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to meet its science requirements , the large synoptic survey telescope ( lsst ) @xcite places stringent requirements on the accuracy of its photometric calibration . the uniformity of photometric zeropoints across the sky is required to be 10 mmag in the grizy bands , and 20 mmag in u. additionally , color accuracy is required to be 5 mmag for colors that do not involve u , and 10 mmag for those that do . a plan for achieving these requirements is presented in @xcite . achieving this plan , and verifying the accuracy of its results , relies on external data in the form of photometric standards . the goal of this paper is to explore the possibility that gaia @xcite can contribute to the external data needed for lsst photometric calibration . the paper is organized as follows . section [ sec : lsstphotocal ] presents an overview of the lsst photometric calibration process , and the role of external data . section [ sec : gaiadata ] is a brief summary of the gaia data products that are relevant to our purpose . section [ sec : directgaiastandards ] presents the transformation of gaia magnitudes into lsst magnitudes , and simulated results for both ms stars and wd . section [ sec : syntheticg ] presents the synthetic g band approach and simulated results for ms stars . section [ sec : conclusion ] summarizes our results , and implications for future work . the overall strategy for lsst photometric calibration is based on @xcite . they proposed _ directly _ measuring the system throughput as a function of wavelength , focal plane position , and time . further , the _ normalization _ of the throughput in each observation ( the gray - scale zeropoint ) and the _ shape _ of the throughput curve ( the wavelength dependent terms ) , are explicitly separated and measured with separate procedures for both the telescope system response and the atmospheric transmission . calibration systems based on this approach are already in use at panstarrs and des . the goal of the calibration process is to produce standard magnitudes , defined by @xmath4 where @xmath5 is the normalized standard bandpass for band b , set a priori for the survey , @xmath6 is the object flux at the top of the atmosphere , and @xmath7 = 3631 jy . to calculate @xmath8 the calibration system must determine , for each observation , the actual realized bandpass from detected electrons to the top of the atmosphere , @xmath9 , and a gray zeropoint @xmath10 . with these quantities in hand , the standard magnitude is calculated by @xmath11 where @xmath12 is the instrumental magnitude ( log of counts ) and @xmath13 is the _ shape _ of the source spectrum , which we will refer to as the sed . several hardware systems are required to implement the approach , which we briefly describe :    * a dome screen projector designed to provide uniform ( @xmath14 variation ) illumination across the field of view , while minimizing stray light . this projector system will have the capability to not only illuminate the screen with broadband white light , but also narrow - band light to measure the system response at individual wavelengths . the narrow - band light will be generated by a tunable laser , capable of producing light from @xmath15  nm and tunable in 1  nm increments . the brightness of the screen is measured with a nist - calibrated photodiode , so that the relative intensity at different wavelengths can be precisely determined . * a 1.2-m auxiliary telescope with an @xmath16 spectrograph , located adjacent to the lsst itself . this auxiliary telescope will obtain spectra of a chosen set of atmospheric probe stars across the sky to determine an atmospheric absorption model . * water vapor monitoring system , consisting of a gps system and a microwave radiometer copointed with the lsst telescope , and monitoring a similar field of view . this supplements the auxiliary telescope spectra , which are unable to track the sometimes rapid variations of water vapor in time and space . lsst uses a form of self calibration , as pioneered by @xcite , to set the zero point for each exposure . the inputs to the self calibration process are the sed - corrected magnitudes of the calibration stars in band @xmath17 , @xmath18 , for the entire survey . these are calculated from the instrumental magnitudes @xmath12 , the object seds @xmath19 , the standard bandpass for the photometric system , @xmath5 , and the system bandpass at the moment of observation , @xmath20 as    @xmath21    self calibration utilizes the facts that the survey observes each calibration star many times over the course of the survey , and that the survey fields overlap , to construct a least squares system for the zero points of each observation . the resulting set of zero points for each band has an arbitrary offset , which makes it impossible to determine object colors without additional information . this additional information can come from a variety of sources :    * photometric standards in individual lsst bands . * photometric standards in a bandpass which can be synthesized from the lsst bands . * locus of a well understood stellar population in color - color space . the need for external information goes beyond simply determining the band zero point offsets , which , in principle , could be obtained from only a single standard . in practice , results from self calibration will contain some level of spatially dependent error , which tends to be greatest for the low - order spatial modes . assessing the magnitude of this error , and correcting it if needed , can only be performed if one or more of the above sources of information is available as a function of position on the sky . this is , of course , the more demanding application , and it is the one we focus on in this paper .    as we show below , gaia data can contribute both to band zero point determination and to verification / correction of self calibration errors through the first two of the approaches listed above . we do not discuss the third approach here , but refer the reader to @xcite as an example of its use . gaia s principal photometric data product @xcite is calibrated magnitudes through a broadband g filter . additionally , it produces low resolution red ( rp ) and blue ( bp ) spectra , and higher resolution ( rvs ) spectra within a narrow band intended mostly to supply radial velocities . it is tempting to use the rp and bp spectra directly to estimate the stellar sed @xmath22 and then use that to calculate the color terms needed to transform from the gaia g band to the 6 lsst bands . the biggest problem with this approach is that rp and bp spectra , due to their low resolution and relatively broad psf , can not be directly inverted to give the incident stellar sed with sufficient accuracy . preliminary simulations showed that systematic errors of order 0.1 mag are incurred , which rules out the direct approach for our purposes . we are better off taking a different approach , and instead using apsis @xcite , gaia s astrophysical parameters inference system . we realized that apsis , which of course utilizes bp / rp as a major input , would lead to smaller errors for our case . there are two reasons for this . first , apsis takes account of data beyond bp / rp , in particular the parallax to determine absolute luminosity . second , apsis has built in the knowledge that these are not arbitrary spectra , but the spectra of stars . this prior knowledge significantly reduces the errors in the color terms @xmath23 ( defined in section [ sec : transforming ] ) from our initial naive concept . apsis utilizes all available gaia data for each star to infer the stellar parameters @xmath0 , @xmath1 , @xmath24 $ ] , and @xmath25 . @xmath25 is a stellar extinction parameter used by apsis , which is within a few percent of the more commonly used extinction at 550 nm , @xmath2 , for values of @xmath26 near 3 . for our purposes the difference between @xmath25 and @xmath2 is inconsequential , and in the remainder of the paper we use the more common @xmath2 . for sufficiently bright stars ( @xmath27 ) , this includes spectra from the rvs instrument , which substantially improves the accuracy of the resulting stellar parameters . the usable range of magnitudes for lsst standards is roughly @xmath28 , with the bright end set by detector saturation , and the faint end set by signal - to - noise for a 15 sec exposure . therefore , we must make use of stellar parameters determined without rvs . table [ table : apsis_accuracy ] shows the predicted accuracy of these parameters . as mentioned in section [ sec : lsstphotocal ] , the errors remaining after self calibration can be assessed and/or eliminated if we have an accurate set of photometric standards in the six lsst bands with a suitable distribution across the sky . work is in progress @xcite to obtain a small set of suitable white dwarfs observed with hst whose photometry can be accurately transformed to the lsst bands , but their number and spatial distribution may not be adequate to the task addressed here . the minimum number of standards required depends on the spatial power spectrum of the self calibration errors , and is difficult to determine in the absence of actual lsst data . the low order spatial error modes ( @xmath29 ) from self - calibration can be adequately measured over the roughly @xmath30 steradians observed by lsst with roughly ten standards . clearly , it would be desirable to have more , if only to verify that the power spectrum is negligible at higher @xmath31 values . as an upper limit , 2000 standards would provide one standard in each lsst field of roughly 10 square degrees . every lsst observation would then include a standard , and spatial error modes to @xmath32 would be tightly constrained . the number of standards employed in practice will be guided by the error characteristics of the self calibration process , and possibly constrained by the availability of ground - based spectroscopy resources ( see section [ sec : usingwd ] )    each observation of a standard constrains the self calibration equations by a direct measurement of the zero point at that place and time :    @xmath33    in this paper we are concerned with the uncertainties in determining @xmath34 that result solely from uncertainties in @xmath35 . in evaluating these , we make two assumptions :    * @xmath36 is known exactly . this is of course not the case , since it depends on a variety of measurements with finite uncertainty , but this allows us to isolate the uncertainty which is due solely to the precision of the standard . * @xmath36 is sufficiently close to @xmath37 that the difference can be ignored in this context . this is not obviously the case , since @xmath37 is in principle arbitrary , but in the case of lsst we will pick @xmath37 to be statistically as close as possible to @xmath36 . we therefore approximate the calibration errors that result from an imperfectly characterized standard as @xmath38    where @xmath39 is the difference between the actual and assumed sed of the standard .      to make use of gaia data for lsst photometric standards , we must transform gaia g magnitudes into lsst magnitudes in each of the six lsst filter bands . as figure [ fig : bandpasses ] shows , the gaia g band is very broad compared with the lsst bands , and has an overlap with each of them . this suggests that an accurate transformation might be possible given an achievable level of knowledge of the stellar sed . the transformation process is as follows : @xmath40 where @xmath17 denotes one of the six lsst bands , and @xmath41 the gaia g band . @xmath42 we assume that the stellar flux , @xmath6 , can be adequately parameterized by the apsis stellar parameters @xmath43 , which in turn means that the color @xmath23 can also be so parameterized : @xmath44    our main concern in this paper is with the uncertainty in the resulting @xmath45 magnitudes when the stellar parameters are determined by apsis . the uncertainty is given by    @xmath46    the expected uncertainty in gaia photometry , @xmath47 , is presented in @xcite , and reproduced in figure [ fig : gaiaphotometricerrors ] . note that the uncertainties in the figure are for single visit photometry . the expected uncertainty in the end of mission photometry is lower by roughly @xmath48 , yielding uncertainties of roughly 0.4 mmag at g=16 and 0.8 mmag at g=18 . it is clear that @xmath47 makes a negligible contribution to the total uncertainty of 5 mmag required for lsst , as discussed in section [ sec : lsstphotocal ] . therefore we require the uncertainty @xmath49 to be 5 mmag or less . the remainder of the paper is largely occupied with the assessment of @xmath49 .      given the enormous number of stars in both the lsst and gaia catalogs in the relevant magnitude range , we can be highly selective in the stellar types we select as standards . the most useful stellar type for our purpose is not immediately clear , since it depends on the details both of the performance of gaia apsis , and on the shape of the lsst filter bandpasses . we first examine the suitability of main sequence ( ms ) stars .    in order to estimate the precision to which lsst magnitudes can be found using gaia s apsis , we simulate the observed magnitudes of the stars in the phoenix spectral library @xcite for the gaia g band and the lsst u , g , r , i , z and y bands . the phoenix spectra do not include interstellar reddening due to dust , so we redden them with the prescription of @xcite using @xmath50 . we then used the published uncertainties in the stellar parameters determined by apsis ( in @xmath0 , @xmath1 , @xmath51 and @xmath2 ) to find the standard deviation of the colors @xmath23 for objects whose parameters are within the apsis uncertainties . in this fashion , the standard deviation and other statistics concerning the predicted precision of gaia measurements of stars can be found . we neglect the covariance between the various stellar parameters , which surely results in an overestimate of the dispersion in the resulting colors . this assumption is necessary due to our lack of knowledge of the covariance terms , which have not yet been published . the overestimate is likely in the range of 2 to 4 . it is necessary to interpolate between the phoenix data points because the phoenix spectra used as input are widely spaced in stellar parameter space , so that typically only a few spectra are within the range of uncertainty of the apsis stellar parameters . the interpolation program linearly interpolates in @xmath52 and @xmath2 between the magnitudes calculated from the available spectra . the program selects the interpolated points drawn from a 4-dimensional normal distribution with variances determined by the expected apsis errors , and then calculates the lsst magnitudes for each point . it then finds the standard deviation of the @xmath23 colors of the indistinguishable points , in order to find the predicted error from gaia data . .accuracy of astrophysical parameter estimation ( internal rms errors ) with the apsis algorithm aeneas using bp / rp spectra and parallaxes , but without rvs data @xcite . [ table : apsis_accuracy ] [ cols="^,>,>,^,^,^ " , ]     [ table : wd_sd ] in this section we turn to the second approach mentioned in section [ sec : lsstphotocal ] , that the six lsst bands can be combined into an approximate gaia g band , and then compared directly with gaia g band photometry , a possibility suggested by @xcite . looking again at figure [ fig : bandpasses ] , it seems plausible that the gaia g bandpass can be reasonably approximated by linear combination of the lsst bandpasses @xmath53 which leads to @xmath54 where @xmath55 is the flux of an object with sed @xmath56 integrated over the bandpass @xmath57 . if the approximation were an equality , then we could directly compare , for all objects in the gaia catalog within the lsst magnitude range , this synthetic g band flux to the actual measured g band flux . any discrepancies beyond the expected random scatter would indicate a systematic problem with the lsst photometric calibration process . the spatial dependence of these discrepancies , and their dependence on stellar parameters , particularly @xmath0 , would give clues to what aspects of the calibration process were at fault . additionally , it might also be possible to correct the lsst fluxes to minimize the discrepancies , while maintaining consistency with our knowledge of stellar populations in color - color space , thereby improving the lsst calibration . in practice , there will be a significant difference between @xmath58 and @xmath59 , so that @xmath60    leading to a systematic flux error @xmath61 figure [ fig : gaiagsynthresid ] shows @xmath62 as a function of @xmath63 , expressed in magnitudes , evaluated over the entire phoenix sed set restricted to stars with @xmath64 . this restriction corresponds roughly to @xmath65 . cooler stars show much larger residuals . as shown in the figure , if the @xmath66 magnitudes are corrected with a third order polynomial in @xmath63 , then the remaining scatter is @xmath67 mmag , negligible in the lsst context . note that the same approach could be used for wd , with a smaller scatter , but also a much smaller sample of stars . it is clear from this that @xmath66 magnitude residuals will be diagnostic of any systematic errors in lsst calibration that exceed the 5 mmag level required by the srd . if a significant level of systematic error is found , the @xmath66 data can be employed to diagnose the causes , though not as directly as with the direct standards discussed in section [ sec : directgaiastandards ] . could we use the @xmath66 data to directly constrain , and thereby improve , the self calibration process , as we can with the direct standards ? the main limitation is that , given an error @xmath62 for some particular star , there is not a unique way to decompose it into the errors in the individual lsst bands . in the six dimensional lsst photometry space , we can move it anywhere on the five dimensional hyperplane given by @xmath68 to remove this degeneracy , we can additionally require that the star must remain on the color - color track for its stellar population . to the extent that this track is scatter free , the corrections to the individual lsst magnitudes can be uniquely assigned . again , wd may be more valuable than ms stars for this purpose . a quantitative assessment of the resulting accuracy is beyond the scope of this paper . we investigated two different approaches to using gaia data to provide accurate photometric standards for wide - field surveys such as lsst . in the first , gaia g - band photometry is combined with stellar parameters derived from the gaia apsis pipeline ( for ms stars ) or from ground based spectroscopy ( for wd ) to determine the transformation from g - band to the six lsst bands . the accuracy achieved with ms stars falls well short of lsst requirements , though with potential for improvement through the use of ground based spectroscopy . direct standards derived from wd have the potential to meet lsst requirements , particularly if they are restricted to high galactic latitudes . additional work will be required to realize this potential in practice . possible systematic errors in wd sed models must be carefully assessed , and the accuracy of interstellar reddening derived from ground - based spectroscopy or from gaia rp / bp spectroscopy needs to be more carefully quantified . both of these tasks will become possible in the near future using multi band hst photometry in conjunction with ground - based spectroscopy . it is worth noting that the ground - based observations required to realize the potential of gaia wd for photometric calibration may require substantial investment of telescope resources if a large number of standards is desired . a spectrum of a single wd in our magnitude range requires roughly two hours of exposure time on a 6 meter class telescope to reach the necessary signal - to - noise . similar considerations apply to the use of ms stars .    in the second approach , the six lsst bands are linearly combined into a synthetic g - band which approximates the gaia g - band . although the two g - bands have significant wavelength dependent differences , the resulting two sets of magnitudes have very small differences for most stars with @xmath69 . this allows a useful check on the spatial dependence of lsst calibration accuracy , and within currently unknown limits , the correction of such systematic errors . it seems worthwhile to actively pursue both approaches during the next years , as gaia science data is released , and lsst nears operation . we wish to thank drs . abhijit saha ( noao ) , edward olszewski ( u. arizona ) , and eljko ivezi ( u. washington ) for many useful discussions which have improved the paper .
we consider the possibility that the gaia mission can supply data which will improve the photometric calibration of lsst . after outlining the lsst calibration process and the information that will be available from gaia , we explore two options for using gaia data . 
 the first is to use gaia g - band photometry of selected stars , in conjunction with knowledge of the stellar parameters @xmath0 , @xmath1 , and @xmath2 , and in some cases @xmath3 , to create photometric standards in the lsst u , g , r , i , z , and y bands . 
 we consider both main sequence ( ms ) stars and da white dwarfs ( wd ) . 
 the accuracies of the resulting standard magnitudes from ms stars are found to be insufficient to satisfy lsst requirements using gaia data alone , but with the potential to do so when supplemented with ground based spectroscopy . 
 the accuracies of the wd derived standards are generally adequate , but also require ground based spectroscopy . 
 the second option is combine the lsst bandpasses into a synthetic gaia g band , which is a close approximation to the real gaia g band . 
 this allows synthetic gaia g photometry to be directly compared with actual gaia g photometry at a level of accuracy which is useful for both verifying and improving lsst photometric calibration .