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synoptic surveys for gamma - ray bursts ( grbs ) , and subsequent ground - based observational follow - up at radio through optical wavelengths , has highlighted the importance of transient celestial phenomena ( masetti 2001 ) . the new parameter space of the transient cosmos has been emphasized in the design of future telescopes , such as the optical large synoptic survey telescope ( tyson & angel 2001 ) , and the radio square kilometer array ( van haarlem 1999 ) . while it is well documented that flat spectrum radio sources can be variable ( aller et al . 1985 ) , the areal density of such sources has not been well quantified through multi - epoch , wide field blind surveys . at high flux density levels ( @xmath8 mjy at 1.4 ghz ) , one can make a rough estimate of the areal density of variable radio sources by simply assuming that all flat spectrum sources are variable . for instance , the areal density of all sources @xmath9mjy is @xmath10 arcmin@xmath2 , and the fraction of flat spectrum sources is about 10@xmath11 , implying an areal density of variable radio sources of @xmath12 arcmin@xmath2 ( gruppioni et al . 1999 ; white et al . 1997 ; windhorst et al . 1985 ; hopkins et al . 2000 , 2002 ) . this number is consistent with the ( null ) results of frail et al . ( 1994 ) in their search for highly variable mjy - level sources associated with grbs . source populations at these high flux densities are dominated by agn . below about 1 mjy the slope of the source counts flattens , and star forming galaxies are thought to dominate the faint source population ( windhorst et al . 1985 ; georgakakis et al . 1999 ; hopkins et al . 2000 , 2002 ) . hence , when considering the areal density of variable sub - mjy radio sources , one can not simply extrapolate the results from high flux density source samples to low flux densities . knowledge of the areal density of variable sub - mjy radio sources is critical for setting the back - ground , or ` confusion ' , level for studies of faint variable source populations , such as grbs ( frail et al . . a recent comparison of the nvss and first surveys by levinson et al . ( 2002 ) sets a conservative upper limit of @xmath13 arcmin@xmath2 to the areal density of ` orphan ' grb radio afterglows ( i.e. grbs for which the @xmath14-ray emission is not beamed toward us ) with s@xmath15 mjy . they also argue that the areal density of radio supernovae will be considerably smaller . while the area of the sky covered by levinson et al . ( 2002 ) was much larger that the study presented herein , their flux density limit was higher than any grb radio afterglow yet recorded . in this paper we present a smaller area study , but we consider variable sources at flux density levels ( @xmath16mjy ) applicable to typical grb radio afterglows .    in general , variability of radio sources at the sub - mjy level is an essentially unexplored part of parameter space  a part of parameter space which may fundamentally drive the design of future radio telescopes , such as the ska ( carilli et al . herein we present the first study to delve into this part of parameter space , by exploring systematically the variability of the sub - mjy radio source population at 1.4 ghz . we examine variability on timescales of 17 months and 19 days . note that the lsst will probe similar variability timescales in the optical , with sampling on weekly to yearly timescales . observations were made using the using the vla at 1.4 ghz in the b configuration ( maximum baseline = 10 km ) . the region observed is in the lockman hole centered at : ( j2000 ) 10@xmath17 52@xmath18 56.00@xmath19 , 57@xmath20 29@xmath21 06.0@xmath22 . table 1 summarizes the observations . column 1 gives the observing date , column 2 gives the observed hour angle range , and column 3 gives the rms in the final image . these observations are part of a larger multiwavelength program to study the evolution of dusty star forming galaxies ( bertoldi et al . in prep ) . standard wide field imaging techniques were employed in order to generate an unaberrated image of the full primary beam of the vla ( fwhm = 32@xmath21 ) . the absolute flux density scale was set using 3c286 . we then generated a clean component model of the field using self - calibrated data taken on sept . the data from all days were then self - calibrated in amplitude and phase without gain renormalization using this model . this process should ensure that all the data are on the same flux scale . images made before and after this process showed that the absolute flux scale changed by at most 1@xmath11 . we also checked to see if variable sources could be removed ( or added ) due to using a single model to self - calibrate data from all the different days . components at the 0.1 to 0.2 mjy level were added to the self - calibration model at random positions in the field , and the self - calibration process was repeated . in no case was a new source generated . this gives us confidence that the self - calibration process is robust to small perturbations in the model , i.e. that the problem is over - constrained and that the input self - calibration model is dominated by non - varying sources . images for each day were generated using the wide field imaging capabilities in the aips task imagr ( perley 1999 ) . to remove problems with ` beam squint ' ( slightly different pointing centers for right and left circular polarizations ) the right and left polarizations were imaged separately . the images were then summed , weighted by the rms on each image . the final image using data from all the observing days is shown in figure 1 . the rms noise on this image is 7@xmath4jy beam@xmath23 and the restoring clean beam is circular with fwhm = 4.5@xmath22 . we searched for source variations over 17 months by comparing images made in april 2001 with those made in august / september 2002 , and also on timescales of 19 days by comparing images made on august 17 and september 5 , 2002 . we searched for variable sources out to the 10@xmath11 point of the primary beam , and we limited the analysis to sources with variations @xmath24 . images from all epochs were convolved to 6@xmath22 resolution to mitigate differences that might occur due to the non - linear process of deconvolution . the rms on the images used for variability analysis was 12.5 @xmath4jy for the 17 month comparison , and 17@xmath4jy for the 19 day comparison . the images from different epochs were both subtracted and averaged . a fractional variability image was then generated by dividing these two images , blanking at the 5@xmath25 level ( in absolute value ) . for the 17 months variability analysis @xmath25 was 17@xmath4jy on the differenced image , and 9@xmath4jy on the averaged image . the corresponding numbers for the 19 day analysis were 26@xmath4jy and 15@xmath4jy , respectively . the difference image for the 17 month analysis is shown in figure 2 . some artifacts are seen around the brighter extended sources arising from differences in deconvolution and residual calibration errors . beyond these artifacts , the difference images are remarkably free of sources . this result gives us confidence that the imaging process ( self - calibration and deconvolution ) does not generate spurious sources at the @xmath26 level , and tells us right away that the radio sky is not highly variable at the 100@xmath4jy level . variable sources were identified in the divided image . five variable sources were found at the @xmath27 level in the 17 month comparison , while four sources were found in the 19 day analysis . we then returned to the original images to find the flux densities of the sources at each epoch . table 2 lists source positions ( columns 1 and 2 ) , flux densities at the two epochs ( columns 3 and 4 ) , and the distance from the phase center ( column 5 ) for the 17 month analysis , and table 3 lists the corresponding values for the 19 day analysis . note that the source j1051 + 5734 was seen to vary on both timescales . in fact , this source varied between september 5 and september 9 from 1.53 mjy to 0.71 mjy . the september 2002 value listed in table 2 is the weighted average of these two measurements . we next consider the sensitivity of our observations to variable sources at some absolute level , @xmath28s . the analysis is complicated by the roughly gaussian roll - off of the primary beam of the vla , with fwhm @xmath29 . identification of variable sources was done using the non - primary beam corrected maps in order to have uniform noise across the field . of course , the final flux densities and noise levels for the variable sources were corrected for the primary beam attenuation .    given a @xmath28s , one can calculate the maximum primary beam correction , f , for which a change in flux density @xmath28s could be detected at the 5@xmath25 level , where @xmath25 is the noise at the field center on the difference /@xmath28s . the value of f then sets the distance from the pointing center , r , to which such variation could have been detected , given the primary beam shape of the vla . values for @xmath28s , f , and r are listed in columns 1 , 2 , and 3 , respectively in table 4 . column 4 lists the number of variable sources over 17 months that meet these criteria . column 5 lists the number of sources within the specified radius with flux density , s @xmath30 @xmath28s/2 . this flux density sets the limit for 100@xmath11 variability , e.g. a source could be 100@xmath4jy on day 1 , and 0 @xmath4jy on day 2 , leading to a value on the difference image of 100 @xmath4jy and a value on the average image of 50@xmath4jy . column 6 lists the areal density of sources with s@xmath30@xmath28s/2 on our lockman hole image , while column 7 lists the corresponding values from the recent study of fomalont et al . ( 2002 ) comprised of a number of different fields . we have investigated the source sizes using the b array observations at 4.5@xmath22 resolution . two of the sources are partially resolved . for j1051 + 5708 gaussian fitting to the profile yields a peak surface brightness , i@xmath31 mjy beam@xmath23 , a total flux density , s@xmath32 mjy , and a ( deconvolved ) size of @xmath33 , with major axis position angle ( pa ) = @xmath34 . the corresponding numbers for j1055 + 5718 are : i@xmath35 mjy beam@xmath23 , s@xmath36 mjy , and @xmath37 at pa @xmath38 . the rest of the sources are unresolved , with upper limits between 1@xmath22 and 2@xmath22 depending on signal - to - noise . the nature of this analysis is such that we are not sensitive to very rapid variations , e.g. timescales of minutes or less . for instance , a 50 mjy flare of 1 min duration would average down to about 100@xmath4jy over 7 hours . while our nominal sensitivity would be adequate to detect such an event , the existence of such a transient source in the visibility data would lead to errors in the self - calibration and imaging process which would manifest themselves clearly on the images . of course , it is possible that such a bright , short timescale event was mis - identified as interference in the data editing process , and removed . for the analysis on 17 month timescales we could have detected sources with @xmath28s @xmath0jy out to a radius of 7.8@xmath21 , but none were detected . this sets an upper limit to the areal density of such sources of @xmath39 arcmin@xmath2 . another interesting point is that within this radius there are 46 sources between 50 and 100 @xmath4jy on the averaged image . hence , we set an upper limit of about 2@xmath11 to the fraction of sources in this flux density regime that are highly ( @xmath40 ) variable . these results are grossly consistent with the idea that below 1 mjy at 1.4 ghz the radio source population is dominated by star forming galaxies , as compared to agn which dominate at high flux densities ( hopkins 2000 , 2002 ) . the exact distribution of agn vs. starbursts vs. other source types as a function of radio flux density is not fully determined at this time , and is an area of active current research ( hopkins et al . 2002 ; georgakakis et al . 1999 ; richards 2001 ; fomalont et al . 2002 ) . as a rough guide we consider the models of hopkins et al . they suggests that at @xmath30 10 mjy the source population is 90@xmath11 steep spectrum radio sources , 10@xmath11 flat spectrum radio sources , and @xmath41 star forming galaxies . again , at high flux density levels the flat spectrum sources correspond to the variable radio source population . at 100 @xmath4jy the models suggests that the proportions change to roughly 80@xmath11 star forming galaxies , 15@xmath11 steep spectrum agn , and 5@xmath11 flat spectrum agn . these models have not considered truly transient source populations , such as grb radio afterglows , which have timescales of 10 s of days . the statistics of radio afterglows of grbs are such that in an area of 7.8@xmath21 radius one would expect to see @xmath42 sources above 0.1 mjy at 1.4 ghz at any given time , assuming grbs are highly beamed ( see below ) . hence , the fact that we did not see such a source is not surprising . more importantly , the results presented herein allow us to set a limit on the variable source confusion level for grb radio afterglow searches , again at a flux density level relevant to the observed population . for example , radio searches are needed for localizing an important subclass of afterglows known as ` dark grbs ' ( e.g. , djorgovski et al . 2001 ) , for which optical emission from the grb is not detected . it has been suggested that the absence of optical emission from these sources is the result of either dust obscuration , or the gunn - peterson effect , i.e. ly@xmath43 absorption by the neutral intergalactic medium . this latter effect would place the sources at @xmath44 ( fan et al . an upper limit to the areal density of @xmath39 arcmin@xmath2 for variable sources at the 100@xmath4jy level implies a lower limit of 200 arcmin@xmath45 to the area that can be safely searched at 1.4 ghz before one such source is expected to be detected by chance . for comparison , the typical grb error circle is @xmath4630 arcmin@xmath45 , but larger error circles are not uncommon . this result can also be used to derive a rough limit on the mean beaming angle for grbs ( perna & loeb 1998 ) . from a sample of 25 radio afterglows ( frail et al . in prep ) we estimate that 10 - 25% will be visible above the 100 @xmath4jy level at 1.4 ghz , with an average lifetime of one month . the grb event rate is approximately 600 per year ( fishman & meegan 1995 ) so we expect to find only @xmath47 arcmin@xmath2 . however , if grbs are highly beamed , as recent studies seem to suggest ( frail et al .  2001 ) , then our upper limit to areal density of @xmath39 arcmin@xmath2 implies a beaming factor @xmath48 , or a mean jet opening angle @xmath49 . this value is not very constraining compared to existing limits , but to our knowledge this is the first time a survey for variability has been done at the appropriate flux level for radio afterglows . more stringent limits will require sensitive , larger area surveys with existing or planned instruments ( totani & panaitescu 2002 ) . a final point we consider is cosmic variance . it is possible that the lockman hole region we have sampled was just statistically - poor in transient sources . a rough indication of the effect of cosmic variance comes from the overall source counts ( columns 6,7 in table 4 ) . the source counts we derive agree to within 20@xmath11 with those found by fomalont et al . ( 2002 ) in other areas of the sky . in general , it has been found that for 30@xmath21 fields - of - view the maximum field - to - field scatter in the sub - mjy source counts is about a factor two ( fomalont et al . we consider this an upper limit to the effect cosmic variance has on the areal density of variable source presented herein . the national radio astronomy observatory ( nrao ) is operated by associated universities , inc . under a cooperative agreement with the national science foundation . we thank f. owen , e. fomalont , r. becker , and j. condon for useful comments concerning this work , and the referee for careful and insightful criticism . hopkins , a.m. , afonso , j.m . , chan , b. , cram , l.e . , georgakakis , a. , mobasher , b. 2002 , in `` galaxy evolution : theory and observations '' , eds . v. avila - reese , c. firmani , c. frenk , & c. allen , revmexaa sc . tyson , a. & angel , r. 2001 , in ` the new era of wide field astronomy , asp conference series , vol . 232 , ' eds . roger clowes , andrew adamson , and gordon bromage , ( san francisco : astronomical society of the pacific ) , p.347
we present 1.4 ghz vla observations of the variability of radio sources in the lockman hole region at the level of @xmath0jy on timescales of 17 months and 19 days . 
 these data indicate that the areal density of highly variable sources at this level is @xmath1 arcmin@xmath2 . 
 we set an upper limit of @xmath3 to the fraction of 50 to 100@xmath4jy sources that are highly variable ( @xmath5 ) . 
 these results imply a lower limit to the beaming angle for grbs of 1@xmath6 , and give a lower limit of 200 arcmin@xmath7 to the area that can be safely searched for grb radio afterglows before confusion might become an issue .