they were delayed according to thepulsarÏs measured dispersion (Taylor, Manchester, & Lyne1993 ; Lyne & Graham-Smith 1998) at the center frequencyof each of the 14 bands. Thus, dispersion smears the gates intime by no more than the dispersion across 1 MHz, or\50ks B 0.6 milliperiods. To the extent that structure of thepulsarÏs emission region changes with pulse phase, we can

8 The National Radio Astronomy Observatory is operated by Associ-ated Universities Inc., under a cooperative agreement with the NationalScience Foundation.

No. 2, 2000 VELA PULSAR SIZE 909

regard the gates as sampling di†raction patterns fromsources with di†erent structures, or from a single sourceviewed from di†erent angles.

The Haystack correlator is a lag or ““ XF ÏÏ correlator, sospectral data are a†ected by the fractional bitshift e†ect(Thompson et al. 1986). After correlation, we removed theaverage phase slope introduced by this e†ect for each timesample. Because the pulsar period is short compared to thefractional bit-shift rate, we did not encounter aliasingbetween the fractional bit-shift and the pulse period (Britton1997).

For each baseline, we analyzed all three gates with identi-cal phase models, with identical correlator parameters andthe same phase model removed from each gate. This phasemodel consists of a phase o†set and slopes in time andfrequency, Ðtted to the data in the three gates, for all corre-lated tracks over each 13 minute scan. This process is oftencalled ““ fringing ÏÏ the data. Because we study the distribu-tion of amplitude of correlation in this paper, the phasemodel inÑuences the results only indirectly, through thee†ects of averaging.

After fringing, we averaged the data in time to increasesignal-to-noise ratio. We boxcar-averaged the data by twosamples, to a resolution of 10 s in time. This resolution infrequency and time is Ðner than the di†raction pattern ofthe pulsar, but coarse enough to yield good signal-to-noiseratio.

In this paper we focus on observations on the shortTidbinbilla-Parkes baseline, about 200 km long. The Tid-binbilla and Parkes antennas are both large and have sensi-tive receivers. They can be regarded as nearly identical ; anydi†erence is important only for details of the distribution ofnormalized correlation at high amplitude, discussed in ° 2.5above and ° 4.5 below. Because the characteristic spatialscale of the di†raction pattern is j/h B (8000 km east-west)] (13,000 km north-south), this short baseline e†ectivelymeasures the Ñux density of the scintillation pattern at asingle point in the plane of the observer.

3.2. Instrumental Gain and Noise3.2.1. Gain from a Continuum Source

From observations of a strong continuum source, wefound that gain varies within each recorded band. Figure 5shows the amplitude of a strong continuum source plottedwith frequency, as an example. The source, 0826[373, wasobserved on 1992 November 1 from 0:22 :30 UT to 0 :35 :30UT. The Ðgure shows the band between 2286.99 and2288.99 MHz. The roll-o† of gain at the high- and low-frequency ends of the band arises primarily from the Ðlter-ing required to isolate the recorded band. These gainvariations, and the di†erences in gain between recordedbands, can be measured from observations of a continuumsource and, in principle, corrected. However, the noise doesnot follow the same proÐle, so this correction would distortthe statistics of noise. Because our measurements relyheavily on accurate knowledge of the noise level, we deletethe outer part of each recorded band (the lower 25 and theupper 20 of the 80 channels, as shown in Fig. 5) and use thecentral part without gain calibration. We detect no signiÐ-cant variation of gain with time.

3.2.2. Noise from a Continuum Source

We measured the system noise from the distribution ofamplitude for 0826[373, and compared the result with

FIG. 5.ÈGain plotted with frequency in a typical 2 MHz recordedband. Data are for the extragalactic continuum source 0826[373,observed from 00 :22 :50 to 00 :35 :30 UT on 1992 November 1, between2286.99 and 2288.99 MHz on the Tidbinbilla-Parkes baseline. The datawere averaged coherently for 10 s in time and 25 kHz in frequency and thenaveraged incoherently in time for the entire 13 minute scan. To reducee†ects of gain variations, we use only data from the central portion of theband, between the vertical lines.

other estimates. For a strong continuum source, the dis-tribution of correlation is expected to follow a two-dimensional Gaussian distribution in the complex plane,o†set from the origin by the correlated Ñux density of thesource, with standard deviation equal to the noise level p(Thompson et al. 1986). For a strong source, the distribu-tion of the amplitude of the correlation follows a one-dimensional Gaussian distribution with the same standarddeviation p, centered on the mean amplitude.

We found the distribution of amplitude about the meanfor 0826[373 using the same data shown in Figure 5. Eachamplitude was measured with bandwidth 25 kHz and timeaveraging of 10 s, in the central region of the band, as shownfor a single band in Figure 5. We Ðt a Gaussian distributionto the amplitude found in each of the six correlated fre-quency bands, with parameters of the standard deviation,normalization, and mean amplitude. Table 1 summarizesresults of these Ðts. The noise, given by standard deviation,should be identical in each band. The Ðts show a variationof up to 2.4% in the noise, or up to 2.2 times the standarderrors. Again, we do not regard these di†erences as signiÐ-cant. The mean amplitudes show variations of up to 3.0%,with high signiÐcance, reÑecting variations in instrumentalgain in the di†erent bands. The di†erences in the noise do

910 GWINN ET AL. Vol. 531

TABLE 1

FITS TO NOISE

Noise Average AmplitudeBand (p) oC o6 Normalizationa

1 . . . . . . . . . 78.3 ^ 1.2 1459.9 ^ 1.2 4414 ^ 602 . . . . . . . . . 76.8 ^ 1.1 1479.8 ^ 1.0 4387 ^ 523 . . . . . . . . . 75.7 ^ 1.5 1510.4 ^ 1.5 4405 ^ 784 . . . . . . . . . 74.6 ^ 0.8 1524.8 ^ 0.8 4386 ^ 435 . . . . . . . . . 76.6 ^ 1.2 1499.8 ^ 1.1 4384 ^ 586 . . . . . . . . . 73.4 ^ 1.0 1498.5 ^ 0.9 4392 ^ 49Allb . . . . . . 74.3 ^ 1.0 1499.6 ^ 1.1 26260 ^ 320

a Actual number of data : 4386 in individual bands, 26316 in allbands.

b Mean of each frequency band subtracted, then 1500 added, so thatdistribution reÑects noise rather than variations in gain.

not correlate with the di†erences in gain. We could correctfor the gain variations, but prefer to preserve identical con-ditions in all bands.

Figure 6 shows the distribution of correlated Ñux densityfor all the recorded bands. So that the histogram reÑectsnoise, rather than the variations in gain between bands, weremoved the mean amplitude for each band, as given inTable 1, and added the overall mean of 1500. The distribu-tion of noise is nearly Gaussian, as expected. A Ðt to thiscomposite distribution yields noise level p \ 74.3^ 1.0.This value is quite consistent with that found from otherscans, and with the values for individual recorded bands, asTable 1 shows.

We can also compare this noise level with that found inobservations that failed to detect any fringes, which shouldproduce purely noise, and in observations in a gate o† thepulsar pulse. In either case the distribution function in thecomplex plane is a circular Gaussian, centered at the origin(Thompson et al. 1986) ; the resulting distribution of ampli-tude follows a Rayleigh distribution. In earlier work, wedescribed the second approach (Gwinn et al. 1997), appliedto observations made at the same epoch, but for a di†erentscan on the pulsar, correlated on a di†erent date and with aslightly di†erent correlator conÐguration. Expressed in thesame units, this procedure yielded a noise level ofp \ 74.0^ 0.5. This di†ers by less than the standard errorfrom the value we measure here. We adopt p \ 74.3.

FIG. 6.ÈHistogram of noise. Data are the same as in Fig. 5. To removethe e†ects of gain variations on the relatively high amplitude of the source,we subtracted the average amplitude given in Table 1 from the data in eachband, and then added the overall average amplitude of 1500. Thus, thehistogram reÑects noise rather than gain di†erences among the bands. Thesolid curve shows the best-Ðtting Gaussian distribution, with parametersgiven in Table 1.

3.2.3. Noise for Pulsar Observations

Because the pulsar gate reduces the duty cycle, we mustincrease the noise level to reÑect the shorter e†ective inte-gration time, for comparisons with gated pulsar obser-vations. Gates 1 and 2 have duty cycles of 0.013, and Gate 3

TABLE 2

FITS TO DISTRIBUTIONS OF INTENSITY OF THE VELA PULSAR

Parameter Gate 1 Gate 2 Gate 3

Fixed Parameters

Number of data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74492 69305 74076Bin width . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 75 25Assumed noise levela . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 558.6 507.7 474.9

Fitted Parameters

Size parameter (kMhm p6s)2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0.173 ^ 0.005 0.163 ^ 0.005 0.149 ^ 0.005

Amplitude (S0) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4395 ^ 56 6185 ^ 104 2826 ^ 23Normalizationb . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76804 ^ 537 72441 ^ 742 75631 ^ 340Standard deviation of residualsc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.29 20.49 17.99Size parameter after correction for averagingd (kMhmps

)2 . . . . . . 0.091 ^ 0.009 0.070 ^ 0.012 0.020 ^ 0.020

a Corrected for e†ects of source amplitude as discussed in ° 3.2.3.b Normalization larger than the number of data points reÑects the correction for normalization of the correlation function

(eq. [23]), not included in the Ðt. See ° 4.5 below.c Di†erences in the standard deviations of residuals among gates reÑect the di†erent widths and populations of bins.d Assuming decorrelation bandwidth of *l\ 66 kHz and decorrelation time of sec. See ° 2.2 and Fig. 8.tISS\ 26

No. 2, 2000 VELA PULSAR SIZE 911

has a duty cycle of 0.021 ; the noise increases by a conse-quent factor of 8.77 or 6.32, respectively.

Quantization during analog-to-digital conversion a†ectsthe noise level. Qualitatively, the presence of strong signalsin the recorded band reduces the noise level in frequencychannels where the signal is weak or absent (Gwinn et al.1999). The ““ dithering ÏÏ employed to enhance dynamicrange in some commercial analog-to-digital conversions is aclosely related phenomenon (Bartz 1993). Quantitatively,the noise is not distributed evenly over the band, but isdistributed according to the autocorrelation spectra at thetwo antennas. However, for identical antennas, thesummed, squared noise remains the same as for zero corre-lation, or a source without spectral variation. We computethe expected noise level for weak signals for each timesample of the scintillation spectrum of the pulsar, and afterexcising outliers, use the mean value for the expected noiselevel. The resulting values, used in our analysis, are given inTable 2. Relatively more conservative or more liberal cutshad little e†ect on the results of the Ðts for size, describedbelow.

4. DISTRIBUTION OF INTENSITY FOR THE VELA PULSAR

4.1. Observed Intensity Distribution and FitsWe used the distributions of amplitude observed for the

pulsar, in the three gates across the pulse, in combinationwith the measured system noise and the models discussed in° 2, to Ðnd the size of the pulsar in the three gates. We usedthe ratio (Gwinn et al. 1997) to set the ratio ofhm/hg \ 1.65the scales and for the Ðts. JustS1m\ kMhm p

sS1m\ kMhg p

sas for the continuum source 0826[373, the data were aver-aged for 10 s in time to yield high signal-to-noise ratio whileremaining within the speckle limit, as discussed in ° 1.2above. Extrapolation from previous, published measure-ments yields a decorrelation bandwidth for the pulsar of*l\ 60 kHz at our observing frequency of 2.3 GHz, and ascintillation timescale of s (Backer 1974 ; RobertstISS\ 15& Ables 1982 ; Cordes, Weisberg, & Boriako† 1985). Fromour observations, as discussed in ° 4.2 below, we infer adecorrelation bandwidth of *l\ 66 kHz and a timescale of

s.tISS \ 26Figure 7 shows the observed distribution of correlation in

the three gates. To each histogram, we Ðt a model for thesize of the pulsar, of the form given by equation (18). Weused the Levenburg-Marquardt algorithm for the nonlinearÐt (see, for example, Press et al. 1989). We Ðtted for threeparameters, corresponding to the overall normalization, theexponential constant and the size of the source scaled byS0,scattering parameters, We searched broad(kMhm p

s)2.

ranges of initial values for the parameters, including therange from 0 to 0.3 for the scaled size of the source. Table 2summarizes the results of the Ðts. We found that the best-Ðtting model has nonzero size for the pulsar, with size thatdecreases across the pulse. The results for the normalizationparameter were approximately consistent with the numberof values in the histogram, as expected for a good Ðt.

4.2. Corrections for Averaging in Frequency and T imeWe can Ðnd the e†ects of averaging in time and frequency

by theoretical calculations and by empirically comparingresults for data with di†erent degrees of averaging. To Ðndresults of averaging, we boxcar-averaged the interferometriccorrelation over 1, 2, or 3 of the 25 kHz channels output by

the correlator in frequency, and over 1, 2, or 3 of the 5 ssamples in time. We then Ðt a model as described in ° 4.1,with a free parameter for source size, and with the strengthof noise as found in ° 3.2, scaled appropriately by the inversesquare root of the averaging time and frequency. Figure 8shows the results for gate 2, the gate in which the pulsar wasstrongest. Results for the other gates are similar, althoughthe extrapolated size of the pulsar with zero averaging aredi†erent. Averaging in time or frequency increases the best-Ðtting size, as expected.

The curves in Figure 8 show the best-Ðtting forms for theexpected variation of size as a function of averagingp6

sbandwidth and time interval, as given by equations (4), (7),and (8), Ðtted to the data points. Parameters of the Ðts arethe decorrelation bandwidth *l and scintillation timescale

and the sizes of the source in the three gates in thetISS,absence of averaging. The best-Ðtting decorrelation band-width is *l\ 66 ^ 1 kHz, and the best-Ðtting scintillationtimescale is s. These are comparable to thosetISS\ 26 ^ 1extrapolated from previous measurements, reported in theliterature (see ° 4.1). We can remove e†ects of time andfrequency averaging to Ðnd the true sizes for the source inthe three gates, without averaging in time or frequency,inferred from these Ðts. Table 2 gives the results.

Figure 7 shows both the best-Ðtting size, and the distribu-tion predicted for a source of zero size, averaged in fre-quency and time by the same amount. In gates 1 and 2 amodel with size greater than zero Ðts remarkably better.This indicates the signiÐcance of the Ðtted size after correc-tion for time averaging : about 10 standard deviations ingate 1, and about 6 standard deviations in gate 2. In gate 3,a model with size zero Ðts about as well as the best-Ðt, withkMhmps

\ 0.020.

4.3. Gain and Flux Density V ariationsWe measure variations of as much as 3.5% in gain across

the bands, as Figure 5 shows ; and we Ðnd variations of gainbetween bands of as much as 3.0%, as shown in Table 1.These are the maximum departures ; the distribution isactually much more concentrated than these Ðgures indi-cate. The discussion in ° 2.4 shows that this level of gainvariations a†ects the analysis by a completely negligibleamount.

Individual pulses from Vela show signiÐcant variability(Krishnamohan & Downs 1983). These intensity variationsare almost completely random in time. A measure of thisvariability is the intrinsic, single-pulse modulation index,

where is the Ñux of themi1\ SS12T[ SS1T2/SS1T2, S1pulsar in a single pulse. The index is 0.4 for the averagem

i1intensity of the pulse. It varies from 0.3 to 1.3 in individualbins of 1¡ in pulse longitude, with the largest variations onlyat the beginning and end of the pulse, where the averageintensity is low (Krishnamohan & Downs 1983). Our timeaveraging over 10 s averages together about 112 pulses. Ourgates cover or of pulse longitude. The modulation4d.7 9d.0index for an average of N pulses, assuming that the mea-surements are uncorrelated, is We thusm

iN\ (m

i1)/JN.expect the modulation index of our 112 pulse average to beno more than 0.15, at the very most ; and we expect theoverall average of to be closer tom

iN0.4/J112 \ 0.04.

We adopt 15% as a conservative upper bound for the nete†ect of gain variations and intrinsic variability. Figure 3indicates that, even if the distribution of intensity follows aÑat distribution with this width, e†ects of gain variations

912 GWINN ET AL. Vol. 531

FIG. 7.ÈHistograms of observed amplitude in three gates across the pulsar pulse. For each gate, the right panel shows the histogram of amplitudes withthe best-Ðtting model (solid curve) and the model with zero size (dotted curve). Inset shows the pulse proÐle with the gate. The left panel shows residuals fromthe model with zero size, for the data (histogram), and for the best-Ðtting model (solid curve). The decrease in pulse intensity, as the gates progress across thepulse, appears as a decrease in the scale of the exponentials, from the top panel (gate 1) to the bottom panel (gate 3).

and intrinsic variability might be barely detectable, over 3orders of magnitude in the probability. They will not signiÐ-cantly a†ect the measurements of pulsar size.

4.4. Self NoiseFor our observations, the product of bandwidth and inte-

gration time (including the duty cycle of the pulsar gate) isfor gates 1 and 2, and for gate 3. TheN

q\ 3250 N

q\ 6250

average amplitude of the source, is between 12.1 and 4.9S0,times the noise level, In these cases, as Figure 2TS/(!JN

q).

indicates, self-noise has a negligible e†ect on the distribu-tion, to our accuracy. In other words, the distributions are

indistinguishable from those for Self-noise hasNq] O.

negligible impact on the distribution function.

4.5. Normalized CorrelationNormalization of the correlation detectably a†ects the

distribution of amplitude for the Vela pulsar. Figure 9shows the distributions of amplitude in the three gates, witha logarithmic vertical axis. A purely exponential distribu-tion would follow a straight line on this plot, but theobserved distribution falls below this line at both small andlarge amplitudes. The Ðgure shows the best-Ðtting modelsincluding e†ects of normalization of the correlation (eq.

TABLE 3

FITS FOR NORMALIZED CORRELATION

Parameter Gate 1 Gate 2 Gate 3

Conversion factor ( oC12 o /(o0/g)) . . . . . . (5.6 ^ 1.4) ] 10~6 (5.6 ^ 1.4) ] 10~6 (5.6 ^ 1.4) ] 10~6Normalizationa . . . . . . . . . . . . . . . . . . . . . . . . . . 77640 ^ 1212 73430 ^ 746 78756 ^ 1086Scale (!S0/Ts

) . . . . . . . . . . . . . . . . . . . . . . . . . . . 0.0233 ^ 0.0002 0.0324 ^ 0.0004 0.0152 ^ 0.0001

a Discrepancy from actual number of data, as given in Table 2, reÑects e†ects of noise and pulsar size at lowamplitudes.

No. 2, 2000 VELA PULSAR SIZE 913

FIG. 8.ÈFitted size parameter, plotted as a function of averaging intime and frequency, in gate 2. The size is normalized to the resolution ofthe scattering disk acting as a lens, expressed as The ordinate is(kMhmps

).the averaging bandwidth : 25 kHz as produced by the correlator, andboxcar averages to 50 and 75 kHz. Circles show Ðts for data averaged by 5s in time, squares by 10 s, and triangles by 15 s. The four curves show thepredictions of the best-Ðtting model with parameters of decorrelationbandwidth *l\ 66 kHz and scintillation timescale s. The di†er-tISS \ 26ent curves show predictions for averaging in time by 0, 5, 10, or 15 s.Section 2.2 describes the model.

[23]) for each gate, which accurately describe the distribu-tions at large amplitude. Table 3 summarizes results of theÐts of the distribution expected for a normalized correlationfunction, given by equation (23), to the data in Figure 9. Toimprove the sensitivity of the Ðt to the high end of thedistribution, we Ðt to the logarithm of the histogram. The

FIG. 9.ÈHistogram of distribution of amplitude in three gates, with thebest-Ðtting distribution for the distribution of normalized correlation, asgiven by eq. (23) and shown in Fig. 4. Table 3 gives parameters of the Ðt.The distributions are concave downward because of correlator normal-ization, as discussed in ° 2.5.

Ðtted parameters agree well with expectations. The normal-ization is greater than the number of data in the histogram,probably because of the reduction in the distribution atsmall amplitudes by e†ects of noise and Ðnite source size.

Models including e†ects of Ðnite size and noise (eq. [18])accurately describe the distributions at low amplitudes,where these e†ects become important, as Figure 7 shows. Ifwe compare the parameters of the two distributions at theintermediate amplitudes where both take a nearly exponen-tial form, using the approximations in equations (17) and(24), unsurprisingly we Ðnd that they agree very well, asFigure 9 would suggest.

5. DISCUSSION

5.1. Size of the PulsarÏs Emission RegionOur analysis indicates a decreasing size across the pulse.

Table 4 summarizes our results, expressed both in terms of

TABLE 4

SIZE OF THE VELA PULSAR

Parameter Gate 1 Gate 2 Gate 3

Size corrected for averaging (kMhmps) . . . . . . 0.091 ^ 0.009 0.070 ^ 0.009 0.020 ^ 0.020

Size (FWHM)a (J8 ln 2ps) . . . . . . . . . . . . . . . . . . . 440 ^ 90 km 340 ^ 80 km 100 ^ 100 km

a Uncertainties dominated by uncertainty in magniÐcation factor M.

914 GWINN ET AL. Vol. 531

the size parameter, and as the FWHM of the best-ÐttingGaussian distribution, in kilometers. To calculate the size inkilometers from the results of the Ðt requires the additionalparameters k, M, and The angular broadening of thehm, hg.pulsar is mas) ] (2.0^ 0.1 mas), withhm ] hg\ (3.3^ 0.2the major axis at a position angle of 92¡ ^ 10¡ (Gwinn et al.1997).

We use comparison of our measured decorrelation band-width from ° 4.2, *l\ 66 ^ 1 Hz, with the angularbroadening and the pulsar distance of 500 ^ 100 pc (Tayloret al. 1993) to obtain the characteristic distance of scatteringmaterial (Desai et al. 1992 ; Gwinn, Bartel, & Cordes 1993).We Ðnd that the fractional distance of the scatteringmaterial from the Earth to the pulsar is D/(D] R) \0.60^ 0.05. Here recall that D is the distance from observerto scatterer, and R is the distance from scatterer to pulsar.We thus obtain the magniÐcation M \ 1.5^ 0.3.

We then use the Ðtted values for the size parameterto Ðnd the size in km in each gate, given in Table(kMhm p

s)

4. Note that the uncertainty in the magniÐcation factor Mdominates the quoted uncertainties of the size. The uncer-tainty in the magniÐcation factor stems, in turn, primarilyfrom uncertainty in the distance to the pulsar.

The Ðtted size is in reasonable agreement with our earlierresults for gate 1 (Gwinn et al. 1997). The major di†erencesin the analysis are the accounting for e†ects of averaging intime and frequency, for the e†ects of spectrally-varyingsignals on noise, and our use of the measured decorrelationbandwidth to determine the magniÐcation factor, in thispaper.

5.2. Size and Emission MechanismAmong the four classes of processes for pulsar emission

discussed by Melrose (1996), the measured size of about 440km rules out only models in which the observed emissioncomes from close to the polar cap. This region has a size ofless than 1 km, much smaller than the observed size. If theemission originates at this location, but is transported to alarger region, such models are still viable.

Perhaps because pulsar radiation is collimated into abeam, many pictures of pulsar emission treat the radiationas emerging in only a single direction from each point of theemitting surface, perhaps along the local magnetic Ðeld.However, for the emission region to have the size wemeasure, the observer must receive radiation from points atthe emission surface separated by D400 km. The observermeasures the separations between these points as the size ofthe emission region. Figure 10 illustrates this fact. Ofcourse, di†erent emitting regions could contribute at di†er-ent times, so that the emitting region is a point at anyinstant, and the Ðnite size is observed only by averagingover many points. Our measured size is averaged over 10 sof observations, or 112 pulses, and over a range of pulsephase.

Aberration can signiÐcantly a†ect the observed size of theemission region. At a fraction of the radius of the lightcylinder, the pulsarÏs magnetosphere travels at a fraction oflight speed. This can a†ect the interpretation of the mea-sured size. In particular, the measured size can include theheight of the emission region. Figure 10 shows an example.Aberration is particularly important in models where radi-ation is emitted at a range of heights.

Models for emission at a small fraction of the radius ofthe light cylinder, km, agree naturally with ourR

L\ 4250

FIG. 10.ÈUpper panel : Observed size of a source of beamed radiation.If the source radiates in a single direction from each point, an observer seesonly a single point. For example, if the source radiates only along the solidarrows, an observer at top sees only point ““ cÏÏ. If the source emits in arange of directions at each point, the observer measures a Ðnite size. Forexample, if each point radiates into the angle shown by the dotted lines, anobserver at top sees radiation from points ““ b ÏÏ through ““ d ÏÏ and measuressize L for the emission region. L ower panel : E†ects of aberration on mea-sured source size. The emission region rotates with the pulsar, at left, atangular frequency ). Points on the surface of the emission region radiateinto narrow cones toward the right, in the rotating frame. Aberrationredirects radiation so that it travels along the dotted paths from the loweremitting point, and dashed paths from the upper emitting point. An obser-ver above the source measures a size for the emission region comparable toits height.

observation. For a simple dipole Ðeld, the open Ðeld lineshave a cross-section of about 440 km at an altitude of about450 km. However, sources that emit at a range of altitudescan have measured sizes that include e†ects of the spread inaltitude, so that emission could arise at altitudes of 0 to 440km, for example. Models for emission from near or at thelight cylinder could also produce the observed size,although in that case some physical process must limit theemission to a region less than 10% of the diameter of thelight cylinder in size.

The size, and its decrease across the pulse, are in approx-imate agreement with the results of Krishnamohan &Downs (1983). They concluded that the emission early inthe pulse arose from a larger region than that late in thepulse, and that the spread in altitude of emission was about400 km. Although their methodology was quite di†erent,their picture of the emission region has remarkably similardimensions.

The measured size is large for ““ core ÏÏ emission, which isbelieved to arise near the stellar surface (Rankin 1990),because it statistically reÑects the opening angle of thedipole Ðeld lines at the polar cap. If VelaÏs pulse does showcore emission, as some suggest, this leads to an apparentparadox. The third gate, with size consistent with zero,might represent the core emission, with the leading edge of

No. 2, 2000 VELA PULSAR SIZE 915

the pulse one side of a cone. Studies of size of the emissionregion as a function of frequency, and imaging of the source,may resolve this question. Measurements of size for otherpulsars would also be extremely valuable. Such obser-vations are now in progress.

Arons & Barnard (1986) pointed out that radiation trav-eling nearly along pulsarsÏ magnetic Ðeld lines, with polar-ization in the plane deÐned by the pulsarÏs magnetic Ðeldand the radiationÏs wavevector (the ““ O-mode ÏÏ), interactswith the outÑowing electron-positron wind. This radiationcan be ““ ducted ÏÏ along Ðeld lines (Barnard & Arons 1986).In principle, such ducting can lead to a large apparent emis-sion region even though the original source of radiation,perhaps near the pulsarÏs polar cap, is quite compact. Lyuti-kov (1998b) points out that microphysics can amplify, aswell as refract, these waves.

Magnetospheric refraction takes place for only onepolarization. The Vela pulsar is heavily linearly polarized,so this picture and the size measurements suggests that thedominant polarization represents the O-mode. However,the less-dominant polarization mode is present, and shouldshow zero size if magnetospheric refraction sets the size.Thus, polarimetric size measurements suggest a simple testof whether magnetospheric refraction is responsible for theobserved size. Such measurements should be little more dif-Ðcult than those described here, and are presently beingpursued.

6. SUMMARY

We present measurements of the size of the radio-emission region of the Vela pulsar from the observed dis-tribution of correlation amplitude on a short interferometerbaseline. In strong scintillation, this distribution is exponen-tial if the source is pointlike, and is the sum of three expo-nentials if the source is small but extended (Gwinn et al.1998). The e†ects of Ðnite size on this distribution are con-centrated at small amplitudes. We Ðnd that averaging infrequency and time has e†ects similar to those of Ðnite size.We calculate the distribution including e†ects of Gaussiannoise in the observing system, when an interferometerobserves the source. This e†ect is likewise greatest at thelowest amplitudes, with a di†erent functional form. We cal-culate the expected e†ects of variations in pulsar Ñuxdensity and in instrumental gain : these e†ects tend to makethe distribution more sharply peaked at zero amplitude andÑatter at high amplitudes, and are small unless the varia-tions in gain approach 100%. We calculate the e†ects ofself-noise, which likewise make the distribution functionhigher at low amplitudes and Ñatter at high amplitudes, and

are important when the number of independent samples,is small. We Ðnally consider the e†ect of normalizationN

q,

of the correlation function, and Ðnd that this makes thedistribution function fall more rapidly than the exponentialextrapolated from small amplitudes, when the correlationapproaches 100%.

We compare model distributions of amplitude with thatobserved for the Vela pulsar on the Parkes-Tidbinbillabaseline at j \ 13 cm. We measure the noise level, and thevariation in gain with frequency channel, from observationsof a strong continuum source. Models including a Ðnite sizefor the pulsar in all three gates provide good Ðts. We correctthe Ðtted sizes for the e†ects of averaging in frequency andtime. The Ðtted size parameters are signiÐcant at about 10times the standard errors, depending on the gate. Table 2summarizes the results. The size decreases across the pulse.We Ðnd that e†ects of gain and pulsar Ñux-density varia-tions, self-noise, the Van Vleck correction, and averaging intime and frequency are expected to be undetectably smallfor our observation. E†ects of normalization of the corre-lation function are detected, but do not signiÐcantly a†ectthe size estimates. The linear size of the pulsarÏs emissionregion is about 440 km in the Ðrst gate, 340 km in thesecond, and less than 200 km in the third.

We discuss size measurements for sources that, likepulsars, radiate into a collimated beam. The measured sizeis the size of the region on the source visible to the observer ;that size may not be identical from all lines of sight, so thatthe measured size may easily change as the source rotates,as we observe. Aberration e†ects may also a†ect the mea-sured size, so that the range in altitude of the emissionregion, as well as its width, may contribute to the size.

Our measured size is much larger than the sizes of postu-lated polar-cap emission regions, so that such models,without additional physics, are ruled out for this pulsar.However, magnetospheric refraction can duct radiationfrom a compact emission region to produce a much largersize (Arons & Barnard 1986 ; Barnard & Arons 1986). Theobserved size is on the order of postulated emission regionsin the lower open-Ðeld-line region, suggesting that this is alikely location for pulsar emission. It is much smaller thanthe outer open-Ðeld-line region or the light cylinder, butemission might arise in only parts of these regions, or mightbe beamed toward the observer from only certain parts.

We thank B. Rickett for useful discussions and J. Cordesfor sharing an unpublished manuscript. We thank the U.S.National Science Foundation for Ðnancial support (AST97-31584).

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