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ABSO LUTE PRO PERTIES O F THE SPO TTED ECLIPSING BINARY STAR CV BO �O TIS

G uillermo T orres1,Luiz Paulo R .Vaz

2,and C laud H .Sandberg Lacy

3

D raftversion A pril23,2013

ABSTRACT

W e present new V -band di�erentialbrightness m easurem ents as wellas new radial-velocity m ea-surem ents ofthe detached,circular,0.84-day period,double-lined eclipsing binary system CV Boo.Thesedataalongwith otherobservationsfrom theliteraturearecom bined toderiveim proved absolutedim ensions ofthe stars for the purpose oftesting various aspects oftheoreticalm odeling. Despitecom plicationsfrom intrinsicvariability wedetectin thesystem ,and despitetherapid rotation ofthecom ponents,we are able to determ ine the absolute m asses and radiito better than 1.3% and 2% ,respectively. W e obtain M A = 1:032� 0:013M � and R B = 1:262� 0:023R � forthe hotter,larger,and m orem assiveprim ary (starA),and M B = 0:968� 0:012M � and R B = 1:173� 0:023R � forthesecondary.Theestim ated e�ectivetem peraturesare5760� 150K and 5670� 150K .Theintrinsicvari-ability with aperiod � 1% shorterthan theorbitalperiod isinterpreted asbeing dueto m odulation byspotson oneorboth com ponents.Thisim pliesthatthe spotted star(s)m ustbe rotating fasterthanthesynchronousrate,which disagreeswith predictionsfrom currenttidalevolution m odelsaccordingto which both starsshould be synchronized. W e also �nd thatthe radiusofthe secondary islargerthan expected from stellarevolution calculationsby � 10% ,a discrepancy also seen in other(m ostlylower-m ass and active) eclipsing binaries. W e estim ate the age ofthe system to be approxim ately9 G yr. Both com ponentsare nearthe end oftheirm ain-sequence phase,and the prim ary m ay havestarted the shellhydrogen-burning stage.

Subjectheadings: binaries:eclipsing | stars:evolution | stars:fundam entalparam eters| stars:individual(CV Boo)| stars:spots

1. IN TRO D U CTIO N

CV Boo (= BD + 37 2641 = G SC 2570 0843; � =

15h 26m 19:s54, � = + 36� 58053:004, J2000.0; V � 10:8,SpT = G 3V) was discovered as a possible eclipsing bi-nary star by Penicheetal.(1985). Busch (1985) con-�rm ed itasan eclipsing binary oftypeEA and found itsperiod to be0.8469935days.In hislastpublished paper,astudy of4lowerm ain sequencebinaries,Popper(2000)determ ined a spectroscopic orbit for CV Boo. Popperwaspessim istic aboutthe prospectsfordeterm ining ac-curate absolute properties ofthem because \It appearsunlikely thatde�nitive photom etry willbe obtained forthese stars,partly because ofintrinsic variability." Re-cently,a lightcurveand radialvelocity study ofthesys-tem were done by Nelson (2004b),resulting in the �rstestim atesofitsabsoluteproperties.The param eters of CV Boo m ake it potentially in-

teresting as the m ost evolved system am ong the well-studied double-lined eclipsing binarieswith com ponentsnear1M � (seeFigure1),aregim ewheresom ediscrepan-cieswith theoreticalm odelshave been pointed out. W edescribe in the following ourextensive new photom etricand spectroscopicobservationsoftheobjectintended toim prove our knowledge ofthe system . The presence ofstarspotsdoesin factlim itsom ewhatourabilitytodeter-m ine highly accurateabsolute propertiesforthisbinarystar,butthe resultsare stillaccurate enough form ean-

1 H arvard-Sm ithsonian Center for A strophysics, 60 G ardenStreet,Cam bridge,M A 02138,e-m ail:gtorres@ cfa.harvard.edu

2 D epto.deF��sica,ICEx-U FM G ,C.P.702,30.123-970 Belo H or-izonte,M G ,Brazil,e-m ail:lpv@ �sica.ufm g.br

3 D epartm ent of Physics, U niversity of A rkansas, Fayetteville,A R 72701,e-m ail:clacy@ uark.edu

ingfultests of current stellar m odels. As we describehere,CV Boocontributessigni�cantly to thebody ofev-idenceconcerning thedi�erenceswith theory m entionedabove.

2. O BSERVATIO N S A N D R ED U CTIO N S

2.1. Di�erentialand absolute photom etry

New di�erentialbrightness m easurem ents ofCV Boowereobtained with thefacilitiesavailableatthe K im pelO bservatory (ursa.uark.edu). They consist ofa M eade10-inch f/6.3 LX-200 telescopewith a Santa Barbara In-strum entsG roup ST8 CCD cam era (binned 2� 2 to pro-duce765� 510pixelim ageswith 2.3arcsecsquarepixels)inside a Technical Innovations Robo-Dom e, controlledautom aticallyby an AppleM acintosh G 4com puter.Theobservatory is located on top of K im pel Hall on theFayetteville cam pusofthe University ofArkansas,withthecontrolroom directly beneath theobservatory insidethebuilding.Sixty-second exposuresthrough aBessellV�lter(2.0m m ofG G 495and 3.0m m ofBG 39)werereadout and downloaded by Im ageG rabber (cam era controlsoftware written by J.Sabby) to the controlcom puterover a 30-second interval,and then the next exposurewasbegun. The observing cadence wastherefore about90sperobservation.The variablestarwould frequentlybe m onitored continuously for 4{8 hours. CV Boo wasobserved on 89 nightsduring partsoftwo observing sea-sonsfrom 2001 Decem ber1 to 2003 June 9.The im ageswere analyzed by a virtualm easuring en-

gineapplication written by Lacy that at-�elded theim -ages,autom atically located thevariableand com parisonstars in the im age, m easured their brightnesses, sub-tracted the corresponding sky brightness,and corrected

2 Torresetal.

Fig.1.| M ain-sequence eclipsing binaries in m ass range ofCV Boo with accurate determ inations of their absolute proper-ties (m asses and radiigood to better than � 2% ). D ata are takenfrom A ndersen (1991) and updates from the literature. Prim aryand secondary com ponentsareconnected with solid lines.CV Boois represented with open circles. The dashed line shows the solar-m etallicity zero-age m ain sequence from the m odels by Y ietal.(2001),forreference.

for the di�erences in airm ass between the stars. Ex-tinction coe�cients were determ ined nightly from thecom parison star m easurem ents. They averaged 0.20m ag/airm ass.CV Boo isalso known asG SC 2570 0843.The com parison stars were G SC 2570 0511 (\com p",V = 10:26, as listed in the Tycho Catalogue), andG SC 25700423(\ck").Both com parison starsarewithin8 arcm in of the variable star (\var"). The com pari-son starm agnitudedi�erenceshcom p� ckiwereconstantat the level of 0.013 m ag (standard deviation withina night),and 0.007 m ag for the standard deviation ofthe nightly m ean m agnitude di�erence. The di�eren-tialm agnitude hvar� com piofthe variable starwasref-erenced only to the m agnitude ofthe com parison star,com p.Theresulting 6500 V -band m agnitudedi�erenceshvar� com piarelisted in Table1and plotted in Figure2.Thetypicalprecisionofthevariablestardi�erentialm ag-nitudesisabout0.013 m ag perm easurem ent.In addition to our own, di�erential photom etry of

CV Boo was obtained by Nelson (2004b) in V andCousins I between 2003 M arch and June (253 and 265m easurem ents,respectively). The com parison star wasG SC 2570 0869,and thecheck starwasG SC 2570 0511,which isthesam estarweused asthecom parison.Theseobservationsareincorporated into ouranalysisbelow.Absolute photom etry of CV Boo is available in the

literature from several sources, and color indices canbe used to estim ate a m ean e�ective tem perature forthe com bined light of the system (assum ing no inter-stellar reddening). The results are collected in Ta-ble 2. W e used the color/tem perature calibrations ofRam��rez& M el�endez(2005) for dwarfstars for allbutthe Sloan g� r index; for that color we used the cali-bration ofG irardietal.(2004).In allcaseswe assum ed

Fig.2.| The top panelshowsourV -band lightcurveofCV Boo(+ sym bols),consisting of6500 points,together with the V (� )and I (� ) light curves from N elson (2004b),m arked \�V n" and\�In". N elson’s light curves are shifted as indicated,for clarity.O ur theoreticalsolution without spots is overplotted (continuousgrey lines;x3.1). The lower panels show the O � C residuals fromthese �ts,and in the upper leftcorner,the standard deviation fora single m easurem ent.

solarm etallicity. The value ofJohnson V is thatlistedin theTycho Cataloguewith no uncertainty given there.W e have assum ed a conservative error of0.10 m ag forV . The tem perature estim ate from the Johnson B � V

index usesthe valueofthatindex aslisted in theTychoCatalogue,with itslisted error.Thetem peraturevaluesestim ated in these waysagree quite well,exceptfortheestim atesfrom B �V and B T�VT ,which happen to havethe largest form alerrors. The weighted average ofthe7 estim ates is 5706� 60K ,where the uncertainty doesnotaccountforpossible system aticerrorsin the variouscalibrations.Thiscolor-index-basedtem peratureisquiteconsistentwith spectroscopicestim atesdiscussed below,and thissuggeststhattheinterstellarreddening value,ifany,isvery sm all.CV Boo is identi�ed as a strong X-ray source in the

RO SAT catalog (Vogesetal.2000).Thisispresum ablydue to an active chrom osphere/corona associated withitsspotactivity (see below).

2.2. Ephem eris

Photoelectric or CCD tim es of m inim um light forCV Boo carried outoverthe pastdecade have been re-portedbyanum berofauthors(Agerer& H�ubscher2002,2003; Bakisetal. 2003; Diethelm 2001; Dogru etal.2006; H�ubscher 2005; H�ubscheretal. 2005, 2006;K im etal.2006;Lacy 2002,2003;M aciejewski& K arska2004;Nelson 2000,2002,2004a). Additionaltim es ofeclipseincluding oldervisualand photographicm easure-m entsreaching back to 1957 werekindly provided by J.M .K reiner(seeK reiner,K im & Nha2000)ortaken from

CV Boo 3

theliterature(Locher2005;M olik 2007).Separateleast-squares�tsto the98 availableprim ary and 50secondarym inim a yielded ephem erideswith virtually thesam epe-riod within the errors.A sim ultaneous�tto allm inim awas then perform ed assum ing a circular orbit. Uncer-taintieswere initially adopted aspublished,orassignedby iterationsand by typeofobservation so asto achievea reduced �2 ofunity.Theresulting linearephem erisis

M in I(HJD)= 2;452;321:845322(50)+ 0:846993420(69)E :(1)

where the �gures in parentheses represent the uncer-tainty in units ofthe lastdecim alplace. No signi�canttrendsindicativeofperiod changesareseen in theO � Cresiduals. A test solving for separate prim ary and sec-ondaryepochswith acom m on period yielded aphasedif-ference between the eclipsesof�� = 0:49991� 0:00013.This is consistent with 0.5, supporting our earlier as-sum ption ofa circularorbit.

2.3. Spectroscopy

CV Boowasobserved spectroscopicallywith an echelleinstrum enton the1.5m Tillinghastre ectorattheF.L.W hippleO bservatory(M t.Hopkins,Arizona).A totalof66 spectra were gathered from 1991 June to 2005 April,

each ofwhich covers a single echelle order (45 �A) cen-

tered at5188.5 �A and wasrecorded using an intensi�edphoton-counting Reticon detector. The strongest linesin this window are those ofthe M g I b triplet. The re-solving power ofthis setup is �=�� � 35;000,and theobservationshave signal-to-noise ratiosranging from 13to 36 perresolution elem entof8.5 km s�1 .Radial velocities were obtained using the two-

dim ensional cross-correlation algorithm TO DCO R(Zucker& M azeh 1994). Tem plates for the cross cor-relations were selected from an extensive library ofcalculated spectra based on m odelatm ospheresby R.L.K urucz4 (see also Nordstr�om etal.1994;Latham etal.2002). These calculated spectra cover a wide range ofe�ectivetem peratures(Te�),rotationalvelocities(vsiniwhen seen in projection),surface gravities (logg),andm etallicities. Experience has shown that radialveloc-ities are largely insensitive to the surface gravity andm etallicity adopted forthetem plates,aslongasthetem -peratureischosen properly.Consequently,theoptim umtem plate for each star was determ ined from extensivegrids ofcross-correlationsvarying the tem perature andthe rotationalvelocity,seeking to m axim ize the averagecorrelation weighted by the strength ofeach exposure.Solarm etallicity wasassum ed.Theresults,interpolatedto surface gravities oflogg = 4:25 for both stars (see

x3),are Te� = 5800 K and vsini= 73 km s�1 for the

prim ary star,and Te� = 5650 K and vsini= 67 km s�1

for the secondary. Estim ated uncertainties are 200 Kand 10 km s�1 for the tem peratures and projectedrotationalvelocities,respectively. Tem plate param etersnear these values were selected for deriving the radialvelocities. Typical uncertainties for the velocities are5.6 km s�1 for the prim ary and 5.9 km s�1 for thesecondary,which areconsiderably worsethan usualwiththis instrum ent because of the signi�cant rotationalbroadening ofboth stars.

4 A vailable at http://cfaku5.cfa.harvard.edu.

The stability ofthe zero-pointofour velocity systemwas m onitored by m eans ofexposures ofthe dusk anddawn sky,and sm allrun-to-run correctionswereappliedin the m annerdescribed by Latham (1992). Additionalcorrectionsforsystem aticswereapplied to thevelocitiesas described by Latham etal.(1996) and Torresetal.(1997) to account for residualblending e�ects and thelim ited wavelength coverage ofour spectra. These cor-rectionsarebased on sim ulationswith arti�cialcom pos-ite spectra processed with TO DCO R in the sam e wayasthe realspectra. The �nalheliocentric velocitiesarelisted in Table 3.Thelightratiobetween thecom ponentswasestim ated

directly from the spectra following Zucker& M azeh(1994). After corrections for system atics analogous tothose described above,we obtain ‘B =‘A = 0:71 � 0:04

atthe m ean wavelength ofourobservations(5188.5 �A),wherewerefertostarA asthem orem assiveone(thepri-m ary)and to theotherasstarB.Thisvalueisin reason-able agreem entwith estim ates by Popper (2000) basedon the relative strength ofthe Na ID linesin CV Boo.G iven thatthestarshaveslightly di�erenttem peratures(see below),a sm allcorrection to the visualband wasdeterm ined from syntheticspectra integrated overtheVpassband and the spectralwindow ofour observations.Thecorrected value is(‘B =‘A )V = 0:73� 0:04.RadialvelocitiesforCV Boo have been reported pre-

viously also by Popper(2000),who observed the staraspartofhisprogram focusing on binary system sofspec-traltypeF to K .His45 m easurem entsfrom 1988 Febru-ary to 1997 Junewith theHam ilton spectrom eterattheLickO bservatorypartiallyoverlap in tim ewith ours,andareofexcellentquality.However,they requirea num berofadjustm ents before they can be com bined with ours.O ne ofthese adjustm entshasto do with correctionsheapplied to hisraw velocities. The raw velocities(whichhereferred to as\O bserved")werereported forCV Booalongside \O rbital" velocitieswhich di�erfrom the rawones by the application oftwo corrections. The �rst isanalogousto thecorrectionsforsystem atice�ectsweap-plied to ourown velocities,and wasderived in a sim ilarm anner using synthetic binary spectra (for details seePopper& Jeong 1994). The second correction accountsfordistortionsand m utualirradiation in the closeorbit,and was com puted by Popper (2000) using the form al-ism developed by W ilson (1990) asim plem ented in theW ilson-Devinney (W D)program thatwealso usebelow,and added to the velocities. G iven our plan to use theW D program to com binethelightcurveswith theveloc-ity m easurem entsin a sim ultaneoussolution,the lattercorrectionsin the data by Popper(2000)need to be re-m oved priorto use orthey would be applied twice. W eestim ated these correctionsfrom a prelim inary solutionwith W D (presum ably em ulating Popper’s procedure),and applied them with theoppositesign tothe\O rbital"velocities.Thesecorrectionsarenolargerthan 1km s�1 ,which issm allerthan theform aluncertaintiesin theve-locities (2.7 km s�1 for the prim ary and 2.1 km s�1 forthe secondary,from the residualsofprelim inary �ts).A second adjustm ent we found necessary to apply to

Popper’s m easurem ents is to correct for an o�set of1.60 km s�1 between his prim ary and secondary veloc-ities,asindicated by prelim inary sine-curve�ts(see Ta-

4 Torresetal.

ble6byPopper2000,wheretheo�sethe�ndsissim ilar).E�ectivelythetwostarsyield di�erentcenter-of-m assve-locities. W e found no such o�set in our own m easure-m ents,butexperienceindicatesitcan som etim esappearwhen thereisasigni�cantm ism atch betweentheadoptedtem platesand the realstars,and ifnotcorrected itcanbiasthesem i-am plitudeswhen enforcing a com m on cen-terofm assin the�t.W ehavethusadded � 1:60 km s�1

to Popper’ssecondary velocities.Finally,a third adjust-m entisto bring Popper’soverallvelocity zero pointintoagreem ent with ours. From trialorbital�ts we foundthis required a shift of+ 0.37 km s�1 to his velocities.5

Popper’scorrected velocitiesare listed in Table 4. Sep-arate �ts to his data and ours give sim ilar values forthesem i-am plitudes,and yield m assesthatdi�erby lessthan twice their com bined uncertainties. W e thereforeproceed to m ergethe two data setsbelow.A third setofradialvelocitiesforCV Boowasreported

by Nelson (2004b),butthey wereobtained atlowerres-olution,they are few in num ber(12),and show a m uch

largerscatterthan thetwo otherdata sets(� 15 km s�1 )so they areoflittle useforourpurposes.

3. M O D ELIN G O F TH E PH O TO M ETR IC O BSERVATIO N S

The overallshape ofCV Boo’slightcurves(Figure 2)showsratherm oderateproxim ity e�ectsdespitethesys-tem ’s relatively short period of slightly m ore than 20hours, with the curvature between the m inim a beingm ostly due to the deform ation ofthe com ponents and,to a sm allerdegree,to them utualillum ination.A num -ber ofsm all-scale features are obvious to the eye thatare possibly due to spots,other intrinsic variability,oreven instrum entale�ects,especially in the sm allerdatasets ofNelson (2004b). O ther features described beloware revealed through a m ore detailed exam ination,andintroducesom ecom plicationsinto the analysis.

3.1. Initialsolutionswithoutspots

To begin with, we chose to m odel all the obser-vations together in order to obtain a baseline solu-tion against which to com pare m ore com plex solutionsthat attem pt to account for the features m entionedabove.W eused a version oftheW ilson-Devinney (W D)m odeling program (W ilson & Devinney 1971; W ilson1979, 1993) with extensive m odi�cations as describedin Vaz,Andersen & Claret (2007) and references giventherein. The m odi�cations pertinent to CV Boo in-cludethecapability to usem odelatm ospheres(now alsoavailablein thedistributed versionsofW D),consistencychecksbetween variousparam eters,and theabilitytousethe downhillsim plex algorithm (Nelder& M ead 1965)instead of di�erential corrections. W e com bined ourown V -band lightcurve with the sparserV and I lightcurves ofNelson (2004b) in order to im prove the con-strainton the e�ective tem perature ratio,and with ourradialvelocities from x2.3 as wellas those of Popper(2000).Thuswesolved sim ultaneously3lightcurvesand4 radial-velocity curves. The param etersadjusted were

5 W hile in principle the latter two adjustm ents (o�sets) couldbe accounted forin our com bined photom etric and radialvelocitysolution described below by sim ply adding free param eters to the�t,lim itationsin the currentversion ofthe W D code do notallowthis,so we have applied the o�sets externally.

theorbitalinclination i,thesecondaryTe� ( ux-weightedm ean surface tem perature), the bandpass-speci�c pri-m ary lum inosity (see W ilson 1993),both stellarsurfacegravitationalpseudo-potentials (related to the stellarradii),the center-of-m assradialvelocity ,the m assra-tio q� M B =M A ,and an arbitrary phase shift.The sim -ilar depths ofthe two m inim a in both V and I (Fig-ure2)im ply thatthecom ponentsm usthaverathersim -ilartem peratures,consistentwith indicationsfrom spec-troscopy. The prim ary tem perature washeld �xed atavalue determ ined from ourresultsbased on photom etryand spectroscopy,as follows. A photom etric estim ateofthe prim ary tem perature wasderived from the m eansystem tem perature(x2.1)using approxim atevaluesfortheradiusratio and tem peratureratio from prelim inarylight-curvesolutions.The result,5755� 60 K ,wasthencom bined with the spectroscopic value of the prim arytem perature(x2.3),givingaweighted averageforStarAof5760� 150 K ,which weadopt.The re ection albedosforboth com ponentswere held

�xed atthevalue0.5,appropriateforstarswith convec-tive envelopes,and the gravity-brightening exponents�were com puted internally in W D using the localvalueofTe� for each point on the stellar surface taking intoaccount m utual illum ination, following Alencar& Vaz(1997) and Alencaretal.(1999). The ux from eachof the com ponents is represented by NextG en atm o-sphere m odels based on the PHO ENIX stellar atm o-sphere code (Allard & Hauschildt 1995; Allard etal.1997;Hauschildtetal.1997a,b). The lum inosity ofthesecondary iscalculated by theprogram from itssizeandTe�. The lim b-darkening coe�cients for both com po-nents,xA and xB ,were taken from the tablesby Claret(2000),and interpolated using a bi-linearschem eforthecurrentvaluesofloggand Te� ateach iteration.W ehaveonly considered thelinearlaw herein view ofthedistor-tions in the light curve,which willtend to overwhelmthe rathersubtle e�ectoflim b-darkening. Although wehaveno evidenceofanotherstarin the system ,thepos-sibility ofthird light (‘3) was explored carefully for itspotentialin uence on the geom etric param eters,partic-ularly in the solutions described below in x3.2, whichinclude spots. Achieving convergence when solving forthird lightwasfound to be very di�cultdue to the in-trinsic variability and the large num beroffree param e-ters,even when considering m ultiple param etersubsets(W ilson & Bierm ann 1976).W e found thatthe solutionwas not im proved,and ‘3 was not considered further.W e estim ate a conservative upper lim it to ‘3 of� 1% ,which does not produce signi�cant changes in the geo-m etric param eters. A circular orbit has been assum edin thefollowing,based on ourinvestigation oftheeclipsetim ingsin x2.2,and thelackofanyindication in thelightcurvesofadisplacem entofthesecondary m inim um fromphase0.5.Thisisconsistentwith expectationsfrom tidaltheory foran orbitwith such ashortperiod (seex6).W ehavealsoassum ed heretidalforceshavesynchronized thecom ponents’rotation with theorbitalm otion ofCV Boo.In theseinitialcalculationsweapplied both least-squaresdi�erentialcorrections and/or the sim plex m ethod be-tween successive iterations. The lim b-darkening coe�-cients,norm alization m agnitudes,surface gravities,andindividualvelocity am plitudeswereallupdated betweenconsecutive runsto correspond to the solution from the

CV Boo 5

Fig.3.| Prim ary (� ) and secondary (+ ) radialvelocity m ea-surem ents collected at CfA (Table 3) along with the theoreticalcurvesobtained with W D and no spots(top panel). Velocities areshifted so thatthecenter-of-m assvelocity isatzero (dashed line).The large deviation from K eplerian m otion in the predicted veloc-ity around both conjunctions is due to the R ossiter-M cLaughline�ect (see Schlesinger 1909a,b;R ossiter 1924;M cLaughlin 1924),caused by partial eclipses of the rotating stellar surfaces (an ef-fect built into the W D m odel). The O � C residuals are shown atthe bottom . The standard deviation ofthe unweighted residualsis �rv = 6:09 km s�1 for both com ponents and is shown in theupper left corner. The reduced �2 values were 1.000 and 0.998,respectively.

previousiteration.Thesolutionsareshown in Figures2,3and 4,with the

corresponding residuals. The residuals from the radialvelocity �tsm atch the quality ofthe observations. Thephotom etry isreasonably wellrepresented on averagebythe theoreticalcurves,butthe residualsforourV -bandobservationsshow an rm s scatterof0.0196 m ag thatism uch largerthan them ean internalerror(� 0.013 m ag).This im m ediately suggeststhere m ay be unm odeled ef-fects. The intrinsic errorsofNelson’sobservationswerenotreported in the originalpublication.

3.1.1. Study ofthe light-curve residuals

Partoftheextra scatterisno doubtdueto featuresinthelightcurvealluded toearlierthatareseen in Figure2,such aschangesin the lightlevelfrom nightto nightatthe sam e orbitalphase (e.g.,near phase 0.1),or othershort-term deviations(e.g.,nearphase0.8),both in ourdata and in Nelson’s. W e investigated the residuals ofourm orenum erousV -band photom etryfurthertosearchforperiodicsignalsthatm ightadditionally contributetotheexcessscatter.O urinitialexploration ofpossiblesig-nalsnear the orbitalperiod using the La er& K inm an(1965)m ethod revealed severalsim ilarperiodicitiesthatappear signi�cant. W e then extended the search to am uch widerrangeoffrequenciesbycom putingtheLom b-Scargle periodogram ,and found other signals. This isshown in the top panelofFigure 5. W e refer to thisas the \dirty" power spectrum , since it is a�ected by

Fig.4.| Prim ary (� ) and secondary (+ ) radialvelocity datafrom Popper(2000),with the sam e lim itson the verticalaxesasinFigure 3.The standard deviations ofthe unweighted residualsare�rv;A = 3:14 km s�1 and �rv;B = 2:14 km s�1 forthe prim ary and

secondary velocities,respectively (shown in the upper left corner

ofthe lowerpanels). The reduced �2 values were 1.005 and 1.002.

the particular tim e sam pling ofthe observations (win-dow function). The highest peak corresponds to a pe-riod of� 0.837 days,which is shorter than the orbitalperiod of0.846993420days.Thetwo nexthighestpeaks(indicated with arrows)turn outto be 1-day aliases.Toillustrate this, we have applied the CLEAN algorithmas im plem ented by Robertsetal.(1987) to rem ove thee�ectsofthewindow function.Thesecond panelofFig-ure 5 shows that only the 0.837-day peak survives thisprocess,suggesting itisa realsignal.In the third panelan enlargem entofthedirty powerspectrum in thevicin-ity ofthe m ain peak reveals�ne structure thatwasalsoseen with the La er& K inm an (1965) m ethod. How-ever,noneofthesepeaksagreewith thefrequency corre-sponding to the orbitalperiod,which isrepresented forreferencewith a dotted line.Thetwo m ain sidelobesin-dicated with arrowsare1-yearaliasesofthe m ain peak.O nceagain they disappearafterapplication ofCLEAN,as seen in the bottom panel,supporting the reality ofthe rem aining signal. The statisticalsigni�cance ofthissignalwasestim ated by num ericalsim ulation.W egener-ated 100;000arti�cialdata setsusing theactualtim esofobservation and thevarianceoftheoriginalresidualsas-sum ing a G aussian distribution oferrors,and com putedtheLom b-Scarglepowerspectrum forthesedatasetsoverthe sam e frequency intervalconsidered above. W e thenselecting the highest peak in each case. None ofthemcam e close to the height ofthe peak we see in the realdata,indicating a false alarm probability sm aller than10�5 .The precise frequency ofthis signalwas m easured in

the CLEANed spectrum , and its uncertainty was es-tim ated from the half width at half m axim um of thepeak. The corresponding period is 0:83748 � 0:00052

6 Torresetal.

Fig.5.| (a) Lom b-Scargle power spectrum ofthe residuals ofourV -band K im pelO bservatory observationsofCV Boo from theno-spot solution described in the text. The arrows indicate 1-day aliases of the central peak; (b) CLEA N ed power spectrumof the sam e m easurem ents using the algorithm of R oberts etal.(1987)to rem ove the e�ectsofthe tim e sam pling;(c)Enlargem entof panel (a), with the 1-year aliases of the m ain peak indicatedwith arrows. The dotted line represents the orbitalfrequency;(d)Enlargem ent ofpanel(b). The period corresponding to the signalis0:83748 � 0:00052 days.

days,which is di�erent from the orbitalperiod at the18� level. A plot of the photom etric residuals foldedwith this period is shown in Figure 6,and indicates apeak-to-peak am plitude ofabout0.04{0.05 m ag.TheK im pelO bservatorydatacovertwoobservingsea-

sons. Separate Lom b-Scargle power spectra show thatthe sam e signalis present in both seasons,along withthe 1-day aliases,indicating the phenom enon is persis-tentfrom one yearto the next.Itisnotseen asclearly,however,in the residualsfrom ourbaseline�tofthe ob-servations of Nelson (2004b), which are m uch sparserthan ours(and span only 71 daysinstead of549 days).HisV -band data show a hintofthem ain peak and its1-day aliases,butnottheI-band data,which havea largerscatter.W e carried out a sim ilar power spectrum analysis of

thehcom p� ckidi�erentialm agnitudesfrom K im pelO b-servatory,to explorethepossibility thateitherthe com -parison or the check star m ight be the source of thisvariation. No signi�cant periodicity was seen. Thus,the phenom enon isintrinsic to CV Boo.Possibleexpla-nations for this variation include stellar pulsation,andstar spots on one or both com ponents. Neither ofthestars appear to be in an evolutionary state that favorspulsationalinstability.Forexam ple,theabsolutedim en-sionsderived below place both com ponentswelloutside

Fig.6.| R esidualsofourV -band K im pelO bservatory observa-tionsofCV Boo shown asa function ofphase,using the period of0.83748 days inferred from the power spectrum analysis (see Fig-ure 5).The tim e origin has been setarbitrarily to H JD 2;450;000.

the Cepheid or � Sct instability strips in the H-R dia-gram indicated by K j�rgaard etal.(1983).O n theotherhand,CV Boo isa known strong X-ray source detectedby RO SAT (Vogesetal.2000),with an X-ray lum inosity

oflogLX = 30:656 thatissom e4000tim esstrongerthanthe Sun. In term sofitsbolom etric lum inosity CV Boohas logLX =Lbol = � 3:39, which is near the high endforactive binaries. Thusitseem slikely thatthe under-lying reason for the 0.837-day periodicity is related tospotson the surfaceofoneorboth com ponents,and weproceed underthisassum ption. Itisinteresting to notethat these features on CV Boo seem to have lasted foran unusually long tim e (1.5 yearsin our case),at leastcom pared to sunspots,although even m ore extrem e ex-am pleshave been docum ented in the literature. O ne isthewell-known activebinaryHR 1099(Vogtetal.1999),with surfacefeaturespersisting foratleast11 years.An im portant im plication of this spot hypothesis is

that the com ponent having spots would appear to berotating slightly m ore rapidly than synchronously withthem otion in thecircularorbit,which isunexpected forsuch ashort-periodbinary.W ediscussthisin m oredetailbelow.

3.2. Solutionswith spots

An accuratem easurem entofthevsinivaluesforbothcom ponentswould allow fora directtestofourhypoth-esisofnon-synchronousrotation,and could even distin-guish which ofthe stars is the culprit (or ifboth are).Unfortunately,however,the quality ofourspectroscopicm aterialis insu�cient for that purpose. In principle,m odern light-curve m odelssuch asW D enable the userto solve for variousparam etersthatdescribe the spots.

6 LX isin unitsoferg s�1 ,and wasdeterm ined from theRO SATcount rates and hardness ratios,the distance estim ate in x4,andthe energy conversion factor ofFlem ing etal.(1995).

CV Boo 7

However,with only photom etricdata atourdisposalforCV Boo,and m ost ofit in a single passband,it is es-sentially im possible to tellwhich star has the spots,orwhether both com ponents have them . This is a well-known di�culty in light-curve m odeling. O ther inver-sion techniques such asDoppler im aging are m uch bet-tersuited to m apping surfaceinhom ogeneities,althougheven they are not without their lim itations. M oreover,even ifweknew which starhasthespots,thedeterm ina-tion oftheirparam etersfrom lightcurvesalone isa no-toriously ill-posed problem ,on which there isabundantliterature discussing issues of indeterm inacy and non-uniqueness in the presence oflim ited data quality (see,e.g.,Eker1996,1999,and num erousreferencestherein).Having photom etry in m ultiple passbandsm ay aleviatetheproblem som ewhat,butitdoesn’tsolveitand strongdegeneraciesarelikely torem ain with othersubtlee�ectsin the light-curves. Therefore,while we cannothope toobtain an accuratepictureofthedistribution ofany sur-facefeatureshere,theconsequencesofspotson thelightcurveare fairly clearin CV Boo (to the extentthatourhypothesis is true), and we m ake an e�ort in the fol-lowing to atleastrem ove som e ofthose distortionsandstudy theirin uenceon thegeom etricparam etersofthesystem ,which areofm oreim m ediate interest.In orderto perm itthe num ericaltreatm entofsurface

featuresin thiscase,we introduced m odi�cationsin theW D code to allow fora precise tracking ofthe spotpo-sition ata period di�erentthan the orbitalone. In thisschem e,wespecify thespotpropertiesatacertain Juliandate and,through the speci�ed intrinsic rotation rate,the code keeps track ofthe spot m otion,with its lon-gitude following the com ponent’s rotation,and its co-latitude,size,and e�ective tem perature rem aining oth-erwise constant. In view ofthe am biguities m entionedabove regarding the location ofspots in CV Boo,andourinability to telliftherem ighteven bem ultiplespotson one or both stars,we have taken our light-curve �tfrom x3.1 asourstarting pointand investigated thefol-lowing three sim ple cases separately: (a) a single spoton the prim ary;(b)a single spoton the secondary;and(c) one spot on each com ponent. M ore com plex con-�gurationsbecom e increasingly di�cultto study due toconvergenceproblem sin thesolutions,and itisnotclearthey arejusti�ed with the data available.The in uence ofspotson the radialvelocity curvesis

very sm allcom pared to our errors,so that those dataarenotofvery helpfulforstudying surfacefeatures.W euse only ourm ore extensive V -band photom etry in thestudy ofthese three cases,although the solutions werechecked usingNelson’sV and I lightcurves.Asthepho-tom etric coverage doesnotnecessarily overlap with theradialvelocity coverage,wehaveadopted forthespottedcasesthem assratioqobtained from a no-spot�tsim ilarto thatin x3.1 thatassum esasynchronousrotation (seebelow),and held it�xed.Forlack ofotherphysicalcon-straints,and given thatthe starsare quite sim ilarin alltheir properties,we assum ed that both com ponents ro-tate slightly super-synchronously,ata rate given by theratio between the orbitalperiod and the residualperiodfound in the previoussection,equalto 1.0114.Cases (a) and (b) converged quite rapidly to sim ilar

con�gurations,in which theco-latitude,size,and theef-fective tem peraturesofthe spotsare com parable,while

Fig.7.| D i�erence between the light curves with spots andthose without spots (shown in Figure 2),for the �rst cycle ofourobservations. The thin horizontal line shows the norm alizationlevelofthetheoreticallightcurves(phase0.25,�rstorbiting cycle).

their longitudes are such that the spots present alwaysthe sam e position relative to the center ofthe spottedcom ponentand theobserver.Thiscan beseen in Table5,wherethelongitudesofthespotsin cases(a)and (b)areseparatedbynearly180�.Anothersim ilarityisthatbothspotscoverthecom ponents’polarregions.Although at-tem pted,no solution could be obtained for\hot" spots(i.e.,with tem peraturefactorslargerthan unity;seebe-low).Solution (c)with onespoton each com ponentdid not

converge aseasily. W hen the param etersofboth spotswere left free to be adjusted,one ofthe spots (usuallythe one on the prim ary) tended to becom e very sm alland cold,with the tem perature factor Tfactor (ratio be-tween the spot tem perature and the photospheric tem -perature)becom ingsm allerthan allowedbytheNextG enatm ospheretablesweused.Thesolution wepresentwasachieved by �rstadjusting som e ofthe spotparam eterswhile holding others�xed,and then alternating and it-erating untilconvergence.The m axim um am plitude ofthe in uence ofthe spots

on the lightcurve is� 0.08m ag,and occursforthe one-spotsolutions,asshown in Figure 7. This�gure corre-spondsto the �rstorbitalcycleofourobservationsand,since the spotsfollow the com ponents’non-synchronousrotation, the dips change place at each orbiting cy-cle. The two-spot solution ofcase (c) gives a slightlysm allerpeak-to-peak am plitude(� 0.06 m ag)thatseem sm arginally largerthan indicated in Figure 6,suggestingthatperhapsa m orecom plex spotcon�guration m ay beneeded.W e report in Table 6 the m odelparam eters we ob-

tain for the solution with no spots and for cases (a),(b),and (c),togetherwith the radiiofthe com ponentsin term s ofthe orbitalseparation. For the reasons de-scribed above,thesolution withoutspotswasperform edby solving sim ultaneously threelightcurvesand 4 radialvelocity curves,whereasthe spotted �tsare based onlyon our V -band light curve. The m ain di�erence in theparam etervaluesisseen in theinclination angle,which isapproxim atelyonedegreehigherforthesolution withoutspots.O therparam eterssuch asthe secondary e�ectivetem peratureand thesizesofthecom ponentstend to dif-ferlessbetween the spotted and unspotted solutions.Figure 8 givesa representation ofthe spotcon�gura-

8 Torresetal.

tion resulting from case (c) with the com ponents’sizeand separation rendered to scale,and seen from the ob-server’s viewpoint at six di�erent orbitalphases. Thestars are welldetached from the corresponding Rochelobes, with �ll-out factors (M ochnacki 1984) that are0.7693and 0.7501fortheprim ary and secondary,respec-tively.W enoted abovethatour�tsyield polarspots,ashas often been found (also from Doppler im aging tech-niques)forotheractivebinariessuch astheRS CVn sys-tem s. There isconsiderable theoreticalsupportforthispreference for high-latitude surface features in rapidly-rotating active system s (see, e.g., Sch�ussler& Solanki1992;G ranzeretal.2000;I�sik etal.2007). A curiousresult from our �ts is that the spots happen to be po-sitioned so as to avoid eclipses,although the reality ofthiscon�guration isdi�cultto assess.Itisneverthelessan indication that the phenom enon responsible for theperiodicbehaviorofthe residualsin the unspotted solu-tion doesnotlead tostrongdiscontinuities,such asthoseresulting from the eclipses.Although the rm s residualis m arginally sm aller for

thesolution obtained in case(c),asindicated in Table6,this�taswellasthe othertwo spotted solutionsarevi-sually indistinguishable from the solution withoutspotsshown in Figure 2. The residuals for case (c) are dis-played in Figure 9 for our V -band light curve as wellasforthe V and I lightcurvesofNelson (2004b). Thepatternsclearly visible in these O � C diagram sare notvery di�erent from those in Figure 2, which m ay givethe im pression thatnotm uch progresshasbeen m ade.7

They certainly indicatethattherearestillfeaturesofthebrightnessvariation thatarenotcom pletely orcorrectlym odeled,possibly due to a m ore com plex spotcon�gu-ration than wehaveassum ed,oreven som ecom binationofspotsand m ulti-m odalpulsations.Problem sofan in-strum entalnature in the photom etry cannotentirely beruled outeither.However,whatisnotim m ediately obvi-ousto theeyeisthatno signi�cantperiodicitiesthatwecan detectrem ain in these residuals. Thisisillustratedin Figure 10,in which the top curve shows our La er-K inm an period study oftheK im pelO bservatoryV -bandresidualsfrom the no-spotsolution,and the lowercurveshows the sam e study for the residuals from case (c).Notethecom m on verticalaxisforboth setsofresiduals,indicating the im provem entin the overallvariance.

4. A BSO LU TE D IM EN SIO N S A N D PH Y SICA L PRO PERTIES

Exam ination ofTable 6 showsthatkey geom etric pa-ram eterssuch astherelativeradii(rA ;vol,rB ;vol)vary byas m uch as 3{4% between the three spotted solutions,with solution (c) generally giving interm ediate results.Asindicated earlier,thisisthe�tthatprovidesform allythe sm allestrm s residual,although the di�erence com -pared to the othertwo spotted solutionsism arginal.Inallfoursolutionsthe m ean lightratio outside ofeclipseaccounting for spots,(‘B =‘A )V ,is quite sim ilar to thespectroscopically determ ined valueof0:73� 0:04 (x2.3).From thee�ectivetem peraturesofCV Boo A and B theconvective turnover tim e for both stars is estim ated tobe � 25 days,following Hall(1994). The Rossby num -

7 N ote,however,thatthosepatternsarem ostobviousin N elson’sdata,which do not actually enter into the �nalsolution adoptedin x4.

Fig.8.| R epresentation ofthe com ponentsofCV Boo atdi�er-entorbitalphasesasindicated on theleft,shown to scalewith theirhigh-latitude spots as m odeled here. These spots resulting fromoursolution (c)are positioned in such a way that they practicallyavoid being eclipsed. This is an indication ofa rather sinusoidalbehaviorofthe disturbing phenom enon causing periodicvariationsin the residualsofthe unspotted solution (see text).

ber (ratio between the rotation period and the convec-tive turnover tim e) is then R 0 � 0:033, which placesboth com ponents in the regim e where starsusually dis-play signi�cant light variations due to spots (see Hall1994,Figure 6). O n the basisofthe above we adopt�t(c) with one spot on each com ponent as the best com -prom ise for CV Boo,but we reiterate that this m odelisstillprobably only a crude approxim ation to the truespot con�guration in the system , assum ing that spotsare the underlying reason for the periodic signalfoundin thelight-curveresiduals.Forcalculating the absolutedim ensionsofthe two starswe havechosen to use m oreconservativeuncertaintiesthan theform alerrorslisted inTable6,to accountforthespread am ong thethreespot-ted solutionsgiven theuncertaintiesin them odeling:wehavecom bined theinternalerrorsquadratically with halfofthe m axim um range in each param eter. The valuesadopted are i = 86:�24 � 0:�33,a = 4:748 � 0:019 R � ,

CV Boo 9

Fig.9.| O � C residuals from the solution with one spot oneach com ponent,based on the �t to our V -band light curve. Them odelused to com pute the residuals for the V and I light curvesof N elson (2004b) in the top panels is based on the sam e light-curve param eters as our V -band �t,except for the m agnitude atquadrature and the wavelength ofthe observations. The standarddeviation ofa single observation for each residualcurve is shownin the upper leftcorner ofeach panel.

Fig.10.| The variance versus trial period in days (followingLa er & K inm an 1965)forthe K im pelO bservatory V -band resid-uals of the no-spot solution (top curve, corresponding to the �tshown in Figure 2),and for the solution with two spots (bottomcurve;see text). The dotted linem arksthe m ostsigni�cantperiodfound forthe O � C ofthe solution with no spots,while the dashedline indicates the orbitalperiod.

q = 0:9378 � 0:0070, rA ;vol = 0:2658 � 0:0047, andrB ;vol= 0:2470� 0:0048,and arebased only on theK im -pelO bservatorym easurem ents.The�nalresultsarepre-sented in Table7,wheretheuncertaintieswereobtainedby propagating allobservationalerrorsin theusualway.Thestarsin CV Boodepartsom ewhatfrom thespher-

icalshape due to tidaland rotationaldistortions. Therelative di�erence between the polarradiusand the ra-dius toward the inner Lagrangian point is 5.5% for theprim ary and 4.8% forthesecondary.Thesystem isnev-ertheless welldetached: the sizes ofthe stars representfractionsof70% and 66% oftheirrespectivem ean Rochelobe sizes. The tem perature forthe secondary from thelight-curve solution is in excellent agreem ent with thespectroscopicvalue (x2.3).Included in Table 7 are the predicted projected rota-

tionalvelocities(vasyncsini)com puted with theadoptedrotation period for the stars (Prot = 0:83748 days =Porb=1:0114; see x3.1.1), as well as the synchronousvalues (vsyncsini), for reference. These m ay be com -pared with them easured vsinivaluesfrom spectroscopy(x2.3). The stellar radiiused forthese calculationsarethosepresented to theobserveratquadrature(which are2.7% and 2.4% largerthan thevolum eradii;seeTable6),sincethatisthephaseatwhich thespectroscopicobser-vations are concentrated. As a proxy for the radius at

quadratureweusethe averageofrpoint and rback.Finally,for com puting the absolute visualm agnitude

M V and distancewehaverelied on theapparentV m ag-nitude listed in the Tycho Catalog,and ignored extinc-tion.CV Boowasnotobserved by theHipparcosm ission(Perrym an etal.1997),so no direct parallax m easure-m entisavailable.

5. CO M PA R ISO N W ITH STELLA R EVO LU TIO N TH EO RY

In this section we com pare the absolute dim ensionsofCV Boo with current stellar evolution m odels fromthe Yonsei-Yale seriesby Yietal.(2001),incorporatingan updated prescription forconvectivecoreovershootingas described by Dem arqueetal.(2004). These m odelsadopta m ixing length param eterof�M L = 1:7432,cal-ibrated against the Sun. In Figure 11 we show evolu-tionary trackscom puted forthe exactm asseswe derivefor each star (see Table 7),for a heavy-elem ent abun-dance equalto thatofthe Sun (which isZ� = 0:01812in these m odels;dotted lines). The uncertainty in thelocation ofthe tracks that com es from our m ass errorsis indicated with the error bar in the lower left. Thetracks show excellent agreem ent with the observations,suggesting the com position isnearsolar.The m easuredtem perature di�erence between the com ponentsisquiteclose to what the m odels predict. A m arginally betterm atch is achieved with a slightly higher abundance ofZ = 0:01955 (corresponding to [Fe/H]= + 0:04,assum -ing no enhancem entofthe � elem ents),shown as solidlines in the �gure. The m odels indicate the prim ary isbeginning itsshellhydrogen-burning phase,and thesec-ondary isnearthe end ofitsm ain-sequence phase. Theage thatbest�tsboth com ponentsin thislogg{Te� di-agram is9:0� 1:8 G yr,and thecorresponding isochroneisshown asa dashed line.W e have also considered a second setofm odels,from

the series by Claret (2004). The physics in these cal-culations is sim ilar though notexactly the sam e as thepreviousones.Forexam ple,thesolarcom position in thiscase is taken to be Z� = 0:020,and the m ixing lengthparam eterthatbestreproducesthe observed propertiesofthe Sun is �M L = 1:68. The com parison with theobservations for CV Boo is shown in Figure 12. Al-though the Claret m odels m atch the m easured proper-ties very well,we �nd as with the Yonsei-Yale m odelsthata slightly higherm etallicity (Z = 0:0225,or[Fe/H]= + 0:05)providesan even better�t. Thisisshown bythe solid linesin Figure12.The age ofthe system fromthese calculationsis9.8 G yr,consistentwith the previ-ous estim ate. Experim ents changing the m ixing lengthparam eter show the sensitivity ofthe best-�t com posi-tion to �M L. In Figure 13 we com pare the observationswith trackscom puted for a lowervalue of�M L = 1:50,which has the e�ect ofyielding lower tem perature pre-dictions.Solar-m etallicity m odelsareindicated with thedotted lines.In thiscasewe�nd thatthebest-�tm etal-licity (Z = 0:0185,or[Fe/H]= � 0:03)is slightly lower

than solar(solid lines).The preceding com parisons m ay give the im pression

thattheobservationsforCV Boo arevery wellm atchedbythepredictionsfrom theory,thatstellarphysicsiswellunderstood,and thatthereforethereisnoreason forcon-cern.However,am orecarefulexam inationindicatesthatthisisnotnecessarilytrue.O fthethreebasicparam eters

10 Torresetal.

Fig.11.| A bsolute dim ensionsforCV Boo com pared with evo-lutionary m odelsfrom theseriesby Y iet al.(2001).Theerrorbarsfor logg are sm allerthan the size ofthe sym bols. M ass tracks forthe exact m asses we m easure are indicated with solid curves forthe best-�tting m etallicity ofZ = 0:01955 (where Z � = 0:01812forthese m odels). Solar m etallicity tracks are shown forreference(dotted curves). The isochrone producing the best sim ultaneousm atch to both com ponents isshown with the short-dash line,andcorresponds to an age of9.0 G yr. The long-dash lines representsm allsectionsofthetwo isochronescorresponding to them axim umand m inim um age allowed by the errors (9:0 � 1:8 G yr). The un-certainty in the location ofthe m ass tracks is indicated with theerror bar below the tracks for the prim ary,and is m uch sm allerthan the tem perature uncertainty.

typically determ ined in eclipsing binaries (M ,R,Te�),thetem peratureisusuallytheweakestsinceitoften relieson externalcalibrations.Figure14displaysthem easure-m entsforCV Booin adi�erentdiagram ,them ass-radiusplane,along with isochronesfrom the Yonsei-Yaleseriesforthesam etwo m etallicitiesdiscussed in Figure11.Nosinglem odelm atchesboth com ponentswithin theerrors,and thesecondary appearsnom inally olderthan thepri-m ary,thedi�erencein agebeing� 25% .Thisisthesam ephenom enon pointed out by Popper (1997) for severalothersystem sincluding FL Lyr,RT And,UV Psc,and� Cen.Anotherwayofinterpretingthisisthatthesecon-dariesin allthesebinariesaretoo largefortheirm asses,com pared to theory or com pared to the prim aries. ForCV Boo the o�set in the secondary radius is � 10% ,which represents a very signi�cant 5� deviation. Sim -ilarradiusdiscrepancieshavebeen described recently byothers(e.g.,Clausen etal.1999a;Torres& Ribas 2002;Ribas2003;L�opez-M orales& Ribas2005;Torres2007),although early indicationsgo asfarback asthe work ofHoxie (1973)and Lacy (1977). The prevailing explana-tion seem s to be that the enlarged radiiofthe secon-daries,which are typically wellunder a solarm ass,arecaused by strongm agnetic�eldsand/orspotscom m onlyassociated with chrom ospheric activity in these system s(see, e.g., M ullan & M acDonald 2001; Chabrieretal.2007,forthetheoreticalcontext).ThesignsofactivityinCV Booarefairly obvious(spottedness,X-rayem ission),and are no doubt associated with the rapid rotation of

Fig.12.| A bsolute dim ensionsforCV Boo com pared with evo-lutionary m odels from the series by Claret (2004) with a value ofthe m ixing-length param eter of�M L = 1:68. M ass tracks for theexact m asses we m easure and for solar com position (Z = 0:020,X = 0:70,in these m odels)are indicated with dotted curves. Thesolid curvesgiving a som ewhatbetter�tcorrespond to m odelswitha slightly higherm etallicity ofZ = 0:023.A n isochrone forlog age= 9.99 isshown forreference (dashed line).The uncertainty in thelocation ofthem asstracksisindicated with theerrorbarbelow thetracks for the prim ary,and is m uch sm aller than the tem peratureuncertainty.

Fig.13.| Sam e asFigure12,butfora m ixing-length param eterof �M L = 1:50. M ass tracks for the m easured m asses and forsolarcom position (Z = 0:020,X = 0:70)areindicated with dottedcurves. The som ewhat better-�tting solid curves correspond toa slightly lower m etallicity of Z = 0:0185 in this case, showingthe in uence of the � M L param eter in the determ ination of thecom position ofCV Boo.

the com ponents.

CV Boo 11

Fig.14.| M ass-radius diagram for CV Boo,showing the m ea-surem ents against isochrones from the Yonsei-Yale series for thesam e two m etallicities displayed in Figure 11. A ges are indicatedalong the top.

6. CO M PA R ISO N W ITH TID A L TH EO RY

The predictions oftidaltheory were com pared withthe observations by com puting the tim e of circular-ization and synchronization for CV Boo using the ra-diative dam ping form alism of Zahn (1977) and Zahn(1989), as well as the hydrodynam ical m echanism ofTassoul& Tassoul(1997),and references therein. Theprocedure followsclosely thatdescribed by Claretetal.(1995) and Claret& Cunha (1997). Both theoriespredict that synchronization and circularization areachievedveryquicklyin thissystem byvirtueoftheshortorbitalperiod,atanageofm erely157M yr(logt= 8:197,orlessthan 2% oftheevolutionaryage).Thefactthatwem easurethe orbitto be circularisthereforenotsurpris-ing.O n theotherhand,theevidencefrom ourphotom et-ricobservations(x3.1.1)suggesting therotation m ay beslightly super-synchronousforatleastoneofthecom po-nentsism oreinteresting,asitisnotpredicted by theory.G iven the nature ofthe system ,an activity-related ex-planation to thisdiscrepancy iscertainly possible.M oreprecisem easurem entsofthe projected rotationalveloci-tiesvsiniforthe com ponentswould be very helpful.

7. D ISCU SSIO N A N D CO N CLU SIO N S

Despite the system ’sintrinsic variability,the absolutedim ensionsforthecom ponentsofCV Boohavenow beenestablished quite precisely. The relative errorsare bet-ter than 1.3% in the m asses and 2% in the radii. Theobjectcan now becounted am ong thegroup ofeclipsingbinaries with well-known param eters. Under di�erentcircum stancesthe large num berand high quality ofthephotom etric observations we have collected m ight haveperm itted a m ore detailed study ofthe lim b darkeninglaws and a com parison with theoretically predicted co-e�cients,but this possibility was thwarted here by theintrinsicvariability.Thisphenom enon isnotitselfwith-out interest. If interpreted as due to the presence ofspots,aswe have done here,itim pliesthatatleastoneofthe starsisrotating about1% m ore rapidly than the

synchronous rate, a result that was unexpected for aclose but welldetached system such as this. W e con-clude that our currentunderstanding oftidalevolutionis stillincom plete,or that other processes are at playin this system that theory does not account for. O neinteresting possibility isdi�erentialrotation. The inter-pretation ofm easurem ents ofthe rotation period ofastarm ade by photom etric m eans,aswe have im plicitlydonehere,usually relieson theassum ption ofsolid-bodyrotation.M oreoften than not,spotsarelocated atinter-m ediatelatitudesratherthan on theequator,orathighlatitudes in m ore active stars,and di�erentialrotationis such that the stellar surface revolves m ore slowly athigher latitudes,at least in the Sun. This willtend tobiasphotom etric rotation m easurem entstowardslongerperiods,ifdi�erentialrotation is signi�cantenough. InCV Boo we see the opposite:the period isshorter thantheequatorialrate,assum ingthatsynchronization holds.Thusdi�erentialrotation can only explain thesignalwehavedetected ifitis\anti-solar",with thepolarregionsrotatingm orerapidly.A handfulofstarsdoindeed showevidence of weak anti-solar di�erential rotation (e.g.,IL Hya,HD 31933,� G em ,UZ Lib;W eberetal.2003;Strassm eieretal. 2003; K ~ov�arietal. 2007; Vida etal.2007).They allhappen to be very active(som eofthemwith high-latitude spots,asin CV Boo),although theytend to be giants or subgiants rather than dwarfs. Itis thought that this phenom enon m ay result from fastm eridional ows(see,e.g.,K itchatinov & R�udiger2004).Further progress in understanding the rotation of theCV Boo com ponentscould be m adewith additionaldif-ferentialphotom etric observationsin severalpassbands,along with sim ultaneoushigh-resolution,high signal-to-noiseratio spectroscopy overa fullorbitalcycle.Another signi�cant discrepancy we �nd with theory

is in the radius ofthe secondary,which appears to besom e10% toolargecom pared with predictionsfrom stel-larevolution m odels. This di�erence is in the sam e di-rection as seen for a num ber ofother low-m ass eclips-ing binaries,as m entioned in x5. In those cases one ofthe explanationsm ostoften proposed isthatthe strongm agnetic �elds associated with activity (which is com -m on in rapidly-rotating K and M dwarfs in close bina-ries)tend to inhibit convective m otions,and the struc-tureofthestaradjustsby increasing itssizeto allow thesurface to radiate the sam e am ount ofenergy. At thesam e tim e,the e�ective tem perature tends to decreasein order to preserve the totallum inosity. Spot cover-age can produce sim ilar e�ects. Theoreticaland obser-vationalevidence forthe conservation ofthe lum inosityin these system s has been presented by Delfosseetal.(2000), M ullan & M acDonald (2001), Torres& Ribas(2002),Ribas(2006),Torresetal.(2006),Chabrieretal.(2007),and others(seealso M oralesetal.2008).W edonot see any obvious discrepancy in the tem perature ofCV Boo B com pared to m odels,although ouruncertain-tiesarelargeenough thatthe e�ectm ay be m asked.If we restrict ourselves to well studied double-lined

eclipsing binaries in which the m ass and radius deter-m inationsare the m ostreliable,deviationsfrom theorysuch asthose described abovehaveusually been seen instars that are considerably less m assive than the Sun,which have deep convective envelopes.However,the re-cent study by Torresetal.(2006) pointed out that the

12 Torresetal.

problem is not con�ned to the lower m ass stars, butextends to active objects approaching 1 M � , such asV1061 Cyg Ab, with M = 0:93 M � . CV Boo B hasan even largerm assof0.968 M � ,and also appearsto beoversized. Sim ilarly with the virtually identicalactivestar FL Lyr B (M = 0:960 M � ). The convective en-velopesoftheseobjectsareconsiderably thinnerthan inK and M dwarfsand representonly a few percentofthetotalm ass,yetthey appearsu�cientform agnetic�eldsto take hold and alterthe globalpropertiesofthe star,ifthatisthecauseofthediscrepancies.Theseexam plesshow onceagain thatourunderstanding ofstellarevolu-tion theory isincom plete,even forstarsnearthem assofthe Sun.

ThespectroscopicobservationsofCV Boo used in thispaperwereobtained with thegeneroushelp ofP.Berlind,M .Calkins,R.J.Davis,E.Horine,D.W .Latham ,J.Pe-ters,and R.P.Stefanik.R.J.Davisisalso thanked for

m aintainingtheCfA echelledatabase.W earegratefulaswellto J.M .K reinerforproviding unpublished tim esofeclipseforCV Boo,to A.Claret,forcalculating speci�cm odelsforthe starsstudied here,and to the referee forhelpfulcom m ents.G T acknowledgespartialsupportforthis work from NSF grant AST-0708229. LPRV grate-fully acknowledges partial support from the BrazilianagenciesCNPq,FAPEM IG and CAPES.Sum m er 2004ArkansasREU studentS.L.W altersisthanked byCHSLforherprelim inary analysisoftheabsolutepropertiesofthis binary star (W alters& Lacy 2004). This researchhas m ade use of the SIM BAD database, operated atCDS,Strasbourg,France,ofNASA’sAstrophysicsDataSystem AbstractService,and ofdata productsfrom theTwo M icron AllSky Survey,which is a jointprojectofthe University ofM assachusetts and the Infrared Pro-cessingand AnalysisCenter/CaliforniaInstituteofTech-nology,funded by NASA and the NSF.

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14 Torresetal.

TA BLE 1

D ifferential V -band

measurements of C V Boo.

H JD � 2;400;000 Phase �V

52250.99816 0.35467 + 0.45252250.99907 0.35575 + 0.41652251.00000 0.35685 + 0.44552251.00091 0.35792 + 0.50852251.00182 0.35900 + 0.433

N ote. | Table 1 is available in itsentirety in the electronic edition oftheA stronom ical Journal. A portion isshown here for guidance regarding itsform and contents.

TA BLE 2

Photometric indices and inferred mean effective

temperature of C V Boo.

Photom etric Index Value Te� (K ) R ef.

Johnson V ................. 10.75 � 0.10 � � � 1Johnson B � V ............. 0.73 � 0.11 5417 � 350 1Tycho-2 B T � VT .......... 0.82 � 0.13 5448 � 329 1

Johnson/2M A SS V � J ..... 1.18 � 0.10 5693 � 103 1,2

Johnson/2M A SS V � H ..... 1.47 � 0.10 5666 � 155 1,2Johnson/2M A SS V � K s.... 1.55 � 0.10 5692 � 157 1,2

Tycho-2/2M A SS VT � K s .. 1.629 � 0.081 5679 � 129 1,2Sloan g� r ................. 0.473 � 0.002 5760 � 100 3

N ote.| R eferences:(1)H �g etal.(2000);(2)Cutrietal.(2003);(3) Sloan D igitalSky Survey data.

CV Boo 15

TA BLE 3

N ew radial velocity measurements of C V Boo.

H JD

�2 440 000Phase

StarAkm =s

(O�C )A

Star Bkm =s

(O�C )B

H JD

�2 440 000Phase

Star Akm =s

(O�C )A

Star Bkm =s

(O�C )B

48408.8881 0.1792 � 127.53 � 2.12 + 144.05 + 13.68 52805.7304 0.2974 � 133.61 � 1.12 + 134.43 � 3.5348428.7637 0.6453 + 113.28 + 6.33 � 121.89 � 4.64 52807.6959 0.6180 + 79.11 � 12.00 � 96.25 + 4.1248435.7679 0.9148 + 70.53 + 1.73 � 80.45 � 4.19 52808.6829 0.7833 + 141.29 + 8.91 � 158.46 � 14.0052336.9667 0.8530 + 101.70 � 6.29 � 121.62 � 3.39 52828.6620 0.3715 � 116.79 � 16.14 + 112.11 + 8.1352362.8251 0.3826 � 96.29 � 2.48 + 97.00 + 0.31 52830.7704 0.8608 + 104.81 + 0.97 � 115.59 � 1.8152391.8515 0.6526 + 115.23 + 4.57 � 126.58 � 5.38 52894.6202 0.2449 � 148.22 � 9.80 + 145.71 + 1.3752395.7780 0.2884 � 126.96 + 7.58 + 131.35 � 8.81 53011.0543 0.7124 + 138.93 + 7.38 � 147.38 � 3.8452419.8759 0.7395 + 133.04 � 1.97 � 156.34 � 9.07 53017.0656 0.8097 + 130.12 + 4.16 � 144.86 � 7.3152420.8446 0.8832 + 80.62 � 9.96 � 102.38 � 2.82 53036.0482 0.2214 � 130.15 + 6.19 + 149.26 + 7.1452424.9263 0.7022 + 119.97 � 9.29 � 137.99 + 3.09 53045.0203 0.8143 + 110.06 � 14.40 � 137.98 � 2.0352481.7239 0.7601 + 131.60 � 3.44 � 147.89 � 0.59 53047.9883 0.3184 � 133.81 � 7.71 + 138.58 + 7.4652537.6015 0.7319 + 145.12 + 10.69 � 153.73 � 7.10 53072.0214 0.6931 + 120.32 � 6.41 � 138.36 + 0.0252657.0283 0.7327 + 136.61 + 2.10 � 143.34 + 3.38 53102.9332 0.1890 � 128.08 + 0.67 + 140.61 + 6.6552681.9996 0.2150 � 139.26 � 3.99 + 149.89 + 8.92 53124.9836 0.2227 � 148.22 � 11.69 + 142.32 � 0.0152687.0564 0.1853 � 126.70 + 0.86 + 125.61 � 7.06 53125.8812 0.2825 � 135.40 + 0.26 + 136.82 � 4.5452688.0019 0.3016 � 136.00 � 4.61 + 134.21 � 2.57 53131.8024 0.2733 � 148.17 � 11.15 + 142.70 � 0.1252690.9568 0.7903 + 137.79 + 6.78 � 152.45 � 9.46 53133.8280 0.6648 + 113.70 � 2.64 � 125.78 + 1.4952712.0278 0.6677 + 117.71 + 0.12 � 132.03 � 3.42 53134.8008 0.8134 + 123.52 � 1.25 � 134.88 + 1.4052718.9482 0.8383 + 103.39 � 11.69 � 126.31 � 0.46 53155.9103 0.7362 + 135.62 + 0.82 � 159.54 � 12.5052720.9903 0.2493 � 142.32 � 3.84 + 135.44 � 8.97 53156.8078 0.7959 + 124.95 � 4.81 � 145.87 � 4.2352721.8866 0.3075 � 131.97 � 2.26 + 135.59 + 0.61 53157.6883 0.8354 + 118.15 + 1.81 � 128.87 � 1.6652743.0017 0.2370 � 138.96 � 0.91 + 141.72 � 2.22 53158.8651 0.2248 � 132.66 + 4.16 + 136.23 � 6.4152743.8681 0.2599 � 138.48 � 0.26 + 136.36 � 7.76 53159.7571 0.2779 � 137.36 � 0.97 + 141.77 � 0.3852745.9296 0.6938 + 127.96 + 1.02 � 152.37 � 13.76 53182.6826 0.3449 � 107.94 + 7.08 + 117.19 � 2.1052748.8870 0.1854 � 124.72 + 2.87 + 133.76 + 1.05 53183.7214 0.5713 + 55.26 � 2.71 � 66.52 + 2.4052751.9548 0.8074 + 122.44 � 4.20 � 141.35 � 3.07 53183.8170 0.6842 + 118.57 � 5.32 � 136.77 � 1.4452752.8104 0.8176 + 118.39 � 4.95 � 137.92 � 3.18 53184.7284 0.7602 + 139.79 + 4.76 � 147.31 � 0.0152769.7356 0.8003 + 123.65 � 5.00 � 141.11 � 0.66 53189.7763 0.7200 + 134.61 + 1.70 � 150.90 � 5.9052771.8062 0.2449 � 139.56 � 1.14 + 137.19 � 7.15 53190.6836 0.7912 + 129.10 � 1.72 � 146.12 � 3.3452773.8521 0.6604 + 103.84 � 10.53 � 124.10 + 1.06 53192.6964 0.1676 � 123.51 � 2.64 + 140.40 + 14.9252800.6851 0.3407 � 116.42 + 0.57 + 128.81 + 7.42 53217.6764 0.6602 + 113.85 � 0.42 � 123.95 + 1.1152802.6741 0.6890 + 120.03 � 5.45 � 138.94 � 1.90 53452.8782 0.3505 � 114.77 � 2.51 + 115.01 � 1.3452804.7941 0.1920 � 127.64 + 2.03 + 145.25 + 10.30 53485.8782 0.3118 � 121.62 + 6.74 + 125.58 � 7.96

N ote. | T he O �C residuals correspond to the solution described in x 3.1.

TA BLE 4

R adial velocities for C V Boo from Popper (2000).

H JD

�2 440 000Phase

Star Akm =s

(O�C )A

Star Bkm =s

(O�C )B

H JD

�2 440 000Phase

Star Akm =s

(O�C )A

Star Bkm =s

(O�C )B

47198.0795 0.6418 + 100.59 � 4.51 � 112.85 + 2.43 49117.9386 0.3175 � 126.53 � 0.09 + 133.81 + 2.3247254.9417 0.7760 + 134.29 + 0.77 � 144.43 + 1.25 49202.6852 0.3733 � 102.13 � 2.52 + 101.37 � 1.4947397.6592 0.2749 � 137.69 � 0.88 + 142.48 � 0.12 49204.7213 0.7772 + 133.70 + 0.35 � 145.93 � 0.4347695.7285 0.1895 � 129.31 � 0.39 + 136.69 + 2.54 49204.7416 0.8012 + 126.62 � 1.79 � 140.09 + 0.1047696.7035 0.3407 � 112.58 + 4.42 + 121.75 + 0.35 49204.7623 0.8256 + 117.42 � 2.96 � 130.86 + 0.6948080.7574 0.7727 + 134.39 + 0.45 � 144.60 + 1.53 49496.9018 0.7392 + 135.56 + 0.57 � 147.61 � 0.3648081.6981 0.8833 + 87.15 � 3.33 � 98.55 + 0.90 49496.9163 0.7563 + 134.71 � 0.49 � 146.30 + 1.1848312.0038 0.7931 + 128.21 � 2.20 � 139.74 + 2.60 49583.6596 0.1695 � 126.79 � 5.13 + 126.42 + 0.0848344.9980 0.7476 + 131.13 � 4.16 � 144.15 + 3.42 49583.6807 0.1944 � 128.99 + 1.40 + 132.75 � 2.9848345.8895 0.8001 + 128.12 � 0.57 � 138.79 + 1.70 49907.6918 0.7371 + 136.77 + 1.91 � 148.30 � 1.1948345.9531 0.8752 + 96.80 + 1.25 � 105.62 � 0.74 49907.7182 0.7683 + 139.29 + 4.87 � 148.07 � 1.4348819.6895 0.1906 � 137.81 � 8.56 + 137.28 + 2.78 49907.7666 0.8254 + 124.72 + 4.28 � 135.86 � 4.2448819.7122 0.2174 � 140.25 � 4.55 + 140.79 � 0.64 49907.7881 0.8508 + 108.25 � 0.84 � 121.30 � 1.8848819.7362 0.2457 � 143.18 � 4.74 + 141.52 � 2.85 50176.9827 0.6746 + 122.91 + 2.53 � 131.47 + 0.1148820.6789 0.3587 � 111.27 � 3.34 + 107.70 � 4.03 50177.0097 0.7065 + 133.62 + 3.34 � 140.19 + 1.9948820.7049 0.3894 � 92.10 � 2.69 + 92.66 + 0.66 50177.9325 0.7960 + 129.82 + 0.09 � 138.96 + 2.6548822.7202 0.7688 + 135.49 + 1.12 � 149.18 � 2.59 50177.9511 0.8179 + 124.78 + 1.57 � 131.51 + 3.0948822.8020 0.8654 + 97.14 � 4.17 � 112.48 � 1.42 50177.9691 0.8392 + 112.79 � 1.87 � 120.04 + 5.3749116.9777 0.1830 � 131.43 � 4.67 + 132.23 + 0.41 50177.9891 0.8628 + 103.32 + 0.58 � 113.06 � 0.4649117.8084 0.1638 � 121.42 � 2.21 + 122.85 � 0.85 50178.0072 0.8842 + 88.65 � 1.30 � 99.35 � 0.4649117.8249 0.1832 � 133.33 � 6.49 + 135.33 + 3.42 50178.0259 0.9063 + 77.68 + 0.27 � 84.92 � 2.0649117.8549 0.2187 � 137.56 � 1.65 + 142.79 + 1.14 50616.7131 0.8409 + 113.60 � 0.27 � 121.75 + 2.8049117.8748 0.2421 � 140.97 � 2.64 + 145.24 + 0.99

N ote. | Sm allcorrections to the published values have been applied as described in x2.3. T he O �C residuals correspond to the

solution described in x3.1.

16 Torresetal.

TA BLE 5

Spot parameters for C V Boo.

Co-latitude Longitude R adius

Case Com ponent (deg) (deg) (deg) Tfactor

a pri 8:062 � 5:47 44:50 0.82� 59 � 90 � 50 � 5

b sec 7:926 170:212 47:76 0.775� 52 � 10 � 69 � 19

c pri 4:589 13:058 35:835 0.5940� 63 � 43 � 46 � 55

c sec 4:665 151:43 55 0.881� 43 � 38 � 1 � 11

N ote. | T he spot co-latitude is m easured from the pole visible to

the observer, and the longitude is m easured from the line joining the

com ponents’centers and increasing in the direction of orbital m otion.

T he radius is m easured as seen from the center ofeach com ponent,and

the tem perature factor is relative to the unspotted photosphere. T he

uncertaintieslisted are in unitsofthe last decim alplace and correspond

to the internalerrors from the least-squares m ethod.

TA BLE 6

Light-curve solutions for C V Boo based on our V -band photometry.

Param eter N o spots Case (a) Case (b) Case (c) Param eter N o spots Case (a) Case (b) Case (c)

i(�)........... 87.651 86.891 86.650 86.237 rA ;pole.......... 0.26533 0.26646 0.25798 0.26028� 42 � 34 � 33 � 32 � 67 � 49 � 47 � 46

A ............ 4.6752 4.6591 4.7844 4.7495 rA ;point......... 0.28105 0.28268 0.27188 0.27478� 76 � 48 � 49 � 46 � 90 � 67 � 61 � 62

B ............ 4.9014 4.9614 4.8186 4.8707 rA ;side .......... 0.27032 0.27167 0.26254 0.26501� 78 � 50 � 47 � 43 � 73 � 53 � 51 � 50

Te� ;B (K )..... 5632.8 5628.1 5656.1 5672.6 rA ;back ......... 0.27726 0.27877 0.26870 0.27142� 1:6 � 1:4 � 2:4 � 4:3 � 82 � 62 � 58 � 57

a (R � )........ 4.757 (4.748) (4.748) (4.748) rA ;vol........... 0.27189 0.27252 0.26327 0.26577

� 12 (�xed) (�xed) (�xed) � 74 � 55 � 51 � 51

(km s�1 ).... � 15:877 (� 15:889) (� 15:889) (� 15:889) rB ;pole.......... 0.24054 0.23699 0.24573 0.24247

� 31 (�xed) (�xed) (�xed) � 98 � 86 � 84 � 81

q � M B =M A ... 0.9376 (0.9378) (0.9378) (0.9378) rB ;point......... 0.25177 0.24757 0.25825 0.25423

� 24 (�xed) (�xed) (�xed) � 117 � 101 � 101 � 97

(‘B =‘A )V ...... 0.741 0.746 0.769 0.778 rB ;side .......... 0.24410 0.24042 0.24972 0.24624� 10 � 4 � 21 � 14 � 104 � 91 � 90 � 86

(‘B =‘A )V ;0:25 .. 0.734 0.699 0.829 0.802 rB ;back ......... 0.24935 0.24535 0.25548 0.25168

�V (m ag)...... 0.0196 0.0148 0.0147 0.0146 � 111 � 94 � 96 � 93

x�bolo;A

........ 0.428 0.429 0.428 0.428 rB ;vol........... 0.24483 0.24107 0.25049 0.24697

x�bolo;B

........ 0.437 0.437 0.435 0.434 � 105 � 91 � 90 � 87

x�V ;A

.......... 0.715 0.715 0.715 0.715 ��A.............. 0.378 0.378 0.378 0.378

x�V ;B

.......... 0.724 0.724 0.722 0.721 ��B.............. 0.390 0.390 0.388 0.386

N ote. | In allspotted solutions both com ponentswere assum ed to rotate at a rate 1.0114 tim es faster than the orbitalm otion (see x3.1.1).

Forthe no-spot solution the rotation is assum ed to be synchronous. T he linearlim b-darkening coe� cients(x)as wellas the gravity-brightening

exponents(�) are m arked w ith an asterisk to indicate that they were changed dynam ically during the iterations as Teff and logg changed. T he

gravity-brightening exponents varied over the m utually illum inated stellar surfaces follow ing A lencar & Vaz (1997) and A lencar et al.(1999),

and the valuespresented here are forthe non-illum inated hem ispheres.Teff;A was held � xed at 5760K .T he quantity (‘B =‘A )V ;0:25 corresponds

to the V -band light ratio at the � rst quadrature w ithout considering the e� ect ofspots,and (‘B =‘A )V is the m ean light ratio outside ofeclipse

accounting for the spots and proxim ity e� ects. T he uncertaintiesgiven on the left-hand side ofthe table (in units ofthe last decim alplace)are

the form alinternalerrors ofthe m inim ization procedure,w hile the ones on the right for the com ponent radiiaccount for the uncertainties of

the gravitationalpseudo-potentials as wellas the m ass ratio. T he quantities rA ;vol and rB ;vol represent the \volum e radius" for each star,i.e.,

the radius ofa sphere w ith the sam e volum e as the distorted stars.

CV Boo 17

TA BLE 7

Physical parameters of C V Boo.

Param eter Prim ary Secondary

A bsolute dim ensions

M ass(M � )................. 1.032 � 0.013 0.968 � 0.012

R adius(R � )............... 1.262 � 0.023 1.173 � 0.023logg (cgs).................. 4.249 � 0.016 4.285 � 0.017

M easured vsini(km s�1 )... 73 � 10 67 � 10

vasync sini(km s�1 )........ 78.5 � 1.1 72.7 � 1.1

vsync sini(km s�1 )......... 77.6 � 1.1 71.9 � 1.1

R adiative and other properties

Te� (K ).................... 5760 � 150 5670 � 150

logL=L� ................... 0.197 � 0.048 0.107 � 0.049

M bol (m ag)................ 4.24 � 0.12 4.46 � 0.12M V (m ag).................. 4.32 � 0.12 4.57 � 0.13

LB =LA ..................... 0.81 � 0.13

D istance (pc)............... 259 � 16

N ote. | M V and M bol were com puted using bolom etric corrections

from Flower (1996) along w ith M�bol

= 4:732. T he predicted asyn-

chronous projected rotational velocities vasync sin i correspond to the

valuesassum ing the rotationalperiod isPorb =1:0114 forboth stars (see

x3.2),w hile the vsync sin ivalues give the resultifProt = Porb . In both

cases we use the radius ofthe stars at quadrature.