Derivation of veiling, visual extinction and excess flux from spectra of T Tauri stars

Transcription

1 Atron. Atrophy. 34, ) ASTRONOMY AND ASTROPHYSICS Derivation of veiling, viual extinction and exce flux fro pectra of T Tauri tar A. Chelli Laboratoire d Atrophyique, Obervatoire de Grenoble, U.J.F./B.P. 53, F-3804 Grenoble Cedex 9, France Received 9 Deceber 997 / Accepted 7 October 998 Abtract. Thi work ai to analye within a rigorou fraework, veiling, viual extinction and exce extraction fro the pectra of T Tauri tar. We invetigate further the ethod of Hartigan et al. 989) for veiling etiate fro all pectral bandwidth of a few ten of Angtro. The calculated veiling value i enitive to the etiated noie ratio and to pectral iatche between the object and the reference. We how that an incorrect input noie ratio together with low contrat pectra in noie unit can lead to iportant biae and we propoe olution to iniize thi proble. In cae of pectral iatche and for large contrat pectra copared to the reidual of the veiling equation, the relative veiling bia i doinated by the apparent veiling of the reference with repect to the correct underlying object tellar pectru. The veiling error i found to be proportional to the quare of the veiling when the latter becoe larger than unity. If we are liited by the tatitical noie, it i little dependent on the pectral reolution. Becaue of yteatic error, however, it will be difficult to etiate the veiling in a very all bandwidth at pectral reolution of a few hundred. For viual extinction and exce etiate, we generalize the dicrete ethod of Gullbring et al. 998) by a continuou approach. Thi new approach, which ue the pectra a a whole through a continuou odelling, ha been uccefully teted on iulated data. The viual extinction error i proportional to the veiling when the veiling becoe larger than unity and to a function which depend on the input reference pectru. Thi function decreae with increaing pectru contrat, which ean going fro earlier to later pectral type. If we are liited by the tatitical noie, it i, like the veiling, little dependent on the pectral reolution. For very active T Tauri tar or when the exce i doinated by eiion line, however, it will be difficult to handle very low pectral reolution, becaue of yteatic error. The real enitivity to biae and the perforance of the algorith are to be tudied experientally. Neverthele, an efficient ue of all the inforation contained in the pectra through the propoed continuou approach, together with a better undertanding of the ource of bia, can greatly help to derive the viual extinction and the exce on object uch fainter than thoe o far tudied. Send offprint requet to: A. Chelli Key word: ethod: data analyi technique: pectrocopic tar: circutellar atter tar: pre-ain equence. Introduction In the lat decade it ha been hown that the viible pectru of Claical T Tauri tar CTT) i copoed of a noral tellar pectru and a oothly varying continuu with eiion line uperpoed Walker 987; Hartigan et al. 989, HHKHS hereafter). Thi continuu exce exce hereafter) ake the photopheric aborption line to appear veiled, i.e. le deep than thoe of a tar with the ae pectral type. The exce i bluer than the noral tellar continuu and ee to be related with oe echani of the accretion proce: eiion fro an active photophere Calvet et al. 984), fro a boundary layer Lynden-Bell & Pringle 974; Bertout et al. 988), or fro the bae of agnetopheric accretion colun Hartann et al. 994). In order to odel a CTT, it i neceary to know the aount of exce, or equivalently, the aount of veiling defined a the ratio between the exce and the local continuu of the underlying noral tar). Several ethod have been developped to etiate the veiling. The ot popular one i that propoed by HHKHS ee alo Bari & Batalha 990) which conit in coparing, within a all wavelength interval of a few ten of angtro where the veiling and the extinction toward the object can be aued contant), high reolution pectra of a CTT with a tandard tar of the ae pectral type. More recently, Gullbring et al. 998, GHCC hereafter) introduced a new ethod to derive both the viual extinction and the exce pectral hape fro pectrophotoetrically calibrated data by veiling analyi in a liited nuber of optiized photopheric aborption line. Thi article intend to tudy within a rigorou fraework, veiling, viual extinction and exce extraction fro the pectra of CTT. Sect. invetigate further the ethod of HHKHS for veiling etiate. We introduce a non linear leat quare fit which allow to deal with non contant noie, and we derive a foral expreion for the reulting error of the veiling. We alo dicu the ain ource of bia and the influence of the pectral reolution on the veiling etiate. In Sect. 3, we generalize the

2 764 A. Chelli: Derivation of veiling, viual extinction and exce flux fro pectra of T Tauri tar ethod of GHCC for the deterination of the viual extinction and exce pectral hape by introducing a continuou ethod which i teted uccefully on iulated data. The originality of thi new approach i that it ue all the avalaible inforation contained in the pectra. We derive a foral expreion for the viual extinction error and we conclude thi work by a brief dicuion of the reult.. Derivation of veiling In thi ection, we conider a pectral range all enough for the exce and the extinction to be contant. The reference pectru S i noralized to it local continuu and the object pectru O i corrected fro any reidual continuu lope. In the following for iplicity we do not ake the ditinction between a paraeter and it etiated value and the reference pectru i aued to atch exactly the underlying object tellar pectru, unle clearly tated... The forali In the abence of noie, by definition O and S are related by: [ ] Oλ)wλ) =p 0 Sλ)+r wλ), ) where λ i the wavelength, p 0 a caling factor, r the veiling, and wλ) a weight which i equal to 0 at unuable wavelength range like eiion line or partially filled line), and to elewhere. In the following, we aue that the pectra are apled at the Shannon frequency at value λ i for i =,...,, and the entire acquired pectru i uable, wλ i )=. Now we want to etiate the paraeter, p 0 and p = r. Auing that the eaureent error are gauian, the axiu likelihood etiate of the paraeter i iply obtained via a leat quare fit, non linear in thi cae. To perfor the fit, we firt define the vector Q, whoe coponent Q i are: Q i = O i p 0 S i + r), ) σ i where the index i tand for λ i, σ i =σo,i + p 0σ,i )/, and σo,i and σ,i repreent the variance of O i and S i, repectively. For tatitically independent eaureent, the correct value of the p are obtained by iniizing the quare odulu of the vector Q with an iterative procedure, olving at each tep the yte of equation ee Knoechel & Heide 978): t AAδP = t AQ, 3) where δp i a vector whoe coponent are δp 0 and δp = δr, i.e. the increent to vector P whoe coponent are p 0 and p = r; t A i the tranpoe atrix of the derivative of Q with repect to the p, the upper bar over Q tanding for the expected value. The eleent i, j) of atrix A i written a: A i,j = Q i. 4) p j For the calculation, Q i can conveniently be approxiated by Q i. At each tep, the atrix A and the vector Q are then recoputed. The procedure i very robut and it generally converge after a few iteration, if the initial value of the p are cloe to the correct one. At the end of the iterative proce, the variance of the p are the diagonal eleent of the atrix [ t AA]. The reduced chi-quare value i iply given by t QQ/ ) and hould be cloe to for a tatitically correct fit... Error analyi The detailed error calculation i given in Appendix A. Generally, the reference tar i uch brighter than the object ource, and it contribution to the noie can be neglected. Indeed, it i iportant to have a et of good quality reference tar which can be ued in any veiling tudy. Under thi auption, for both photon and contant additive liited noie, within a very good approxiation, the tandard deviation error hereafter) of the veiling i given by: + r) σr) ɛ o, 5) / and the relative error on the caling factor can be written a: σp 0 ) +r p 0 ɛ o, 6) / where ɛ o i the ignal to noie ratio on the object total pectru flux and i the variance of the noralized reference pectru expected value S. eaure the quare of the pectru contrat and i defined by: = S i S ) i. 7) Clearly, for veiling value greater than, the veiling error increae a the quare of the veiling which illutrate the difficulty to etiate large veiling value. Thee reult are dicued in Sect..4. The relative veiling error can alo be written a: σr) +r. 8) / q o Thi forula can be ueful to etiate rapidly the veiling error fro the data and allow u to introduce the contrat pectru in noie unit q o = / o /σ o rep. q ), which i a central quantity in bia tudie..3. Bia proble There are two ain ource of bia in veiling calculation. The firt i due to a bad etiate of the noie ratio between the object and the reference. The econd i due to iatche between the object and the reference pectra..3.. Uncorrect noie etiate Let u firt tudy the noie proble. We aue for iplicity contant additive noie σ o for the object and σ for the reference. Eq. ) can then be rewritten a: σ o Q i = O i p 0 S i + r) + p, 9) 0 ξ 0 )/

3 A. Chelli: Derivation of veiling, viual extinction and exce flux fro pectra of T Tauri tar 765 where ξ 0 = σ /σ o. Clearly, Eq. 9) how that the etiated veiling r depend on the input noie ratio between the reference and the object. If thi noie ratio i not correctly etiated, the veiling will be biaed. We derive an analytical expreion of thi bia in Appendix B. We find that it i a function of the veiling r, q o and q the contrat of the object and the reference pectra in noie unit) and of the quantity f = ξ/ξ 0, where ξ = σ /σ o i the input noie ratio between the reference and the object. The expected value of the etiated veiling r lie between two extree value, r ax and r in, which define the range of peritted veiling value copatible with the data, and correpond to σ i = σ o f =0) and σ i = p 0 σ f infinite) in Eq. ), repectively. Defining the extree relative veiling biae δr ax = r ax r and δr in = r in r, it follow that ee Appendix B): δr ax S + r = q, 0) and δr in S + r = +qo, ) where the bracket tand for the ean value. Thi bia ha a very iple explanation. Underetiating the noie ratio ξ 0 i equivalent to underetiate the reference noie σ, or equivalently, to overetiate the object noie σ o. Let u concentrate for exaple on the reference. If σ i underetiated, the algorith will interpret part of the reference noie in ter of high frequency ignal, it will ee the reference aborption line apparently deeper than they really are and will tend to overeretiate the object veiling. On the contrary, if σ i overetiated, the algorith will interpret part of the true ignal in ter of noie, it will ee the reference aborption line le deep than they really are and will then tend to underetiate the object veiling. It i not alway eay to etiate the true noie ratio between the reference and the object. But if one of the two ource ha a very good ignal to noie ratio, uually the reference, then q i large and r ax r, a hown in Eq. 0). It i then convenient to et σ i = σ o in Eq. ): the derived veiling i biaed, but the bia i negligible. On the other hand, if q o i large, r in r [ee Eq. )]: in thi cae it i better to et σ i = p 0 σ in Eq. ). If both q and q o are all, the bia can be very iportant even if the input noie ratio i not correct by only a factor of a hown in Fig. Ba and Bb). In cae of doubt, one can iniize the bia by underetiating or overetiating the noie ratio ξ 0, depending if q i larger or aller than q o, repectively. To conclude thi ection, we generally recoend to filter the pectra before veiling calculation, but the choice of the working pectral reolution ut be exained cae by cae. The pectra between 580 Å and 50 Å preented in Fig. of Hartann & Kenyon 990) are typical exaple. Indeed, we can ee the preence of high frequency noie uperpoed on lower frequency tructure. Filtering thee data by a factor of a few will coniderably reduce the noie without affecting ignificantly the pectru contrat, which in turn will greatly decreae the poible bia due to a bad noie etiate at a negligible cot in ter of the veiling error. Fig.. Spectru of the K7V tar HD 009, firt noralized to it local continuu at R = and filtered at R = Miatche between the object and the reference pectra The other ource of bia, regard iatche between the object and the reference pectra, and it i treated in detail in Appendix B. By iatche, we ean pectral difference between the reference pectru and the true underlying object tellar pectru T and/or yteatic error of any kind. We how that if the contrat of the object and the reference pectra are large copared to the reidual of the veiling equation, then the bia i doinated by the apparent veiling r of S with repect to T. Defining the bia by δr = r c r, where r c i the calculated veiling, the relative bia i written: δr S + r = r. ) In the cae of non noiy data, the validity of Eq. ) can be checked by coputing the extree veiling value r M and r obtained fro ξ =0and ξ infinite, repectively. If thee two value coincide, then Eq. ) i correct. Otherwie, we ut add to Eq. ) another bia which depend on the input ξ value. The ditance between r M and r i a good indicator to evaluate the iportance of iatche between S and T, which cannot be interpreted in ter of veiling. It alo probably give a good order of agnitude of the reulting bia..4. The effect of the pectral reolution In thi ection, we tudy the effect of the pectral reolution on the veiling value and the aociated noie. For thi purpoe, we ue a high quality pectru of the K7V tar HD 009, obtained at the.93 telecope of the Obervatoire de Haute Provence France) with the intruent ELODIE, at R = ee Baranne et al. 995) and for which S S and / /. Fig. how thi pectru between 4000 Å and 6800 Å, noralized to it local continuu at R = and then filtered at R = 500. Defining a we did o far, the veiling a the ratio between the exce and the local tellar continuu, doe not provide an abolute quantity, becaue the level of the local continuu i reolution dependent. To tudy the effect of the pectral reo-

4 766 A. Chelli: Derivation of veiling, viual extinction and exce flux fro pectra of T Tauri tar lution on the veiling value, we have elected a pectral band of 40 Å centered at 500 Å fro our K7V tar, noralized to it local continuu to iulate the reference, and added a veiling of to the reference to iulate the object. We then degrade the pectral reolution of both the object and the reference by gauian filtering, and etiate the veiling, renoralizing firt the reference pectru to it new local continuu. The veiling i found to vary by about 0% when the pectral reolution varie fro to 500. However, the product of the veiling with the pectrophotoetrically calibrated local continuu of the reference pectru, which eaure the exce, will obviouly reain unchanged. Defining the veiling a the ratio between the exce and the ean reference flux ee HHKHS) would reult in a quantity little dependent on the pectral reolution. The veiling error, Eq. 5), i inverely proportional the product ɛ 0 / which i enitive to the pectral reolution R. Fig. how / between 4000 Å and 6600 Å, coputed fro our K7V pectru for variou pectral reolution, every 50 Å in a wavelength interval of 00 Å. It ha two axia, the firt around 4300 Å and the econd, lightly higher around 500 Å. Thi explain why veiling tudie are often perfored around 500 Å. Indeed, thi wavelength cobine a high pectru contrat / with a good experiental repone. However, the region around 4300 Å can alo be intereting if the ignal to noie ratio i high enough. At ot wavelength / varie only by a factor of to 4 fro R = 500 to R = For a given integration tie and wavelength interval, the ignal to noie ratio on the object total pectru flux ɛ o i obviouly independent on R for photon and background liited noie. However, for CCD readout liited noie, it i inverely proportional to the quare root of the nuber of pixel, i.e. the quare root of R. Hence, if we take only into account the tatitical noie, we conclude that within a factor of a few, the veiling error i independent on the pectral reolution and can even increae with the latter for readout liited noie. However in practice, it will be difficult to etiate the veiling in a 00 Å interval at pectral reolution a low a a few hundred. A firt reaon i that, excluding the unuable feature, the ueful pectru could be reduced to only a few point, leading to a high veiling error. More iportant, at uch low pectral reolution the pectru contrat i generally all of the order of a few%) and can becoe coparable to yteatic local error probably of the order of one to a few %) leading to iportant biae. For oderatly veiled T Tauri tar, thee difficultie can be overcoe by working on large tructure of high contrat like the one extending fro 4900 Å 0.). Although on uch a large pectral bandwidth, the forali of Sect.. cannot be applied, the proble can be olved by iple polynoial odel fitting Chelli et al. 997; Chelli, in preparation). to 550 Å in Fig. / 3. Derivation of viual extinction and exce pectral hape In the lat decade, it ha been realized that becaue of the exce, the viual extinction could not be deduced directly fro the color of T Tauri tar. Hartigan et al. 99) propoed to Fig.. Variance, for variou pectral reolution, of a K7V reference pectru noralized to it continuu and coputed every 50 Å over a 00 Å bandwidth. The trong Na doublet in aborption around 5893 Å, often partially filled in CTT, ha been excluded fro the calculation. etiate the extinction toward T Tauri tar, cobining color and veiling eaureent in the V band. Recently, GHCC extended the previou ethod by the ue of pectrophotoetry and veiling analyi of a few photopheric aborption line panned over a large pectral bandwidth. Then, cobining the extinction and the calibrated pectra, they were able to deduce the exce pectral ditribution. We propoe here to generalize the dicrete ethod of GHCC by a continou approach which ue all the inforation contained in the pectra. In thi ection, the whole pectru of our K7V tar between 4000 Å and 6800 Å, except the deep Na doublet around 5893 Å, i ued a the reference pectru. A the forer wa firt noralized to it local continuu, the exce and the veiling are repreented, without any lo of generality, by the ae quantity. 3.. The forali We aue that the object and the reference pectra, Oλ) and Sλ), are oberved at a pectral reolution R, are already calibrated pectrophotoetrically, and that the reference pectru ha been corrected for extinction. In the abence of noie, Oλ) and Sλ) are trictly related by the following equation ee GHCC): Oλ)wλ) =p fλ)Av[ Sλ)+Eλ) ] wλ), 3) where A v i the viual extinction toward the object, fλ) = A λ /A v decribe the extinction law, and Eλ) i the continuu exce flux the other quantitie are defined in Sect..). It i ueful to firt exaine the approach of GHCC. Let n be the nuber of fit photopheric line at the wavelength λ k, k =,..., n. The output of their dicrete ethod conit in a et of local caling factor {p 0,k } and excee {E k }.Inthe abence of noie, each p 0,k i related to A v and to the overall caling factor p 0 by: p 0,k = p f ka v. 4)

5 A. Chelli: Derivation of veiling, viual extinction and exce flux fro pectra of T Tauri tar 767 In practice, A v and p 0 are obtained by iniizing a et of n relation via Eq. 4), which in turn, allow to derive the exce by inverting Eq. 3): Eλ) = 00.4fλ)Av Oλ) Sλ). 5) p 0 The ethod of GHCC i very intereting and it baic idea i innovating. However, becaue of it dicrete nature, it dicard an iportant part, perhap ot, of the inforation contained in the pectra, in particular their low frequency tructure. In the general cae, it i not poible to olve directly Eq. 3) becaue we do not know the functional dependence of Eλ). However, what we know i that it i a ooth function of the wavelength and thi inforation can conveniently be ued in a continuou odelling. Let u now introduce our approach. In a given pectral interval λ, any well-behaving phyical function like the exce can be decopoed into the u of a traight line joining it end point and the Fourier erie of a periodical function over λ. The exce being ooth, the Fourier erie can be truncated at a certain axiu order n. Under thee condition, Eλ) can be expreed in the for: Eλ) =aλ + + n b j co j=0 n c j in j= πj λ λ πj λ λ ) ). 6) The proble reduce to the etiate of the n +4paraeter p 0,A v,a,{b}, {c}), with {b} =b 0,..., b n ) and {c} = c,..., c n ), but the forali of Sect.. can be generalized to any nuber of paraeter. Auing that the wavelength i apled at the Shannon frequency at value λ i and that wλ i )=for i =,..., ), we define the vector Q, of coponent Q i, whoe quare odulu i to be iniized, with: Q i = O i p fiAv [ Si + E i a, {b}, {c}) ] σ o,i + p fiAv σ,i )/, 7) and: E i a, {b}, {c}) =ai + n j=0 b j co πj i ) n + c j in πj i ). 8) j= The exce cut-off frequency i controlled by the paraeter n and the aociated pectral reolution R e i iply given by: R e = n R. 9) If R e i aller than R, then n <, our proble i well defined and the olution of the fit i unique. The excee deterined by GHCC on 7 T Tauri tar how that thi i in practice alway the cae. Indeed, they vary very oothly with a cut-off frequency aller than a few ten, which correpond to R e /R < 0., even at pectral reolution a low a a few hundred. 3.. How doe it work? To undertand how the algorith work, let u go back to the dicrete approach of GHCC. A v and p 0 are derived fro a leat quare fit of the et of local caling factor {p 0,k }. Auing a in Sect. that the noie on the reference pectru i negligible, the error on each p 0,k, given by Eq. 6), i inverely proportional the the pectru contrat. For contant veiling and ignal to noie ratio ɛ 0, the procedure will weight ore the region of high contrat with repect to the region of low contrat which, incidentally, are alo ore enitive to biae. Now, if intead of fitting individual line, we ue the whole pectru divided into equal interval where the extinction and the exce can be conidered contant, the ae analyi applie. By extrapolation, the ae arguent i valid for our continuou approach, which can be conidered a the liit when the length of each interval tend to zero. A we do not know a priori the exce cut-off frequency, we have to vary the paraeter n increaing it and coputing at each pectral reolution the output of the fit, until the viual extinction and the overall caling factor tabilize within the error bar. Thi will happen when the true exce cut-off frequency i reached. In order to tet the algorith, we have perfored a nuber of iulation at low pectral reolution R = 500) with the reference pectru of Fig.. The object pectra were generated by cobining the reference pectru with a large nuber of ooth exce or veiling) hape, viual extinction value, and white noie. R e wa varied fro all value to 50. We find that in all cae, the coputed viual extinction and overall caling factor tabilize around the correct one, within the error bar, after a critical pectral reolution ha been reached which depend on the exce cut-off frequency). The bet etiate of the paraeter correpond to thoe where the error are the allet., i.e. when the paraeter begin to tabilize. Fig. 3 and 4 how two exaple of iulation. The extree exce hape of Fig. 3a olid line) wa generated fro a rando function of unit ean and variance, oothed by gauian filtering at a pectral reolution of 50. The aociated object pectru of Fig. 3b exhibit a viual extinction of and an average ignal to noie ratio of 00. We ee in Fig. 3c and 3d that the coputed A v begin to tabilize to the correct value and that the reduced chi-quare value t QQ/ n 4) reache and reain contant, at about R e /R =0. or, equivalently, at R e =50. The optiu viual extinction value, taken at the beginning of the plateau, i A v =.08 ± 0.0. The correponding exce, deduced fro Eq. 5) dotted line in Fig. 3a) how a quai perfect agreeent with that in input. Fig. 4a repreent a ore realitic exce olid line) of unit ean value generated by uing up a contant and an exponential function. The aociated object of Fig. 4b exhibit a viual extinction of and an average ignal to noie ratio of 50. Here, the coputed A v begin to tabilize at R e /R =0.05 or R e =5, for which A v =.0 ± 0.5. The pectral hape of the correponding exce dotted line in Fig. 4a) how an excellent agreeent with that in input, only 0% aller. The reduced chi-quare value of Fig. 4d varie fro 0.94 to 0.90, clearly howing that the goodne of the fit

6 768 A. Chelli: Derivation of veiling, viual extinction and exce flux fro pectra of T Tauri tar Fig. 3. a Solid line: odel of the exce; dotted line: recontructed exce at R e =50. b Object pectru generated fro the exce odel and the reference pectru at a pectral reolution R = 500; the input viual extinction i A v =and the average ignal to noie ratio N i 00. c Calculated viual extinction a a function of R e/r, A v begin to tabilize, within the noie, at the correct value, fro R e/r =0.or R e =50. d Reduced chi-quare a a function of R e/r. Fig. 4. a Solid line: odel of the exce; dotted line: recontructed exce at R e =5. b Object pectru generated fro the exce odel and the reference pectru at a pectral reolution R = 500; the input viual extinction i A v =and the average ignal to noie ratio N i 50. c Calculated viual extinction a a function of R e/r, A v begin to tabilize, within the noie, at the correct value, fro R e/r =0.05 or R e =5. d Reduced chi-quare a a function of R e/r ee text). ha not to be judged on the bai of the reduced chi-quare value, but on the exitence of a plateau in which the coputed viual extinction reain, within the noie, contant a a function of R e Error analyi In thi ection we derive a foral expreion for the error on the viual extinction and the overall caling factor. A careful

7 A. Chelli: Derivation of veiling, viual extinction and exce flux fro pectra of T Tauri tar 769 Fig. 5a and 5b how the function gr) and hr e /R). Aexpected, gr) i a decreaing function of R wherea hr e /R) i an increaing function of R e /R approxiatively equal to for R e /R < 0.. The viual extinction error i baically controlled by the function gr), which depend only on the input reference pectru. Coparion of Eq. 6), 0) and ) how that the higher the reference pectru contrat, the aller the function gr). Hence, we can eaily infer that for M tar, whoe pectra are generally ore contrated that thoe of K tar, the correponding gr) function i aller than that hown in Fig. 5a. On the other hand, it will be larger for G tar or earlier pectral type. GHCC pointed out that to work their ethod needed aborption line panning over a large pectral bandwidth. A well, in our continuou approach, we need different region of high contrat, otherwie the function gr) would be prohibitively large. Therefore, it i illuory trying to etiate any viual extinction fro a pectru by uing only one aborption line and an arbitrary aount of continuu Dicuion and concluion Fig. 5. a gr) a a function of the pectral reolution R, b hr e/r). The error on the viual extinction i proportional to the product of thee two function ee text). exaination of the atrix [ t AA] aociated to the dicrete cae how that: σp 0 ) σa v ), 0) p 0 and that within a good approxiation: σa v ) + r N, ) where r and N repreent the average veiling and the average ignal to noie ratio on the object pectru, repectively. Eq. 0) and ) are alo valid in the liit of our continuou approach. To evaluate the coefficient of proportionality in Eq. ), we perfored a nuber of iulation with our reference pectru. The object pectra were generated by adding a contant veiling to the reference and introducing variou viual extinction. The pectral reolution, R and R e, were controlled by filtering and by varying the nuber of paraeter n, repectively. Nuerical calculation of the diagonal eleent of the correponding atrix [ t AA] how that the coefficient of proportionality i, in a firt approxiation, independent on the viual extinction. It increae by only a factor of two when increaing the extinction fro 0 to 5, and can be well approxiated by the product of two function, gr) and hr e /R). Hence, the error on the viual extinction can be written a: σa v ) gr)hr e /R) + r ) N A pecific proble of the continuou approach could be the ue of large pectral bandwidth becaue of wavelength calibration error. For exaple, iatche of pectral line between the object and the reference ay introduce puriou high frequency noie in the calculated exce. The ooth excee derived by GHCC in a bandwidth larger than 000 Angtro how that, at leat here, thi doe not ee to be a evere liitation ee in particular the cae of DS Tau). Alo, it ay occur that the calculated viual extinction never tabilize when increaing the veiling pectral reolution. For a tatitically correct fit, the abence of convergence ean that either the teplate i not adequate or the object ha a coplex pectru which cannot be fit by a iple extinction and exce odel. Eq. ) can be ued to tudy the variation of the viual extinction error a a function of the pectral reolution. Auing photon or background liited noie, for a given integration tie N i inverely proportional to the quare root of R. The error σa v ) i repreented in Fig. 6 a a function of R, for zero veiling and viual extinction, R e =50and N = 00 at R = 500.Iti practically proportional to gr)r / and decreae fro 0.04 at R = 500, by about a factor of at R = 000, and a further factor of 4 going to R = Taking only into account the tatitical noie, there i not uch gain at high pectral reolution when uing the whole pectru in the continuou approach propoed here. Moreover, the factor 4 quoted above i indeed an upper liit, becaue in reality higher reolution correpond to aller optical traniion due to the ore ophiticated experiental et-up. The error given in Eq. ) repreent, however, what we can hope for in the abence of yteatic error. In practice, at high pectral reolution the contrat of the pectra i higher, and conequently the output paraeter are le enitive to biae than at low pectral reolution. It i alo eaier to identify eiion line and local pectral iatche, and hence, the uable pectral bandwidth i larger. Therefore, it will probably be dif-

8 770 A. Chelli: Derivation of veiling, viual extinction and exce flux fro pectra of T Tauri tar Making the approxiation S i + r +r, the relative error on the caling factor i written: σp 0 ) p 0 = σr) +r. A3) The value of p 0 i iply obtained by applying Eq. ) to the continua: Fig. 6. Error on the etiated viual extinction for a given integration tie, for zero veiling and viual extinction, derived fro the pectru of a K7V tar between 4000 Å and 6800 Å, a a function of the pectral reolution R. The exce pectral reolution i R e =50and the average ignal to noie ratio on the object pectru i N = 00 at the pectral reolution R = 500. The doinant ource of error wa aued to be photon or background noie. ficult to handle reolution of a few hundred, for highly veiled T Tauri tar or when the exce i doinated by eiion line. The tatitical error given in Eq. ) i uch aller than the experiental error etiated by GHCC who pointed out that they are liited by yteatic error. Hence, if the yteatic error are not coupled to the tatitical noie, we conclude that it i in principle poible to tudy object uch fainter than thoe o far tudied by uing the continuou approach over a large pectral bandwidth. Thi concluion i further reinforced by the fact that there i no need to correctly iolate oe individual photopheric aborption line for veiling calculation, a it i the cae with the dicrete ethod. Thi i an iportant advantage which can greatly help to work on noiy pectra. There are oe reaining quetion about our continuou approach: What i the bet pectral reolution? What i the enitivity to yteatic error? What are the perforance of the propoed ethod with repect to other? All thee quetion have to be adreed experientally. Appendix A: veiling error Fro Eq. ) and 4), we can eaily copute the atrix A and then the atrix [ t AA]. The veiling variance σ r) i written: σ r) = p 0 Si+r σ i ) Si+r σ i ) σi and that on the caling factor i given by: σ p 0 )=p 0σ r) σ i S i+r ), A) σ i Si+r σ i ), A) where the upper bar over S i tand for the expected value, r i the veiling of the object and σ i =σ o,i + p 0σ,i )/, where σ o,i and σ,i repreent the variance of O i and S i, repectively. p 0 = O c +r, A4) where O c i the expected value of the object continuu. In the following, we aue, for iplicity and whithout lo of generality, that the noie of the reference pectru i uch aller than that of the object pectru and can be neglected, σ i σ o,i. Now we conider contant additive and photon liited noie. A.. Contant additive noie Setting σ i = σ o, inerting Eq. A4) into Eq. A) and uing the relation: O i = S i + r O c +r, A5) it coe after oe tranforation: + r) σr) =a ɛ o. A6) / i the variance of the reference pectru expected value S, given by: = S i S ) i, A7) ɛ 0 = / O /σ o i the ignal to noie ratio on the object total pectru flux, O = O i and a i written: a = S i + r +r Si + r +r ) )/. A8) A rapid exaination of Eq. A8) how that the coefficient a i alway cloe to. A.. Photon liited noie Here, σ i = O / i. Inerting Eq. A4) into Eq. A) and uing Eq. A5), it coe: + r) σr) =a ɛ o, A9) / where ɛ 0 = / O / i the ignal to noie ratio on the object total pectru flux and a i given by: a = / S + r +r +r S i + r) ) / S i + r. A0)

9 A. Chelli: Derivation of veiling, viual extinction and exce flux fro pectra of T Tauri tar 77 Nuerical calculation uing our K7V pectru how that the coefficient a i generally cloe to for any veiling value. Note that uing the relation / o O / / + r), the relative veiling error for additive and photon noie take the very iple for: σr) +r, / q o A) where q o = / o /σ o i the contrat of the object pectru in noie unit. Appendix B: bia calculation B.. Uncorrect noie etiate We aue contant additive noie σ o for the object and σ for the reference. Under thee condition, the etiated veiling value r ha the following analytical expreion ee HHKHS): r = O S, B) C with: C = B) + ξ o + ξ o + ξ 4 o +4ξ o) / ξ. o O and o rep. S and ) are defined in Appendix A, o = Oi S i Oi Si and ξ i the input noie ratio σ /σ o between the reference and the object. The expreion and are yet averaged quantitie over the nuber of point and o, are cloe to their expected value. For exaple, O O σ o / / and o o σo/ /. Hence, a good approxiate of the expected value r of r can be obtained by taking the expected value of the erie developent to the econd order of r around the expected value of the and the ee Papouli 965). We find that, within a relative preciion of, r i given by: r = O S, B3) C where C i derived fro C by replacing the by their expected value. Let r be the correct veiling, we define the bia by δr = r r. Fro Eq. B3), uing the relation o = o + σo, = + σ and o = o, and after oe iple tranforation, the relative bia δr/ S + r) i written: δr S + r ) + f q qo f + 4 B4) f + q ) ) /, q q + o f qo f where q o rep. q ) i defined in Appendix A, and f = ξ/ξ 0, with ξ 0 = σ /σ o. It can be verified that for f =, there i no bia, i.e. δr =0. We alo checked the general validity of Fig. Ba and b. Relative bia on the etiated veiling a a function of the input error ratio between the reference and the object, noralized to the true error ratio, ξ/ξ 0: a for q = / /σ =3and variou value of q o = / o /σ o, b for q =0. Eq. B4) through iulation, uing the pectru S of Sect..4 with R=40000, centered at 500 Å and of width 40 Å. Two liiting cae are particularly intereting. The firt one, f =0, aue that the noie aociated with the reference i zero, σ i = σ o in Eq. ). The derived veiling expected value r ax i axiu, the relative bia i poitive and i given by: δr ax S + r = q. B5) The econd cae correpond to f infinite, it aue that the noie aociated with the object i zero, σ i = p 0 σ in Eq. ). The derived veiling expected value r in i iniu, the relative bia i negative and i given by: δr in S + r = +qo. B6) Fig. Ba and Bb how the relative bia a a function of f for variou value of q and q o.

10 77 A. Chelli: Derivation of veiling, viual extinction and exce flux fro pectra of T Tauri tar B.. Miatche between the object and the reference pectra We aue here that the eaured object and reference pectra O and S are not noiy, i.e. σ = σ o =0, and for iplicity that the object pectru i unbiaed. Let u tudy for exaple the proble of iatche between the reference pectru S and the underlying object tellar pectru T. In the abence of yteatic error, we define r a the apparent veiling of the reference with repect to T calculated for exaple for the liit ξ =0) and γ the reidual function of the veiling equation. S can be written a follow: S = T + r + γ, B7) +r with γ = γ + γ, where γ repreent the yteatic error function. If S i an exact veiled verion of T, i.e. γ =0, then the object veiling with repect to S ut be independent of any input noie ratio ξ. Let u exaine the extree object veiling value r M and r obtained for the liit ξ =0and ξ infinite, repectively. Fro Eq. B) and B), r M and r have very iple analytical expreion: r M = O S, o and B8) r = o O S. B9) o Let r be the correct object veiling with repect to T, obtained for exaple by replacing S by T in Eq. B8). We define the biae δr M and δr by δr M = r M r and δr = r r. Cobining Eq. B7), B8) and B9) and after oe tranforation, the relative veiling biae are written: δr M T + r = r +r +r T + r and δr T + r = r +r ) + γt ++r ) γ T ++r ) γt, ) +r T + r γ T + r γ T + r + γt T. B0) B) They are the u of three ter and differ only through the third ter. The firt ter i due to the apparent veiling of the reference S with repect to T. Auing +r and T + r +r, it can be approxiated by r. The econd and the third ter are due to real pectral iatche and to yteatic error. The econd ter i in general negligible, becaue we expect the ean value of γ to be cloe to zero for local iatche of the order of a few% only. The third ter i caled by the ratio γt / T and γ / T and i alo negligible if the pectru contrat / T i large copared to the reidual function γ. Under thee condition, the bia due to iatche between the object and the reference i doinated by the apparent veiling of the reference. Setting δr = δr M = δr and aking the approxiation T S, the relative bia i iply given by: δr S + r = r. B) To invetigate further the third ter in Eq. B0) and B), it i intereting to calculate the relative difference between r M and r.itigivenby: r M r T + r =+r ) γ T Γ γt ++r ) δ γt T ), B3) where Γ γt i the correlation coefficient between γ and T. Eq. B3) clearly how that the two veiling value r M and r are equal only if the γ i the null function or if it i a linear function of T. A the latter poibility i unlikely, the equality between r M and r iplie necearily that the γ function i null and conequently that Eq. B) i valid. On the other hand, if r M and r are ditinct, the third ter in Eq. B0) and B) cannot either be neglected. It introduce another additive bia which depend on the input ξ value. The ditance between r M and r i a good indicator to evaluate the iportance of iatche between S and T which cannot be interpreted in ter of veiling, and probably give alo a good order of agnitude of the reulting bia. Acknowledgeent. The author would like to thank the referee Dr. P. Hartigan for hi helpful coent that iproved the clarity of the paper and Dr. C. Ceccarelli for her careful englih reviion of the paper and helpful dicuion. Reference Baranne A., Queloz Q., Mayor M., et al., 995, A&AS 9, 373 Bari G., Batalha C., 990, ApJ 363, 654 Bertout C., Bari G., Bouvier J., 988, ApJ 330, 350 Calvet N., Bari G., Kuhi L.V., 984, ApJ 77, 75 Chelli A., Cruz-Gonzalez I., Sala L., et al., 997, Modulated teporal variability in DF Tauri: a bipolar Brγ eiion tructure. In: Malbet F., Catet A. ed.) Poter Proceeding of the IAU Sypoiu 8. Chaonix France), 63 Gullbring E., Hartann L., Briceño C., Calvet N., 998, ApJ 49, 33 Hartigan P., Hartann L., Kenyon S.J., Hewett R., Stauffer J., 989, ApJS 70, 899 Hartigan P., Kenyon S.J., Hartann L., et al., 99, ApJ 38, 67 Hartann L., Kenyon S.J., 990, ApJ 349, 60 Hartann L., Hewett R., Calvet N., 994, ApJ 46, 669 Herbig G.H., Bell K.R., 988, Lick Ob. Bull. Knoechel G., Von der Heide K., 978, A&A 67, 09 Lynden-Bell D., Pringle J.E., 974, MNRAS 68, 603 Papouli A., 965, Probability, Rando Variable and Stochatic Procee. McGraw Hill, New York, p. Walker M.F., 987, PASP 99, 39