Photometric brown-dwarf classification. I. A method to identify and accurately classify large samples of brown dwarfs without spectroscopy
Nathalie Skrzypek, Stephen J. Warren, Jacqueline K. Faherty, Daniel J. Mortlock, Adam J. Burgasser, Paul C. Hewett
aa r X i v : . [ a s t r o - ph . I M ] N ov Astronomy&Astrophysicsmanuscript no. skrzypek˙090714˙article˙final c (cid:13)
ESO 2018September 11, 2018
Photometric brown-dwarf classification. I.
A method to identify and accurately classify large samples of brown dwarfswithout spectroscopy
Skrzypek, N.1, Warren, S. J.1, Faherty, J.K.2, Mortlock, D. J.1 ,
3, Burgasser, A.J.4, Hewett, P.C.5 Astrophysics Group, Imperial College London, Blackett Laboratory, Prince Consort Road, London SW7 2AZ, UK Department of Terrestrial Magnetism, Carnegie Institution of Washington, Washington, DC 20015, USA Department of Mathematics, Imperial College London, London SW7 2AZ, UK Department of Physics, University of California, San Diego, CA 92093, USA Institute of Astronomy, Madingley Road, Cambridge CB3 0HA, UKReceived 2014 July 9 / Accepted 2014 November 27
ABSTRACT
Aims.
We present a method, named photo-type , to identify and accurately classify L and T dwarfs onto the standard spectral classifi-cation system using photometry alone. This enables the creation of large and deep homogeneous samples of these objects e ffi ciently,without the need for spectroscopy. Methods.
We created a catalogue of point sources with photometry in 8 bands, ranging from 0.75 to 4.6 µ m, selected from an area of3344 deg , by combining SDSS, UKIDSS LAS, and WISE data. Sources with 13 . < J < .
5, and Y − J > .
8, were then classifiedby comparison against template colours of quasars, stars, and brown dwarfs. The L and T templates, spectral types L0 to T8, werecreated by identifying previously known sources with spectroscopic classifications, and fitting polynomial relations between colourand spectral type.
Results.
Of the 192 known L and T dwarfs with reliable photometry in the surveyed area and magnitude range, 189 are recoveredby our selection and classification method. We have quantified the accuracy of the classification method both externally, with spec-troscopy, and internally, by creating synthetic catalogues and accounting for the uncertainties. We find that, brighter than J = . photo - type classifications are accurate to one spectral sub-type, and are therefore competitive with spectroscopic classifications. Theresultant catalogue of 1157 L and T dwarfs will be presented in a companion paper. Key words.
Stars: low-mass – Techniques: photometric – Methods: data analysis – Stars: individual: SDSS J1030 + +
1. Introduction
The first brown dwarfs were discovered by Nakajima et al.(1995) and Rebolo et al. (1995), having been theorised earlierby Kumar (1963a,b) and Hayashi & Nakano (1963). Explorationof the brown dwarf population has proceeded rapidly over thesubsequent two decades, enabled by new surveys in the opti-cal, the near-infrared, and the mid-infrared. This has resulted inthe creation of three new, successively cooler, spectral classesbeyond M: the L (Kirkpatrick et al. 1999; Mart´ın et al. 1999);T (Geballe et al. 2002; Burgasser et al. 2002a, 2006b); and Ydwarfs (Cushing et al. 2011). The temperature sequence has nowbeen mapped all the way down to e ff ective temperatures of ∼
250 K (Luhman 2014). This almost closes the gap to the ef-fective temperature of gas giants in our Solar System ( ∼
100 K).The current paper focuses on L and T dwarfs.One of the fundamental observables that characterises theLTY population is the luminosity function (e.g. Cruz et al. 2007;Reyl´e et al. 2010) i.e. the dependence of space density on abso-lute magnitude (or, equivalently, spectral type). The luminosityfunction can be used to learn about the sub-stellar initial massfunction and birth rate. The study of the luminosity function re-quires a homogeneous sample with well-defined selection crite-ria. DwarfArchives.org provides a compilation of L and T dwarfspublished in the literature. This collection is heterogeneous, hav-ing been culled from several surveys with di ff erent character-istics, including, in particular, the Sloan Digital Sky Survey(SDSS; York et al. 2000) in the optical, the Deep Near-InfraredSouthern Sky Survey (DENIS; Epchtein et al. 1997), the TwoMicron All-Sky Survey (2MASS; Skrutskie et al. 2006) and theUKIRT Infrared Deep Sky Survey (UKIDSS; Lawrence et al.2007) in the near-infrared, and WISE in the mid-infrared.DwarfArchives.org now contains over 1000 L and T dwarfs butthe constituent samples are themselves heterogeneous and so notwell suited for statistical analysis.A good example of the state of the art in obtaining a sta-tistical sample of brown dwarfs is the study of the sub-stellarbirth rate by Day-Jones et al. (2013). They classified 63 new Land T dwarfs brighter than J = .
1, using X-shooter spec-troscopy on the Very Large Telescope (VLT), after selectingthem through colour cuts. The sample targeted a limited sectionof the L and T sequence and required substantial follow-up re-sources: at J ≃ . T8, that reaches similar depth and that is more than an orderof magnitude larger. Such a sample is useful to reduce the un-certainty in the measurement of the luminosity function andalso allows a variety of studies, such as measuring the brown-dwarf scale height (Ryan et al. 2005; Juri´c et al. 2008), the fre-quency of binarity (Burgasser et al. 2006c; Burgasser 2007;Luhman 2012) and, if proper motions can be measured, kine-matic studies (Faherty et al. 2009, 2012; Schmidt et al. 2010;Smith et al. 2014). A large sample can also be used to iden-tify rare, unusual objects (e.g. Burgasser et al. 2003, Folkes et al.2007, Looper et al. 2008).All previous searches for L and T dwarfs have required spec-troscopy for accurate classification. This paper describes an al-ternative search and classification method that uses only exist-ing photometric survey data. We call the method photo-type ,by analogy with photo-z , the measurement of galaxy redshiftsfrom photometric data alone. § § § § § §
3. In § § § photo-type , explaining how to measure the spectral type, andits accuracy, for a source with photometry in all or a subset of the8 bands used here. We summarise in §
5. In Paper II we presentthe catalogue of 1157 L and T dwarfs classified by photo-type ,and quantify the completeness of the sample.The photometric bands used in this study are the i and z bands in SDSS, the Y , J , H , K bands in UKIDSS, and the W W All the magnitudes and colours quoted inthis paper are Vega based.
The
Y JHKW W ff sets tabulated in Hewett et al. (2006) toconvert the SDSS iz AB magnitudes to Vega.
2. The technique χ The overall aim here is to develop a method to classify ev-ery point-source in a multiband photometric catalogue as a star,white dwarf, brown dwarf or quasar, based only on its observedphotometry. This could be achieved most rigorously by usingBayesian model comparison (Mortlock et al. 2012). A formal-ism for doing so is detailed in the Appendix, which also includesthe series of approximations and assumptions which lead fromthe full Bayesian result to the much simpler χ -based classifi-cation we actually use. While some of the approximations be-low are clearly unrealistic, that is unimportant per se – all thatmatters in this context is whether the final classifications remain(largely) unchanged. The e ff ect of our approximations are exam-ined in detail at the end of § / N). At low S / N, the correct ap-proach is to work with fluxes (e.g., Mortlock et al. 2012) where, except in the low-photon regime (e.g., X-ray astronomy), the er-rors are Gaussian. A few T dwarfs in our sample have low S / N inthe SDSS i and z bands; for these we use — asinh — magnitudesand errors (Lupton et al. 1999) as the resultant χ value is closeto that which would have been calculated from the fluxes.The second simplification is to adopt equal prior probabil-ities for each of the di ff erent (sub-)types. In the limited regionof colour-magnitude space that we search for L and T dwarfs,13 . < J < . Y − J > .
8, the main contaminating popula-tion is reddened quasars (see Hewett et al. 2006). At these brightmagnitudes this population is easily distinguished, i.e. the pri-ors are not very important. Similarly, provided the priors varyrelatively slowly along the MLT spectral sequence and the like-lihoods are sharply peaked, the prior probabilities for all the sub-types can be assumed to be equal without a ff ecting the final clas-sifications.A third, related, approximation is to adopt broad, uniformpriors for the magnitude distributions (i.e., number counts) ofeach type. This again is obviously unrealistic, as the numberof L and T dwarfs per magnitude is expected to be N ( m ) ∝ m / , as they are an approximately uniform population in lo-cally Euclidean space. But the form of this prior has minimal im-pact on the classifications because we are operating in the highS / N regime in which the data constrain the overall flux-level ofa source to a narrow range.Applying the above assumptions and approximations to thefull Bayesian formalism, described in the Appendix, results inthe following classification scheme:1. The first requirement is photometric data on the source tobe classified: the measured magnitudes, { ˆ m b } , and uncertain-ties, { σ b } , in each of N b bands (with b ∈ { , , . . . , N b } ) .For most sources considered here N b = izY JHKW W N t source types, eachspecified by a set of template colours, { c b , t } , which give themagnitude di ff erence between band b and some referenceband B for objects of type t (with t ∈ { , , . . . , N t } ). Hence c B , t = m B , is the natural quantity to specify theoverall source brightness. Here we use J as the referenceband as our conservative magnitude cuts in this band ensurethat m B = m J is well constrained.3. The first processing step is, for each of the N t types, to calcu-late the inverse variance weighted estimate of the referencemagnitude, ˆ m B , t = N b X b = ˆ m b − c b , t σ bN b X b = σ b , (1)which, in general, is di ff erent for each template.4. Next, the above value for ˆ m B , t is used to calculate the mini-mum χ value for type t , χ ( { ˆ m b } , { σ b } , ˆ m B , t , t ) = N b X b = ˆ m b − ˆ m B , t − c b , t σ b ! . (2) The braces {} are used to denote a list of values, so that, e.g., { m b } = { m , m , . . . , m N b } is the list of true magnitudes in each of the N b bands.This is not a set, as it is ordered, but neither is it a vector as it is not ageometrical object.2krzypek, N., et al.: Photometric brown-dwarf classification. I.SpT i − z z − Y Y − J J − H H − K K − W W − W Table 1.
The i − z , z − Y , Y − J , J − H , H − K , K − W W − W All photom-etry is on the Vega system.
It is important that χ is calculated by comparing measuredand predicted magnitudes, as opposed to colours. The rea-sons for this are given in the Appendix.5. Finally, the source is classified as being of the type t whichresults in the smallest value of χ ( { ˆ m b } , { σ b } , ˆ m B , t , t ). Theconditions under which this corresponds to the most prob-able type are detailed in the Appendix; a more empiricaldemonstration that this results in a reliable classification isgiven in § The templates used in photo-type fit quasars, white dwarfs, starsand brown dwarfs. The template colours for quasars were takenfrom Maddox et al. (2012). Templates for quasars with weak,typical, and strong lines were included, for a range of redden-ing, E ( B − V ) = . , . , . , . , . , .
50, for the redshiftrange 0 < z < .
4. The template colours for white dwarfs andmain-sequence stars earlier than M5 used in this work are takenfrom Hewett et al. (2006). These are included for completeness,but in fact are not relevant for objects with colour Y − J > . izY JHK andwere used in the initial selection of candidate L and T dwarfs,before the classifications were refined by including the WISEcolours, as explained in § Y band data with UKIDSS had beentaken. Now that UKIDSS is complete it is possible to improveon the colour relations of ultra-cool dwarfs using photometryof known sources within the UKIDSS footprint. These will bemore accurate than colours computed from spectra, in particularfor the z band where the computed fluxes are very sensitive to the Fig. 1.
Plot of i − z colour vs. spectral sub-type for MLT dwarfs inthe UKIDSS LAS DR10 footprint. The error bars plotted showthe random photometric errors. The fitted curve provides thetemplate colours listed in Table 1. In making the fit, a colourerror of 0.07 mag. (i.e. 0.05 mag. in each band) is added inquadrature to the random photometric errors to account for in-trinsic scatter in the colours. The very large errors in the T dwarfregime mean that the curve is poorly defined in this region, asdiscussed in the text. The vertical scale is the same as in Figure2. The outlying blue T0 dwarf is discussed in § All photom-etry is on the Vega system. exact form of the red cuto ff of the band, defined by the decliningquantum e ffi ciency of the CCDs.To compute template colours for ultracool dwarfs we fitpolynomials to colours of known dwarfs as a function ofspectral type, as shown in Figures 1 and 2. We searchedDwarfArchives.org, and additional recent published samples( § § of SDSS,UKIDSS DR10, and WISE, in the magnitude range 13 . < J < .
5, with Y − J > .
8. Of the sample of spectroscop-ically classified L and T dwarfs, 190 appear in our cataloguei.e. are classified as stellar, have reliable matched photometryin izY JHKW W
2, and meet the magnitude and colour selec-tion criteria Our fundamental assumption is that the measuredcolours of these sources are representative of the colours of theL and T populations. Within this sample there are 150 L dwarfsand 40 T dwarfs. We discuss possible bias of this sample in thenext section. We supplemented the sample of L and T dwarfswith 111 cool M stars of spectral type M5 to M9 from the SDSSspectroscopic catalogue (Ahn et al. 2012) in order to tightly con-strain the polynomials at the M / L boundary. The missing sources are discussed in § a a a a a a a i − z –9.251 + + + z − Y –0.942 + + Y − J (¡L7) –0.174 + Y − J (¿L8) + J − H –2.084 + + + H − K –1.237 + + + K − W + + + W − W + + Table 2.
Dependence of colour c on spectral type t defined by polynomials c = P Ni = a i t i , where the correspondence between t andspectral type is 5 − − M9, 10 −
19 represents L0 − L9, and 20 −
28 represents T0 − T8. The polynomials are valid overM5 − T8.
All photometry is on the Vega system.
The 40 T dwarfs are classified on the revised system ofBurgasser et al. (2006b), based on near-infrared spectroscopy,and so our colour fits are anchored to this system for Tdwarfs. Of the 150 L dwarfs, 116 have optical classificationson the system of Kirkpatrick et al. (1999), and of these 16 alsohave near-IR classifications. For these 16 cases we averagedthe two classifications quantised to the nearest half spectralsub-type. The remaining L dwarfs have near-IR classificationsonly. Of the near-IR classifications, close to half use the SpeXprism spectral library (Burgasser 2014) , which is anchored toKirkpatrick et al. (1999), while the remainder use the index sys-tem of Geballe et al. (2002). Since we wish photo-type to be an-chored to the optical classification system we checked for anybias introduced by including the near-IR spectral types. We com-puted template colours, as described below, for the two cases,including and excluding near-IR spectral types for the L dwarfs.We then classified a large sample of L dwarfs (the 1077 de-scribed in §
3) using both sets of templates and looked at thedi ff erences in spectral type for the two sets of templates. The re-sult was a negligible o ff set, µ = .
06 spectral types, and smallscatter, σ = .
23, with no clear trends with spectral type, con-firming that for L dwarfs photo-type is accurately anchored tothe optical system of Kirkpatrick et al. (1999).Although the SpeX prism near-infrared spectral libraryfor L dwarfs is tied to the optical classification scheme ofKirkpatrick et al. (1999), there exist spectrally peculiar sourceswhere the optical and near-IR classifications di ff er, in some casesby more than two spectral sub-types. In the same way, for pecu-liar sources the photo-type classifications may di ff er from the op-tical classifications. One of the virtues of the photo-type methodis that it provides both a best-fit spectral type, and a goodnessof fit statistic (the multi-band χ ) which can be used to identifypeculiar sources. In this sense it provides more information thana simple classification based on optical spectroscopy or near-infrared spectroscopy alone.Denoting colour by c and spectral type by t , with numericalvalues M5 − M9 as 5 −
9, L0 − L9 as 10 −
19 and T0 − T8 as 20 − c = P Ni = a i t i via χ minimisation , over the 8bands.In the development of this analysis it was immediately clearthat the scatter in the colours is larger than the typical photo-metric error i.e. there is an intrinsic scatter in the colours, duee.g. to variations in metallicity, surface gravity, cloud cover, andunresolved binaries in the sample, as well as uncertainty in the hosted at http://pono.ucsd.edu/ ∼ adam/browndwarfs/spexprism/html/mldwarfnirstd.html χ = N X i = c i − f i σ i ! , where c is the colour of the individual source,and f is the colour of the polynomial fit spectral classification. It is important to allow for this scatter infitting the curves, or the fits could be a ff ected by outlying pointswith small photometric errors.We estimated the intrinsic scatter for each colour in an iter-ative fashion as follows. We first guessed the intrinsic scatter byeye, and added this value in quadrature to the photometric erroron each point. We then found the lowest order polynomial thatprovided a good fit to the data. We then re-estimated the intrinsicscatter as the value that, added in quadrature to the photometricerror on each point, and summed over all points, matched themeasured variance about the fitted polynomial, having identi-fied and removed any discrepant outliers. Averaging over all thecolours we measured an intrinsic scatter of 0.07 mag, which weadopted as the intrinsic scatter for all colours. The implementa-tion involved adding 0.05 mag. error (i.e. 0 . / √
2) in quadra-ture to the photometric error in each band for every object. Thismay be viewed as the uncertainty on the templates.Having established a suitable value for the intrinsic scatterin the colours, the polynomials were refit, starting with a linearfit, and then successively increasing the order of the polynomial,only provided a significant improvement in the fit was achieved, ∆ χ >
7, in order to prevent over-fitting . In most cases a fourthor fifth order polynomial was su ffi cient. The fitted polynomialsare shown in Figures 1 and 2. The coe ffi cients of the polynomialsare provided in Table 2, and the template colours are provided inTable 1.An additional source of error that should be accounted forin classifying sources is the uncertainty in the polynomial fitsthemselves. Since some curves are more tightly constrainedby the data than others, by incorporating this uncertainty,and its variation with spectral type, the di ff erent curves arethen correctly weighted. We have established the uncertaintiesfrom the covariance matrices of the polynomial fits, and theresults are plotted in Fig. 4. We have incorporated these errorsin classifying sources, and found that in fact they have nosignificant influence on the classifications. The analysis leadingto this conclusion is presented in § – i-z colour: The i − z colour polynomial is not well definedfor T dwarfs. This is because nearly all the T dwarfs arevery faint in the i band, and so the errors are large, ∼ . i − z colours for this sample in order to better establish the shapeof the curve and the intrinsic scatter, but this is di ffi cult be-cause the z band used will have to match the SDSS passband At each stage, in adding one free parameter, the improvement in χ will be distributed as the χ distribution with one degree of freedom.Then ∆ χ > >
99% significance.4krzypek, N., et al.: Photometric brown-dwarf classification. I.
Fig. 2.
Plot of colours z − Y , Y − J , J − H , H − K , K − W W − W All photometry is on theVega system.
Fig. 3.
The colour curves from Figures 1 and 2 plotted in a singlefigure in order to compare the relative usefulness of di ff erentcolours in classifying di ff erent spectral types in the LT sequence. Fig. 4.
The uncertainties of the fits of the colour polynomials asa function of spectral type. With the exception of the i − z and K − W ffi cultyof measuring the i − z curve accurately in the T dwarf regimeis not critical because, of course, most of the candidates arevery faint in i , and so have large photometric errors, meaningthat the contribution to the total χ from the i band is rela-tively small. We show later ( § i bandimproves the accuracy of the classification of L dwarfs, butnot of T dwarfs. – Y-J colour:
We found we were unable to fit the Y − J curvesatisfactorily with a single polynomial. We attribute this toa discontinuity in the relation near spectral type L7. From inspection of the SpeX spectra of the near-IR spectral stan-dards the jump appears to be associated with the rapid weak-ening of FeH absorption in the Y band between spectral typesL6 and L8 (Burgasser et al. 2002b). To check, and to decidewhere to impose the discontinuity, we computed syntheticcolours from the SpeX near-IR spectral standards. These areprovided in Table 3 ( § Y − J colourbetween L7 and L8. We therefore fit two separate polynomi-als to the data, with a quadratic for types ≤ L7, and a linear fitfor types ≥ L8, joined by a straight line from L7 to L8. Thestep between L7 and L8 is 0.14 mag. In Paper II we show thesame Y − J plot for the new sample of L and T dwarfs, andthe jump is seen more clearly in the larger sample. – J-H, J-K colours:
The flattening and return of the colour toredder values at the end of the T sequence appears to be real,rather than an artefact of the fitting procedure, as it can alsobe seen in the plots in Leggett et al. (2010) and Cushing et al.(2011).Figure 3 illustrates the usefulness of di ff erent colours in theclassification of dwarfs of di ff erent spectral type. The steeper thecurve, the more accurate the classification, with the exception of i − z in the T dwarf region where the errors are larger. Thereforeit is evident that spectral classification will be most accurate inthe region from about T1 to T5, because in this region severalof the colour relations are steep, whereas around L6 several ofthe colour curves are rather flat and classification will be lessaccurate. We quantify the accuracy of classification in § We now return to the assumption of flat priors for all the tem-plates. To check the potential contamination of the sample byquasars, we used the quasar templates as synthetic sources(adding photometric errors as appropriate) and picked out thequasar, from the full set, that provided the best-fit to any dwarfalong the sequence L0 to T8. The best match between the twotemplate sets is a reddened quasar E ( B − V ) = . z = . χ =
92 (for six degrees of freedom). With such a poor fit itis clear that L and T dwarfs will be easily discriminated fromreddened quasars, independent of priors, within reason.In a flux-limited sample early L dwarfs are much more com-mon than late L dwarfs. Because of uncertainty in the classifi-cations, the assumption of flat priors along the LT sequence willlead to a bias in the classifications (akin to Eddington bias). Theextent of the bias depends on the luminosity function and onthe precision of the classifications i.e. how sharp the curve of χ against spectral type is. In Figure 5 we plot χ against spectraltype for four known L and T dwarfs from the sample of 190 pre-viously known dwarfs. These plots show that the χ minimum isvery sharp and unambiguous, minimising any bias in the classi-fication. The Eddington bias will be corrected for in computingthe luminosity function in a future paper. In this section we discuss possible bias in the derived colour re-lations. This issue is closely related to the question of complete-ness, but we defer a detailed discussion of the completeness ofthe new sample to Paper II. As stated in § Fig. 5.
Plot of χ (computed from Equation 2) against spectraltype, in the classification of four known L and T dwarfs, asfollows: SDSS J0149 + J = . ± . + J = . ± . + J = . ± . + J = . ± . χ curves show no degeneracies, and aresharp, indicating accurate classification (quantified in § . < J < . Y JHK footprint, supplemented by the recentsamples of Burningham et al. (2013), Day-Jones et al. (2013),Kirkpatrick et al. (2011), and Mace et al. (2013). A few sourceswith unreliable photometry in any of the eight bands (e.g.blended with a di ff raction spike, or landing on a bad CCD row)were discarded. In addition, three sources were classified asstellar in 2MASS, but appear elongated in UKIDSS becausethey are binaries. This means there is a small bias against findingbinaries where the angular separation of the pair is a few tenthsof an arcsec, but this should not have any significant e ff ect onthe colour relations. The final sample comprises 192 known Land T dwarfs in the surveyed area, with reliable photometry, andclassified as stellar.A further two sources are missing from our sample: – WISEPC J092906.77 + Y and K UKIDSS observations was 2 . ′′
06, which is just greater thanthe 2 . ′′ – SDSS J074656.83 + Y = .
37, and J = .
58, giving Y − J = .
79, meaning it is just bluerthan our selection limit Y − J = .
8. A second epoch J mea-surement gave J = .
60. The source is apparently anoma-lously blue in Y − J , as the typical colour of an L0 dwarf is Y − J = .
04 (Table 1).The inclusion of these two sources would not significantly a ff ectthe colour relations. Therefore whether the colour polynomialsare biased is a question of whether the sample in DwarfArchivescontains significant colour selection biases, or whether there areany significant populations of L and T dwarfs with peculiarcolours that remain undiscovered. The samples of Kirkpatrick et al. (2011) and Mace et al.(2013), contain many late T dwarfs selected with WISE. Tocheck specifically the W − W W − W J − K by ∼ . ffi ciently blue cut in i − z to ensure theyincluded all L dwarfs, and so their J − K colours should be un-biased. This large sample is now included in DwarfArchives, soany bias is much reduced. Schmidt et al. (2010) ascribed the biasin the older DwarfArchives sample to the colour cut J − K > . J − K colour, as well as the random errors. Our own analysis, whichuses the much more precise UKIDSS photometry rather than theoriginal 2MASS photometry, finds a smaller bias between thesetwo samples of 0 .
05 mag over the spectral type range L0 to L4.Our analysis suggests that much of the original bias noted wasdue to random photometric errors, rather than true colour bias.Accounting for the relative sizes of the two samples used in theanalysis, any remaining bias in our J − K colour relation wouldbe . .
01 mag.It remains to consider the possibility that significant popula-tions with substantially di ff erent colour relations are underrep-resented in DwarfArchives. These might include unusually blueobjects, such as SDSS J074656.83 + > . . ff erences between the L and T population and quasars ( § photo-type should pick up unusual L and T dwarfs, which willbe identifiable in the new sample. We reconsider this question inPaper II. A small number of sources are classified as L or T but with large χ . These objects may be peculiar single sources, or could beunresolved binaries. To check whether any might be unresolvedbinaries, we created template colours for all possible L + T bi-nary combinations, over the range of spectral types L0 to T8.We use the relation between absolute magnitude in the J bandand spectral type from Dupuy & Liu (2012) to provide the rela-tive scaling of the two templates, hereafter referred to as S1 andS2. Then for sources with large χ , we compare the improvementin χ achieved by introducing the extra degree of freedom of abinary fit. Objects with ∆ χ > ∆ χ = χ − χ )are accepted as candidate binaries.The search for candidate unresolved binaries is e ff ective onlyover particular regions of the S1, S2 parameter space because,given the colour uncertainties, some binary combinations S1 + S2are satisfactorily fit by the colours of a single source. For exam-
Fig. 6.
Contour plot illustrating sensitivity to detection of unre-solved binaries. The axes correspond to the two components ofa binary. The contours plot the improvement in χ of a binarydwarf solution over the best single dwarf solution.ple, the colour template for the combination L0 + T5 is very sim-ilar to the colour template of a L0 dwarf. This issue is illustratedin Figure 6. Here we tested the improvement of a binary fit overa single fit for di ff erent combinations S1 + S2. Each point in thegrid represents a binary S1 + S2. We created synthetic templatecolours for the binary, adding random and systematic errors asbefore. Then for each artificial binary we found the best fit sin-gle solution and the best fit binary solution, and recorded theimprovement in χ . The contours plot the average improvementin χ achieved with the binary fit, and therefore map out regionswhere photo-type is sensitive to the detection of binaries.In Paper II we present spectra of some sources identified ascandidate binaries using photo-type .
3. Application
In this section we describe the creation of a catalogue of pointsources matched across 3344 deg of SDSS, UKIDSS and WISEthat we search for L and T dwarfs. We also quantify the accuracyof the photo-type method. Our study is concerned with field (as opposed to cluster) browndwarfs and uses survey data at high Galactic latitudes. The start-ing point for the search is the Data Release 10 (DR10) versionof the UKIDSS Large Area Survey (LAS) which provides pho-tometry in the
YJHK bands. Point sources are matched to theSDSS and ALLWISE catalogues, to add, respectively, iz and W1W2 photometry. Sources are then classified using the full izYJHKW1W2 data set. The footprint of the survey is definedby the 3344 deg area of the UKIDSS LAS, where all four ofthe YJHK bands have been observed, contained within the SDSS
DR9 footprint. All sources also possess WISE photometry fromthe ALLWISE catalogue.Creating a matched catalogue requires careful considerationof the image quality, pixel scale, and flux limits of the di ff er-ent surveys. The pixel scales of UKIDSS and SDSS are thesame, 0 . ′′
4, and the two surveys are well matched in terms of im-age quality: the typical seeing in UKIDSS is 0 . ′′ . ′′ . ′′
75, andthe full width at half maximum of the W1 and W2 point spreadfunction (PSF) is 6 ′′ . The larger PSF results in significant blend-ing of images, leading to incorrect photometry. Such cases areidentified by visual inspection of the spectral energy distribu-tions and the images. In the final catalogue, the 7% of sourcesblended in WISE are classified using only the 6-band izYJHK photometry.We chose the UKIDSS J band for defining the flux range ofthe survey, and selected J = . J = . σ detection limit in J is 19 .
6, so at J = . / N of sourcesis about 35. Using the template colours for the L and T sequence( § . ff erent possible J flux limits, and comparedthe synthetic photometry against the detection limits in each ofthe other 7 bands. Moving progressively fainter in J , the firstobjects to fall below any of the detection limits are a small frac-tion of the coolest T dwarfs, by J = .
5, absent from SDSS(i.e. fainter than the detection limit in both i and z ). Therefore toextend the depth of the survey we implemented a procedure toinclude sources undetected in SDSS.The di ffi culty here is compounded by the fact that T dwarfs,being nearby, can have significant proper motions. Given theepoch di ff erences between the optical and near-infrared obser-vations, typically a few years, this requires a search radius ofseveral arcsec. Then, in some cases, the nearest SDSS match tothe UKIDSS target may be the wrong source. The procedure weadopted was to find all SDSS sources within a large search radiusof 10 ′′ about the UKIDSS source. Starting with the SDSS matchnearest to the target we checked whether there was a di ff erentUKIDSS source closer to the SDSS source, and if so eliminatedthe SDSS source as a match to our target, and proceeded to thenext nearest match. Sources that were not matched by this pro-cedure to any source within the 10 ′′ search radius were retained.This allowed us to extend the depth of the survey to J = . J = . YJHK set, classified as stellar, using -4 5, contains 6 775 168sources. The procedure to match to WISE is cumbersome, sowe reduced the size of the catalogue before matching to WISE, The parameter mergedclassstat measures the degree to whichthe radial profile of the image resembles that of a star, quantified by thenumber of standard deviations from the peak of the distribution. while retaining all the L and T dwarfs, as follows. L and T dwarfsare redder in Y − J than all main-sequence stars. So to find L andT dwarfs we can make the problem more manageable by takinga cut in Y − J . In § Y − J = . 04, and that the in-trinsic scatter in colours is ∼ . 07. On this basis we applied a cutat Y − J > . 8, to produce a sample of 9487 sources. From thissample we then visually checked all sources classified as miss-ing in SDSS, eliminating obvious errors (due e.g. to blendingor bad rows). In a small number of cases, where the undetectedobject is just visible, we undertook aperture photometry of theSDSS images using the Image Reduction and Analysis Facility(IRAF; Tody 1986) .The sample at this stage is dominated by late M dwarfs. Toreduce the sample size further, before matching to WISE, wemade a first-pass classification, using the method described in § izY JHK photometry, and then limitedour attention to the 7503 sources classified as cool dwarfs, withclassification M6 or later. All these sources were then matchedto WISE, using a 10 ′′ search radius, to extract the W W photo-type as L1.5 ,was unmatched to WISE, because it is just below the detectionlimit. It is retained, together with its UKIDSS + SDSS classifi-cation. The final, refined, classifications were obtained from thefull izY JHKW W Y JHK catalogue due to objects falling below the de-tection limit in any band. Because of their blue J − K colours,at J = . K ∼ 18, and a few sources inthe tail of the random photometric error distribution might bescattered fainter than the detection limit in K . We undertook afull simulation to quantify the incompleteness, accounting forthe UKIDSS detection algorithm, random photometric errors,the intrinsic spread in J − K colour ( § < . 2% forall spectral types, except T6 and T7 where the incompleteness is0 . 6% and 1 . we have selectedall stellar sources 13 . < J < . Y JHK in theUKIDSS LAS, and produced a catalogue of 7503 sources with Y − J > . 8, matched to SDSS and WISE, that are classified ascool dwarfs, M6 or later, from their izY JHK photometry. This isthe starting catalogue for a search for L and T dwarfs. A hand-ful of sources are undetected in SDSS or WISE, but these areincluded in the catalogue. Incompleteness due to sources not de-tected in any of the YHK bands is negligible.Classification by photo-type using the full izY JHKW W photo-type as ultra-cool dwarfs. The exception is the unusual L2 dwarf 2MASSJ01262109 + In the final catalogue of 1157 L and T dwarfs, there are 10 sourceswithout photometry in both i and z , that were classified using the otherbands. ULAS J211045.61 + the surveyed area and magnitude range, 189 are recovered byour selection and classification method. In this subsection we estimate the accuracy of photo-type . To re-cap, for a source measured in izY JHKW W 2, an intrinsic uncer-tainty of 0.05 magnitude is added in quadrature to the photomet-ric error in each band, and then the χ of the fit to each templateis measured, using Equations 1 and 2 from § χ fit provides the classification. We es-timate the accuracy of the classification in three ways. – In the first case we see how accurately we recoverthe spectroscopic classifications of the 189 sources fromDwarfArchives. – Because this sample is heterogeneous, with classificationsbased on spectra covering di ff erent wavelength ranges, wealso obtained our own follow-up spectra of candidates fromPaper II, to obtain a second assessment of the classificationaccuracy, from a homogeneous spectroscopic sample. – The third estimate uses Monte Carlo methods to create ar-tificial catalogues of colours of all spectral types, over themagnitude range of the catalogue, to estimate the classifica-tion accuracy as a function of spectral type and magnitude.We also investigate by how much the accuracy degrades asdi ff erent bands are removed, to quantify the usefulness ofthose bands.There is good agreement between the various methods discussedin § § § photo-type isaccurate to a root mean square error ( r.m.s. ) of one spectral sub-type over the magnitude range of the sample. In Figure 7 we plot the photo-type classification against thespectroscopic classification for the 189 L and T dwarfs fromDwarfArchives (i.e. excluding the single misclassified source),together with 111 M stars. The vertical scatter in this plot is ameasure of the accuracy of the classification. There is a contri-bution to the variance from the quantisation of the spectral clas-sification. Therefore for this plot the photo-type classificationswere measured to the nearest half sub-type, by interpolating thecolours in Table 1. This reduces the contribution of quantisationto the variance, in order to be able to measure the scatter accu-rately. To account for outliers we estimate the vertical scatter inthis plot using the robust estimator σ = P Ni = | ∆ t | N √ π ∆ t is the di ff erence between the photo-type classificationand the spectroscopic classification. The three outliers markedare discussed below. For the L and T dwarfs we measure, re-spectively, σ L = . σ T = . 2. This may be consideredan upper limit to the uncertainty in the type because there is acontribution to the scatter from the spectroscopic classificationitself. Some of this scatter comes from the fact that spectroscopicclassification is based on a restricted portion of the photometricwavelength range covered in this study (0 . − . µ m). A pecu-liar source might have a di ff erent spectral sub-type if classifiedin the optical or the near-infrared. The photo-type method willsmooth out such di ff erences because of the broad wavelength Fig. 7. Comparison of photo-type classification vs. spectro-scopic classification for the 189 known L and T dwarfs fromDwarfArchives, as well as 111 late M stars. Classifications werecomputed to the nearest half spectral sub-type. Because the clas-sifications are quantised, small o ff sets have been added in orderto show all the points. The dashed lines mark misclassificationby four spectral sub-classes. The three outliers marked by dia-monds are discussed in the text.coverage. This discussion suggests that σ = photo-type . There is an elementof circularity in using the same L and T dwarfs used to definethe colour templates in measuring the classification accuracy. Inprinciple this should not be a concern, since the number of ob-jects used is very much greater than the number of parameters inthe fitting. Nevertheless it motivates checking the classificationaccuracy by other means.There are three sources in the plot, marked by diamonds,where the photo-type classifications di ff er from the spectro-scopic classifications by more than four sub-types. These out-liers are now discussed in detail. – SDSS J1030 + ± O in J ), T1 (H O in H ), T0.5 (CH in H ) and L8 (CH in K ). Using the full 8-band photometry photo-type provides a classification of L4 with χ = . χ = . 43. The two fits are illustrated in Figure 8. The object has i − Y = . ± . 25, which is unusually blue compared tothe template colour i − Y = . 45 for L9.5. Neither model fitsthe W / T transition can show significant vari-ability, which provides a possible explanation for the poor Y − J Res ReferenceVB 8 J16553529–0823401 M7 0.72 120 Burgasser et al. (2008)VB 10 J19165762 + + + + + + + + + + + + + + + + + + + + + + + + Table 3. The spectral standard templates from the SpeX Prism Spectral Library that are used for spectral classification. The Y − J colours computed from the spectra are listed. R.A. (2000) decl. (2000) Date (UT) J ± J err (mag) PhT χ SpT Exp. time (sec) Slit Width ( ′′ ) A0 star00 26 40.46 + 06 32 15.1 29 / / ± + 06 25 59.0 29 / / ± + 01 56 44.3 20 / / ± + 23 16 37.6 20 / / ± 08 49 37.09 + 27 39 26.8 20 / / ± 09 15 44.13 + 05 31 04.0 22 / / ± 10 29 35.20 + 06 20 28.6 20 / / ± + 04 52 22.3 22 / / ± No WISE data, therefore the χ uses only the izYJHK bands Low S / N, therefore there is an error of ± No WISE data, therefore the χ uses only the izYJHK bands Table 4. Details of the eight sources observed with SpeX.fits, since the various photometric data were taken at di ff er-ent dates. – χ = . 09, while theL7 fit gives χ = . 82. The SED of the source and thetwo fits are plotted in Figure 9. Optical spectroscopy of thissource would clearly be useful, as noted also by Geißler et al.(2011). The relatively high χ of the photo-type best fitwould have marked it down as an object worthy of furtherstudy. – ULAS J2304 + photo-type with a satisfac-tory χ = . 82. In contrast, fitting the T0 template yields χ = . 24, a very poor fit. The two fits are comparedagainst the SED of the source in Figure 10. As can be seen,the source is substantially bluer than a T0 in both i − z and W − W 2. The source has i − z = . ± . 13 (Vega), and isvisible as the T0 outlier in Fig. 1. This compares to the ex-pected colour i − z = . 15 for an L3.5 dwarf, and i − z = . iz region would be revealing. Neverthelessgiven the fact that a single L3.5 provides a satisfactory fit tothe photometry, this could be an example of an unresolvedbinary that cannot be identified from colours alone, as pre-figured in § . 4. It provides a warning that if a spectrum overa limited wavelength range provides a substantially di ff erentclassification to the photo-type classification, the source maybe an unresolved binary. A cleaner estimate of the accuracy of photo-type may be ob-tained from uniform high-quality spectra over a wide wavelengthrange, using the same instrumental set-up, to ensure uniformaccurate spectral classifications. For this purpose we selected asample of objects from the catalogue of Paper II for follow-upobservations. We limited the sample to sources with χ < 15, forthe fits to the 8-band photometry, to avoid outliers (objects withlarge χ are considered in detail in Paper II). Other than that weselected objects at random from a range of spectral types.We obtained spectra of the 8 sources listed in Table 4 withthe SpeX instrument on the NASA Infrared Telescope Facilitybetween March and November 2013. The data were reduced us- Fig. 8. SED of the source SDSS J1030 + photo-type classification of L4 (black line), and the spectro-scopic classification of L9.5 (red line). The upper plot uses flux( f λ ), normalised to J, and the lower plot uses magnitudes. Fig. 9. SED of the source 2MASS J1542-0045 compared to the photo-type classification of L2 (black line), and the spectro-scopic classification of L7 (red line). The upper plot uses flux( f λ ), normalised to J, and the lower plot uses magnitudes. Fig. 10. SED of the source ULAS J2304 + photo-type classification of L3.5 (black line), and the spectro-scopic classification of T0 (red line). The upper plot uses flux( f λ ), normalised to J, and the lower plot uses magnitudes.ing the SpeXtool package version 3.4 (Cushing et al. 2004). Thespectra are plotted in Figure 11.The spectra were classified by one of us (JKF), by visualcomparison against the SpeX Prism Spectral Library maintainedby AJB, and without knowledge of the photo-type classifica-tions. The standards are listed in Table 3. The resulting spectro-scopic classifications are listed in Table 4, where they are com-pared to the photo-type classifications. All objects are confirmedas ultra cool dwarfs. Taking the spectroscopic classification to bethe correct classification, the accuracy of photo-type estimatedfrom this small sample is only 0.4 sub-types r.m.s. In all, including peculiar objects (Paper II), so far we haveobserved 20 objects from our list of L and T dwarfs, and all areultra-cool dwarfs. This indicates, at least, that the contaminationof the L and T sample of Paper II is not large. Nevertheless alarger spectroscopic sample would be required to quantify thisaccurately. A third estimate of the accuracy of photo-type was obtained byMonte Carlo methods. For a particular J magnitude and for eachspectral type, we created synthetic data, accounting as appro-priate for the photometric errors and the intrinsic scatter in thecolours of the population (by adding an error of 0.05 magnitudesin each band in quadrature to the photometric error, § J = . 0, and at thesample limit J = . 5. This analysis indicates that the classi-fication method is accurate to better than one spectral type, for Fig. 11. SpeX spectra of eight sources used to estimatethe accuracy of photo-type . The sources are classified asL1 (ULAS J0026 + + + + + + + + J = . 5. The plotshows that the method performs least well, around spectral typeL6, as expected (see discussion in § ff erent photometric bands, by simply removing one ormore bands, and observing the e ff ect on the classification accu- Fig. 12. Estimated accuracy r.m.s. of the photo-type classifica-tion for sources of J = . J = . i band contributes usefully to the classi-fication of L dwarfs, but that for our dataset the i band makes anegligible contribution to the classification of T dwarfs, becausethe photometric errors are so large. The WISE data are useful inclassifying all spectral types. The improvement in the classifi-cation accuracy depends on spectral type and brightness, but onaverage including the WISE data reduces the uncertainty in thespectral type by ∼ ff ect on the classification accu-racy of disregarding the uncertainties of the polynomial fits.We found that including or excluding this error term has anegligible e ff ect on the accuracy of the classification for ourSDSS / UKIDSS / WISE dataset. For most bands and spectraltypes this is because the fit error is smaller than the intrinsiccolour scatter, as shown in Fig. 4. Even the large fit errors forthe i − z and K − W W − W J = . 5, for single normal L andT dwarfs, the accuracy of photo-type classifications is at the levelof one spectral sub-type r.m.s. 4. photo-type cookbook In this section, as a reference, we provide a brief summary ofhow to use photo-type to classify a photometric source, withcomplete or partial photometry from the izY JHKW W χ for each template, and then se-lect the template with the smallest χ as the classification.For the object, add 0.05 magnitude intrinsic scatter inquadrature to the photometric error in each band. Then, for a par-ticular template, setting J = 0, and using the colours in Table 1(or using the polynomial relations), assemble the template mag-nitudes for the bands in which photometry is available. Next,compute the magnitude o ff set that provides the minimum χ match of the template to the object, from equation 1, and calcu-late χ for that template using equation 2. Repeat the procedurefor all templates to obtain the classification, as the template withthe minimum χ . The χ of the best fit provides an indication ofwhether the object is peculiar or not.The accuracy of the classification will depend on the bright-ness of the source and the number of photometric bands. Theaccuracy may be estimated by the Monte Carlo method of theprevious section. Starting with the measured photometry for theobject, create a large number of synthetic objects by adding er-rors in each band drawn from a Gaussian of the appropriate dis-persion (i.e. random + intrinsic scatter). Then classify each ofthese synthetic objects as if real objects, and record the scatter.For this purpose it makes sense to classify to the nearest halfspectral sub-type in order to measure the dispersion more accu-rately. 5. Summary In this paper we have described a new method to identify andaccurately classify L and T dwarfs in multiwavelength 0.75-4.6 µ m photometric datasets. For typical L and T dwarfs the clas-sification is accurate to one spectral sub-type r.m.s. The sampleof 1157 L and T dwarfs, 13 . < J < . 5, selected from anarea of 3344 deg , is provided and described in Paper II. Theprincipal benefit of the photo-type method is the production ofa sample of L and T dwarfs across the entire range L0 to T8,with accurate spectral types, that is an order of magnitude largerthan previous homogeneous samples, and therefore ideal for sta-tistical studies of, for example, the luminosity function, and forquantifying the dispersion in properties for any particular sub-type. The photo-type method can also be used to select unusualobjects, including unresolved binaries, or rare types, identifiedby large χ . An important advantage of the method is that it cov-ers a broad wavelength range, 0.75 to 4.6 µ m, meaning that themethod may identify peculiar objects that look normal in spectrathat cover only a small wavelength range, and so might otherwisebe overlooked. The strengths and weaknesses of photo-type forthe study of unusual L and T dwarfs are discussed in detail inPaper II. Acknowledgements. We would like to thank the anonymous referee for severalconstructive comments that helped improve the paper. The authors are gratefulto Subhanjoy Mohanty for discussions that contributed to this work significantly.We also acknowledge a useful discussion with Nicholas Lodieu who encouragedus to include objects undetected in SDSS in the sample. NS would like to thankSTFC for the financial support supplied.This research has benefited from the SpeX Prism Spectral Libraries, maintainedby Adam Burgasser at http: // pono.ucsd.edu / ∼ adam / browndwarfs / spexprism.This publication makes use of data products from the Wide-field InfraredSurvey Explorer, which is a joint project of the University of California, LosAngeles, and the Jet Propulsion Laboratory / California Institute of Technology,and NEOWISE, which is a project of the Jet Propulsion Laboratory / CaliforniaInstitute of Technology. WISE and NEOWISE are funded by the NationalAeronautics and Space Administration.This research has benefitted from the M, L, T, and Y dwarf compendiumhoused at DwarfArchives.org. The UKIDSS project is defined in Lawrence et al.(2007). UKIDSS uses the UKIRT Wide Field Camera (Casali et al. 2007).The photometric system is described in Hewett et al. (2006), and the calibra-tion is described in Hodgkin et al. (2009). The science archive is described inHambly et al. (2008).Funding for SDSS-III has been provided by the Alfred P. Sloan Foundation,the Participating Institutions, the National Science Foundation, and theU.S. Department of Energy O ffi ce of Science. The SDSS-III web site ishttp: // / . 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E., et al. 2000, AJ, 120, 1579 Appendix A: Bayesian template classification The classification scheme described in § χ statistic, but is motivated by a fully Bayesian approachto photometric template-fitting. The full Bayesian result is de-rived below, after which a series of approximations are madeto obtain the χ classification scheme – one benefit of at leaststarting with a Bayesian formalism is that all assumptions andapproximations must be made explicit.For a target source with photometric measurements { ˆ m b } and uncertainties { σ b } in each of N b passbands ( i.e. , b ∈{ , , . . . , N b } ), the aim here is to evaluate the probability,Pr( t |{ ˆ m b } , { σ b } , N t ), that it is of type t , given that there are N t types of astronomical object (indexed by t ∈ { , , . . . , N t } ) un-der consideration. Under the assumption that the source is of oneof these types, Bayes’s theorem implies thatPr( t |{ ˆ m b } , { σ b } , N t ) = Pr( t | N t ) Pr( { ˆ m b }|{ σ b } , t ) P N t t ′ = Pr( t ′ | N t ) Pr( { ˆ m b }|{ σ b } , t ′ ) , (A.1)where Pr( t | N t ) is the prior probability of the t ’th model ( i.e. , howcommon this type of astronomical object is) and Pr( { ˆ m b }|{ σ b } , t )is the marginal likelihood that the observed photometry wouldhave been obtained for an object of type t .Each type is assumed to be specified by a template of modelcolours ( i.e. , band-to-band magnitude di ff erences), defined rel-ative to some reference passband B . The colour for template t and band b is denoted c b , t , with c B , t = m B , must also beincluded in each model. If this quantity is not of interest (as isthe case here), m B can be integrated out to give the marginallikelihood for the t ’th type asPr( { ˆ m b }|{ σ b } , t ) = Z ∞−∞ Pr( m B | t ) Pr( { ˆ m b }|{ σ b } , m B , t ) d m B , (A.2)where Pr( m B | t ) is the prior distribution of m B for objects of the t ’th type (and so approximately proportional to their observednumber counts) and Pr( { ˆ m b }|{ σ b } , m B , t ) is the likelihood of ob-taining the measured data given a value for m B . The marginal likelihood is sometimes referred to as the model-averaged likelihood or, especially in astronomy, as the (Bayesian) ev-idence. Under the assumptions that the measurements in the N b bands are indepenent and that the variance is additive and nor-mally distributed (in magnitude units ), the likelihood isPr( { ˆ m b }|{ σ b } , m B , t ) = N b Y b = exp h − ( ˆ m b − m B − c b , t ) /σ b i (2 π ) / σ b , (A.3)where m B + c b , t is the predicted b -band magnitude .Equation A.3 can be re-written asPr( { ˆ m b }|{ σ b } , m B , t ) = exp h − χ ( { ˆ m b } , { σ b } , m B , t ) iQ N b b = (2 π ) / σ b , (A.4)where χ ( { ˆ m b } , { σ b } , m B , t ) = N b X b = ˆ m b − m B − c b , t σ b ! (A.5)is the standard χ mis-match statistic. For the purposes of evalu-ating the integral in equation A.2 it is useful to further rearrangeequation A.3 into the formPr( { ˆ m b }|{ σ b } , m B , t ) (A.6) = exp h − χ ( { ˆ m b } , { σ b } , ˆ m B , t , t ) iQ N b b = (2 π ) / σ b exp − m B − ˆ m B , t (cid:16)P N b b = /σ b (cid:17) − / , where ˆ m B , t = P N b b = ( ˆ m b − c b , t ) /σ b P N b b = /σ b (A.7)is the natural inverse-variance weighted estimate of m B forthis combination of source photometry and template, and χ ( { ˆ m b } , { σ b } , ˆ m B , t , t ) is similarly the minimum value of χ .Inserting the above expression for the likelihood into equa-tion A.2 allows the marginal likelihood to be written asPr( { ˆ m b }|{ σ b } , t ) = exp h − χ ( { ˆ m b } , { σ b } , ˆ m B , t , t ) iQ N b b = (2 π ) / σ b × Z ∞−∞ Pr( m B | t ) exp − m B − ˆ m B , t (cid:16)P N b b = /σ b (cid:17) − / d m B , (A.8)illustrating that the goodness of fit and the number counts of thistype play quite strongly separated roles in this problem.The classification statistic defined in § m B prior distribution of the formPr( m B | t ) = Θ ( m B − m B , min ) Θ ( m B , max − m B ) 1 m B , max − m B , min , (A.9) This is not a good approximation for sources that are fainter than thedetection limit in any of the relevant bands; in this case the likelihoodshould be calculated in flux units as described in, e.g. , Mortlock et al.(2012). While the template is specified in terms of colours, at no point areobserved colours of the form ˆ m b − ˆ m b ′ ever calculated. If observedcolours were used then the resultant likelihood would have to incor-porate the correlations induced by the fact that the same measured mag-nitude was used to calculate more than one colour. The resultant likeli-hood could not be expressed in terms of a simple χ statistic as is donehere.14krzypek, N., et al.: Photometric brown-dwarf classification. I. where Θ ( x ) is the Heaviside step function and m B , min and m B , max are taken to be the same for all types (hence the lack of a t sub-script). Provided that m min ≪ ˆ m B , t and m max ≫ ˆ m B , t , the inte-gral in equation A.8 can be approximated analytically to givethe marginal likelihood asPr( { ˆ m b }|{ σ b } , t ) = (cid:16)P N b b = /σ b (cid:17) − / (2 π ) N b / − (cid:16)Q N b b = σ b (cid:17) exp h − χ ( { ˆ m b } , { σ b } , ˆ m t , t ) i m B , max − m B , min , (A.10)where ˆ m B , t is given in equation A.7 and χ ( { ˆ m b } , { σ b } , m B , t )is given in equation A.4. Inserting this expression into equa-tion A.1, the posterior probability that the source is of type t becomes Pr( t |{ ˆ m b } , { σ b } , N t ) = Pr( t | N t ) exp h − χ ( { ˆ m b } , { σ b } , ˆ m B , t , t ) iP N t t ′ = Pr( t ′ | N t ) exp h − χ ( { ˆ m b } , { σ b } , ˆ m B , t ′ , t ′ ) i . (A.11)This probabilistic template matching scheme can be madeabsolute by classifying a source as being of the type withthe maximum posterior probability, which in turn corre-sponds to the maximum value of the numerator of A.11,Pr( t | N t ) exp[ − χ ( { ˆ m b } , { σ b } , ˆ m B , t , t ) / ff erent source types are comparable (or if the templateshave very distinct colours, relative to the photometric noise) thenthe di ff erences in the priors can be neglected, in which case asource would be classified as being of the type t which yieldsthe minimum value of χ ( { ˆ m b } , { σ b } , ˆ m B , t , t ). This is the approachtaken in §2.