Accurate fundamental parameters for lower main-sequence ...

Mon. Not. R. Astron. Soc. 373, 13–44 (2006) doi:10.1111/j.1365-2966.2006.10999.x

Accurate fundamental parameters for lower main-sequence stars

Luca Casagrande,1� Laura Portinari1� and Chris Flynn1,2�1University of Turku – Tuorla Observatory, Vaisalantie 20, FI-21500 Piikkio, Finland2Mount Stromlo Observatory, Cotter Road, Weston, ACT, Australia

Accepted 2006 August 30. Received 2006 August 14; in original form 2006 May 25

ABSTRACTWe derive an empirical effective temperature and bolometric luminosity calibration for G andK dwarfs, by applying our own implementation of the Infrared Flux Method to multibandphotometry. Our study is based on 104 stars for which we have excellent BV(RI )C JHKS

photometry, excellent parallaxes and good metallicities.Colours computed from the most recent synthetic libraries (ATLAS9 and MARCS) are

found to be in good agreement with the empirical colours in the optical bands, but somediscrepancies still remain in the infrared. Synthetic and empirical bolometric corrections alsoshow fair agreement.

A careful comparison to temperatures, luminosities and angular diameters obtained withother methods in the literature shows that systematic effects still exist in the calibrations at thelevel of a few per cent. Our Infrared Flux Method temperature scale is 100-K hotter than recentanalogous determinations in the literature, but is in agreement with spectroscopically calibratedtemperature scales and fits well the colours of the Sun. Our angular diameters are typically3 per cent smaller when compared to other (indirect) determinations of angular diameterfor such stars, but are consistent with the limb-darkening corrected predictions of the latest3D model atmospheres and also with the results of asteroseismology.

Very tight empirical relations are derived for bolometric luminosity, effective temperatureand angular diameter from photometric indices.

We find that much of the discrepancy with other temperature scales and the uncertaintiesin the infrared synthetic colours arise from the uncertainties in the use of Vega as the fluxcalibrator. Angular diameter measurements for a well-chosen set of G and K dwarfs would goa long way to addressing this problem.

Key words: techniques: photometric – stars: atmospheres – stars: fundamental parameters –Hertzsprung–Russell (HR) diagram – stars: late-type – infrared: stars.

1 I N T RO D U C T I O N

Temperatures, luminosities and radii are amongst the basic phys-ical data against which models of stellar structure and evolutionare tested. In this paper, we address one particular area of theHertzsprung–Russell (HR) diagram, that of the G and K dwarfs –focusing particularly on stars for which the effects of stellar evolu-tion have been negligible or nearly negligible during their lifetimes.Somewhat higher mass stars than those considered here have beenextensively studied historically because their luminosities may beused to infer stellar ages. Lower mass stars of late-G to K spectraltypes have been neglected to some extent, probably because therehave been few secondary benefits in getting stellar models right for

�E-mail: [email protected] (LC); [email protected] (LP); [email protected] (CF)

these stars, and the lack of good parallax, diameter and other data.This situation is rapidly changing: first, Hipparcos has providedthe requisite parallax data; secondly, interferometric techniques aremaking the measurement of diameters for such small stars likely tobe routine within a few years; thirdly, we have our own motivationin the form of an ongoing project to follow the chemical historyof the Milky Way from lower mass stars, for which we can inferindirectly the amount of their constituent helium via stellar lumi-nosity (Jimenez et al. 2003). In order to achieve our goals, we needaccurate and homogeneous effective temperatures and luminositiesfor our G+K dwarfs.

The effective temperature of a stellar surface is a measure of thetotal energy, integrated over all wavelengths, radiated from a unit ofsurface area. Since its value is fixed by the luminosity and radius,it is readily calculated for theoretical stellar models, and as oneof the coordinates of the physical HR diagram, it plays a central

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14 L. Casagrande, L. Portinari and C. Flynn

role in discussions of stellar evolution. Most observations, however,provide spectroscopic or photometric indicators of temperature thatare only indirectly related to the effective temperature.

The effective temperature of lower main-sequence stars is noteasy to determine and different measurement techniques are still farfrom satisfactory concordance (e.g. Mishenina & Kovtyukh 2001;Kovtyukh et al. 2003). At present, spectroscopic temperature de-terminations return values that are some 100-K hotter than most ofthe other techniques. Even in a restricted and thoroughly studiedregion like that of the solar analogues, effective temperature deter-minations for the same star still differ significantly, by as much as150 K (Soubiran & Triaud 2004). Models predict values that aresome 100-K hotter than those measured (Lebreton et al. 1999, andreferences therein).

In this work, we apply the Infrared Flux Method (IRFM) to multi-band BV(RI )C JHKS photometry of a carefully selected sample ofG and K dwarf stars. We compare observed with synthetic broad-band colours computed from up-to-date (1D) model atmospheres.Such models are then used to estimate the missing flux needed torecover bolometric luminosities from our photometry. Other thanthe high quality of the observational data, the strength of this workrelies on using very few basic assumptions: these are the adoptedVega absolute calibration and zero-points. This also makes clearerthe evaluation of possible errors and/or biases in the results. Boththe absolute calibration and the zero-points are expected to be wellknown and the latest generations of model atmospheres producerealistic fluxes for a wide range of temperatures, gravities and abun-dances (Bessell 2005) so that the adopted model and calibration areat the best level currently available.

The paper is organized as follows. In Section 2, we describe oursample; we compare different libraries of model atmospheres withobservational data in Section 3 and in Section 4 we present ourimplementation of the IRFM along with the resulting temperaturescale. In Section 5, we test our scale with empirical data for the Sunand solar analogues and with recent interferometric measurementsof dwarf stars. The comparison with other temperature determina-tions is done in Section 6, and in Sections 7 and 8 we give bolometriccalibrations and useful tight relations between angular diameters andphotometric indices. We conclude in Section 9.

2 S A M P L E S E L E C T I O N A N D O B S E RVAT I O N S

The bulk of the targets has been selected from the sample of nearbystars in the Northern hemisphere provided by Gray et al. (2003).Our initial sample consisted of 186 G and K dwarfs with Hipparcosparallaxes for which the relative error is less than 14 per cent. Accu-rate BV(RI )C broad-band photometry has been obtained for the bulkof the stars at La Palma, with additional photometry adopted from

Figure 1. Distribution of log (g) and [Fe/H] for the 104 dwarf stars in the basic sample, collected from the literature. As noted in Sections 3.1 and 4.4.2,adoption of an average log (g) of 4.5 for all the stars is sufficiently accurate for our purposes. All stars but one (HD 108564) with [Fe/H] < − 1 have directα-element measurements.

Bessell (1990a), Reid et al. (2001) and Percival, Salaris & Kilkenny(2003). All our stars have accurate spectroscopic metallicities andfor most of them also α-element abundances. Our sample is neithervolume nor magnitude limited, but it gives a good coverage of theproperties of local G and K dwarfs from low to high metallicity(Fig. 1). The sample has been cleaned of a number of contami-nants, reducing it to a final sample of 104 stars, as described in thefollowing sections.

2.1 Removing double/multiple and variable stars

Particular attention has been paid to removing unresolved dou-ble/multiple stars. These stars primarily affect spectroscopic andphotometric measurements, making the system appear brighter andredder. The Hipparcos catalogue was used to make a prior exclusionof ‘certain’, ‘possible’ and ‘suspected’ (i.e. good, fair, poor, uncer-tain and suspected) multiple systems, on the basis of the quality ofthe MultFlag field in the Hipparcos catalogue.

Some unrecognized double/multiple stars almost certainly remainin the sample, since a few stars are found in a ‘binary main se-quence’ above the bulk of the main sequence, as Kotoneva, Flynn& Jimenez (2002) suspected in such data but were unable to prove.A new technique for identifying binaries/multiples has been intro-duced by Wielen et al. (1999), termed as the �µ method, whichindicates the presence of multiple unresolved systems by using thedifference between the near-instantaneous direction of the propermotion of the star(s) as measured by the Hipparcos satellite, and thedirection of the ground-based proper motion measured over muchlonger time-scales. Most of the suspect stars in the Kotoneva et al.(2002) sample turned out to be �µ multiples, and we have excludedthose stars which are likely �µ multiples also from the present sam-ple (�µ classifications were kindly made for our sample stars byChristian Dettbarn in Heidelberg). This reduced the sample to 134stars.

We have retained widely separated binary systems in the samplewhere components could be studied separately. These were retainedby making a prior cross-checking of our sample with the catalogueof wide-binary and multiple systems of Poveda et al. (1994) andthen verifying the separation of such systems by inspecting our ownor SIMBAD images. 13 such stars have been retained.

Even if dwarfs stars are not expected to show signs of strong vari-ability, the existence of ∼100 or more Hipparcos measurements perstar spread over several years, together with the excellent temporalstability of the magnitude scales, allows the detection of variabilityat the level of few hundredths of a magnitude (Brown et al. 1997)in Hipparcos targets. We use the Hipparcos classification to re-move so-called duplicity-induced variables, periodic variables, un-solved variables and possible microvariables. This removed a total of

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Accurate fundamental parameters for GK dwarfs 15

26 stars (however, eight of these were already removed as non-singlestars).

The final number of stars satisfying the above requirements was129.

2.2 Broad-band photometric observations

To recover accurate bolometric fluxes and temperatures for the stars,we have obtained accurate and homogeneous Johnson–CousinsBV(RI )C and JHKS photometry for all the 186 stars in our initialsample.

2.2.1 Johnson–Cousins photometry

For most of the stars in our sample with declination north of δ =− 25◦, we have made our own photometric observations from Aprilto December 2004. Observations were done from Finland in full re-mote mode, using the 35-cm telescope piggybacked on the Swedish60-cm telescope located at La Palma in the Canary Islands. ASBIG charge-coupled device was used through all the observations.Johnson–Cousins BV(RI )C colours were obtained for all stars.

Standard stars were selected from Landolt (1992) and amongE-regions from Cousins (1980). Although E-regions provide an ex-tremely accurate set of stars, there are some systematic differencesbetween the Landolt and the South African Astronomical Observa-tory (SAAO) system (Bessell 1995). Therefore, we have used theLandolt (1992) standards placed on to the SAAO system by Bessell(1995).

10–20 standard stars were observed each night, bracketing ourprogramme stars in colour and airmass. Whenever possible pro-gramme stars were observed when passing the meridian, in orderto minimize extrapolation to zero airmass. Only if the standarddeviation between our calibrated and the tabulated values for thestandards was smaller than 0.015 mag in each band the nightwas considered photometric and observations useful. In addition,only programme stars for which the final B − V scatter (obtainedaveraging five frames) was smaller than 0.015 mag were consideredusefully accurate. Scatter in the other bands was usually smaller. Weexpect our photometry to have accuracies of 0.010–0.015 mag onaverage.

In addition to our observations, we have gathered BV(RI )C pho-tometry of equivalent or better precision from the literature: Bessell(1990a) (who also includes measurements from Cousins 1980), Reidet al. (2001) and Percival et al. (2003). For stars in common withthese authors, we have found excellent agreement between the pho-tometry, with a scatter of the order of 0.01 mag in all bands, andzero-point shifts between authors of less than 0.01 mag. This is morethan adequately accurate for our study.

2.2.2 Two-Micron All-Sky Survey photometry

Infrared JHKS photometry for the sample has been taken from theTwo-Micron All-Sky Survey (2MASS) catalogue. The uncertaintyfor each observed 2MASS magnitude is denoted in the catalogueby the flags: flags ‘j ’, ‘h ’ and ‘k msigcom’, and is the completeerror that incorporates the results of processing photometry, inter-nal errors and calibration errors. Some of our stars are very brightand have very high errors in 2MASS. We use 2MASS photometryonly if the sum of the photometric errors is less than 0.10 mag (i.e.‘j ’+‘h ’+ ‘k msigcom’<0.10). The final number of stars is thusreduced from 129 to 104. For our final sample, the errors in J and KS

bands are similar, with a mean value of 0.02 mag, whereas a slightlyhigher mean error is found in H band (0.03 mag).

2.3 Abundances

We have gathered detailed chemical abundances for our samplestars from the wealth of on-going surveys, dedicated to investigatethe chemical composition of our local environment as well as to thehost stars of extrasolar planets (see references in Table 1). The inter-nal accuracy of such data is usually excellent, with uncertainties ofthe order of 0.10 dex or less. However, abundance analysis for late-type dwarfs can still be troublesome in some cases (e.g. AllendePrieto et al. 2004). We are aware that, in gathering spectroscopyfrom different authors, the underlying temperature scale used toderive abundances can differ by as much as 50–150 K, whichwould translate into a [Fe/H] error of ∼0.1 dex (e.g. Asplund 2003;Kovtyukh et al. 2003). Therefore, a more conservative error estimatefor our abundances would be ∼0.15 dex. However, as we show inSection 4.4.2 the IRFM, which we employ to recover the funda-mental stellar parameters, is only weakly sensitive to the adoptedmetallicity and uncertainties of ±0.15 dex do not bear heavily onour main results.

The best measured elemental abundance in our dwarf stars isusually iron (i.e. [Fe/H]), whereas for theoretical models the mainmetallicity parameter is the total heavy-element mass fraction,[M/H]. For most of the stars in our sample, the spectra provide mea-surement not only for [Fe/H], but also for the α-elements, whichdominate the global metallicity budget. For stars with α-elementestimates, we compute [M/H] (Yi et al. 2001):

[M/H] = [Fe/H] + log(0.694 fα + 0.306), (1)

where

fα = 10[α/Fe] (2)

is the enhancement factor and [α/Fe] has been computed by av-eraging the α-elements. The older formula by Salaris, Chieffi &Straniero (1993) gives similar results with a difference smaller than0.02 dex in [M/H] even for the most α-enhanced stars ([α/Fe] ∼0.4).

For stars for which the α-elements were not available, we have es-timated their contribution from the mean locus of the [α/Fe] versus[Fe/H] relation from the analytical model of Pagel & Tautvaisiene(1995). There were 34 such stars in our final sample compared to70 for which α estimates were directly available.

2.4 Reddening corrections

Interstellar absorption and reddening must be taken into accountfor a correct derivation of stellar parameters, but these effects arenegligible for our sample stars, as we discuss in this section.

The distance distribution of the sample peaks around ∼30 pc,the most distant object being located at a distance of ∼70 pc.For distances closer than 50 pc, the polarimetric approach is ex-tremely sensitive and can be used as a lower limit even to theexpectedly small amounts of dust (at least for anisotropic parti-cles). Tinbergen (1982) and Leroy (1993b) have confirmed thecomplete depletion of dust within ∼40 pc from the Sun. Usingthe catalogue of Leroy (1993a), we have found polarimetric mea-surements for 21 stars out of the 129 single and non-variable starsselected in Section 2.1. The mean percentage polarization is 31 ×10−5 which using the Serkowski, Mathewson & Ford (1975)conversion factor corresponds to a E(B − V) of 0.0034 mag.


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Tabl

e1.

Obs

erva

ble

and

phys

ical

quan

titie

sfo

rou

rsa

mpl

est

ars.

Nam

eH

IPπ

±δ

πT

eff±

�T

eff

θ±

�θ

mB

olV

B−

VV

−R

C(R

−I)

CJ

HK

S[F

e/H

][α

/Fe]

(mas

)(K

)(m

as)

HD

3765

3206

57.9

0±

0.98

5002

±57

0.42

6±

0.01

07.

063

7.36

00.

940

0.54

00.

440

5.69

±0.

025.

27±

0.05

5.16

±0.

020.

010.

05a

HD

4256

3535

45.4

3±

0.95

4966

±46

0.32

6±

0.00

77.

675

7.99

00.

990

0.56

00.

460

6.31

±0.

025.

87±

0.04

5.74

±0.

030.

34−0

.08

bH

D43

0834

9745

.76

±0.

5657

81±

880.

423

±0.

013

6.44

96.

560

0.65

00.

373

0.35

25.

37±

0.02

5.10

±0.

024.

95±

0.02

−0.4

00.

05c

HD

4747

3850

53.0

9±

1.02

5406

±53

0.37

5±

0.00

87.

002

7.15

50.

790

0.42

00.

400

5.81

±0.

025.

43±

0.05

5.30

±0.

03−0

.21

0.06

dH

D51

3341

4871

.01

±0.

7849

56±

520.

474

±0.

011

6.87

07.

180

0.94

00.

540

0.46

05.

54±

0.03

5.05

±0.

034.

89±

0.03

−0.1

00.

04e

HD

8638

6607

24.7

5±

1.22

5568

±59

0.20

7±

0.00

58.

164

8.30

00.

690

0.38

50.

365

7.00

±0.

026.

68±

0.03

6.62

±0.

02−0

.50

0.12

cH

D10

002

7539

29.7

2±

1.08

5258

±63

0.25

4±

0.00

67.

968

8.14

00.

860

0.45

00.

390

6.68

±0.

026.

34±

0.03

6.21

±0.

020.

200.

00b

HD

1085

382

7543

.37

±1.

3846

38±

780.

267

±0.

009

8.40

58.

910

1.04

00.

625

0.53

76.

93±

0.02

6.44

±0.

036.

32±

0.02

−0.7

40.

21f

HD

1102

083

4625

.79

±1.

2852

88±

310.

172

±0.

002

8.79

08.

980

0.81

00.

443

0.40

57.

55±

0.03

7.13

±0.

057.

06±

0.02

−0.2

80.

21b

HD

1113

085

4337

.33

±0.

9752

19±

440.

270

±0.

005

7.86

88.

070

0.75

80.

430

0.40

36.

60±

0.02

6.20

±0.

026.

11±

0.02

−0.5

70.

15f

HD

1137

388

6745

.16

±1.

0947

38±

580.

301

±0.

008

8.05

28.

483

1.02

00.

607

0.48

96.

61±

0.02

6.11

±0.

046.

02±

0.02

−0.4

50.

12f

HD

1662

312

364

15.1

8±

1.15

5858

±55

0.14

9±

0.00

38.

657

8.76

00.

600

0.33

50.

335

7.62

±0.

027.

30±

0.04

7.28

±0.

04−0

.60

0.13

cH

D17

190

1292

638

.95

±1.

1351

39±

740.

302

±0.

009

7.69

27.

878

0.82

70.

418

0.40

46.

37±

0.02

6.00

±0.

045.

87±

0.03

−0.1

1−0

.00

bH

D18

632

1397

642

.66

±1.

2249

84±

870.

316

±0.

011

7.72

68.

020

0.91

80.

529

0.43

96.

32±

0.02

5.95

±0.

045.

84±

0.02

0.18

−0.0

1g

HD

2119

715

919

66.1

5±

0.95

4577

±60

0.45

3±

0.01

37.

314

7.86

81.

162

0.68

40.

565

5.83

±0.

025.

31±

0.03

5.12

±0.

020.

270.

13j

HD

2287

917

147

41.0

7±

0.86

5930

±67

0.37

9±

0.00

96.

577

6.67

90.

550

0.32

50.

337

5.59

±0.

025.

30±

0.03

5.18

±0.

02−0

.91

0.34

bH

D25

329

1891

554

.14

±1.

0847

96±

620.

279

±0.

008

8.16

48.

519

0.87

60.

495

0.47

86.

77±

0.03

6.30

±0.

046.

20±

0.01

−1.8

00.

49i

HD

2566

519

422

54.1

7±

0.79

4902

±65

0.38

0±

0.01

17.

398

7.75

70.

873

0.56

70.

443

5.99

±0.

035.

54±

0.03

5.46

±0.

02−0

.03

−0.0

0b

HD

2567

319

007

24.2

3±

1.53

5132

±11

30.

145

±0.

007

9.29

19.

532

0.82

30.

467

0.44

37.

96±

0.02

7.65

±0.

047.

46±

0.03

−0.5

30.

15i

HD

2894

621

272

37.3

3±

2.58

5308

±55

0.27

1±

0.00

67.

787

7.94

00.

764

0.39

20.

400

6.52

±0.

026.

16±

0.03

6.07

±0.

02−0

.03

−0.0

4h

HD

3050

122

122

48.9

0±

0.64

5153

±40

0.35

0±

0.00

67.

360

7.58

00.

880

0.48

00.

425

6.06

±0.

025.

64±

0.02

5.53

±0.

020.

13−0

.02

jH

D31

128

2263

215

.55

±1.

2060

89±

810.

118

±0.

003

8.99

69.

135

0.48

90.

309

0.32

58.

03±

0.02

7.80

±0.

037.

74±

0.02

−1.5

40.

41i

HD

3432

824

316

14.5

5±

1.01

6025

±70

0.10

6±

0.00

39.

275

9.42

70.

496

0.31

20.

335

8.32

±0.

028.

05±

0.03

8.00

±0.

03−1

.69

0.38

iH

D37

792

2668

820

.61

±0.

8965

56±

100

0.19

0±

0.00

67.

641

7.69

70.

419

0.24

90.

268

6.83

±0.

026.

68±

0.03

6.57

±0.

02−0

.60

0.23

kH

D39

715

2791

837

.57

±1.

2647

59±

370.

253

±0.

004

8.41

08.

830

1.01

00.

605

0.49

56.

99±

0.03

6.47

±0.

046.

35±

0.02

−0.0

40.

01b

HD

5121

933

382

31.1

5±

0.85

5584

±58

0.30

6±

0.00

77.

303

7.41

00.

699

0.37

70.

357

6.15

±0.

025.

83±

0.03

5.72

±0.

02−0

.09

0.04

lH

D53

545

3414

615

.74

±0.

9163

93±

550.

169

±0.

003

8.00

58.

047

0.45

60.

268

0.27

17.

16±

0.03

6.90

±0.

036.

86±

0.02

−0.2

90.

11i

HD

5392

734

414

44.9

2±

1.43

4882

±35

0.28

8±

0.00

58.

018

8.35

10.

889

0.52

90.

450

6.64

±0.

026.

14±

0.02

6.06

±0.

03−0

.37

0.09

hH

D57

901

3587

239

.91

±1.

1448

65±

480.

311

±0.

007

7.86

78.

224

0.96

20.

564

0.45

26.

46±

0.03

5.98

±0.

035.

89±

0.02

0.00

0.03

fH

D59

374

3649

120

.00

±1.

6658

39±

580.

171

±0.

004

8.37

38.

485

0.54

10.

324

0.32

67.

33±

0.02

7.03

±0.

026.

98±

0.03

−0.8

80.

29i

HD

5946

836

210

44.4

3±

0.53

5625

±57

0.41

1±

0.00

96.

630

6.72

00.

700

0.37

70.

355

5.51

±0.

035.

15±

0.02

5.04

±0.

020.

020.

00b

HD

5974

736

704

50.8

0±

1.29

5063

±54

0.34

9±

0.00

87.

443

7.69

50.

877

0.49

30.

416

6.09

±0.

025.

66±

0.02

5.59

±0.

020.

06−0

.04

bH

D65

486

3893

954

.91

±1.

1546

78±

290.

326

±0.

005

7.93

48.

400

1.05

00.

605

0.53

06.

51±

0.02

5.94

±0.

025.

83±

0.02

−0.2

40.

07e

HD

6719

939

342

57.8

8±

0.58

5148

±44

0.42

5±

0.00

86.

943

7.16

60.

889

0.48

00.

416

5.65

±0.

025.

23±

0.03

5.11

±0.

020.

03−0

.04

bH

D73

524

4229

136

.40

±0.

5759

78±

600.

389

±0.

009

6.48

66.

530

0.60

00.

330

0.31

05.

46±

0.02

5.21

±0.

035.

10±

0.03

0.12

−0.0

0b

HD

8551

248

331

89.6

7±

0.82

4429

±13

40.

557

±0.

034

7.00

87.

670

1.19

00.

720

0.62

25.

45±

0.02

5.00

±0.

034.

72±

0.02

−0.1

80.

06d

HD

8700

749

161

23.2

3±

1.41

5260

±44

0.18

7±

0.00

48.

632

8.80

20.

821

0.44

50.

391

7.35

±0.

026.

97±

0.03

6.89

±0.

040.

270.

00j

HD

8981

350

782

37.3

0±

1.31

5372

±51

0.28

6±

0.00

67.

618

7.76

90.

784

0.42

30.

383

6.40

±0.

036.

03±

0.04

5.91

±0.

02−0

.17

0.06

fH

D90

663

5125

439

.75

±1.

5449

36±

430.

258

±0.

005

8.20

98.

530

0.92

00.

541

0.43

96.

85±

0.02

6.39

±0.

026.

26±

0.02

−0.2

90.

08f

HD

9278

652

470

39.5

6±

0.99

5301

±69

0.26

5±

0.00

77.

841

8.02

00.

750

0.42

70.

385

6.57

±0.

026.

23±

0.02

6.13

±0.

02−0

.29

0.08

fH

D94

028

5307

019

.23

±1.

1360

91±

680.

180

±0.

004

8.07

88.

218

0.48

20.

306

0.33

07.

13±

0.02

6.85

±0.

026.

83±

0.02

−1.4

90.

37i


Dow


ic.oup.com/m



Tabl

e1

–co

ntin

ued

Nam

eH

IPπ

±δ

πT

eff±

�T

eff

θ±

�θ

mB

olV

B−

VV

−R

C(R

−I)

CJ

HK

S[F

e/H

][α

/Fe]

(mas

)(K

)(m

as)

HD

9732

054

641

17.7

7±

0.76

6142

±51

0.18

0±

0.00

38.

042

8.16

10.

481

0.29

50.

312

7.14

±0.

026.

87±

0.03

6.79

±0.

02−1

.28

0.30

iH

D98

281

5521

045

.48

±1.

0054

16±

580.

353

±0.

008

7.12

57.

275

0.74

70.

413

0.37

55.

91±

0.02

5.57

±0.

035.

46±

0.03

−0.2

00.

04b

HD

1030

7257

866

39.4

8±

1.58

5058

±52

0.25

5±

0.00

68.

129

8.40

00.

850

0.50

10.

426

6.78

±0.

036.

36±

0.02

6.27

±0.

02−0

.30

0.08

fH

D10

5671

5929

649

.72

±1.

0046

04±

530.

339

±0.

008

7.91

88.

440

1.13

50.

660

0.54

56.

44±

0.02

5.91

±0.

045.

76±

0.02

0.20

0.00

eH

D10

6156

5957

232

.30

±1.

0254

28±

450.

259

±0.

005

7.78

87.

918

0.80

00.

426

0.36

56.

57±

0.02

6.23

±0.

046.

12±

0.03

0.23

−0.0

3b

HD

1065

1659

750

44.3

4±

1.01

6296

±13

50.

431

±0.

019

6.03

86.

104

0.46

60.

286

0.28

55.

13±

0.02

5.00

±0.

054.

84±

0.02

−0.7

80.

37k

HD

1085

6460

853

35.3

0±

1.20

4660

±44

0.20

3±

0.00

48.

980

9.45

00.

964

0.59

70.

535

7.55

±0.

026.

99±

0.03

6.90

±0.

02−1

.18

0.40

nH

D10

9200

6129

161

.83

±0.

6351

42±

670.

431

±0.

012

6.91

77.

143

0.84

80.

480

0.41

65.

63±

0.03

5.23

±0.

035.

07±

0.02

−0.2

40.

07d

HD

1120

9962

942

38.1

2±

1.44

5082

±60

0.26

7±

0.00

78.

008

8.25

00.

852

0.48

00.

425

6.66

±0.

036.

27±

0.04

6.15

±0.

020.

090.

01f

HD

1140

9464

103

14.4

1±

1.48

5643

±35

0.10

6±

0.00

29.

559

9.66

80.

696

0.38

50.

361

8.43

±0.

028.

09±

0.04

8.02

±0.

03−0

.24

0.06

lH

D11

9173

6681

517

.57

±1.

1159

80±

520.

138

±0.

003

8.73

58.

828

0.55

20.

325

0.32

77.

73±

0.02

7.46

±0.

037.

39±

0.02

−0.6

20.

15i

HD

1204

6767

487

70.4

9±

1.01

4406

±46

0.45

8±

0.01

07.

456

8.17

01.

270

0.75

50.

640

5.92

±0.

025.

32±

0.02

5.16

±0.

020.

240.

17m

HD

1205

5967

655

40.0

2±

1.00

5446

±46

0.25

4±

0.00

57.

816

7.97

10.

659

0.37

60.

379

6.63

±0.

036.

27±

0.04

6.20

±0.

02−0

.94

0.33

iH

D12

0780

6774

260

.86

±0.

9550

46±

410.

413

±0.

007

7.09

27.

370

0.90

00.

495

0.48

05.

78±

0.02

5.32

±0.

025.

19±

0.02

−0.1

70.

06e

HD

1241

0669

357

43.3

5±

1.40

5103

±57

0.30

6±

0.00

77.

694

7.93

80.

869

0.48

00.

426

6.36

±0.

025.

95±

0.02

5.86

±0.

02−0

.10

0.00

bH

D12

6681

7068

119

.16

±1.

4456

30±

460.

129

±0.

002

9.14

39.

296

0.60

10.

359

0.36

88.

04±

0.02

7.71

±0.

047.

63±

0.02

−1.1

40.

32i

HD

1275

0670

950

45.4

6±

1.03

4662

±92

0.28

6±

0.01

28.

234

8.70

21.

029

0.62

40.

501

6.76

±0.

026.

31±

0.04

6.14

±0.

02−0

.40

0.11

fH

D12

9518

7193

914

.31

±1.

4962

72±

610.

124

±0.

003

8.76

08.

809

0.48

30.

288

0.28

77.

87±

0.02

7.62

±0.

047.

54±

0.02

−0.3

40.

09i

HD

1303

0772

312

50.8

4±

1.04

5033

±43

0.34

6±

0.00

77.

487

7.76

10.

896

0.50

00.

435

6.15

±0.

025.

69±

0.02

5.61

±0.

02−0

.16

0.03

bH

D13

0322

7233

933

.60

±1.

5154

26±

310.

246

±0.

003

7.90

18.

046

0.78

60.

423

0.38

16.

71±

0.02

6.32

±0.

036.

23±

0.02

0.03

0.02

hH

D13

0871

7257

732

.53

±1.

5648

45±

420.

213

±0.

004

8.70

69.

073

0.97

20.

553

0.47

77.

31±

0.02

6.80

±0.

036.

72±

0.02

−0.1

60.

06b

HD

1309

9272

688

58.9

6±

1.05

4796

±48

0.39

5±

0.00

97.

409

7.80

41.

014

0.58

30.

486

5.99

±0.

025.

49±

0.02

5.39

±0.

03−0

.06

0.04

dH

D13

2142

7300

541

.83

±0.

6352

26±

590.

311

±0.

007

7.55

57.

770

0.76

90.

449

0.41

76.

27±

0.02

5.89

±0.

025.

80±

0.02

−0.5

40.

24a

HD

1344

4074

234

33.6

8±

1.67

4883

±41

0.17

5±

0.00

39.

099

9.42

00.

869

0.49

70.

478

7.76

±0.

037.

28±

0.04

7.15

±0.

02−1

.45

0.19

iH

D13

6834

7526

639

.35

±1.

3749

55±

580.

285

±0.

007

7.97

68.

281

0.99

20.

548

0.43

86.

60±

0.03

6.18

±0.

046.

04±

0.02

0.19

−0.0

0j

HD

1427

0978

170

68.1

8±

1.02

4526

±52

0.42

5±

0.01

17.

502

8.06

01.

120

0.67

00.

565

6.03

±0.

025.

45±

0.05

5.28

±0.

02−0

.17

0.06

dH

D14

4579

7877

569

.61

±0.

5752

51±

770.

506

±0.

015

6.47

86.

660

0.73

40.

402

0.38

65.

18±

0.02

4.82

±0.

024.

76±

0.02

−0.6

90.

31m

HD

1446

2879

190

69.6

6±

0.90

5083

±40

0.45

4±

0.00

86.

855

7.11

00.

860

0.48

00.

445

5.55

±0.

025.

10±

0.02

4.98

±0.

03−0

.33

0.09

eH

D14

4872

7891

342

.57

±0.

8647

92±

640.

277

±0.

008

8.18

38.

580

0.96

30.

568

0.47

06.

76±

0.03

6.29

±0.

026.

16±

0.02

−0.6

40.

18f

HD

1454

1779

537

72.7

5±

0.82

4907

±36

0.41

1±

0.00

77.

223

7.53

00.

830

0.48

00.

460

5.89

±0.

025.

39±

0.03

5.29

±0.

02−1

.39

0.33

iH

D14

7776

8036

646

.44

±1.

2047

51±

480.

301

±0.

007

8.04

08.

420

0.97

00.

538

0.48

26.

62±

0.02

6.11

±0.

056.

00±

0.02

−0.2

90.

10b

HD

1494

1481

170

20.7

1±

1.50

5090

±50

0.14

4±

0.00

39.

342

9.60

70.

746

0.44

70.

453

8.06

±0.

027.

64±

0.04

7.52

±0.

02−1

.26

0.35

kH

D15

4577

8399

073

.07

±0.

9148

82±

380.

447

±0.

008

7.06

47.

395

0.89

00.

515

0.46

55.

69±

0.02

5.20

±0.

025.

09±

0.02

−0.6

30.

17d

HD

1618

4887

089

26.1

6±

1.16

5093

±72

0.19

6±

0.00

68.

670

8.91

20.

823

0.47

40.

424

7.34

±0.

026.

96±

0.04

6.82

±0.

02−0

.39

0.19

bH

D16

6913

8955

416

.09

±1.

0462

73±

690.

170

±0.

004

8.07

48.

216

0.44

90.

290

0.31

27.

20±

0.02

6.95

±0.

036.

92±

0.02

−1.5

40.

37i

HD

1704

9390

656

53.3

0±

1.05

4671

±98

0.38

7±

0.01

77.

569

8.03

61.

106

0.63

10.

505

6.07

±0.

025.

64±

0.05

5.48

±0.

020.

150.

00d

HD

1799

4994

645

36.9

7±

0.80

6200

±10

40.

407

±0.

014

6.22

96.

240

0.54

00.

307

0.28

85.

30±

0.03

5.10

±0.

044.

94±

0.02

0.22

0.00

hH

D19

0007

9869

876

.26

±1.

0346

37±

510.

523

±0.

012

6.94

67.

460

1.14

00.

670

0.53

55.

48±

0.02

4.96

±0.

034.

80±

0.02

0.15

0.15

mH

D19

3901

1005

6822

.88

±1.

2459

05±

560.

157

±0.

003

8.50

98.

642

0.54

60.

332

0.34

67.

50±

0.02

7.22

±0.

047.

14±

0.02

−1.0

90.

15i

HD

2007

7910

4092

67.3

8±

1.35

4430

±66

0.42

7±

0.01

37.

584

8.27

01.

220

0.74

00.

625

6.04

±0.

025.

49±

0.03

5.31

±0.

020.

060.

23m

HD

2025

7510

5038

61.8

3±

1.06

4741

±35

0.39

2±

0.00

77.

475

7.89

71.

020

0.60

20.

499

6.07

±0.

025.

53±

0.02

5.39

±0.

020.

040.

01b


Dow


ic.oup.com/m



Tabl

e1

–co

ntin

ued

Nam

eH

IPπ

±δ

πT

eff±

�T

eff

θ±

�θ

mB

olV

B−

VV

−R

C(R

−I)

CJ

HK

S[F

e/H

][α

/Fe]

(mas

)(K

)(m

as)

HD

2027

5110

5152

52.0

3±

1.23

4809

±35

0.33

3±

0.00

67.

768

8.16

00.

990

0.58

00.

495

6.37

±0.

025.

85±

0.03

5.74

±0.

02−0

.10

0.08

bH

D21

3042

1109

9664

.74

±1.

0747

21±

580.

447

±0.

012

7.20

97.

640

1.10

00.

626

0.50

25.

77±

0.02

5.28

±0.

055.

11±

0.02

0.19

0.14

jH

D21

4749

1119

6073

.85

±1.

1345

48±

810.

471

±0.

017

7.25

77.

810

1.13

00.

685

0.56

55.

77±

0.02

5.26

±0.

045.

04±

0.02

0.12

0.01

eH

D21

5152

1121

9046

.46

±1.

3148

70±

590.

327

±0.

009

7.75

38.

096

0.96

10.

551

0.45

56.

35±

0.02

5.89

±0.

035.

78±

0.03

−0.1

70.

06f

HD

2155

0011

2245

39.8

2±

0.71

5485

±57

0.31

2±

0.00

77.

338

7.48

00.

717

0.40

20.

369

6.14

±0.

035.

80±

0.04

5.73

±0.

02−0

.34

0.09

fH

D21

6259

1128

7047

.56

±1.

1849

57±

610.

283

±0.

007

7.98

98.

294

0.84

20.

503

0.44

16.

61±

0.02

6.18

±0.

026.

08±

0.02

−0.6

30.

27b

HD

2185

0211

4271

14.3

3±

1.20

6324

±64

0.16

5±

0.00

48.

104

8.25

60.

420

0.28

00.

300

7.26

±0.

027.

02±

0.03

6.97

±0.

02−1

.87

0.43

iH

D21

9538

1148

8641

.33

±0.

9750

51±

580.

295

±0.

007

7.81

88.

051

0.89

30.

462

0.40

96.

47±

0.03

6.05

±0.

045.

96±

0.03

−0.0

4−0

.01

bH

D22

0339

1154

4551

.37

±1.

2550

08±

510.

345

±0.

007

7.51

57.

780

0.89

00.

495

0.45

06.

20±

0.02

5.74

±0.

045.

59±

0.02

−0.3

10.

05b

HD

2212

3911

6005

38.9

8±

1.10

4954

±78

0.27

6±

0.00

98.

047

8.33

10.

900

0.47

60.

440

6.66

±0.

036.

26±

0.04

6.14

±0.

02−0

.46

0.12

fH

D22

2335

1167

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The Stromgen Hβ index offers an alternative method for assess-ing the reddening of individual stars in our sample, especially for themost distant ones; 59 stars in our bona fide single and non-variablesample have Hβ measurements (Hauck & Mermilliod 1998) fromwhich E(b − y) was derived using the intrinsic-colour calibration ofSchuster & Nissen (1989) plus a small offset correction as noted byNissen (1994). The reddening distribution for these 59 stars peaksaround E(b − y) = 0.004, retaining the same mean value (and neverexceeding 0.015 for a single star) when we restrict the sample todistances further than 50 pc. E(B − V) can then be calculated usingthe standard extinction law or the relation E(B − V) = 1.35 E(b −y) derived by Crawford (1975). Once the reddening E(B − V) isknown, the extinction at any given wavelength can be determinedusing the standard extinction law. Given that the standard error inthe Schuster & Nissen calibration is of the order of 0.01 in E(b −y) and observational errors in uvby measurements are possible, theimplied reddening corrections are below the noise level. In addi-tion to this, there are indications (Knude 1979; Vergely et al. 1998)that interstellar reddening is primarily caused by small dust cloudscausing it to vary in steps of 0.01–0.03 mag which confirm that thesmall corrections found are consistent with the assumption of zeroreddening for the sample as a whole.

3 E M P I R I C A L C O L O U R S V E R S U S M O D E LC O L O U R S

Theoretical stellar models predict relations between physical quan-tities, such as effective temperature, luminosity and stellar radius.From the observational point of view, these quantities are, in general,not directly measurable and must be deduced from observed broad-band colours and magnitudes. In this context, empirical colour–temperature relations (Bessell 1979; Ridgway et al. 1980; Saxner& Hammarback 1985; Di Benedetto & Rabbia 1987; Blackwell &Lynas-Gray 1994, 1998; Alonso, Arribas & Martınez-Roger 1996b;Ramırez & Melendez 2005b) and colour–bolometric correctionrelations (Bessell & Wood 1984; Malagnini et al. 1986; Bell &Gustafsson 1989; Blackwell & Petford 1991; Alonso, Arribas &Martınez-Roger 1995) are normally used. Such calibrations wereusually restricted to a specific type and/or population of stars, theonly exception being Alonso et al. (1995, 1996b) who give empiri-cal relations for F0-K5 dwarfs with both solar and subsolar metal-licity. Recently, Ramırez & Melendez (2005b) have improved andextended the Alonso et al. (1996b) colour–temperature–metallicityrelations.

In any case, even empirical calibrations make use of model atmo-spheres to some extent, though the model dependence is expectedto be small (see Section 4.3). For example, the work of Alonso,Arribas & Martınez-Roger (1996a) was based on Kurucz (1993)spectra, whereas the recent Ramırez & Melendez (2005a) calibra-tion is based on the original Kurucz spectra (i.e. without empiricalmodifications) taken from Lejeune, Cuisinier & Buser (1997).

In this section, we compare a number of stellar models (ATLAS9,MARCS, BaSel 3.1 and BaSel 2.1.) to our empirical data, mainlyin two colour planes. In general, colour–colour plots of the starsagainst all the model sets look quite satisfactory.

3.1 ATLAS9, MARCS and BaSel spectral libraries

We have tested synthetic spectra from the ATLAS9–ODFNEW(Castelli & Kurucz 2003), MARCS (Gustafsson et al. 2002), BaSel2.1 and 3.1 (Lejeune et al. 1997; Westera et al. 2002, respectively)libraries. All grids are given in steps of 250 K. Note that, since we

are working with dwarf stars, we assume log (g) = 4.5 throughout.This assumption is reasonable given that determinations of log (g)have a typical uncertainty of 0.2 dex or more either by requiringFe I and Fe II lines to give the same iron abundance or by usingHipparcos parallaxes (e.g. Bai et al. 2004). This uncertainty coversthe expected range in log (g) on the main sequence (see Fig. 1).In any case, a change of ± 0.5 dex in the surface gravity impliesdifferences that never exceed a few degrees in derived effective tem-perature, as will be seen (Section 4.4.2).

3.1.1 ATLAS9–ODFNEW models

The ATLAS9–ODFNEW models calculated by Castelli & Kuruczinclude improvements in the input physics, the update of the solarabundances from Anders & Grevesse (1989) to Grevesse & Sauval(1998) and the inclusion of a new molecular line lists for TiO andH2O. The metallicities cover −2.5 � [M/H] � +0.5 with solar-scaled abundance ratios. A microturbulent velocity ξ = 2 km s−1

and a mixing length parameter of 1.25 are adopted. The extensionof these models to include also α-enhanced chemical mixtures is on-going. However, we do not expect large differences among differentmodels as a result of α-enhancement, as explained in Section 4.3.In the remainder of the paper, we refer to the ATLAS9–ODFNEWmodel simply as ATLAS9.

3.1.2 MARCS models

The new generation of MARCS models (Gustafsson et al. 1975;Plez, Brett & Nordlund 1992) now includes a much improved treat-ment of molecular opacity (Gustafsson et al. 2002; Plez 2003),chemical equilibrium for all neutral and singly ionized species aswell as some doubly ionized species, along with about 600 molec-ular species. The atomic lines are based on Vienna Atomic LineDatabase (VALD; Kupka et al. 1999). The models span the metal-licity range −5.0 � [Fe/H] � +1.0. We have used models withsolar relative abundances for [Fe/H] � 0 and enhanced α-elementsabundances for [Fe/H] < 0. The fraction of α-elements is given bythe model and [M/H] has been computed using equation (1).

3.1.3 The BaSel libraries

The BaSel libraries are based on a posteriori empirical correctionsto hybrid libraries of spectra, so as to reduce the errors of the derivedsynthetic photometry. BaSel 2.1 (Lejeune et al. 1997) is calibratedusing solar metallicity data only, and it is known to be less accurateat low metallicities ([Fe/H] < 1), especially in the ultraviolet (UV)and infrared (IR) (Westera et al. 2002). The colour calibration hasrecently been extended to non-solar metallicities by Westera et al.(2002). Surprisingly, they found that a library that matches empiricalcolour–temperature relations does not reproduce Galactic globular–cluster colour–magnitude diagrams, as a result of which they pro-pose two different versions of the library. In what follows, we haveused the library built to match the empirical colour–temperaturerelations. For both BaSel libraries, it is obvious that the transforma-tions used do not correct the physical cause of the discrepancies.However, it is worthwhile to check whether they do give betteragreement with the empirical relations.

All libraries have been tested in the range 4250 � Teff � 6500 Kand −2.0 � [M/H] � +0.5, the intervals covered by our sample ofstars.


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3.2 Comparison of model and empirical colours

All our colour–colour comparisons are as a function of (V − KS).This approach highlights the behaviour of colours as a functionof temperature and also makes more straightforward the compari-son with the empirical colour–temperature relations in Section 3.3.(V − KS) has a large range compared to its observational uncertaintyand it is known to be metallicity insensitive (e.g. Bell & Gustafsson1989 and Fig. 13).

All models are found to be in good overall agreement with theobservations. In particular, the improvement in the input physics ofthe latest ATLAS9 and MARCS models shows excellent agreementwith the optical data (Figs 2 and 3) and as good as the semi-empiricallibraries (Fig. 14) in the other bands, so that we do not find anyspecific reason to prefer the use of semi-empirical libraries.

We expect the IR colours to be very important both in observa-tional terms and in theoretical modelling. Recently, Fremaux et al.(2006) have presented a detailed comparison of theoretical spectra(from the NeMo library) in H band with observed stellar spectra atresolution around 3000. They show that in the IR range, althoughthe overall shape of the observed flux distribution is matched rea-sonably well, individual features are reproduced by the theoreticalspectra only for stars earlier than mid-F type. For later spectraltypes, the differences increase and theoretical spectra of K-typestars have systematically weaker line features than those found inobservations. They conclude that these discrepancies stem from in-

Figure 2. Optical and IR colours for the ATLAS9–ODFNEW from Castelli & Kurucz (2003) compared to empirical colours for G and K dwarfs. The coloursare shown as a function of (V − KS) for [M/H] equal to +0.5 (black line), +0.0 (red line), −0.5 (cyan line), −1.0 (green line), −1.5 (blue line), −2.0 (yellowline). Points correspond to observed colours for the sample stars in the range [M/H] > 0.25 (black), −0.25 < [M/H] � 0.25 (red), −0.75 < [M/H] � −0.25(cyan), −1.25 < [M/H] � −0.75 (green), −1.75 < [M/H] � −1.25 (blue). The metallicities given for the model are solar-scaled, whereas [M/H] for the starshas been computed using equation (1). A typical error bar for the points is shown in the upper left of each plot. The model and empirical colours are generallyin very good agreement.

complete data on neutral atomic (and to a minor extent to molecular)lines.

The synthetic optical colours agree well with observed ones andmodels also predict the correct metallicity dependence. In the IR,models fit satisfactorily the overall trend given by the observedcolours; however, the scatter in the data prevents firm conclusions.In J − H and J − KS, the model colours seem to be slightly offsetby ∼0.05 mag, whereas in H − KS the disagreement is somewhatsmaller though it depends on the models considered and it seems toincrease going to cooler temperatures.

3.3 Empirical colour–temperature relations compared toobserved and synthetic photometry

We now compare empirical colour–temperature calibrations withsynthetic and observed photometry. To this purpose, we use thecalibration recently proposed by Ramırez & Melendez (2005b).

In Fig. 5, we plot the difference between the temperature of themodel atmospheres (BaSel 3.1 and ATLAS9) and the temperatureexpected for their colours according to the empirical Ramırez &Melendez relations. Interestingly, all models fail to set on to the em-pirical temperature scale being at least 100-K hotter in optical andIR colours. The disagreement is particularly strong at low metal-licity as noticed by Ramırez & Melendez. The semi-empirical li-braries of Lejeune et al. (1997) and Westera et al. (2002), though inprinciple optimized to reproduce colour–temperature relations, pro-vide only a marginal improvement with respect to the latest purely


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Figure 3. Same as Fig. 2 but for MARCS models, although the metallicity range is different, since MARCS model are ‘α-enhanced’. Lines of constant [M/H]correspond to +0.5 (black line), +0.0 (red line), −0.35 (cyan line), −0.69 (green line), −1.19 (blue line), −1.69 (yellow line). Points correspond to observedcolours for the sample stars in the range [M/H] > 0.25 (black), −0.175 < [M/H] � 0.25 (red), −0.52 < [M/H] � −0.175 (cyan), −0.94 < [M/H] � −0.52(green), −1.44 < [M/H] � −0.94 (blue) and [M/H] � −1.44 (yellow). Note that MARCS model is α-enhanced for metallicities below the solar and [M/H]for stars and models have been computed using equation (1).

theoretical ATLAS9 and MARCS libraries. The temperature offsetcan be translated into a shift in colours needed to set the models onto the empirical scale. This is done for the ATLAS9 model in thelast row of Fig. 5; in terms of colours, the mismatch corresponds to∼0.1 mag.

Empirical calibrations depend on metallicities and colour indicesonly. The adopted absolute calibration cannot be the main cause ofthe disagreement since all colours (in the optical) scale accordingly.In the optical, the adopted Vega model that sets the absolute calibra-tion has been accurately tested by Bohlin & Gilliland (2004) (seeAppendix A). Although we adopt a different absolute calibration inthe optical and in the IR (see Appendix A), this could at most affectthe colour–metallicity–temperature relations in V − J, V − H andV − KS. Even using in the IR, the same absolutely calibrated Vegamodel adopted in the optical does not eliminate the disagreementand offsets of 100 K still persist.

We have also verified that the choice of zero-points plays a negli-gible role: differences in the optical zero-points based on Vega or onthe average of Vega and Sirius (see Appendix A) lead to mean tem-perature differences of 10–30 K. Besides being much smaller thanthe uncertainty in the colour–temperature relations, such differencesgo in the direction of worsening the disagreement.

As to the IR zero-points, we checked their effects by generatingIR synthetic magnitudes in the TCS and Johnson system (wherethe Vega zero-points are different from those deduced from 2MASSphotometry) and compared the model predictions to the Alonso et al.(1996b) empirical calibration in these bands; typical discrepanciesof more than 100 K still persist.

The causes for the discrepancy must therefore be model deficien-cies and/or inaccuracies in the adopted empirical relations.

In fact, we caution that also empirical relations may hide someinadequacies. For instance, the Ramırez & Melendez calibrationpredicts slightly bluer colours for the Sun than the recent com-pletely empirical determination of Holmberg, Flynn & Portinari(2006). We can also combine and invert the Ramırez & Melendez’colour–temperature calibrations, to derive the corresponding em-pirical colour–colour relations and compare them to the observedcolours for our sample of stars. Interestingly, the agreement is notalways good: the solar metallicity is slightly offset with respect tothe data in (R − I)C band and in the IR tends to oscillate (see Fig. 6).This underlines that there is still room for improvement also in theempirical relations.

4 A N I M P L E M E N TAT I O N O F T H E I R F M

In this section, we have used BV(RI )C JHKS photometry as the basicobservational information to derive bolometric fluxes and effectivetemperatures of our 104 G and K dwarfs. Our approach follows theIRFM used in the extensive work of Alonso et al. (1995, 1996a).Ours is a new independent implementation of the IRFM, applied toour 104 G- and K-type dwarfs.

Our implementation differs from Alonso et al. (1995, 1996a), aswe base its absolute calibration on a synthetic spectrum of Vega (asdescribed in detail in Appendix A) rather than on semi-empiricalmeasurements (Alonso, Arribas & Martınez-Roger 1994). The useof absolutely calibrated synthetic spectra rather than ground-based


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Figure 4. Comparison of empirical and theoretical colours for G+K dwarfs. Same schema as Fig. 2 but for the BaSel 3.1 models. The BaSel 2.1 modelscompare similarly to the empirical data, but are not shown for the sake of brevity (contact the author if interested).

Figure 5. Effective temperatures recovered by means of the Ramırez & Melendez empirical calibration are compared with the effective temperature of thecorresponding BaSel 3.1 (first row) and ATLAS9 (second row) models used to generate synthetic photometry. ζ − ξ indicates the colours B − V , V − H andV − KS. The notation �T (ζ − ξ ) means temperature found by means of the (ζ − ξ ) calibration minus the effective temperature of the model atmosphereused to generate the synthetic ζ and ξ colours. The following metallicities are plotted: +0.0 (red), −0.5 (cyan), −1.0 (green), −1.5 (blue) and −2.0 (yellow).The dashed area is the standard deviation of the empirical calibration in the corresponding colours. The third row shows the shift in colours required to set theATLAS9 models on the empirical relation.


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Figure 6. Lines represent the colour–colour relations derived from the em-pirical colour–temperature calibrations of Ramırez & Melendez (2005b).Points correspond to our sample of stars. Colours and metallicity ranges forlines and points are same as in Fig. 2 except that metallicities of +0.5 havebeen excluded since the empirical relations do not hold in that range. Inthe optical, the empirical relations fit the observed colours with an accuracycomparable to the theoretical ones (compare with Fig. 2). However, in the IR,for metallicities around the solar, empirical relations oscillate in reproducingthe observed trend. This is especially interesting in light of the puzzling IRcolours of the Sun and the shifts required to cool down the temperature scaleas discussed in Section 5.

measurements traces back to Blackwell et al. (1990) and Blackwell& Petford (1991), who concluded that atmospheric models of Vegaoffered higher precision than observationally determined absolutecalibrations. This approach has been adopted in the extensive workof Cohen and collaborators and also Bessell, Castelli & Plez (1998),who concluded that model atmospheres are more reliable than thenear-IR absolute calibration measurements. Models are nowadayssophisticated enough to warrant detailed comparisons between ob-servation and theory and the latest space-based measurements con-firm the validity of the adopted absolute calibration (see AppendixA). We also differ from the recent work of Ramırez & Melendez(2005a) in using the most recent model atmospheres and absolutecalibration available. Furthermore, we use only direct observationaldata, explicitly avoiding the use of colour calibrations from previousstudies to infer the bolometric luminosity.

Following Alonso et al. (1995), the flux outside the wavelengthrange covered by our photometry has been estimated using modelatmospheres. Since the percentage of Fbol measured in the directlyobserved bands range from ∼70 to ∼85 per cent depending on thestar, the dependence of the estimated bolometric flux on the modelis small.

In Sections 4.1 and 4.2, we give a detailed description of ourimplementation of the IRFM, mainly following the formulation andterminology adopted by Alonso et al. (1995, 1996a).

Our procedure can be summarized as follows. Given the metal-licity, the surface gravity (assuming log (g) = 4.5 for all our dwarfs,see Section 3.1) and an initial estimate for the temperature of astar, we have interpolated over the grid of model atmospheres tofind the spectrum that best matches these parameters. This spec-trum is used to estimate that fraction of the bolometric flux out-side our filters [BV(RI )C JHKS], i.e. the ‘bolometric correction’.The bolometric flux is determined from the observations, includingthe bolometric correction. A new effective temperature, Teff, canbe computed by means of the IRFM. This temperature is used fora second interpolation over the grid, and the procedure is iterateduntil the temperature converges to within 1 K (typically within 10iterations).

We have tested the results using ATLAS9, MARCS, BaSel 2.1and 3.1 libraries. A detailed discussion on the dependence on theadopted model is given in Section 4.3.

In the following sections, we describe our implementation of theIRFM effective temperature scale.

4.1 Bolometric fluxes

As mentioned above, we use grids of synthetic spectra to bootstrapour IRFM. The grids of synthetic spectra all have a resolution of250 K in temperature, whereas the resolution in metallicity dependson the library, but with typical steps of 0.25 or 0.50 dex.

For any given star of overall metallicity [M/H] (cf. equation 1), weinterpolate over the grid of synthetic spectra in the following way.First, we use the Teff: (V − K) calibration of Alonso et al. (1996b)to obtain an initial estimate of the effective temperature Teff,0 ofthe star. Then we linearly interpolate in log (Teff) bracketing ourtemperature estimate Teff,0 at two fixed values of metallicity, whichbracket the measured [M/H] of the star. A third linear interpolationis finally done in metallicity in order to obtain the desired syntheticspectrum.

Having obtained the spectrum F(λ) that better matches the phys-ical parameters of the star, for any given band ζ (running from B toKS), we convolve it through the transmission curve Tζ (λ) of the filterand associate the resulting Fζ (model) with its effective wavelengthλeff (see Appendix B).

We then compute the flux covered by the passbands B to KS forthe model star [FB−KS (model)]. When the latter is compared withthe bolometric flux for the same model star [FBol(model)], we canobtain an estimate of the fraction of the flux C encompassed by ourfilters:

C = FB−KS (model)

FBol(model). (3)

We now have the correction factor 1/C for the missing flux of agiven star, we use its observed BV(RI )C JHKS magnitudes (mζ ) tocalculate the flux as it arrives on the Earth

Fζ (Earth) = F stdζ (Earth)10

−0.4(

mζ −mstdζ

), (4)

where F stdζ (Earth) is the absolute calibrated flux on the Earth of

the standard star and mstdζ is its observed magnitude. The observed

magnitudes and the absolute calibration of the standard star arethose given in Tables A1 and A2, respectively, and play a key rolein determining both bolometric flux and effective temperature as wediscuss in Section 4.4.3.

The flux at the Earth for each band is once again associated withthe corresponding effective wavelength of the star, and a simpleintegration leads to FB−KS (Earth). The latter is then divided by theC factor as defined in equation (3) in order to obtain the bolometricflux measured on the Earth, FBol(Earth).

4.2 Effective temperatures

The effective temperature Teff of a star satisfies the Stefan–Boltzmanlaw FBol = σ T 4

eff, where FBol is the bolometric flux on the surfaceof the star. Only the bolometric flux on the Earth is measurable, andwe must take into account the angular diameter (θ ) of the star. Thus,

FBol(Earth) =(

θ

2

)2

σ T 4eff. (5)

The way to break the degeneracy between effective temperatureand angular diameter in the previous equation is provided by the


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IRFM (Blackwell & Shallis 1977; Blackwell, Shallis & Selby 1979;Blackwell, Petford & Shallis 1980). The underlying idea relies onthe fact that, whereas the bolometric flux depends on both angu-lar diameter and effective temperature (to the fourth power), themonochromatic flux in the IR at Earth – FλIR (Earth) – depends onthe angular diameter but only weakly (roughly to the first power) onthe effective temperature:

FλIR (Earth) =(

θ

2

)2

φ(Teff, g, λIR), (6)

where φ(Teff, g, λIR) is the monochromatic surface flux of the star.The ratio FBol(Earth)/FλIR (Earth) defines what is known as the ob-servational R-factor (Robs), where the dependence on θ is eliminated.In this sense, the IRFM can be regarded as an extreme exampleof a colour method for determining the temperature. By means ofsynthetic spectra, it is possible to define a theoretical counterpart(Rtheo) on the surface of the star, obtained as the quotient betweenthe integrated flux (σT4

eff) and the monochromatic flux in the IRFλIR (model).

The basic equation of the IRFM thus reads

FBol(Earth)

FλIR (Earth)= σ T 4

eff

FλIR (model)(7)

and can be immediately rearranged to give the effective temperature,eliminating the dependence on θ .

The monochromatic fluxFλIR (Earth) is obtained from the IR pho-tometry using the following relation:

FλIR (Earth) = q(λIR) F stdλIR

(Earth) 10−0.4(

mIR−mstdIR

), (8)

where q(λIR) is a correction factor to determine monochromaticflux from broad-band photometry, mIR is the IR magnitude of thetarget star, mstd

IR and F stdλIR

(Earth) are the magnitude and the absolutemonochromatic flux of the standard star in the same IR band (seeTables A1 and A2). For the standard and the target star, λIR refersto their respective effective wavelength in the given IR band, as wediscuss in more detail, together with the definition of the q factor,in Appendix B.

The IRFM is often applied using more than one IR wavelength anddifferent Teff are obtained for each wavelength. Ideally, the derivedtemperatures should of course be independent of the monochromaticwavelength used. In our case, we have used J, H, KS photometry andthe corresponding effective wavelengths.

We have tested the systematic differences in temperatures, lumi-nosities and diameters obtained when only one IR band at a timeis used for the convergence of the IRFM. The results are shown inFig. 7.

The J band returns temperatures that are systematically coolerthan those found from KS, which translates into greater angulardiameters. The H band returns temperatures that are systematicallyhotter and therefore smaller angular diameters than in KS. Thesesystematic differences do not appear to depend on the temperaturerange, at least between 4500 and 6500 K.

The systematic offsets could be traced back to the absolute cal-ibration in different bands. We further point out that for the sakeof this test, the convergence on Teff in the IRFM is required in oneband only and in Fig. 7 the differences are thus exaggerated. Thediscrepancy can reach a few per cent in temperatures and diameters,whereas for bolometric luminosities it is usually within 2 per centthough it increases with increasing temperature (Fig. 7). This is dueto the fact that the derived bolometric luminosity is constrained fromthe full optical and IR photometry and absolute calibration, whereas

Figure 7. Comparison of effective temperatures, angular diameters andbolometric luminosities obtained in the three IR bands used. Angular diam-eters have been scaled using the stellar parallaxes from Hipparcos, althoughthe parallaxes do not affect the comparison since all bands scale accordingly.The luminosities of the stars are in good agreement across all the IR bands(because luminosity is also tied down by the optical data as discussed in thetext). Offsets of this size are likely to stem from the difficulties in setting thezero-points of the absolute calibration. Horizontal and diagonal lines withslope 1 are intended to guide the eye.

for temperature and angular diameter, when relying on one IR bandonly, the absolute calibration in that band strongly affects the results.

Alonso et al. (1996a) concluded that the consistency of the threeIR bands they used (J, H, K) was good over 4000 K, but decidedto use only H and K below 5000 K and only K below 4000 K andbasically the same is done by Ramırez & Melendez (2005a). Wehave decided to use the temperature returned in all three bands alsobelow 5000 K, since systematic differences are not too large evenbelow 5000 K, the cooler temperatures in J band being compensatedby the hotter ones in H band.

At each iteration, the temperature used for the convergence isthe average of the three IR values weighted with the inverse oftheir errors ‘j ’, ‘h ’ and ‘k msigcom’. This temperature is thenused to select a new model interpolating over the grid of syntheticspectra as described in Section 4.1 and a new bolometric flux andtemperature are then computed. The procedure is iterated until theaverage temperature converges within 1 K. As can be appreciatedfrom Fig. 8, the systematic differences between the three bands aremuch reduced. Ideally, the ratio of the temperatures determined inthe three bands should be unity. The three mean ratios T(KS)/T(J),T(KS)/T(H) and T(H)/T(J) together with their standard deviations,are 1.0069 (σ = 0.77 per cent), 0.9943 (σ = 0.82 per cent) and1.0126 (σ = 0.89 per cent) and this also confirms the quality ofthe adopted absolute calibration. The mean standard deviation forthe three IR temperatures is 39 K, reflecting the different bandsensitivity to the temperature, the effect of the absolute calibra-tion in different bands as well as the observational errors in the IRphotometry.


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Figure 8. Differences as a function of temperature between the final adoptedvalues and those obtained in each ζ = J, H, KS band when the convergenceis done averaging the three IR values.

4.3 Internal accuracy and dependence on the adopted library

We have probed the accuracy and the internal consistency of theprocedure described in Sections 4.1 and 4.2 generating syntheticmagnitudes in the range 4250 � Teff � 6500 and −2.0 � [Fe/H] �0.5 and testing how the adopted method recovers the temperaturesand luminosities of the underlying synthetic spectra. The accuracyis excellent given that in all three IR bands the IRFM recoversthe right temperature within 1 K and the bolometric luminosity(i.e. the theoretical value σT4

eff) within 0.1 and 0.04 per cent forthe ATLAS and MARCS models, respectively, i.e. at the level ofthe numerical accuracy of direct integration of the bolometric fluxfrom the spectra. This confirms that the interpolation over the gridintroduces no systematics and even a poor initial estimate of thetemperature does not affect the method.

We expect the dependence of the results on the adopted syntheticlibrary to be small, since most of the luminosity is actually observedphotometrically and the model dependence for the q factor and Rtheo

(see Section 4.2) is weak because we are working in a region ofthe spectrum largely dominated by the continuum. The method is infact more sensitive to the adopted absolute calibration that governsRobs.

The principal contributor to continuous opacity in cool stars isdue to H−. Blackwell, Lynas-Gray & Petford (1991) have shownthat by using more accurate opacities with respect to the previouswork, temperatures increased by 1.3 per cent and angular diametersdecreased up to 2.7 per cent, the effect being greatest for cool stars.Even though the dependence of the results on the adopted syntheticlibrary is small, the use of absolute calibrations derived using themost up-to-date model atmospheres (see Appendix A) is of primaryimportance. Though we use a great deal of observational informa-tion, the new opacity distribution function (ODF) in the adoptedgrid of model atmospheres is particularly important. With respectto the more accurate but computationally time consuming opacitysampling (OS), older ODF models can underestimate the IR flux byfew per cent, translating into cooler effective temperatures (Grupp2004b). Likewise, the better opacities in the UV lead to an increasedflux in the visual and IR region. This directly affects the ratio be-tween the bolometric and monochromatic flux used for the IRFM(see Section 4.2), so that the latest model atmosphere have to bepreferred (Megessier 1994, 1998).

For all our 104 G–K dwarfs, we have tested how the recoveredparameters change by using ATLAS9, MARCS, BaSel 2.1 and 3.1

Figure 9. Comparison between final effective temperatures, bolometric lu-minosities and angular diameters for our sample stars, when different syn-thetic libraries (with respect to ATLAS9) are used.

libraries. The comparison in temperatures, luminosities and angulardiameters is shown in Fig. 9.

The ATLAS9 and MARCS models show a remarkably goodagreement through all the temperature range, with differences inthe temperature that never exceed a few degrees (or 0.2 per cent)and those in the bolometric luminosities and the angular diametersbeing always within 1 per cent. Though not a proof of the validityof the models, it is very encouraging that the sophisticated physicsimplemented in independent spectral synthesis codes shows such ahigh degree of consistency.

The BaSel models show bigger differences with respect toATLAS9. The BaSel 2.1 model tends to give slightly hotter tem-peratures, higher luminosities and smaller angular diameters whencompared to ATLAS9, though the systematic is oscillating (reflect-ing the underlying continuum adjustments of the semi-empiricalBaSel libraries).

We conclude by noting that in the range of temperatures and lumi-nosities studied the dependence of our results on the adopted modelnever exceeds a few per cent and for the latest models (MARCS andATLAS9) the agreement is well within 1 per cent. Hence we willonly present and discuss results based on ATLAS9 in the following.

4.4 Evaluation of the errors: rounding up the usual suspects

Though the IRFM is known to be one of the most accurate waysof determining stellar temperatures without direct angular diame-ter measurements, its dependence on the observed photometry andmetallicities, on the absolute calibration adopted and on the libraryof synthetic spectra make the evaluation of the errors not straightfor-ward. In this work, we have proceeded from first principles and madeuse of high-accuracy observations only, thus facilitating the estima-tion of errors and possible biases. Since we made use of photometricmeasurements in many bands, we expect the dependence on pho-tometric errors to be rather small, small random errors in differentbands being likely to compensate each other. We have also shownin the previous section that when the latest model atmospheres areadopted, the model dependence is below the one per cent level. Theeffect of the absolute calibration can be regarded as a systematicbias; once the absolute calibration in different bands is chosen, all


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temperatures scale accordingly. As we show in Appendix A, theadopted absolute calibration has been thoroughly tested.

4.4.1 Photometric observational error

The photometric errors in different bands as well as errors in metal-licity are likely to compensate each other. In order to check this,for each star we have run 1000 Monte Carlo simulations, assigningeach time random errors in BV(RI )C JHKS and [Fe/H]. The errorshave been assigned with a normal distribution around the observedvalue and a standard deviation of 0.01 mag in BV(RI )C and of ‘j ’,‘h ’ and ‘k msigcom’ in JHKS. Errors in [Fe/H] are those given inthe papers from which we have collected abundances.

The standard deviation of the resulting temperatures obtainedfrom the Monte Carlo for each star is typically 11 K and neverexceeds 19 K. Bolometric luminosities have a mean relative errorof only 0.5 per cent and never exceeding 0.8 per cent; for angulardiameters, the mean error sets to 0.4 per cent and never exceeds 0.5per cent.

4.4.2 Metallicity and surface gravity dependence

The interpolation within the grid of synthetic spectra is as a functionof [M/H]. We have checked the effect of α-elements by makingthe interpolation as a function of [Fe/H] instead, finding very littledifferences in the derived fundamental parameters around the solarvalues. At lower metallicities, the effect of the α-elements start tobe important as expected (Fig. 10). The effect is in any case, alwayswithin 20 K in Teff and within 2 per cent in bolometric luminosityand θ .

Spectroscopic metallicities have typical errors well within0.10 dex. As we show in Fig. 11, a change of a few dex in metallicityhas almost no effect on the resulting temperatures around the solarvalues, but at lower metallicities systematic errors of 0.10–0.20 dexin [Fe/H] can introduce biases up to 40 K. This is an important pointwhen using large surveys with photometric metallicities that havetypical errors above 0.10 dex.

The dependence on the adopted surface gravity is very mild fordwarf stars. We have verified that a change of ±0.5 dex in log (g)

Figure 10. Differences between temperatures, bolometric luminosities andangular diameters when the [M/H] or [Fe/H] are used.

Figure 11. Effect of errors in metallicities on the resulting temperatures.

implies differences that never exceed ∼30 K in temperature andwell within 1 per cent for bolometric fluxes and angular diameters.

4.4.3 Systematic error in the absolute calibration

The errors in the adopted magnitudes and absolute calibrations ofVega in different bands can be regarded as a systematic bias, sinceonce they are selected, the recovered temperatures, luminosities andangular diameters scale accordingly. Errors in the observed BV(RI )C

magnitudes of Vega are around 0.01 mag, whereas the uncertaintiesin JHKS magnitudes given by Cohen, Wheaton & Megeath (2003)are within a few millimag. As for the random errors on the pho-tometric zero-points, it is very likely that uncertainties in Vega’smagnitudes compensate each other.

On the other hand, the uncertainties in the adopted absolute cal-ibration mainly come from the uncertainty in the flux of Vega at5556 Å. Since the absolute calibration in all bands scales accord-ingly (even though we have used slightly different approaches inoptical and IR, see Appendix A), we evaluate the systematic er-ror in temperatures for the worst-case scenario, i.e. when all errorscorrelate to give systematically higher or lower fluxes. The uncer-tainties adopted are those given in Table A2. The results are shownin Fig. 12.

Finally, a detailed comparison with data from satellites validateswithin the errors the adopted calibration, though it seems to sug-gest that IR fluxes should be brightened by 1 per cent at most (seeAppendix A). If so, the resulting temperatures would cool down by10–30 K and luminosities and angular diameters would increase onaverage by 0.2 and 0.7 per cent, respectively.

4.4.4 The final error budget

The primary estimate for errors in Teff is from the scatter in the tem-perature deduced from J, H and KS bands that reflects photometricerrors as well as differences in the corresponding absolute calibra-tion (see Section 4.2). As regards the uncertainties discussed inSections 4.4.2 and 4.4.1, they are of the same order and we quadrat-ically sum them to the standard deviation in the resulting J, H andKS for a fair estimate of the global errors on temperatures.


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Figure 12. Effect of the absolute calibration being systematically brighter(circles) or fainter (squares) when uncertainties given in Table A2 correlate.Translated into magnitudes, it corresponds to a mean shift of Vega’s zero-points of ±0.01 mag in the optical and ±0.02 mag in the IR.

Table 2. Mean accuracy of the derived fundamen-tal stellar parameters. Note that either the systematicsgiven here, or those shown in Fig. 12 are determinedaccording to Table A2 and do not account for possi-ble (and only indicative) shifts in Vega’s zero-pointsas a consequence of its rapidly rotating nature (seeAppendix A).

Internal accuracy Systematics

Teff 0.8 per cent 0.3 per centFBol 0.5 per cent 1.2 per cent

θ 1.7 per cent 0.7 per cent

The uncertainties of the bolometric luminosities mostly dependon the photometry that accounts for ∼70–85 per cent of the resultingluminosity. The errors from the observations have been estimatedby summing in quadrature those from the Monte Carlo simulation(Section 4.4.1) to those due to a change of ±0.5 dex in log (g).

Finally, for the resulting angular diameters we propagate the er-rors in temperature and luminosity from equation (5). The meaninternal accuracy of the resulting temperatures is of 0.8 per cent,that of the luminosities of 0.5 per cent and that of the angular diam-eters of 1.7 per cent.

There are possible systematic errors coming from the adoptedabsolute calibration. As can be seen from Fig. 12 such uncertaintiesgive an additional mean error of 0.3 per cent in temperature and1.2 per cent in luminosity that translate into an additional system-atic error of 0.7 per cent in angular diameters. A summary of theuncertainties is given in Table 2.

4.5 Colour–temperature–metallicity fitting formulae

To reproduce the observed relation Teff versus colour and to take intoaccount the effects of different chemical compositions, the followingfitting formula has been adopted (e.g. Alonso et al. 1996b; Ramırez

& Melendez 2005b; Masana, Jordi & Ribas 2006):

θeff = a0 + a1 X + a2 X 2 + a3 X [Fe/H]

+ a4[Fe/H] + a5[Fe/H]2, (9)

where θ eff = 5040/Teff, X represents the colour and ai (i = 1, . . . , 5)are the coefficients of the fit. In the iterative fitting, points departingmore than 3σ from the mean fit were discarded (very few pointsever needed to be removed). All our relations were adequately fit bysimple polynomials, and we did not need to got to higher orders inX to remove possible systematics as function of the metallicity (asin Ramırez & Melendez 2005b, who covered a much wider rangein temperature than we do here).

Fig. 13 shows the colour–temperature relations in different bands.The coefficients of the fits, together with the number of stars used,the range of applicability and the standard deviations are given inTable 3. Note that the very small scatter in the relations reflects thehigh quality and homogeneity of the input data.

5 T H E T E M P E R AT U R E S C A L E : S O M E L I K EI T H OT

In this section, we test our IRFM in a number of ways. First, wecompare its predictions to empirical data for the Sun and solar ana-logues. Secondly, we compare our predicted angular diameters torecent measurements with large telescope interferometers for a smallsample of G and K dwarfs. Thirdly, we compare our system to othertemperature scales for G and K dwarfs in the literature.

Overall, we find that our scale is in good agreement with otherscales though some puzzles remain. We make some suggestions forfurther work.

5.1 The Sun and solar analogues

While the temperature, the luminosity and the radius of the Sun areknown with great accuracy, its photometric colours can only be re-covered indirectly. Recently, Holmberg et al. (2006) have providedcolour estimates for the Sun based on those of solar analogues.Another way to obtain colours of the Sun levers synthetic photom-etry, convolving empirical or model spectra of the Sun with filterresponse functions (e.g. Bessell et al. 1998). Both methods lead toestimated colours that have typical uncertainties of a few 0.01 mag.Though this prevents the use of the Sun as a direct calibrator ofthe temperature scale, it is still useful to compare to its estimatedcolours.

5.1.1 The colours of the Sun

We have computed synthetic solar colours from the solar referencespectrum of Colina, Bohlin & Castelli (1996) which combines abso-lute flux measurements (from satellites and from the ground) witha model spectrum longward of 9600 Å. This spectrum, togetherwith those of three solar analogues, is available from the CALSPEClibrary.1 According to Colina et al. (1996), the synthetic opticaland near-IR magnitudes of the absolutely calibrated solar referencespectrum agree with published values to within 0.01–0.03 mag.

Recently, two new composite solar spectra extending up to 24 000Å have been assembled by Thuillier et al. (2004) based on the mostrecent space-based data. The accuracy in the UV-visible and near-IR is of the order of 3 per cent. The two solar spectra correspond

1 http://www.stsci.edu/hst/observatory/cdbs/.calspec.html


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Figure 13. Empirical colour–temperature relations. Circles are for stars with [Fe/H] � −0.5, crosses for stars with [Fe/H] > − 0.5. Red (cyan) continuouslines are our (Ramırez & Melendez 2005b) fitting formulae at solar metallicity. Dotted lines are for [Fe/H] =−1. In the IR, for stars with Teff below 5000 Kthe Ramırez & Melendez (2005b) calibration shows a larger metallicity dependence than indicated by our data.

Table 3. Coefficients and range of applicability of the colour–temperature relations.

Colour Metallicity range Colour range a0 a1 a2 a3 a4 a5 N σ (Teff)

B − V [− 1.87, 0.34] [0.419, 1.270] 0.5121 0.5934 −0.0618 −0.0319 −0.0294 −0.0102 103 54V − RC [− 1.87, 0.34] [0.249, 0.755] 0.4313 1.5150 −0.7723 −0.0950 0.0179 −0.0033 101 55(R − I)C [− 1.87, 0.34] [0.268, 0.640] 0.2603 2.3449 −1.4897 −0.1149 0.0641 0.0023 104 70V − IC [− 1.87, 0.34] [0.517, 1.395] 0.3711 0.8994 −0.2467 −0.0545 0.0393 −0.0010 101 49V − J [− 1.87, 0.34] [0.867, 2.251] 0.4613 0.4118 −0.0473 −0.0356 0.0535 0.0012 104 30V − H [− 1.87, 0.34] [1.144, 2.847] 0.4797 0.3059 −0.0252 −0.0196 0.0426 0.0036 99 25V − KS [− 1.87, 0.34] [1.124, 3.010] 0.4609 0.3069 −0.0263 −0.0145 0.0275 0.0006 103 21

Notes. N is the number of stars employed for the fit after the 3σ clipping and σ (Teff) is the final standard deviation of the proposed calibrations.

Table 4. Magnitudes and colours of the Sun compared to theoretical colours and those deduced using our temperature scale.

V U − B B − V V − RC (R − I)C V − J V − H V − KS Ref.

– 0.173 ± 0.064 0.642 ± 0.016 0.354 ± 0.010 0.332 ± 0.008 1.151 ± 0.035 1.409 ± 0.035 1.505 ± 0.041 (a)−26.742 0.131 0.648 0.373 0.353 1.165 1.483 1.555 (b)−26.743 0.146 0.635 0.374 0.358 1.189 1.528 1.613 (c)−26.740 0.101 0.645 0.358 0.349 1.162 1.495 1.556 (d)−26.746 0.120 0.658 0.370 0.350 1.166 1.483 1.555 (e)−26.753 0.188 0.633 0.360 0.345 1.154 1.485 1.547 (f)

– – 0.651 0.356 0.330 1.150 1.459 1.546 (g)

Notes. (a) Holmberg et al. (2006); (b) Colina et al. (1996); (c) Thuillier et al. (2004) (d) ATLAS9 ODFNEW with Grevesse & Sauval solar abundances; (e)Kurucz 2004 model with resolving power 100 000; (f) MARCS; (g) our temperature scale.

to moderately high and moderately low solar activity conditions,although the effect of activity on the resulting synthetic coloursis at or below the millimag level, except for U − B (0.003 mag).Such small differences could be due to the solar spectral variabil-ity (which increases towards shorter wavelengths), the accuracy ofthe measurements or both. In what follows, we adopt therefore thepragmatic approach of averaging the colours returned by the twospectra and we generically refer to the results as the Thuillier et al.(2004) spectrum.

We have also computed the theoretical magnitudes and coloursof the Sun predicted by the latest Kurucz and MARCS syntheticspectra. Magnitudes and colours computed from the aforementionedsolar spectra are given in Table 4 together with the empirical coloursof Holmberg et al. (2006).

Although we have not looked at stellar U magnitudes in this study,for completeness with the work of Holmberg et al. (2006) we have

generated synthetic magnitudes in this band too. However, in whatfollows we focus our discussion from B to KS band (for a discussionof the theoretical solar U − B colour, see e.g. Grupp 2004a andreferences therein). We have used only Vega to set the zero-points.The differences in the resulting colours when also Sirius is used forthe optical bands are given in Appendix A and tend to make thesolar B − V even redder. Assuming no deficiencies in the syntheticspectra, the uncertainties in the derived colours are entirely dueto the uncertainties in the adopted zero-points, i.e. of the order of∼0.01 mag.

The CALSPEC and Thuillier et al. (2004) spectra are already ab-solutely calibrated for a distance of 1 au. For the absolute calibrationof the Kurucz and MARCS solar spectra, we also adopt a distanceof 1 au. The synthetic spectra have a temperature T = 5777 K andwe adopt L = 3.842 × 1033 erg s−1 (Bahcall, Serenelli & Basu

2006) from which we deduce R =√

L/4πσ T4 to be used for


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Table 5. Recovered temperature and luminosity of the Sun.

Teff (K) Luminosity ( LL ) Ref.

5864 – (a)5802 1.004 (b)5720 1.008 (c)5770 0.992 (d)5794 1.003 (e)5791 1.005 (f)

Notes. (a) Holmberg et al. (2006); (b) Colina et al. (1996);(c) Thuillier et al. (2004); (d) ATLAS9 ODFNEW withGrevesse & Sauval solar abundances; (e) Kurucz 2004model with resolving power 100 000; (f) MARCS.

the absolute calibration.2 Proceeding this way, we can immediatelycompare the recovered temperatures and luminosities by means ofthe IRFM (see Table 5) with those given above. The effect of theabsolute calibration of the solar spectra is immediately seen in thededuced value of V and reflects in the recovered luminosity, whereasthe temperature depends on the colours only.

An influential direct V measurement of the Sun is that of Stebbins& Kron (1957) to which Hayes (1985) claimed a further 0.02-magcorrection for horizontal extinction. Using recent standard V magni-tudes for the Stebbins & Kron (1957) comparison G dwarfs returnsfor the Sun V = −26.744 ± 0.015, which becomes V = −26.76 ±0.02 after accounting for the Hayes correction (Bessell et al. 1998).The values deduced using composite and synthetic spectra are closerto the measurements of Stebbins & Kron but fully within the errorbars of the Hayes’ value. Note that through the year the variationof the solar distance due to the ellipticity of the Earth’s orbit corre-sponds to a V difference of ±0.035 mag.

In the optical bands, synthetic and composite colours show a re-markably good agreement with the recent empirical determination,differences being at most 0.01–0.02 mag and therefore within theerror bars. Synthetic and composite spectra also confirm the red-der B − V colour with respect to the determination of Sekiguchi& Fukugita (2000) (B − V = 0.626) and Ramırez & Melendez(2005b) (B − V = 0.619), though the Thuillier et al. (2004) spec-trum is slightly bluer than the Neckel & Labs (1984) measurementsused in the Colina et al. (1996) composite spectrum. Using the em-pirical value of B − V = 0.642 in the Ramırez & Melendez (2005b)colour–temperature relation implies a temperature of the Sun of5699 K, therefore suggesting the need of a hotter temperature scalefor this colour. Such an offset is of the same order of those found inSection 3.3 for optical and IR bands.

In the IR, the only directly measured spectrum is that of Thuillieret al. (2004). It is much redder than the empirical determinationof Holmberg et al. (2006) and also the model spectra. As concernsthe other models, J band shows good agreement with the empiricaldetermination. Though a systematic offset of ∼0.01 mag appears, itis fully within the error bars. The H and KS bands show larger sys-tematic differences of the order of 0.08 and 0.05 mag, respectively,as Fig. 5 already suggests. Such large differences are not easilyunderstood in terms of model failures. Interestingly, an offset of0.075 mag in the H band zero-point given by the 2MASS has alsobeen claimed by Masana et al. (2006) to constrain a set of solaranalogues to have just the same temperature of the Sun. The rea-

2 The deduced value θ = 0.009 301 79 rad compares well with that reportedin Schaifers & Voigt (1982) θ = 0.009 304 84 rad.

son of such a difference remains unclear as the adopted temperaturescale, the model and observational uncertainties all might play a role.Among other reasons, we mention that historically the H filter wasnot defined by Johnson and its values are not easily homogenized,so that further uncertainties might be introduced when comparingto various sources, though 2MASS now allows one to cope with thisproblem. According to Colina et al. (1996), the IR colours deducedfrom the composite spectrum agree within 0.01 mag with the solaranalogues measurements of Wamsteker (1981) and within 0.03 magwith the solar analogues measurements of Campins, Rieke &Lebofsky (1985). The synthetic colours obtained using the Colinaet al. (1996) solar spectrum also agree very closely with those ob-tained when the latest model atmospheres are used.

Using the colours in Table 4, the temperature of the syntheticKurucz and MARCS spectra is always recovered within 7–17 K(see Table 5), differences with the underlying values likely due tothe fact that synthetic solar spectra are tailored to match the observedabundances, log (g) and turbulent velocity whereas the interpolationwhen applying the IRFM is done over a more generic grid. Also,the temperature of 5802 K deduced when using the synthetic mag-nitudes from Colina et al. (1996) solar spectrum agrees very wellwith the known value of 5777 K, whereas the cooler temperature re-turned by the Thuillier et al. (2004) colours is a direct consequenceof the much redder IR colours. The temperature of the Sun obtainedwhen using the empirical colours of Holmberg et al. (2006) is 87-Khotter. Changing the empirical V − H from 1.409 to 1.480 lowersthe recovered temperature to 5811 K; when also V − KS is reddenedfrom 1.505 to 1.550 the recovered temperature then goes to 5774 K.Therefore, as expected, the temperature recovered critically dependson the IR magnitudes in H and KS bands. The H band is known tobe troublesome in both spectral modelling and observations, but adiscrepancy in the models ∼0.08 mag is difficult to understand inlight of the comparison with the observations in Section 3.

The empirical colours of the Sun of Holmberg et al. (2006) are de-duced interpolating in temperature and metallicity a sample of Sun-like stars with temperatures from Ramırez & Melendez (2005a).This scale is some 100-K cooler than our own and such a differencecan easily account for the differences in the IR colours. In compar-ison, the optical colours are almost unaffected by the temperatureused for the fit since they are primarily dependent on the adoptedmetallicity to fit the solar analogues.

5.1.2 The solar analogues

Absolutely calibrated composite spectra of three solar analogues,namely P041C, P177D and P330E (Colina & Bohlin 1997; Bohlin,Dickinson & Calzetti 2001), constitute a set of secondary flux stan-dards in addition to the white dwarf stars with pure hydrogen at-mospheres adopted for the Hubble Space Telescope (HST) UV andoptical absolute calibration. The three solar analogues also deter-mine the absolute flux distribution of the NICMOS filters.

The solar analogues’ flux distribution in the UV and optical re-gions is based on HST Faint Object Spectrograph (FOS) and SpaceTelescope Imaging Spectrograph (STIS) measurements, whereaslongward of 10 020 Å a scaled version of the Colina et al. (1996)absolute flux of the Sun is used.

The STIS solar analogues’ flux distribution in the opticaland longer wavelengths is expected to have uncertainties within2 per cent (Bohlin et al. 2001). Even if the near-IR fluxes of thesolar analogues have been constructed from models, a thoroughcomparison with the high-accuracy JHK IR photometry of Persson


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Table 6. Synthetic and observed 2MASS photometry of HST solar analogues.

J J (synth) H H (synth) KS KS (synth) �mJ �mH �mK Star

10.864 10.869 10.592 10.552 10.526 10.479 −0.010 −0.058 −0.005 P041C12.245 12.253 11.932 11.925 11.861 11.843 −0.007 −0.025 0.024 P177D11.781 11.796 11.453 11.471 11.432 11.390 – – – P330E

Notes. P041C = GSC 4413-304; P177D = GSC 3493-432; P330E = GSC 2581-2323. �m values refer to Bohlin’s model−2MASS difference relative toP330E, as explained in the text.

Table 7. Magnitudes, colours and temperatures of the solar analogues. Temperatures have been deduced using the given colours into the IRFM.

V U − B B − V V − RC (R − I)C V − J V − H V − KS Teff (K) Star

12.005 0.135 0.611 0.346 0.346 1.137 1.454 1.526 5828 P041C13.356 0.139 0.607 0.350 0.347 1.139 1.454 1.528 5831 P177D12.917 0.055 0.604 0.353 0.352 1.148 1.463 1.538 5822 P330E

Notes. Magnitudes and colours of P177D and P330E have been dereddened with the value of E(B − V) given in Bohlin et al. (2001) and using the standardextinction law of O’Donnell (1994) and Cardelli, Clayton & Mathis (1989) in the optical and IR, respectively. The higher temperatures of these solar analoguesare in agreement with the bluer colours and the recovered temperature of P041C agree very well with the estimated value of 5900 K given in Colina & Bohlin(1997). For the other two solar analogues, the reddening is used to adjust the continuum in the IR to match the UV–optical observations thus avoiding anyneed to estimate the temperature (Bohlin, private communication).

et al. (1998) gives confidence in the reliability of the adopted spec-tra. The differences between the observed magnitudes and the syn-thetic ones obtained by using the spectra of the three solar analoguesagree within 0.01 mag in J band. The differences in K band are lessthan 0.025 mag, whereas in H band the models are from 0.03 to0.07 mag brighter. Such differences can partly reflect difficulties inmodelling this spectral region; however, since such a problem is alsopresent when H band synthetic magnitudes of the white dwarf cal-ibrators are studied, an alternative or complementary explanationcould simply arise from errors in the adopted zero-point (Bohlinet al. 2001; Dickinson et al. 2002). For the three solar analogues,IR magnitudes are also available from 2MASS. The observed2MASS and model magnitudes for each of the solar analogue aregiven in Table 6. As done by Bohlin et al. (2001) for the Perssonphotometry, the �m values compare the relative observed and modelmagnitudes to the same difference for P330E (the primary NICMOSstandard):

�m = [msynth(star) − mobs(star)]

−[msynth(P330E) − mobs(P330E)]. (10)

Table 8. Temperature scale applied to solar analogues.

[Fe/H] V B − V J H KS T�eff (K) Teff (K)

HD 168009 −0.04 6.30 0.64(1) 5.12(0) 4.83(6) 4.75(6) 5801 5784HD 89269 −0.23 6.66 0.65(3) 5.39(4) 5.07(4) 5.01(2) 5674 5619HD 47309 +0.11 7.60 0.67(2) 6.40(3) 6.16(1) 6.08(6) 5791 5758HD 42618 −0.16 6.85 0.64(2) 5.70(1) 5.38(5) 5.30(1) 5714 5773HD 71148 −0.02 6.32 0.62(4) 5.15(8) 4.87(8) 4.83(0) 5756 5822HD 186427 +0.06 6.25 0.66(1) 4.99(3) 4.69(5) 4.65(1) 5753 5697HD 10145 −0.01 7.70 0.69(1) 6.45(0) 6.11(2) 6.06(3) 5673 5616HD 129357 −0.02 7.81 0.64(2) 6.58(3) 6.25(7) 6.19(2) 5749 5686HD 138573 −0.03 7.22 0.65(6) 6.02(7) 5.74(2) 5.66(2) 5710 5739HD 142093 −0.15 7.31 0.61(1) 6.15(8) 5.91(0) 5.82(4) 5841 5850HD 143436 0.00 8.05 0.64(3) 6.88(4) 6.64(9) 6.54(1) 5768 5813

Notes. Solar analogues drawn from Soubiran & Triaud (2004) and King et al. (2005). [Fe/H] and T�eff are those reported in the two

papers. For the stars from Soubiran & Triaud (2004), we have adopted the mean metallicity and effective temperature as given in theirtable 4. Teff is the final effective temperature obtained by averaging the temperatures given by our (B − V), (V − J), (V − H), (V −KS) calibration. For HD 168009 and HD 186427, (V − J) has not been used because of the poor quality flag associated with J magnitudes.

The uncertainties in the �m values depend only on the repeata-bility of the photometry and not on uncertainties in the absolutecalibration of the IR photometric systems.

The IR region of the solar analogues’ spectra is a scaled versionof that used for the Sun and the IR synthetic magnitudes of the solaranalogues have been proved to be reliable. The colours of the solaranalogues are bluer than those obtained from the solar spectra ofColina, Kurucz and MARCS and they are consistent with slightlyhigher temperatures for the solar analogues with respect to the Sun(Table 7). None the less, V − H and V − KS are still redder by ∼0.05and ∼0.03 mag, respectively, and the disagreement is even worseif we consider that we are actually comparing solar analogues withhotter temperatures (and bluer colours) than the Sun.

Finally, we test the temperature scale of Section 4.5 via 11 ex-cellent solar analogues for which accurate B, V , J, H, KS coloursare available. Seven are drawn from the top 10 solar analogues ofSoubiran & Triaud (2004) and four from the candidate solar twinsof King, Boesgaard & Schuler (2005). The results are shown inTable 8.

The agreement between our temperatures and those reportedin the two papers is outstanding, the mean difference being only


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�Teff = −7 ± 50 K our temperatures being only slightly cooler.Also, the mean temperature of the 11 solar analogues set to 5742 K,thus suggesting that our scale is well calibrated at the solar value.

5.2 Angular diameters

There are several approaches to deriving effective temperatures ofstars. Except when applied to the Sun, very few of them are gen-uinely direct methods which measure effective temperature empir-ically.

The direct methods rely on the measurement of the angular di-ameter and bolometric flux of the star. These fundamental methodsare restricted to a very few dwarf stars, although interferometric(e.g. Kervella et al. 2004) and transit (Brown et al. 2001) obser-vations have recently increased the sample. None the less angulardiameters obtained from both of the aforementioned methods haveto be corrected for limb darkening by using some model. Hencethe procedure is still partly model dependent. Observational un-certainties stem from systematic effects related to the atmosphere,the instrument and the calibrators used (e.g. Mozurkewich et al.2003). As to giants and supergiants, where more measurements areavailable, the comparison of 41 stars observed on both NPOI andMark III optical interferometers has shown an agreement within0.6 per cent, but with a rms scatter of 4.0 per cent (Nordgren, Sudol& Mozurkewich 2001).

The different limb-darkened values collected by Kervella et al.(2004) for the same stars give an idea of the uncertainties that mightstill hinder progress. For dwarf stars in particular, there are stillvery few measurements, so that a large and statistically meaningfulcomparison cannot yet be done. For example, the limb-darkenedmeasurements of the dwarf stars HD 10700 and HD 88230 obtainedby Pijpers et al. (2003) and Lane, Boden & Kulkarni (2001), re-spectively, are ∼5 and ∼9 per cent smaller than those of Kervellaet al. (2004) and those are ultimately used by Ramırez & Melendez(2005a) to check their scale. Besides, most of the limb-darkeningcorrections are done using 1D atmospheric models that rely on theintroduction of adjustable parameters like the mixing length.3D models for the K dwarf α Cen B provide a radius smaller byroughly 1σ stat (or 0.3 per cent) compared with what can be obtainedby 1D models (Bigot et al. 2006). However, for hotter stars the cor-rection due to 3D analysis is expected to be larger as a consequenceof more efficient convection. Interestingly, using parallax to convertthe smaller angular diameter obtained from 3D models into lin-ear radius returns 0.863 R, which is in better agreement with theasteroseismic value of 0.857 R by Thevenin et al. (2002). For Pro-cyon, the difference between 1D and 3D models amounts to roughly1.6 per cent, implying a correction to Teff of the order of 50 K(Allende Prieto et al. 2002).

If temperatures are to be deduced from these diameters, furtherdependence on bolometric fluxes, often gathered from a variety ofnon-homogeneous determinations in the literature, enters the game.All these complications eventually render ‘direct’ methods to benot as direct, or model independent, in the end. In any case, directangular diameter measurements are still limited to the nearest (andbrightest) dwarfs with metallicity around solar.

In our case, the comparison with direct methods is hamperedsince the angular diameter measurements available so far are ofnearby and bright dwarfs that have unreliable or saturated 2MASSphotometry. Since we do not have any star in our sample with directangular diameter measurements, we can only perform an indirectcomparison by using the stars we have in common with Ramırez& Melendez (2005a). For the common stars, our angular diameters

are systematically slightly smaller (∼3 per cent) than those foundby Ramırez & Melendez (2005a) which agree well with the directdeterminations of Kervella et al. (2004). Nevertheless, for individualmeasurements all determinations agree within the errors.

Finally, we mention another indirect way to determine angu-lar diameters. This is provided by spectrophotometric techniques,comparing absolutely calibrated spectra with model ones via equa-tion (A3) (Cohen et al. 1996, 1999). In the context of the absolutelycalibrated spectra assembled by Cohen and collaborators (see Ap-pendix A), such a procedure has shown good agreement with directangular diameter measurements of giants, even though spectropho-tometry leads to angular diameters systematically smaller by a fewper cent. Considered the aforementioned scatter for angular diam-eters of giants, the difference is not worrisome. However, since theabsolute calibration we have used for the IRFM is ultimately basedon the work of Cohen and collaborators (see Appendix A), such adifference is interesting in light of our results (see also Section 6.1).

6 C O M PA R I S O N W I T H OT H E RT E M P E R AT U R E S C A L E S

Most ways to determine effective temperatures are indirect methodsand to different extent all require the introduction of models. Otherthan via the IRFM, for the spectral types covered by this study, ef-fective temperatures may be determined via (i) matching observedand synthetic colours (Masana et al. 2006), (ii) the surface bright-ness technique (e.g. Di Benedetto 1998), (iii) fitting observed spec-tra with synthetic ones (Valenti & Fischer 2005), (iv) fitting of theBalmer line profile (e.g. Mashonkina et al. 2003), (v) from spectro-scopic conditions of excitation equilibrium of Fe lines (e.g. Santos,Israelian & Mayor 2004; Santos et al. 2005) and (vi) line depth ratios(e.g. Kovtyukh et al. 2003).

In what follows, we compare our results with those obtained viathese indirect methods. Only single and non-variable stars with ac-curate photometry according to the requirements of Sections 2 and2.2 have been used for the comparison.

6.1 Ramırez & Melendez sample

The widest application of the IRFM to Pop I and II dwarf starsis that of Alonso et al. (1996a). Recently, Ramırez & Melendez(2005a) have extended and improved the metallicities in the sampleof Alonso et al. (1996a) recomputing the temperatures. As expected,the updated temperatures and bolometric luminosities do not sig-nificantly differ from the original ones, since the absolute IR fluxcalibration (i.e. the basic ingredients of the IRFM) as well as thebolometric flux calibration used are the same as of Alonso et al.(1994, 1995). The difference between the old and new temperaturescale is not significant, though in the new scale dwarf stars are some40-K cooler.

For 18 stars in common between our study and that of Ramırez& Melendez (2005a), we find an average difference �Teff = 105 ±72 K, our scale being hotter. This translates into a mean Teff differ-ence of 2.0 per cent, luminosities brighter by 1.4 per cent and angulardiameters smaller by 3.3 per cent (Fig. 14). Though not negligible,such differences are within the error bars of current temperaturedeterminations.

The strength of their temperature scale is that the absolute calibra-tion adopted was derived demanding the IRFM angular diameters tobe well scaled to the directly measured ones for giants (Alonso et al.1994). The absolute calibration of Vega was tuned to return angulardiameters that matched the observed ones; hence it depends on the


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Figure 14. Comparison between effective temperatures, bolometric lumi-nosities and angular diameters obtained in this work (TW) and from Ramırez& Melendez (RM). We have checked for dependencies in all the scales on[Fe/H], Teff, θ and bolometric luminosities and found no significant depen-dencies.

input angular diameters and IR photometry that were expected tobe very accurate. Interestingly, the method applied to hot (Teff >

6000 K) and cool (Teff < 5000 K) stars returned absolute calibra-tions that differed by about 8 per cent in all IR photometric bands,thus suggesting that effective temperatures for cool and hot starsand those determined from the IRFM were not on the same scale.The cause was some sort of systematic error affecting direct angu-lar diameters and/or model inaccuracies in predicting IR fluxes; thefinal adopted absolute calibration was then a weighted average ofthat returned for hot and cool stars.

The comparison of the angular diameters derived by Ramırez &Melendez (2005a) for dwarf stars with the 13 recent interferometricmeasurement of subgiant and main-sequence stars (Kervella et al.2004) seems to imply that the adopted absolute calibration holdsalso in this range (Ramırez & Melendez 2004, 2005a). They findgood agreement with direct angular diameter measurements, butone should be reminded that a well defined angular diameter scalefor dwarf stars is not yet available (see Section 5.2). Also, the starswe have in common with Ramırez & Melendez (2005a) have allangular diameters θ < 0.5 mas and differences between our andtheir values lie all below 0.02 mas i.e. below the uncertainties ofcurrent measurements (see table 4 of Ramırez & Melendez 2005a).

We strongly suspect that our hotter temperature scale is the re-sult of the 2MASS versus Alonso absolute calibration as the al-most constant offset in Fig. 14 suggests. When we apply our IRFMtransforming first 2MASS photometry into the TCS system used byAlonso, by means of the relations given by Ramırez & Melendez(2005a) and adopting the TCS filters with the absolute calibrationand Vega’s zero-points in the IR given by Alonso, our temperaturescale sets on to that of Ramırez & Melendez within 20 K, with dif-ferences in temperatures, luminosities and angular diameters well

below 1 per cent. The confidence in our adopted absolute calibrationand zero-points for Vega comes from the extensive comparison withground and space based measurements (see Appendix A). Further-more, we also prefer to avoid any transformation from the 2MASSto the TCS system since that would introduce further uncertaintiesof 0.03–0.04 mag in the photometry (Ramırez & Melendez 2005a).

Finally, we mention another extensive application of the IRFM,that of Blackwell & Lynas-Gray (1998). For 10 common stars, ourmean temperatures are hotter by 60 ± 67 K (1.0 per cent), ourluminosities are brighter by 1.3 per cent and our angular diameterssmaller by 1.4 per cent.

6.2 Masana et al. sample

Recently, Masana et al. (2006) have derived stellar effective tem-peratures and bolometric corrections by fitting V and 2MASS IRphotometry. They calibrate their scale by requiring a set of 50 solaranalogues drawn from Cayrel de Strobel (1996) to have on averagethe same temperature as the Sun. As a result, they find significantshifts in the 2MASS zero-points given by Cohen et al. (2003). Re-markably, the shift in H band is found to be 0.075 mag, i.e. ofthe same order of the discrepancy between synthetic and empiricalcolour (once the Ramırez & Melendez scale is adopted) found inthis band for the Sun (see Section 5.1).

For 64 common stars, our Teff scale is cooler by 50 ± 80 K(1.0 per cent), our luminosities are fainter by 6.9 per cent and ourangular diameters are smaller by 1.5 per cent (see Fig. 15).

Masana et al. (2006) found agreement within 0.3 per cent betweentheir angular diameters and the uniform disc measurements collectedin the CHARM2 catalogue of Richichi & Percheron (2005), with astandard deviation of 4.6 per cent. Since the standard deviation isof the same order of the correction between uniform to limb dark-ened disc, for the comparison they adopt the questionable choiceof using the uniform disc rather than the proper limb-darkenedone.

Figure 15. Same as Fig. 14, but between our work (TW) and the Masana,Jordi & Ribas scale (MJR).


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From 385 stars in common with Ramırez & Melendez (2005a),Masana et al. (2006) found that their temperature scale is on average58-K hotter than Ramırez & Melendez. From stars in common, wefind our temperatures are on average 50-K cooler than that of Masanaet al. (2006), leading us to expect we would be on the same scale asRamırez & Melendez (2005a), yet from stars in common (differentstars) we find we are 105-K hotter than Ramırez & Melendez, avery puzzling result! We worked very hard to impute this inconsis-tency to one or other of the scales, but were unable to resolve theproblem. This section demonstrates that temperatures, luminositiesand angular diameters calibrations for our stars, retain systematicsof the order of a few per cent, despite the high internal accuracy ofthe data.

6.3 Di Benedetto sample

Another way to determine effective temperatures and angular diam-eters of stars is via the surface brightness technique. We have ninestars in common with the extensive work of Di Benedetto (1998).On average, our stars are 50 ± 50 K hotter, the luminosities arebrighter by 0.9 per cent and the angular diameters are smaller by1.4 per cent (see Fig. 16).

Comparing to the angular diameters from the IRFM sample ofBlackwell & Lynas-Gray (1998), Di Benedetto (1998) found in theF–G–K spectral range an overall agreement well within 1 per centthough with an intrinsic scatter as large as 2 per cent.

6.4 Valenti & Fischer sample

Valenti & Fischer (2005) have presented a uniform catalogue ofstellar properties for 1040 nearby F, G and K stars that have beenobserved by the Keck, Lick and AAT planet search programmes.Fitting observed echelle spectra with synthetic ones, they have ob-tained effective temperatures, surface gravities and abundances forevery star. We have 47 stars in common. Except for two of them

Figure 16. Same as Fig. 14, but between our work (TW) and the DiBenedetto scale (diB).

(HD 22879 and HD 193901) which depart from the comparison by242 and 497 K, respectively, we obtain an excellent average agree-ment �Teff = 6 ± 60 K. We could not single out a reason of such alarge discrepancy with the two outliers; however, we are confident inour values since they agree within 80–150 K with the temperaturesof Ramırez & Melendez (2005a) and Masana et al. (2006).

6.5 Santos et al. sample

Santos et al. (2004, 2005) have carried out a detailed spectroscopicanalysis for 119 planetary-host stars and 95 single stars. They ob-tained spectroscopic temperatures based on the analysis of severalFe I and Fe II lines. The comparison done by Ramırez & Melendez(2004, 2005a) seems to imply that such temperatures are hotter byabout ∼100 K. However, for stars with direct angular diameter mea-surements Santos et al. (2005) found an agreement within 7 K withthe direct measured temperatures reported in Ramırez & Melendez(2004).

After removing a star that departs 280 K (HD 142709), the dif-ference between our and Santos Teff scales is 15 ± 81 K, our scalebeing cooler. The reason of the large departure for the one outlieris not clear, but our value agrees better with that of Masana et al.(2006).

Another study with temperatures derived using strictly spectro-scopic criteria (based on the excitation equilibrium of the iron lines)is that of Luck & Heiter (2005). We have only five stars in common;however, the average agreement is good (�Teff = 11 ± 142 K). Eventhough such a result is dominated by the scatter, it further suggeststhat our IRFM temperatures agree very well with spectroscopic de-terminations.

6.6 Mashonkina et al. sample

Balmer line profile fitting allows a very precise determination of stel-lar effective temperature for cool stars (Fuhrmann, Axer & Gehren1993, 1994). Mashonkina & Gehren (2000, 2001) and Mashonkinaet al. (2003) have extensively used such a technique to derive effec-tive temperatures to be used for their detailed abundance analysis.We have nine stars in common and the mean difference is 37 ±64 K, our temperatures being hotter.

6.7 Kovtyukh et al. sample

The line depth ratios technique (Gray 1989, 1994; Gray & Johanson1991) applied to high signal-to-noise ratio (S/N) echelle spectrais capable of achieving an internal precision as high as 5–10 K.Kovtyukh et al. (2003) have applied such a technique to a set of 181F–K dwarfs and adjusted the zero-point of the scale on the solarreflected spectra taken with ELODIE, leading to the uncertainty inthe zero-point of the order of 1 K. Unfortunately, we only have threestars in common (in the range 5000–5500 K), too few to draw anymeaningful conclusions. Nevertheless, we find good agreement witha mean difference of 37 ± 43 K, our stars being cooler. The scale ofKovtyukh is in fact hotter by about 90 K when compared with thestars in the same range from Ramırez & Melendez (2005a). Since thezero-point of Kovtyukh et al. (2003) is calibrated on the Sun, one oftheir conclusion is that either the IRFM of Alonso et al. (1996a) andBlackwell & Lynas-Gray (1998) or the surface brightness techniqueof Di Benedetto (1998) could actually predict a too low temperaturefor the Sun and the solar-type stars. As we have seen in Section 5.1, ahotter temperature scale for the solar analogues could indeed solve


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Table 9. Comparison with other works.

Ramırez & Blackwell & Masana Di Benedetto Valenti & Santos Luck & Mashonkina KovtyukhMelendez Lynas-Gray et al. Fischer et al. Heiter et al. at al.

Method (a) (a) (b) (c) (d) (e) (e) (f) (g)Stars in common 18 10 64 9 45 (2) 14 (1) 5 9 3�Teff (K) 105 60 −50 50 6 −15 −11 37 −37σ (K) 72 67 80 50 60 81 142 64 43�L (per cent) 1.4 1.3 −6.9 0.9 – – – – –� θ (per cent) −3.3 −1.4 −1.5 −1.4 – – – – –

Notes. (a) IRFM; (b) spectral energy distribution fit; (c) surface brightness technique; (d) fitting observed spectra with synthetic ones; (e) excitation equilibriumof Fe lines; (f) fitting of the Balmer line profile; (g) line depth ratios. The number of stars in the bracket refers to those not counted because of large departuresin temperature. Differences are computed our work−others.

the discrepancies between observed and computed colours in theIR.

6.8 Summary: systematic error remains the problem

Our temperature scale is slightly hotter than other scales based on theIRFM, typically by 50–100 K. Our scale agrees closely to the tem-perature scale established via spectroscopic methods. The strengthof our work is that our IRFM was done completely from first prin-ciples (from the multiband photometry and an adopted absoluteflux calibration) with the best quality data available. The bolomet-ric fluxes of the stars are close to completely observationally con-strained (only 15–30 per cent of the stellar flux lies outside ourBV(RI )C JHKS filters). Despite this effort, comparison with manyother temperature scales forces us to conclude that external uncer-tainties in the temperature scale of lower main-sequence stars re-mains of the order of ±100 K. The external uncertainty dominatesour high internal error, which is of the order of only 40 K.

A summary of the comparisons of our scale with the others men-tioned above is given in Table 9.

7 E M P I R I C A L B O L O M E T R I C C O R R E C T I O N S

In this section, we derive bolometric corrections for our stars. Thedefinition of apparent bolometric magnitude is

mBol = −2.5 log(FBol) + constant, (11)

where FBol is the bolometric flux received on the Earth as definedin equation (5). The usual definition of bolometric correction in agiven band

BCζ = mBol − mζ = MBol − Mζ , (12)

where BC is to be added to the magnitude in a given band ζ to yieldthe bolometric magnitude.

Although the definition of bolometric magnitude is straightfor-ward, there can be some confusion resulting from the choice ofzero-point. Originally, bolometric corrections where defined for theV band only and it had been accepted that BCV should be negativefor all stars. From the spectral energy distribution, we expect thebolometric correction in the V band to be largest for very hot and forvery cool stars. The minimum of BCV then occurs around spectraltype F, which then set the zero-point of the bolometric magnitudes.This implied a value BCV for the Sun of between −0.11 (Aller1963) and −0.07 mag (Morton & Adams 1968). However, with thepublication of a larger grid of model atmospheres, the smallest bolo-metric correction in Kurucz’s grid (1979) implied BCV = −0.194.The original choice of the zero-point for the bolometric corrections

has thus proved troublesome. A better method is to adopt a fixedzero-point.

We define the absolute bolometric magnitude of the Sun to beMBol, = 4.74 in accordance with Bessell et al. (1998). By adoptingits measured apparent magnitude V = −26.76 (see Section 5.1),the absolute MV magnitude is thus 4.81 and the V bolometric cor-rection is then BCV = 4.74 − 4.81 = −0.07.

The absolute bolometric magnitude for a star with luminosity, L,radius, R, and effective temperature, Teff, is

MBol = −2.5 log(

L/L) + MBol,

= −2.5 log(

R2T 4eff/R2T 4

eff,) + MBol,, (13)

where L = 4 πR2 σT4eff and L = 3.842 × 1033 erg s−1 (Bahcall

et al. 2006).Using equation (13) and the definition of absolute magnitude

Mζ = mζ + 5 log (π ) − 10, where π is the parallax (in mas), thebolometric correction (equation 12) is

BCζ = −5 logRπ

R− 10 log

Teff

Teff,+ MBol, − mζ + 10

= − 5 logK θ/2

R− 10 log

Teff

Teff,+ MBol, − mζ + 10,

(14)

where K is the conversion factor for the proper unit transformation,θ = 2Rπ/K is the angular diameter (in mas) determined via IRFMand R is the solar radius in cm as given in Section 5.1. From gridsof model atmospheres, bolometric corrections can be calculated ina similar manner, where the dependence on the radius of the stareliminates in the difference between the absolute bolometric andin-band magnitudes (e.g. Girardi et al. 2002).

The comparison between the empirical bolometric corrections indifferent bands computed via equation (14) and those predicted bymodel atmospheres (ATLAS9) is shown in Fig. 17.

This plot can be regarded as the theoretical counterpart of Fig. 2.Again, the agreement between model atmospheres and empiricaldata is very good and it should be remembered that the modeldependence in deducing empirical bolometric correction from ourimplementation of the IRFM is small (only few 10 per cent, seeSection 4).

A complementary way of deriving stellar integrated fluxes is viaphotometric indices. The integrated flux of a star, FBol(Earth), de-pends primarily on its apparent brightness (especially in RC andIC bands, see later), which may be measured by its magnitude indifferent bands. Of lesser importance is its temperature, which is afunction of a colour index and metallicity, φ (X, [Fe/H]). FollowingBlackwell & Petford (1991), we expect a relation of the form

FBol(Earth) = 10−0.4mζ φ(X , [Fe/H]). (15)


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Figure 17. Bolometric corrections calculated from Castelli & Kurucz (2003) ATLAS9 are represented as function of Teff. [M/H] equal to +0.5 (black line),+0.0 (red line), −0.5 (cyan line), −1.0 (green line), −1.5 (blue line), −2.0 (yellow line). Points correspond to our IRFM bolometric corrections for the samplestars in the range [M/H] > 0.25 (black), −0.25 < [M/H] � 0.25 (red), −0.75 < [M/H] � − 0.25 (cyan), −1.25 < [M/H] � − 0.75 (green), −1.75 < [M/H]� − 1.25 (blue). The metallicities given for the model are solar scaled, whereas [M/H] for the stars has been computed using equation (1).

The function φ (X, [Fe/H]) is illustrated in Fig. 18, in whichthe reduced flux in ζ band, FBol(Earth) 100.4 mζ , is plotted againstdifferent colour indices. For the integrated stellar flux, we have fittedexpressions of the form

FBol(Earth) = 10−0.4mζ

(b0 + b1 X + b2 X 2 + b3 X 3

+b4 X [Fe/H] + b5[Fe/H] + b6[Fe/H]2). (16)

We tried different fitting formulae but this form proved themost satisfactory. There are a few fits of this type in the literature(Blackwell & Petford 1991 for a subset of the colours used here, andAlonso et al. 1995 for the IR). Blackwell et al.’s fit has no metallicitydependence because metallicities were not available. Alonso et al’sformula gives fits which are good in the IR but do not reproduce theobserved trend in the optical bands. Our fitting formulae work wellfrom optical to IR. As for the temperature–colour fitting, at everyiteration points departing more than 3σ from the mean fit were dis-carded. The coefficients of the fits, together with the number of starsused, the range of applicability and the standard deviation of the dif-ferences between the measured fluxes and the fluxes calculated fromthe fitting formula are given in Table 10.

The scatter in Fig. 18 of the reduced flux in RC and IC bands is onlyapparently large, due to the small range (in vertical scale) covered bythe reduced fluxes in these two colours (compare also with Fig. 17).For the sake of the proposed calibration, the small range coveredby the reduced flux in these two bands allows for a very accurate(at 1 per cent level or below) calibration. Interestingly, for thesetwo bands tight relations also exist between bolometric fluxes andmagnitudes, as seen in Fig. 19.

8 T H E A N G U L A R D I A M E T E R – C O L O U RR E L AT I O N

Limb-darkened angular diameters can then be readily obtained viaequation (5) from the temperature and bolometric flux calibrationsgiven in Sections 4 and 7. In particular, when using J magnitudeswith various colour indices, we have found very tight and simplerelations, as can be appreciated from Fig. 20. Analogous relationsin other bands are significantly less tight.

We have fitted relations of the form

θ = c0 + c1

√φ(m J , X ), (17)

where

φ(m J , X ) = 10−0.4m J X (18)

for a given colour index X.As can be appreciated from Table 11, these relations show re-

markably small scatter (at the few per cent level). In particu-lar, the upper range covered in angular diameters can be usedto build a network of small calibrators for future long-baselineinterferometric measurements from readily available broad-bandphotometry.

The calibration in J − KS colour is particularly appealing, sincein this photometric index the effect of interstellar extinction is neg-ligible.

The given angular diameter scale can be tested with the G dwarfHD 209458 A for which the linear radius has been calculated fromplanetary transit (Brown et al. 2001). We have taken V magnitudefrom Hipparcos and J, H and KS from 2MASS. Our predicted value


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Figure 18. Reduced flux in different ζ -bands [FBol(Earth) 100.4 mζ ] plotted as function of different colour indices. Circles are for stars with [Fe/H] � − 0.5,crosses for stars with [Fe/H] > −0.5.

of θ = 0.227 ± 0.003 mas is in excellent agreement with that ofθ = 0.228 ± 0.004 mas (Table 12) obtained by Kervella et al. (2004)using surface brightness relations calibrated by interferometry. Thedifference between the two values is only 0.4 per cent and thereforewell below the possible ∼3 per cent offset previously discussed.Unfortunately, both this comparison and those in Sections 5 and6.1 have their own limitations since they are indirect, or sensitiveto photometric uncertainties (for this star, Kervella has to convertthe 2MASS KS magnitude into the Johnson system used to fit hisrelations). When our angular size is translated into linear radius viaHipparcos parallaxes, we obtain R = 1.150 ± 0.056 R slightlysmaller than that of Kervella et al. (2004) but in better agreementwith the direct estimate of R=1.146± 0.050 R obtained by Brownet al. (2001) via HST time series photometry (although all determi-nations agree well within the error budget). Unfortunately, in thiscase the uncertainty in parallax dominates the bulk of the ∼5 per centuncertainty on the radius.

9 C O N C L U S I O N S

We have used the IRFM to deduce fundamental stellar param-eters of effective temperature, bolometric magnitude and angu-lar diameter for a sample of 104 G and K dwarfs. This semidi-rect method is mostly based on empirical data obtained from ourown observations or carefully selected from the literature in or-der to achieve the highest accuracy feasible. All our stars have ex-cellent BV(RI )C JHKS photometry, excellent parallaxes and goodmetallicities.

Most of the bolometric flux of our stars is seen in the optical andIR bands covered BV(RI )C JHKS. For the remaining 15–30 per centof the flux lying outside these bands, model atmospheres have beenused. Good to very good agreement is found for the colours of thestars and the synthetically derived colours of a number of sets ofmodel atmospheres in the literature (ATLAS9, MARCS and BaSel3.1).


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Table 10. Coefficients and range of applicability of the absolute flux calibrations. All calibrations are valid for the metallicity range −1.87 � [Fe/H] � 0.34.

Reduced Colour Colour b0 b1 b2 b3 b4 b5 b6 N σ (per cent)flux range

B B − V [0.419, 1.270] 1.6521 8.6205 −13.7586 12.5042 −0.7734 −0.0644 −0.0882 100 1.7V − RC [0.249, 0.755] 0.4475 19.0419 −32.6586 44.0700 −0.8436 0.4652 0.1417 98 2.3(R − I)C [0.268, 0.640] 11.3466 −69.2615 184.7907 −105.3337 4.8308 −1.1543 0.1516 101 3.5V − IC [0.517, 1.395] 3.9334 −4.2082 7.7254 0.7865 0.5466 −0.1401 0.1045 99 1.8V − J [0.867, 2.251] −0.5405 9.0544 −6.6975 2.5713 0.5042 −0.2272 0.1393 99 2.5V − H [1.016, 2.847] −1.3883 8.9784 −5.3419 1.4980 0.5184 −0.3228 0.1759 99 2.4

V − KS [1.124, 3.010] −0.3850 6.4601 −3.5378 1.0324 0.4693 −0.4049 0.1425 102 2.0

V B − V [0.419, 1.270] 2.9901 −1.9408 1.6146 0.9227 −0.0857 −0.3010 −0.0800 96 1.5V − RC [0.249, 0.755] 2.4783 −0.6395 2.4817 3.3947 −0.0222 −0.1769 −0.0100 103 1.2(R − I)C [0.268, 0.640] 6.0078 −28.2448 69.7676 −44.6415 1.2198 −0.5151 0.0052 99 1.7V − IC [0.517, 1.395] 3.7933 −5.2461 6.3043 −1.4363 0.1987 −0.2959 −0.0118 100 0.9V − J [0.867, 2.251] 2.9726 −0.9034 0.3570 0.1879 0.1641 −0.3144 0.0062 96 0.9V − H [1.016, 2.847] 2.5911 −0.0458 −0.1431 0.1563 0.1532 −0.3338 0.0121 99 0.9

V − KS [1.124, 3.010] 3.0690 −0.8886 0.3413 0.0497 0.1410 −0.3528 0.0024 102 0.9

R B − V [0.419, 1.270] 2.6213 −2.0936 1.6873 −0.1037 −0.0267 −0.1361 −0.0229 100 1.0V − RC [0.249, 0.755] 2.4703 −2.9446 4.5916 −1.0001 0.0889 −0.1652 −0.0075 103 1.2(R − I)C [0.268, 0.640] 3.7090 −12.3326 26.6386 −16.4346 0.5423 −0.3234 −0.0055 101 0.9V − IC [0.517, 1.395] 2.9332 −3.1362 2.9790 −0.7098 0.1182 −0.2203 −0.0090 103 1.0V − J [0.867, 2.251] 2.9077 −1.7416 0.9045 −0.1007 0.0983 −0.2532 −0.0046 95 0.6V − H [1.016, 2.847] 2.6787 −0.9882 0.3538 −0.0127 0.0790 −0.2506 −0.0041 91 0.6

V − KS [1.124, 3.010] 2.8723 −1.2499 0.4812 −0.0383 0.0689 −0.2430 −0.0070 93 0.6

I B − V [0.419, 1.270] 2.6225 −3.6174 3.4760 −1.1216 0.0101 −0.0438 −0.0060 98 1.0V − RC [0.249, 0.755] 2.7401 −7.1587 12.3123 −7.0236 0.1591 −0.1251 −0.0077 100 1.0(R − I)C [0.268, 0.640] 3.2531 −11.0246 21.2495 −13.5880 0.4144 −0.2412 −0.0038 102 1.0V − IC [0.517, 1.395] 2.9439 −4.3412 3.9408 −1.1868 0.1287 −0.1717 −0.0061 103 1.0V − J [0.867, 2.251] 2.9412 −2.6212 1.4468 −0.2658 0.0881 −0.1967 −0.0077 97 1.1V − H [1.016, 2.847] 2.8196 −1.8941 0.8205 −0.1187 0.0645 −0.1866 −0.0077 96 1.1

V − KS [1.124, 3.010] 2.8475 −1.8308 0.7520 −0.1031 0.0540 −0.1688 −0.0078 100 1.0

J B − V [0.419, 1.270] 2.4518 −4.0671 3.2357 −0.9315 −0.0658 0.1077 0.0185 93 2.2V − RC [0.249, 0.755] 2.8867 −10.0254 15.7145 −8.5556 0.0763 −0.0217 0.0067 93 2.0(R − I)C [0.268, 0.640] 3.7432 −15.8775 27.7771 −16.6189 0.3220 −0.1709 0.0065 94 2.5V − IC [0.517, 1.395] 3.2376 −6.2189 5.1663 −1.4808 0.0931 −0.0901 0.0061 90 2.1V − J [0.867, 2.251] 2.7711 −2.8212 1.3163 −0.2179 0.0628 −0.1140 0.0022 96 0.9V − H [1.144, 2.847] 2.8830 −2.4146 0.9164 −0.1227 0.0382 −0.0962 0.0003 91 1.7

V − KS [1.124, 3.010] 2.7876 −2.1525 0.7601 −0.0949 0.0294 −0.0732 0.0039 90 1.5

H B − V [0.419, 1.270] 2.3593 −4.3075 3.3103 −0.9179 −0.1030 0.1659 0.0263 97 2.9V − RC [0.249, 0.755] 2.7616 −10.2963 15.6654 −8.2985 0.0264 0.0186 0.0097 95 2.8(R − I)C [0.268, 0.640] 3.5683 −15.8279 26.9524 −15.7470 0.3107 −0.1504 0.0105 95 3.2V − IC [0.517, 1.395] 3.0862 −6.2854 5.0826 −1.4230 0.0780 −0.0624 0.0083 95 2.8V − J [0.867, 2.251] 2.6182 −2.8623 1.2906 −0.2070 0.0518 −0.0823 0.0053 99 2.5V − H [1.016, 2.847] 2.4555 −2.0593 0.7245 −0.0916 0.0386 −0.0839 0.0028 100 1.0

V − KS [1.124, 3.010] 2.5220 −2.0243 0.6752 −0.0804 0.0246 −0.0505 0.0049 97 2.0

KS B − V [0.419, 1.270] 2.2158 −4.0992 3.1957 −0.9165 −0.0680 0.1228 0.0208 102 2.8V − RC [0.249, 0.755] 2.6509 −10.0609 15.4526 −8.3102 0.0587 −0.0096 0.0049 97 2.9(R − I)C [0.268, 0.640] 3.2130 −13.9314 23.1287 −13.2435 0.3450 −0.1801 0.0049 100 2.9V − IC [0.517, 1.395] 2.8573 −5.7901 4.6467 −1.3010 0.0958 −0.0924 0.0028 103 2.7V − J [0.867, 2.251] 2.5152 −2.8220 1.2970 −0.2136 0.0645 −0.1172 −0.0001 100 2.6V − H [1.016, 2.847] 2.4184 −2.1148 0.7671 −0.1005 0.0453 −0.1096 −0.0014 95 2.2

V − KS [1.124, 3.010] 2.4209 −1.9857 0.6720 −0.0819 0.0323 −0.0795 0.0005 102 1.0

Notes. N is the number of stars used for the fit after the 3σ clipping and σ (per cent) is the standard deviation of the percentage differences between themeasured fluxes and the fluxes calculated from the fitting formula. The coefficients of the calibrations bi are given in units of 10−5 erg cm−2 s−1.

The zero-points of our temperature, luminosity and angulardiameter scales (Sections 4, 7 and 8) depend only on our adoptedabsolute calibration of Vega. The advantages and disadvantages ofour use of Vega are discussed in Appendix A. The main advantage

is that Vega’s absolute calibration has been extensively studied overa wide wavelength range via satellite measurements. This allowsus to put firm constraints on the systematic uncertainties. A likelydisadvantage is that Vega is known to be pole-on and a fast rotator


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Figure 19. Left-hand panel: relation between apparent magnitudes in ζ = RC (circles) and ζ = IC (crosses) band and bolometric fluxes on the Earth(in erg cm−2 s−1). Right-hand panel: relation between absolute magnitudes in the same RC and IC bands and absolute bolometric magnitudes. The fit is in theform γ = A + Bζ , σ ( per cent) is the standard deviation of the percentage differences between the measured values and those calculated from the fit and N isthe number of stars employed for the fitting after the 3σ clipping.

Figure 20. Empirical angular diameter–colour relations for our sample of stars.


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Table 11. Coefficients and range of applicability of the angular diameter calibrations.

Colour Colour range c0 c1 N σ (per cent) σ (mas)

B − V [0.419, 1.270] 0.006 47 6.186 57 102 3.4 0.008V − RC [0.249, 0.755] 0.003 40 8.262 44 103 2.7 0.008(R − I)C [0.268, 0.640] −0.003 85 8.977 58 103 2.0 0.006V − IC [0.517, 1.395] −0.000 45 6.094 87 102 1.6 0.005V − J [0.867, 2.251] 0.000 57 4.631 47 103 1.6 0.005V − H [1.016, 2.847] −0.000 22 4.136 07 102 1.3 0.004

V − KS [1.124, 3.010] 0.000 98 4.013 35 103 1.2 0.003J − KS [0.257, 0.759] 0.003 00 7.960 76 100 2.8 0.007

Notes. N is the number of stars used for the fit after the 3σ clipping. σ (per cent) is the standarddeviation of the percentage differences between the angular diameters obtained via IRFM andthose calculated from the fitting formula. σ (mas) is the standard deviation in terms of mas.

Table 12. Predicted angular diameter for HD 209458 A.

V J H KS θV −J θV −H θV −KS θ

7.65 6.591 6.366 6.308 0.230 0.225 0.224 0.227±0.01 ± 0.020 ± 0.038 ± 0.026 ± 0.003

Notes. θξ−ζ are the angular diameters obtained when (V − J), (V − H) and(V − KS) colour indices are used. Since the accuracy of the calibration inthese bands is very similar (see Table 11), the final angular diameter is theaverage of the three values weighted by the photometric errors.

which prevents the use of a unique Vega model atmosphere in fittingthe observations.

Our calibration of effective temperature is found to be some 100-K hotter than similar determinations in the literature, while it is alsofound to be in close agreement with effective temperature scalesbased on spectroscopic methods. To reduce our temperature scaleby 100 K, the required changes to Vega’s 2MASS zero-points and/orabsolute calibration are significantly larger than the known uncer-tainties in these quantities. Neither can the model atmospheres beblamed, because we find such good agreement between the modelatmosphere colours and the empirical data. We worked hard to im-pute inconsistency to one or other of the scales, but were unable toresolve the issue beyond any reasonable doubt. Our angular diam-eters are smaller by about 3 per cent with respect to other indirectdeterminations, but they seem more in line with the predictionsof asteroseismology and 3D model atmospheres. We conclude thattemperatures, luminosities and angular diameters calibrations forlower main-sequence G and K dwarfs retain systematics of the or-der of a few per cent.

The high quality and homogeneity of the data produce very tightempirical colour–metallicity–temperature and angular diameter–colour calibrations. In particular, the relation between angular diam-eters and magnitudes in the J band is remarkable and it indicates ahigh sensitivity of this band to angular diameters. Since many lowermain-sequence stars are known to host planets, our relations can beused to accurately determine the physical properties of the parentstar thus allowing to effectively break the degeneracy between theproperties of the stars and that of the planets when extremely high-precision data are unavailable (e.g. Sato et al. 2005; Bakos et al.2006).

Future interferometric measurements of angular diameters wouldgo a long way to addressing the uncertainties we have shown aswell as to test our findings. In particular, the shape of the visibilityfunction in the second lobe would directly probe limb-darkeningcorrections (e.g. Bigot et al. 2006) for our stars. Likewise, tying

the absolute calibration and the observed colours to a solar twinwould probably reduce many of these uncertainties. At present goodcandidate solar twins are HD 146233 (Porto de Mello & da Silva1997) and HD 98618 (Melendez, Dodds-Eden & Robles 2006). Ac-curate interferometric measurements together with high-precisionphotometry for a set of nearby solar-like stars would permit to setthe absolute calibration via solar analogues directly (e.g. Campinset al. 1985). Besides, extremely high-precision multiple bandpassstudies of transiting extrasolar planets would provide direct angulardiameter estimates for our stars, and will further test our temperaturescale.

AC K N OW L E D G M E N T S

We are very grateful to Andrei Berdyugin for instruction in the useof the remotely operated telescope at La Palma. We thank SteveWillner for very useful discussions on the absolute calibrations andJohan Holmberg and Jorge Melendez for many constructive com-ments and a careful reading of the manuscript, as well as the refereefor the same kindness. LC and CF acknowledge the hospitality of theResearch School of Astronomy and Astrophysics at Mount Stromlowhere part of this work was carried out. We are very grateful to theAcademy of Finland for considerable financial support (grants nos206055 and 208792). LC acknowledges the support of the MagnusEhrnrooth Foundation and a CIMO fellowship. LP further acknowl-edges the support of a EU Marie Curie Intra-European Fellowshipunder contract MEIF-CT-2005-010884. This research has made useof the SIMBAD database, operated at CDS, Strasbourg, France. Thispublication makes use of data products from the 2MASS, which isa joint project of the University of Massachusetts and the InfraredProcessing and Analysis Centre/California Institute of Technology,funded by the National Aeronautics and Space Administration andthe National Science Foundation.

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A P P E N D I X A : B ROA D - BA N D S Y N T H E T I CP H OTO M E T RY A N D A B S O L U T EC A L I B R AT I O N

The choice of the zero-points is of critical importance to set syn-thetic and real photometry on the same scale. Historically, thezero-points of the UBV system were defined ‘in terms of unred-dened main-sequence stars of class A0 . . . with an accuracy suffi-cient to permit the placement of the zero-point to about 0.01 mag’(Johnson & Morgan 1953). The pioneering work of Johnson wascontinued and extended to other bands by Johnson himself over theyears; in the Southern hemisphere Cousins (1976, 1978, 1980) andastronomers at the SAAO refined its accuracy and colours range(Menzies et al. 1989; Kilkenny et al. 1998). Similar work was car-ried out in the Northern hemisphere by Landolt (1992) at the CTIO.Subtle differences exist between different works (see Bessell 2005,for a review) and especially between the SAAO and the CTIO sys-tem (Bessell 1995). Nevertheless, the BV(RI )C system is nowadaysa well-defined one whose main characteristics and bandpasses aregiven in Bessell (1990b).

The basic equation of synthetic photometry3 in a given band ζ

reads as follows (e.g. Girardi et al. 2002):

mζ = −2.5 log

(R

d

)2∫ λζ f

λζiF(λ)Tζ (λ)dλ∫ λζ f

λζiTζ (λ)dλ

+ ZPζ , (A1)

where F(λ) is the flux of the synthetic spectrum (given inerg cm−2 s−1 Å−1), Tζ (λ) is the transmission curve of the filtercomprised in the interval (λζi , λζ f ) and ZPζ is the zero-point for theζ band. The ratio between the radius R of the star and its distanced from us is known as the dilution factor and relates to the angulardiameter θ (corrected for limb darkening) via

θ = 2R

d, (A2)

where θ is in radians. We also note that when all quantities in thelogarithmic term of equation (A1) are known that gives the abso-lute calibration in ζ -band (see Table A2) for a star with spectrumF(λ).

While the zero-points of observational photometry are definedon an ensemble of well measured stars, for synthetic photometrya common choice is to set the zero-points using a reference star

3 The formulation given is suited for the energy integration that characterizestraditional systems based on energy–amplifier devices like the Johnson–Cousins. Nevertheless, since the 2MASS transmission curves are alreadymultiplied by λ and renormalized (Cohen et al. 2003), equation (A1) canalso be used for the 2MASS photo-counting integration. Notice that even ifour BV(RI )C measurements have been done with a photo-counting device(CCD), we are placing our photometry on a system of standard stars definedwith the use of photomultiplier and therefore energy integration is the mostappropriate.


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Table A1. Observed magnitudes for Vega.

B V RC IC J H KS Ref.

0.02 0.03 0.039 0.035 Bessell (1990a)−0.001 +0.019 −0.017 Cohen et al. (2003)

Table A2. Absolute calibration and effective wavelength of the ground-based optical–IR photometry ofVega used in this work. Quantities tabulated correspond to the definition of the zero magnitude in eachfilter.

Band λeff Monochromatic absolute flux Uncertainty Ref.

Å erg cm−2 s−1 Å−1 erg cm−2 s−1 Å−1

B 4362 6.310E−9 6.310E−11 This paperV 5446 3.607E−9 3.607E−11 This paper

RC 6413 2.153E−9 2.153E−11 This paperIC 7978 1.119E−9 1.119E−11 This paperJ 12285 3.129E−10 5.464E−12 Cohen et al. (2003)H 16385 1.133E−10 2.212E−12 Cohen et al. (2003)KS 21521 4.283E−11 8.053E−13 Cohen et al. (2003)

Notes. The effective wavelengths associated with each filter are computed in accordance with AppendixB. The error in BV(RI )C bands are computed assuming an arbitrary uncertainty of 1 per cent to theabsolute flux. Lacking the actual uncertainties in the measurement of the filters’ transmission curve, theadopted uncertainty is consistent with a wavelength-independent filter uncertainty in addition to the 0.7per cent absolute uncertainty at 5556 Å from Megessier (1995).

for which all the physical quantities in equation (A1), i.e. the spec-trum F(λ) (usually synthetic), the dilution factor and the observedapparent magnitudes or colour indices are known in detail.

This star is usually Vega, for which we have used the magnitudesand colours given in Table A1. The optical colours are from Bessell(1990a), whereas for the 2MASS system they come from Cohenet al. (2003). The latter were determined post-facto, comparing theobserved 2MASS magnitudes of 33 stars with the values predictedfrom the absolutely calibrated templates built by Cohen et al. (2003)within the ‘Cohen–Walker–Witteborn’ (1992b) framework.

In principle, we could adopt a unique spectrum for Vega and use itto compute zero-points from B to KS; in this way, the dependence onthe absolute calibration would cancel out when computing colourindices, since all colours scale in the same way with the adoptedabsolute calibration. In practice we use a slightly different approachusing two models, one for the absolute calibration in the optical andone in the IR – each consistent with the corresponding data source.

The 2MASS magnitudes for Vega are deduced in the absolutelycalibrated system that is ultimately defined on the Kurucz spectrafor Vega and Sirius used by Cohen et al. (1992a). Therefore, forthe JHKS bands we have adopted the zero-magnitude fluxes pro-vided by Cohen et al. (2003). The zero-magnitude fluxes proposedby Cohen for Vega and eight of the primary and secondary stars inthe calibration network of ‘Cohen–Walker–Witteborn’ has been re-cently confirmed in the 4–24 µm range by the Midcourse SpaceExperiment (MSX) flux calibration (Price et al. 2004) to be ac-curate around 1 per cent and thus well within the global error of1.46 per cent quoted by Cohen, even though it seems that thefluxes of Cohen in the IR should be brightened by 1 per cent. Also,Tokunaga & Vacca (2005) have shown the Vega absolute calibrationof Cohen et al. (1992a) and the model-independent measurementsof Megessier (1995) to be identical within the uncertainties in therange ∼1–4 µm.

For the absolute calibration in the optical bands, we have adopteda Kurucz (2003) synthetic spectrum for Vega with Teff = 9550 K,

log (g) = 3.95, [M/H] = −0.5 and microturbulent velocity ξ =2 km s−1. The resolving power used has been 500. This model hasrecently shown excellent agreement with the STIS flux distributionover the range 0.17–1.01 µm (Bohlin & Gilliland 2004). In par-ticular, in the regions of the Balmer and Paschen lines the STISequivalent widths differ from the pioneering work of Hayes (1985)(and used by Colina et al. 1996 to assemble a composite spectrumof Vega) but do agree with the predictions of the Kurucz model.

Once the synthetic spectrum of Vega in the optical bands is cho-sen, we need to scale it in order to match the absolute flux of Vegaas measured on the Earth at a certain fixed wavelength (λ). Themonochromatic flux given by the model Fmodel(λ) is simply relatedto the same monochromatic flux as measured on the Earth FEarth(λ)by

FEarth(λ) =(

R

d

)2

Fmodel(λ), (A3)

where the dilution factor (R/d)2 is the ratio between the radius ofVega and its distance. In principle, the dilution factor can be deducedfrom direct measures of Vega’s angular diameter. In practice, sincesuch measures are more uncertain than direct measures of the flux,we proceed the other way around. Taking the flux value at 5556 Åfrom Megessier (1995) FEarth(5556 Å) = 3.46 × 10−9erg cm−2s−1

Å−1 and at the same wavelength from the Kurucz (2003) Vegamodel Fmodel(5556 Å) = 5.501 5572 × 107erg cm−2s−1 Å−1 we ob-tain (R/d)2 = 6.289 1286 × 10−17. This value implies and angulardiameter of 3.272 mas for Vega, which compares very well with theinterferometric angular diameter measurements of 3.24 ± 0.07 mas(Code et al. 1976), 3.28 ± 0.06 mas (Ciardi et al. 2001) and3.225 ± 0.032 (Mozurkewich et al. 2003).

We chose to use the Megessier flux in accordance with the ab-solute scale adopted for the STIS (Bohlin & Gilliland 2004) ratherthan the value of 3.44 × 10−9 erg cm−2 s−1 Å−1 ± 1.45 per centfound in Hayes (1985) and used by Cohen to tie his absolute cali-bration in the IR. A thorough discussion of the differences in the IR


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is given by Bohlin & Gilliland (2004); here we just mention that thischoice together with the Kurucz (2003) model in the IR would givea flux density of Vega about 2 per cent lower than in Cohen, thusworsening the agreement with MSX. On the other hand, the STISabsolute calibration in the optical is expected to be better than 2 percent (Bohlin 2000). In this range, the increase of 0.6 per cent rel-ative to Hayes translates in absolute fluxes that are higher by thesame factor and in differences for the BV(RI )C zero-points of0.006 mag.

Recent work by Gulliver, Hill & Adelman (1994), Peterson et al.(2004, 2006) and Aufdenberg et al. (2006) indicates that Vega ispole-on and fast rotator. Thus standard model atmospheres are notappropriate for Vega, as discussed by Bohlin & Gilliland (2004).None the less STIS in the optical and MSX in the IR have confirmedthe adopted absolute calibrations to be accurate at the per cent level.It is interesting to note that Peterson et al. (2006) give an estimateof the shift in Vega’s zero-points as a consequence of the higherIR flux due to its rapidly rotating nature. Though only indicative,as the authors say, the shifts for their latest non-rotating syntheticspectrum in the IR amount to 0.05–0.07 mag. The net effect of suchshifts would be to reduce the temperatures recovered via IRFM.Considering that in the IR we are adopting the absolute calibrationgiven by Cohen (already some ∼2 per cent brighter as comparedto the latest Kurucz model) this reduces the size of the requiredshifts to about 0.02–0.04 mag. The agreement within 1 per cent withMSX would, however, require a shift of only 0.01 mag. Since theproposed shifts are only indicative and the measurements confirmthe adopted calibration within the errors, we adopt the pragmaticchoice of using the model that at any given range better agrees withthe data. It is evident from the above discussion that there is noconsistent published atmospheric model for Vega at both visibleand IR wavelengths and this justifies our adoption of the Kurucz(2003) or Cohen models at different wavelengths. Likewise, theuse of composite model atmospheres for Vega is also advocatedfrom Peterson et al. (2006). For synthetic photometry, the use ofcomposite spectra is often avoided on the reason that fixing the zero-points, possible physical deficiencies in the synthetic spectrum ofVega are likely to be present in the synthetic library and thereforethey would cancel out (e.g. Girardi et al. 2002). In our case, thesituation is not so clear, since synthetic spectra give now excellentcomparison with the real ones and the deficiencies for Vega arelikely to be due to the particular nature of this star.

Sirius is often used as an additional fundamental colour standard(e.g. Bessell et al. 1998), allowing one to control the possible prob-lems of putative variability and an IR excess for Vega. First seen byAumann et al. (1984) beyond 12 µm, Vega’s variability at shorterwavelengths has long been a matter of discussion (see e.g. Bessellet al. 1998; Ciardi et al. 2001), though the rapidly rotating modelcould actually resolve the controversy (Peterson et al. 2006). Themost recent data from MSX (Price et al. 2004) and Spitzer (Su et al.2005) support models where Vega’s IR excess and variability isdue to a cold dust disc. The IR excess increase steadily longward of12 µ m, whereas at 4 µm the differences between model atmosphereand measurements are entirely consistent within the uncertaintiesand therefore should not affect the optical and near-IR region (0.36–2.4 µm) we are working with. For the sake of completeness, we havetested the differences in the derived BV(RI )C zero-points when in-cluding Sirius in the calibration. Adopting the latest Kurucz modelavailable for it and the colours from Bessell (1990a), the differencesin the derived zero-points between the use of Vega only or Vega plusSirius range from 0.008 mag in B − V to 0.002 mag in (R − I)C. Suchdifferences are below the uncertainties in magnitude and colours.

In addition, since absolute calibration and magnitudes for Sirius inthe 2MASS system are not available, we have decided to adopt onlyVega as our standard star. This also makes clearer the comparisonbetween synthetic colours and the IRFM that strongly depends onthe Vega absolute calibration.

The transmission curves used come from Bessell (1990b) for theJohnson–Cousins BV(RI )C system and from Cohen et al. (2003) forthe 2MASS JHKS system.

The 2MASS transmission curves carefully incorporate the ef-fect of the optical system, the detector quantum efficiency and thesite-specific atmospheric transmissions. The effect of the telescopeoptics on the estimated BV(RI )C passband are not counted, but inthe V band this has been proved to cause uncertainty below few mil-limag (Colina & Bohlin 1994), and thus smaller than any possibleobservational error we are going to compare with.

We note that all our photometry is reduced to zero airmass andwhen we refer to measurements on the Earth, we always mean atthe top of the Earth’s atmosphere.

A P P E N D I X B : C H A R AC T E R I S T I CPA R A M E T E R S O F A P H OTO M E T R I C S Y S T E M

In the present work, we have reduced broad-band (heterochromatic)measurement in a band ζ to a monochromatic flux (Fζ ) by meansof the following relation:

Fζ =∫ λζ f

λζiF(λ)Tζ (λ)dλ∫ λζ f

λζiTζ (λ)dλ

, (B1)

where the integration is done in the interval λζi , λζ f that comprisea given passband Tζ (λ). We remark that the above formulation issuited for both energy integration in the optical and photocountingin the IR, provided that we are dealing with the Bessell (1990b) andCohen et al. (2003) transmission curves (see Appendix A).

Assuming that the function F(λ) is continuous and that Tζ (λ)does not change sign in the interval λζi , λζ f , the generalization ofthe mean value theorem states that there is at least one value of λ inthe interval λζi , λζ f such that

F(λi)

∫ λζ f

λζi

Tζ (λ)dλ =∫ λζ f

λζi

F(λ)Tζ (λ) dλ. (B2)

Rearranging this, we obtain

Fζ = F(λi) =∫ λζ f

λζiF(λ)Tζ (λ) dλ∫ λζ f

λζiTζ (λ) dλ

, (B3)

where λi is the isophotal wavelength. The isophotal wavelength isthus the wavelength which must be given to the monochromaticquantity F(λi) obtained from a heterochromatic measurement.

Stellar spectra do not necessarily satisfy the requirements of themean value theorem for integration, as they exhibit discontinuities.Although the mean value of the intrinsic flux is well defined (andwe have extensively used it), the determination of the isophotalwavelength becomes problematic because spectra contain absorp-tion lines and hence the definition can yield multiple solutions.

Several authors avoid using the isophotal wavelength and intro-duce the effective wavelength defined by the following expression:

λeff =∫ λζ f

λζiλF(λ)Tζ (λ)dλ∫ λζ f

λζiF(λ)Tζ (λ)dλ

. (B4)


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This is the mean wavelength of the passband as weighted by theenergy distribution of the source over the band. The effective wave-length is thus an approximation for λi; however, since the monochro-matic magnitude at the effective wavelength is almost identical tothat at the isophotal wavelength (Golay 1974), we have used theeffective wavelength through all our work.

We have verified that the detailed choice of the wavelength as-sociated with a monochromatic flux is not crucial for the derivedcorrection of the C factor (equation 3).

When dealing with photometric data, a way to obtain monochro-matic flux from heterochromatic measurements is provided by the qfactor. Ideally, the q factor (equation 8) should be determined fromspectroscopic data but in practice we rely on a grid of models. Sincethe absolute monochromatic flux for a given band is determined asdescribed in Appendix A, the definition of the q factor for a starwith spectra F(λ) is now

q(λIR) =∫ λ f

λiTIR(λ) dλ

1F(λIR)

∫ λ f

λiF(λ)TIR dλ

. (B5)

This definition slightly differs from that given by Alonso et al.(1994, 1996a) to reflect the different absolute calibration we haveadopted (i.e. argument of the logarithm in equation A1). How-ever, the different definition of the q factor has only minor ef-fects on the recovered stellar parameters. Using the definition ofAlonso et al. (1996a) would go in the direction of making tem-peratures hotter by 20–30 K on average, the difference increasingwith increasing temperature. On average also luminosities wouldbe brighter by 0.14 per cent and angular diameters smaller by0.9 per cent.

This paper has been typeset from a TEX/LATEX file prepared by the author.


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