The Nature of Radio Emission from Distant Galaxies The 1.4 G

a r X i v :a s t r o -p h /9908313v 1 27 A u g 1999The Nature of Radio Emission from Distant Galaxies:The 1.4GHz Observations E.A.Richards National Radio Astronomy Observatory 1and University of Virginia Received

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ABSTRACT

We have conducted a deep radio survey with the Very Large Array at 1.4GHz of a region containing the Hubble Deep Field.This survey overlaps previous observations at8.5GHz allowing us to investigate the radio spectral properties of microjansky sources to?ux densities greater than40μJy at1.4 GHz and greater than8μJy at8.5GHz.A total of371sources have been catalogued at1.4GHz as part of a complete sample within20′of the HDF.The di?erential source count for this region is only marginally sub-Euclidean and is given by n(S)=(8.3±0.4)S?2.4±0.1sr?1Jy?1.Above about100μJy the radio source count is systematically lower in the HDF as compared to other?elds.We conclude that there is clustering in our radio sample on size scales of1′-40′.

The1.4GHz selected sample shows that the radio spectral indices are preferentially steep(

α8.5=0.35), with less than15%inverted.We argue that we may be observing an increased fraction of optically thin bremsstrahlung over synchrotron radiation in these distant star-forming galaxies.

Subject headings:galaxies:evolution–galaxies:active–galaxies:starburst–cosmology:observations–radio continuum:galaxies

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1.Introduction

Deep radio surveys show a surface density of radio objects approaching60arcmin?2 down to1μJy(as inferred from?uctuation analyses on the most sensitive VLA images at 1.4and8.4GHz),or equal to the surface density of B=27magnitude galaxies(Fomalont et al.1991,Windhorst et al.1993,Richards1996,Richards et al.1998,hereinafter Paper I).Despite their modest average luminosity of L?1022.5W/Hz,the sheer number of microjansky sources implies that they dominate the radio luminosity budget of the universe at centimeter wavelengths.However,surprisingly little is known about the physical origin of these objects or their nature.

Some generalizations from studies involving deep5and8.5GHz VLA?elds imaged with HST and ground-based telescopes are possible(Hammer et al.1995,Windhorst

et al.1995,Fomalont et al.1997,Richards et al.1998).At least half of microjansky sources are associated with morphologically peculiar,merging and/or interacting galaxies with evidence for active star-formation(blue colors,infra-red excess,HII-like optical spectra).The remaining identi?cations are composed of low-luminosity FR Is,ellipticals, Seyferts,LINERs,and luminous star-forming?eld spirals.Thus a variety of physical mechanisms may be driving the observed evolution among microjansky radio sources, including non-thermal radiation from AGN activity,synchrotron emission associated with di?use supernova remnants,and thermal emission from HII regions.

Another clue is available from the observed distribution of radio spectral indices. While at millijansky levels,the average spectral index for radio sources is aboutα～0.8 (Donnelly et al.1987),microjansky sources selected at high frequencies(ν≥5GHz) have a surprisingly?at spectra ofα=0.3±0.2(Fomalont et al.1991,Windhorst et al. 1993).Several explanations for this observed?attening compared with sources selected at higher?ux density levels are possible,including,free-free absorption,an increasing number

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of synchrotron self-absorbed AGN among the microjansky population,and/or a rising component of thermal radiation from active star-formation.

The purpose of this study is to enlarge the faint radio sample and investigate the radio spectral and morphological properties for a statistically signi?cant sample of microjansky sources.In addition,examination of the optical properties of the identi?cations may shed insight into the nature of the radio emission in these sources.To further this goal we have observed a region of the sky at1.4GHz and8.5GHz centered on the Hubble Deep Field, where excellent wide?eld optical data are available.In Paper I(Richards et al.1998)we presented the optical identi?cations for a complete sample of twenty-nine8.5GHz selected radio sources in our radio survey of the Hubble Deep Field.The principal conclusion is that the majority(～70%)of the identi?cations are with relatively bright(R～22mag.) disk systems,many at moderate redshifts(z=0.4-1).In this paper(Paper II)we present the1.4GHz observations.In section two we describe our observations and data reduction techniques.Section three presents our source list,while in section four we calculate the spectral index distribution.In§5we discuss the spatial clustering of sources in our catalog. Finally,in§6we summarize our?ndings and give conclusions.

2.The1.4GHz Radio Observations and Data Reduction

In November1996,we observed a?eld centered on the Hubble Deep Field(α=

12h36m49.4s andδ=62?12′58.00′′(J2000))for a total of50hours at20cm in the

A-con?guration of the VLA.In order to minimize chromatic aberration,we observed

in a pseudo-continuum,spectral line mode with7×3.125MHz channels centered on intermediate frequencies1365MHz and1435MHz,frequency windows known to be relatively free of radio frequency interference.Each frequency channel was composed of two independent circular polarizations.All knowledge of linear-polarized intensities was lost

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due to the limited number of VLA correlator channels.Visibility data were recorded every 3.3sec from the correlator.

We monitored the point source1313+675(S1.4=2.40Jy),located6.5degrees from the HDF,every40minutes to provide amplitude,phase,and bandpass calibration.Daily observations of3C286with assumed?ux densities of14.91Jy at1385MHz and14.62Jy at 1435MHz provided the absolute?ux density scale.

The calibrator observations allowed us to identify baselines with systematic phase or amplitude errors.A few baselines were found to have recurrent amplitude and/or phase closure errors greater than?ve percent and/or?ve degrees.All data associated with these suspicious baselines were discarded.To remove bursts of radio frequency interference,we excised all visibilities with?ux densities greater than about10σabove the expected rms value of0.08Jy.This amounted to about2%of the data.

After time averaging the(u,v)data from3.3to13sec,we made preliminary 10′′resolution maps which cover the?eld out to the?rst sidelobe of the primary beam about0.8?from the phase center.We searched these images for bright,confusing sources whose sidelobes might contaminate the noise properties of the inner portion of?eld.All sources above0.5mJy were catalogued.

We then imaged and heavily CLEANed these sources using the full unweighted(u,v) data set.Because the primary beam response changes signi?cantly over our44MHz bandpass(about3%),it was necessary to deconvolve each of the confusing sources using each3.125MHz channel.In addition,the confusing sources were independently deconvolved in each of the circular polarizations(right and left)to account for the’beam squint’of the VLA antenna.Their CLEAN components were Fourier transformed and then subtracted from the visibility fd4f14ee4afe04a1b071de03ing these”strong source subtracted data sets”,we then imaged the inner few arcmin of the?eld.With this procedure the rms noise was found to be about

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50%higher than expected from receiver noise alone.In particular,a few sidelobes from particularly strong sources(S>10mJy)located near the half power point and?rst null of the primary beam were still apparent.

By examining the images made from30minute segments of the data,we isolated a few time intervals where the visibility data appeared to be corrupted,possibily due to low level interference.For any of these30minute snap-shot images with a a rms noise greater than50%of the mean value,the corresponding visibility data were deleted from the?nal analysis.These data amounted to about seven hours of time;thus,in all,about42hours of of high quality data were used to construct the?nal images.The?nal combined data set has a rms noise of7.5μJy,whereas we expected a noise closer to5μJy.

2.1.Construction of the1.4GHz Images

Our goal is to map the40′?eld of the1.4GHz observations out to the20%response of the VLA antennas.There are two complications which make this a di?cult task.

First is the sky curvature.While in practice the VLA is often treated as a two dimensional array,in reality the instrumental response to radio emission from the sky is a three dimensional complex function.For observations of short duration,small?elds of view, or low resolution,most sources can be adequately deconvolved without reference to the so called3-D e?ect.However,the isoplanic assumption fails for sources located further than θ?1/n syn radians from the phase tracking center,where n syn is the phase center distance in synthesized beam widths.For the A-array at1.4GHz this corresponds to a patch size of about18arcmin across.

Thus we chose to approximate the one degree primary beam of our observations by using a number of independent and equidistant facet images.Each facet is constructed from

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the Fourier transform of the data which has been phase shifted to its tangent point on the celestial sphere.In all,16facets were used.This technique is known as polyhedral imaging and is discussed at length by Cornwell&Perley(1992).

One further complication comes from the practical limitation of the extensive computations needed to reduce these these wide?eld observations.Our entire calibrated and edited data set consists of over107discrete complex visibilities(after13sec time averaging)which must be directly Fourier transformed to compute the sky brightness distribution over approximately the same number of pixels.Because we are interested in radio emission over the entire primary beam,we must properly deconvolve sources within the entire?eld of view.Although only sources fainter than0.5mJy remain in the visibility data after the subtraction of the confusing sources,their collective sidelobes can limit the dynamic range of the?nal images.

The most accurate method for deconvolution of the faint sources is to CLEAN each of the16images in parallel,subtracting each source’s sidelobe contribution from all the image facets simultaneously.In this manner,one would recover the sky brightness function free from sidelobe contamination over the entire primary beam.However,in practice,this is a prohibitive computational task.Therefore we opted to CLEAN each of the16facets in series using the much more e?cient Clark-Hogbom algorithm(Clark1980).The price paid is that sidelobes from the multitude of sources less than0.5mJy are only properly removed locally,within the inpidual image facet which contains each source.

In order to examine what e?ect this might have on the rms noise near the center of the image,we performed a simple test.First we made an image of the inner seven arcmin of the?eld center,heavily CLEANed using the Clark-Hogbom method.Then we made a similar image twice the size of the former,again heavily fd4f14ee4afe04a1b071de03parison of the two images,one relatively free of sidelobe contamination from the far?eld sources with S≤0.5

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mJy,the other not,yields a?rst order approximation of the unCLEANed sidelobes left in the center of the?eld due to our deconvolution method outlined above.About0.8μJy of ?ux density remains per independent beam,and thus increases the noise about10%over the thermal noise.

We produced the?nal image by CLEANing each of the sixteen2048×2048pixel?elds (0.4′′pixels)with10,000iterations and a10%loop gain.This produced an image with a natural resolution of2.0′′×1.8′′and P.A.=?86?.The rms noise near the center of the?eld was7.5μJy,approximately30%higher than the value expected from thermal noise alone based on our system temperature(37K on average),integration time,and bandwidth.The noise in each of the16images was found to be fairly uniform,although it increased to as much as10μJy in regions near the brightest millijansky sources.

Residual ringlike structure from unCLEANable sidelobes around these strong sources was evident,suggesting that our images are dynamic range limited at about5000:1.This is typical of blank?eld deep imaging where the data cannot be e?ectively self-calibrated. We clipped out regions immediately around these sources,as well as a larger region around one particularly strong source(S=35mJy)located at the half-power point of the primary beam.We attribute the dynamic range limitation in this image to pointing?uctuations, which are typically15′′rms for the VLA.This e?ect causes amplitude?uctuations in the apparent brightness of the radio sources,which are of order1%at the half power point,and hence induces unCLEANable sidelobes into the image.

Examination of the stronger deconvolved sources in the?eld with S p>

5mJy showed

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evidence of a radial sidelobe oriented towards the phase tracking center,or center of our images.These sidelobes appeared to be symmetric about these stronger millijansky sources in the?eld and with?rst order amplitude of about10%.Their width was approximately that of the delay beam.

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These sidelobes were also apparent during our test observations of1400+621,and only appear around the o?axis observations.Other observations taken at the VLA have not shown similar artifacts and our suspicion is that the problem was an online system recording error with the3.3sec visibility data that we chose to use.Examination of VLA data of the same test?eld taken in an identical observing mode,except with10sec visibility integration,showed no sign of the radial sidelobes.

The cause of these artifacts is unknown.They do not appear to be present around the near?eld,weaker sources(less than about1mJy).As a test of whether these sidelobes might adversely e?ect our measurements of the?ux densities of the stronger millijansky sources,we examined the o?-axis?ux density measurements of1400+621in images with and without the presence of the radial sidelobes.The measurements agree to within the standard?ux density errors.It is more di?cult to assess if these radial sidelobes adversely a?ect the general rms noise in the?nal1.4GHz images.

3.The Complete Source List

In order to minimize the introduction of spurious radio sources into our sample,we examined the distribution of the negative pixels in the images to estimate its completeness limit.In general most regions appeared to re?ect well behaved Gaussian noise with the most negative peak being about?ve times the local rms noise.However,one region with a local rms of7.5μJy contained an unresolved negative feature with S p=-63μJy.This feature appeared to be quite isolated and located several arcmin from any of the brighter millijansky confusing sources.No other strange artifacts(e.g.,rings,streaks,residual sidelobes)were apparent in the vicinity of this negative’source’;hence its presence remains enigmatic.The next most negative pixel value of-40μJy is located near a strong confusing source.We therefore adopt40μJy to be the formal completeness limit over our entire one

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degree?eld.Even if there are positive counterparts to the-63μJy’source’,there should be no more than a few if the negative and positive pixel amplitude distribution are fairly symmetric about zero.However,we note that the probability of?nding a–9σsource within our?eld is much less than one percent,and demonstrates that the noise properties of this image are not entirely Gaussian.

Next we searched our images for all pixels with S≥40μJy and?t these sources with elliptical Gaussians to determine their peak?ux densities and positions using the automated AIPS task SAD.In all314sources were found within20arcmin of the center of the image(20%power contour).

3.1.Determination of Discrete Source Angular Sizes and Flux Densities

Gaussian?tting routines such as SAD are subject to noise dependent biases which cause signi?cant overestimation of source sizes and?ux densities(Windhorst et al.1984, Condon1997).In order to estimate the e?ects of population and noise bias in our images, we performed a series of Monte Carlo simulations.Our basic technique was to inject a number of point sources(100)of known?ux density into the CLEANed sky images of this ?eld.Then these sources were recovered from the images using SAD and their measured peak and integrated?ux densities compared to the input model.We used the ratio S p/S i to determine if the simulated source was signi?cantly broadened by population and/or noise bias.

Because our goal is to set a resolution criteria in the presence of these?tting biases, we performed this simulation as a function of?ux density,or alternatively,signal to noise. By examining the distribution of S p/S i at di?erent?ux densities,we obtained some idea about at what level we could con?dently believe that a given source was resolved in our real

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sky images.For a source of intrinsic dimensionθmaj×θmin resolved by a beam of extent a maj×a min,the integrated and peak?ux densities are related by S i=S p(1+θmaj×θmin

whereσsnr is the signal to noise ratio of the radio source.Thus σsnr

at the detection limit of our images(40μJy),a source must have a S p/S i value less than 0.57to satisfy our resolution criterion,or an angular size greater than about2.7′′.It follows that we can resolve only those sources stronger than70μJy integrated?ux in our complete sample.

Next,we used these same simulations with the sources of angular size that we could have just detected at each signal to noise ratio(e.g.,2.7′′at S p=40μJy).Comparison of the input to recovered peak and integrated?ux densities,and angular sizes yields an estimate of the bias induced by Gaussian?tting techniques in the presence of population and noise bias.We corrected our real sky source parameters to account for these biases. In general the peak?ux densities of the?tting routines were in good agreement with the models,while the integrated?ux densities and angular sizes from the Gaussian?ts were overestimated by up to a factor of1.5.

As a check on the integrated?ux densities for apparently resolved sources in

our complete sample obtained from the Gaussian?tting algorithms,we examined the distribution of peak?ux densities as measured in various resolution images.Images were constructed at1′′,3.5′′,and6′′resolution in addition to our nominal2′′naturally weighted image.Table1gives the parameters of each image.For those sources which satis?ed our initial resolution criterion,we checked that the peak?ux density of the source increased with decreasing resolution consistent with the?tted angular size.Those sources which did

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not were considered unresolved and their adopted peak and integrated?ux densities were measured on the the2′′image.Figure1shows a greyscale of the inner10′of the1.4GHz image.In Figure2,we show the greyscale of the8.5GHz mosaic image(see§6).

We also searched the lower resolution3.5′′and6′′images for resolved sources not detected in the untapered2′′image above our completeness limit of S p=40μJy.This search yielded57additional sources in the tapered images above completeness limits of50μJy at3.5′′resolution and75μJy at6′′resolution.Because the angular sizes are uncertain in these low signal to noise ratio detections(and in many cases may be instrumentally broadened),the adopted peak and integrated?ux densities were set equal to the peak pixel values in the tapered images.These additional sources were added to the complete source list which in total contains371sources.The complete sample of all radio sources detected at1.4GHz within20′of the phase center is presented in Table2.

A description of Table2is as follows.All uncertainties are given at the one sigma level.

Column(1)—The Right Ascension in J2000coordinates with one sigma uncertainty.

Column(2)—The Declination in J2000coordinates with one sigma uncertainty.

Column(3)—The deconvolved(FWHM)major axis of the best Gaussian?t to the source,Θ,is given in arcsec.

Column(4)—The signal to noise ratio of the detection calculated from S p/σwhere S p is the peak?ux density as measured in either the2′′,3.5′′,or6′′image,andσequals the rms noise in that image.

Column(5)—The integrated sky?ux density(S1.4)after correction for the instrumental gain,(see§3.2),

with corresponding one sigma errors.

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3.2.Instrumental Corrections

We must also correct the derived source parameters for various instrumental e?ects.In order to measure the o?axis response of the VLA to a point source,we observed the4.2Jy point source1400+621in each of the cardinal directions at positions of5,10,15,20and25 arcmin from the nominal phase reference center.

There are four principal e?ects which reduce point source response as a function of radial distance from the phase center.In their degree of increasing importance they are 1)3-D smearing,2)?nite time visibility sampling(time delay smearing),3)chromatic aberration(bandwdith smearing),and4)primary beam attenuation.

1.We have approximated the curvature of the celestial sphere with a number of two-dimensional facets.However,smearing may still be present at a reduced level near the edges of each of our inpidual facet images.The e?ect of a small amount of3-D smearing are such that the integral?ux density of a given source is preserved while its peak signal is reduced by an amount dependent on distance from the tangent point on the celestial sphere. In order to determine the amplitude of3-D smearing in our data,we performed a series of simulations inserting point sources at a variety of distances from the phase tracking center into visibility data with our exact(u,v)coverage.These data were then imaged in the same manner as the true sky images.In this manner we determined the amount of point source degradation as a function of distance from the celestial sphere tangent point or image facet center.At the maximum possible distance of a source from a tangent point(～600′′)the peak degradation is less than20%.All values of the peak?ux density(S p)were corrected for this e?ect according to our empirically?t polynomial.

2.The calibrated(u,v)data set used to construct our images consisted of13sec averaged visibility data.Because the actual radio sources rotate in the sky during this sampling time,their?ux density is smeared in the image plane.The analytic calculation

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of this smearing is complicated by the aspects of the observing geometry.However,at the North Celestial Pole the e?ect reduces to a tangential smearing in the image plane and its amplitude is given by S p/S t=1–2.06×10?9θ/θsyn where S t is the integrated?ux density of a source located an angular distanceθfrom the phase tracking center andθsyn is the size of the synthesized beam.We used this approximation to correct the peak?ux densities as measured in our images.

3.Although we planned our observations of the HDF?eld to minimize the e?ects of chromatic aberration,for far-?eld sources this e?ect can still be important.It is especially crucial to understand the e?ects of smearing on the completeness level of the images.We measured the o?-axis response of1400+621due to chromatic aberration by examining the ratio of the peak to integrated?ux density,S p/S i,as a function of distance from the image phase center.Bandwidth smearing is governed by the observing frequency,width and shape of the bandpass and the(u,v)coverage,all of which de?ne the synthesized beam.In theory, these are known functions and the?nal beam response can be calculated analytically as a function of position in the image.However,in practice such uncertainties as non-uniform (u,v)coverage,central intermediate frequency o?sets,and imperfect bandpass?lters make this impossible.We chose instead to?t a function to the empirical data of the form S p/S i =(1+(r/k)2)?0.5where r is the distance from the phase center and k is a constant which absorbs the uncertainties discussed above.Our least squares?t to the empirical data is shown in Figure3where k=16.19arcmin.This is within3%of the theoretical value assumingδν/ν=3.125/1400.5and aθbeam=2.1′′.For purposes of correcting S p in our images,we use the above equation scaled by the appropriate beam size.The integrated?ux density,S i is preserved and needs no correction for smearing.

4.The primary beam attenuation at1.40GHz has been measured to about1% (Condon1997)to the?rst sidelobe(-10dB).Each of the peak and integrated?ux densities

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in the source list were corrected for the primary beam attenuation.The uncertainty in the correction is the standard rms pointing error of the VLA elements(about15′′)multiplied by the di?erential log of the primary beam response.

In Figure3we plot the point source response of a source as a function of distance from the phase center.We de?ne the instrumental gain factor as the product of bandwidth smearing,time averaging smearing,and the primary beam correction.

3.3.Positional Accuracy

The assumed position of our phase calibrator1400+621isα=14h00m28.6526s and δ=62?15′38.526′′(J2000).We observed systematic,monotonically increasing shifts in both RA and DEC of comparable amplitude as a function of r.It is a small e?ect and the di?erence between the actual radial distance from the?eld center and that measured in the image plane isδr=-0.042arcsec at25arcmin distance from the?eld center.

This small systematic term can be explained by the so called annual aberration e?ect (Fomalont et al.1992).The predicted scale contraction from annular aberration in the direction of the HDF during the Julian epoch1996.6is0.9999783.The good agreement between the observed and predicted scaling factor for1400+621yields con?dence that our images are free from signi?cant distortions due to asymmetric bandpasses or IF o?sets.

We also corrected all the radio source positions in our catalog for position o?sets induced by the3-D e?ect as discussed in§3.2.By phase shifting the image tangent points to the location of inpidual sources,their true angular positions on the sky were measured. The di?erence between the apparent position as measured on the nominal sky maps and the corrected positions is typically only0.2′′in the north-south direction.

After correction for annular aberration and the3-D term,the relative positional

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accuracy for sources across the?eld in the limit of in?nite signal to noise should approach about0.02′′.The absolute positional accuracy depends on the translation of our phase calibrator position to that of the HDF and is about0.02′′.Thus we estimate our radio catalog to be within0.03′′of the J2000/FK5coordinate grid.The single coordinate rms position errors as given in Table2are de?ned asσ=

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size greater than2.7′′(the angular size detection limit of the weakest radio sources in our sample as de?ned in§3.1),we?nd twice as many resolved sources with S i>250μJy(52%) as opposed to those with S i≤250μJy(25%).This suggests thatθmed may be a decreasing function of?ux density.Figure4shows our measurements of angular size as a function of source intensity.In order to estimate the mean angular size of this sample,we applied the survival analysis techniques of Feigelson&Nelson(1985)using the statistical package ASURV(Rev.1.2;Isobe&Feigelson1992).This technique incorporates upper limits in the calculation of the mean of a distribution,which is particularly important in our sample which is dominated by non-detections(i.e.we are measuring the tail of the angular size distribution).The technique assumes a symmetric Gaussian model,hence the mean and median are equal.At S1.4=370μJy,θmed=2.6′′±0.4′′,and at S i=100μJy,θmed= 1.6′′±0.3′′.The errors are based on the number of angular size measurements(not limits).

In Fig.5,we compare our determinations ofθmed with previous measurements made at1.4GHz.Because of the uncertain selection e?ects inherent in the higher frequency deep radio surveys,particularly their bias towards?at-spectrum,compact AGN,we chose not to include these points.With the notable exception of the discrepant Condon& Coleman(1985)point,there is general agreement amongst the di?erent data.Because the median angular size is known to change rather sharply below a few millijansky at1.4GHz (presumably due to the emergence of an increasing population of starburst galaxies among radio sources)from～10′′to a few arcsec,we suggest that the high?ux density point of Oort(1988)is too low,possibly due to resolution biases in his A-array snapshots.

We believe the decrease in angular size at lower?ux densities to be real.Thus for the purposes of modeling the median angular size-?ux density relationship,we?t a function of the form:

θ=frac34×0.175S0.51.4+frac14×0.1arcsec

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This?t is also shown in Fig.5.For completeness,we also plot a straight line,withθ=2.0′′and independent of?ux density.

fd4f14ee4afe04a1b071de03pleteness

In order to investigate the combined e?ects of noise,population,and resolution bias on the completeness level of our survey,we used a series of Monte Carlo simulations similar to the ones described in§3.1.In particular we want to determine how many sources with S i≥40μJy,but with S p<40μJy we missed based on our peak?ux density detection limit. We randomly populated our sky images with100sources of?nite angular size assuming an angular size distribution as found in§4.1.This simulation was repeated at a variety of ?ux densities from40to1000μJy.In each?ux density interval the ratio of the number of sources recovered from the images with S p≥40μJy to the number originally injected in the model was tabulated.This was taken to be the e?ective correction factor needed to account for the combined e?ects of resolution,population and noise bias in our images (although resolution bias is always the dominant term).At80μJy which is the average?ux density source detected in our survey(weighted by S?2.4)and where the count will most accurately be determined,this correction factor is1.05.As resolution bias is the dominant source of incompleteness in our survey,we estimate that we have detected approximately 95%of the microjansky sources in the HDF region to this?ux density limit.The principal uncertainty in the correction factor is the uncertainity in the angular size distribution of the microjansky radio sources.If we had assumed a constantθ=2.0′′model,our corrections would have increased by over a factor of two.

From the complete source list of Table2,we then binned sources in?ux density intervals such that each bin had at least50radio sources(except for the highest?ux density bin).The di?erential count was then calculated based on the number of sources in each bin

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interval.Bandwidth smearing,time averaging smearing,and the primary beam response decreases the e?ective area over which a source of given S p can be detected.Thus when counting the number of sources in each bin,care must be taken to weight the contribution of each source to the count by the e?ective area over which it could have been detected. This factor can be calculated by solving for the image radius where a source of amplitude S p would have just been missed by our peak detection limit(Katgert1973).Table3presents the di?erential source counts for our complete?ux limited sample.The counts from this survey are compared to other microjansky surveys in Figure6a and6b(Mitchell&Condon 1985,Oort&Windhorst1985,Hopkins et al.1998)normalized to a Euclidean geometry n/n o=n(s)/S?2.5.In general the agreement is reasonable and in agreement with the errors and possible?eld to?eld variations.The best?t to the source counts in this?eld in the range40-1000μJy is n(S)=(8.25±0.42)S?2.38±0.13ster?1Jy?1.

The counts in the HDF appear systematically lower than those of other?elds above 100μJy.This e?ect could be due either to1)real?eld to?eld variations on the degree scale as a result of large-scale clustering of radio sources,or2)survey incompleteness due to the?nite angular size of the radio sources.Without complementary,low resolution observations,it is di?cult to discriminate between these two possibilities.We note that if the mean angular size does not decrease signi?cantly below100μJy,the radio sky will become forever naturally confused at the level of a few hundred nanojansky,perhaps providing a natural limitation to the sensitivity of the next generation of centimeter radio telescopes(Windhorst et al.1993).

5.Spatial Clustering

In order to test for the presence of two dimensional spatial clustering among the radio sources in the Hubble Deep Field,we calculated the two-point correlation function for the

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sources in Table2.First,we compiled a table of angular separations by considering the separation of each inpidual source with all other sources in the catalog.These provide our DD estimate(Peebles,1980).Next,we generated random catalogs of sources according to the source count of§4.2,and distributed randomly across a40′VLA primary beam.The peak?ux densities of these sources have been attenuated by the instrumental corrections discussed in§3.2.Angular pairs were calculated for these catalogs and form the basis of our RR measurement.We de?ne the correlation function of our catalog to be w(θ)=DD/RR -1.The correlation function of our catalog on scales of0′-40′is presented in Table4.

We calculated the errors in our clustering measurement by following the bootstrap method of Ling et al.(1986).These agree well with the Poissonian error estimate ofδw(θ) =1+w(θ))/N DD where N DD is the number of independent data pairs in a given bin.We ?nd evidence for an excess of radio sources on scales of approximately1-10′,while on scales much larger than this there are fewer radio sources in our catalog than expected from a random distribution.Figure7shows the correlation function for radio sources in the HDF, compared to the correlation function of a somewhat shallower1.4GHz survey of Oort (1987)complete to100μJy.The amplitudes are comparable in the two separate surveys.

Spatial clustering at higher?ux density levels and lower amplitudes has been reported by Cress et al.(1996)and Magliocchetti et al.(1998).More recently,Hopkins et al.(1999) claim?uctuations in?eld to?eld source counts at similar completeness levels to ours, possibly indicating the presence of large-scale radio source spatial variations.Thus it is plausible that there are both fewer radio sources in the HDF region than the average?eld, and that these sources are clustered on arcmin scales amongst themselves.

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6.Radio Spectral Indices

The HDF has been observed previously with the VLA at8.5GHz to a one sigma sensitivity of1.8μJy(Paper I).In June1997we observed the HDF region for an additional 40hours at8.5GHz.We mosaiced an area de?ned by four separate pointings o?set

2.7′from the center of the HDF(the half-power scale of the primary beam response)in each of the cardinal directions for about10hours duration each.The observing technique and data reduction are discussed in Paper I.The?nal combined8.5GHz images have an e?ective resolution of

3.5′′and a completeness limit of8μJy.The sensitivity of this mosaic to sky emission is a sharp function of distance from the nominal pointing center because the observations were heavily weighted towards imaging the central HDF region.

Because the size of the VLA primary beam scales inversely with frequency,our sensitivity at8.5GHz is limited to the inner6.6arcmin(HWHM)of the1.4GHz?eld. This is the point where the maximum beam attenuation at8.5GHz is equal to0.2(while at1.4GHz the attenuation is only0.9).Within this region there are109sources contained in the1.4GHz complete sample.We measured the8.5GHz?ux density at the location of each of these sources.When a source was not clearly detected(S p<3σat8.5GHz),we calculated a conservative upper limit to its8.5GHz?ux density equal to three times the rms noise corrected by the antenna gain.If a1.4GHz radio source had a peak?ux value 3σ

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