2 E. Masana et al.: Effective temperature scale and bolometric corrections
angular semi-diameters, radii and BCs for 10999 dwarfs and
subdwarfs in the Hipparcos catalogue ESA (1997). Such large
sample has allowed us to construct simple parametric calibra-
tions as a function of (V − K)0, [m/H] and log g. Note that a
preliminary version of the method presented here was already
successfully applied to the characterization of the properties of
planet-hosting stars (Ribas et al. 2003).
The present paper is organized as follows. Section 2
presents the method and explains in detail the procedure to ob-
tain Teff and angular semi-diameters, including the fitting al-
gorithm, zero point corrections and error estimates. The com-
parison of our temperatures with several previous works, both
based on photometric and spectroscopic techniques, is de-
scribed in Sect. 3. In Sect. 4 we present simple parametric
calibrations of Teff and BC as a function of (V − K)0, [m/H]
and log g valid for dwarf and subdwarf stars. The sample of
10999 stars used to build the calibrations is also described in
this section together with a detailed explanation of the different
contributors to the final uncertainties. Finally, the results are
discussed in Sect. 5 and the conclusions of the present work
are presented in Sect. 6.
2. The Spectral Energy Distribution Fit (SEDF)
method
The use of infrared (IR) photometry to determine effective tem-
peratures was initially proposed by Blackwell & Shallis (1977).
Their so-called Infrared Flux Method (IRFM) uses the ratio be-
tween the bolometric flux of the star and the monochromatic
flux at a given infrared wavelength, both measured at Earth,
as the observable quantity. This ratio is then compared with
a theoretical estimate derived from stellar atmosphere models
to carry out the determination of the effective temperature. The
IRFM has been widely used by a number of authors, being most
noteworthy the work by Alonso et al. (1995, 1996a,b).
The Spectral Energy Distribution Fit (SEDF) method that
we propose here follows a somewhat different approach,
namely the fit of the stellar spectral energy distribution from the
optical (V) to the IR (JHK) using synthetic photometry com-
puted from stellar atmosphere models. Unlike the Alonso et al.
(1996a) implementation of the IRFM, which averages temper-
atures derived individually for each IR band, our method takes
into account the four bands simultaneously (and naturally).
In addition, and also unlike the IRFM, the bolometric flux is
not required a priori by the SEDF method but results self-
consistently with the temperature. The fitting algorithm (see
Sect. 2.2) minimizes the difference between observed and syn-
thetic photometry by tuning the values of the effective temper-
ature and the angular semi-diameter. The BC can be obtained
from these two parameters, and then, when the distance to the
star is known, the luminosity is computed from the BC and the
absolute magnitude in a given photometric band. The uncer-
tainties of the derived parameters (Teff , angular semi-diameter
and BC) are estimated from the errors in the observed and syn-
thetic photometry as well as in the assumed [m/H], log g and
AV .
From a theoretical point of view, the SEDF method could
be applied to stars of any spectral type and luminosity class.
However, the IR flux becomes very sensitive to metallicity and
surface gravity for stars hotter than 8000 K so that small uncer-
tainties in these parameters translate into large uncertainties in
the effective temperature. In such situation the SEDF approach
becomes inadequate. At the cold end, the accuracy of stellar
atmosphere models limits the use of the method to stars hotter
than 4000 K (molecular opacity plays an important role below
this temperature). These limitations restrict the applicability of
the SEDF to FGK type stars. Fortunately, these stars are very
common in the Galaxy and dominate the content of most of
survey catalogues. They are crucial for several key astrophys-
ical topics, such as the study of the structure and evolution of
the Galaxy, both the disk and the halo, and the characterization
of planet-hosting stars, among others.
2.1. Calculation of synthetic photometry
The calculation of the synthetic photometry requires a well-
characterized photometric system, an accurate flux calibra-
tion and suitable synthetic spectra. The work by Cohen et al.
(2003a,b) provides consistent absolute flux calibrations in both
the visible (V) (Landolt system) and IR (2MASS JHK) bands.
The calibration given by Cohen et al. is computed from a set
of calibrated templates, using the synthetic Kurucz spectrum
of Vega of Cohen et al. (1992). In the case of the IR photom-
etry, they consider the transmission of the camera and filters,
the detector properties and the Earth’s atmosphere character-
istics. From the comparison between observed and synthetic
photometry for a set of 9 A-type stars and 24 cool giants, the
authors infer the need to introduce a zero point offset in the syn-
thetic photometry to match the observed 2MASS photometry:
0.001±0.005 mag (J); −0.019±0.007 mag (H); 0.017±0.005
mag (K). The calculation of such values is not exempt of some
difficulty since the dispersions of the differences between both
photometries (synthetic and observed) are of the same magni-
tude as the zero point offsets themselves.
To compute the syntheric magnitudes we made use of the
no-overshoot Kurucz atmosphere models grid (Kurucz 1979)
taken from http://kurucz.harvard.edu/grids.html:
misyn(Teff, log g, [m/H]) = 2.5 log






F i
cal
Fi(Teff, log g, [m/H])






(1)
where F i
cal is the absolute flux calibration given by Cohen et al.
(2003b) (for mi
cal = 0) and Fi(Teff, g, [m/H]) is the flux in the
i band computed from the integration of the model atmosphere
convolved with the transmission function (filter, detector and
Earth’s atmosphere) from Cohen et al. (2003b):
Fi(Teff, log g, [m/H]) =
∫ ∞
0
φ(Teff, log g, [m/H], λ)Ti(λ)dλ (2)
where φ(Teff, log g, [M/H], λ) is the flux given by the stellar
atmosphere model andTi(λ) the effective transmission function
in the i band normalized to a peak value of unity.
2.2. Fitting algorithm
The fitting algorithm is based on the minimization of the χ2
function defined from the differences between observed (cor-

E. Masana et al.: Effective temperature scale and bolometric corrections 3
rected for interstellar extinction) and synthetic VJHK magni-
tudes, weighted with the corresponding error:
χ2 =
(V − AV − Vsyn
σV
)2
+
( J − AJ − Jsyn
σJ
)2
+
+
(
H − AH − Hsyn
σH
)2
+
(
K − AK − Ksyn
σK
)2
(3)
This function depends (via the synthetic photometry) on Teff,
log g, [m/H] and a magnitude difference A, which is the ratio
between the synthetic (star’s surface) and the observed flux (at
Earth) (A = −2.5 log Fstar/FEarth). A is directly related to the
angular semi-diameter by the following expression:
θ = 10−0.2A (4)
Although the synthetic photometry depends implicitly on grav-
ity and metallicity, in practice, the spectral energy distribution
in the optical/IR for our range of temperatures is only weakly
dependent on these quantities. This fact makes it possible to
obtain accurate temperatures even for stars with poor determi-
nations of log g and [m/H].
As can be seen, the χ2 function depends also on the inter-
stellar absorption AV (the absorption in the other bands can be
computed using the extinction law of Schaifers & Voigt (1982):
AJ = 0.30AV , AH = 0.24AV and AK = 0.15AV). In principle,
it is possible to consider AV as a free parameter. However, the
strong correlation between Teff and AV , especially for the hotter
stars, decreases the precision in the determination of both pa-
rameters, with resulting typical uncertainties of 4% in Teff and
0.25 mag in AV . Thus, for best performance, AV should only be
considered as a free parameter when its value is suspected large
and no other method for its estimation is available. In general,
the best approach is to fix the value of AV in Eq. (3) from the
estimation by photometric calibrations, for instance.
Therefore, the only two adjustable parameters by the SEDF
method in the present work are Teff and A, whereas log g,
[m/H] and AV are fixed parameters. To minimize Eq. (3)
with respect to these two parameters we use the Levenberg-
Marquardt algorithm (Press et al. 1992), which is designed to
fit a set of data to a non-linear model. In all our tests, conver-
gence towards the minimum value of χ2 was reached rapidly
and unequivocally.
2.3. Calibration of the SEDF method using solar
analogues
The standard procedure for the calibration of an indirect
method to determine effective temperatures is based on the
comparison of the results with accurate temperatures from di-
rect methods for a set of stars. In this way, the list of stars with
empirical effective temperatures and angular semi-diameters
from Code et al. (1976) has been widely used for calibration
purposes. This list has been recently increased with the the
works of Mozurkewich et al. (2003) and Kervella et al. (2004).
Other authors use well-studied stars, such as the Sun, Vega or
Arcturus, to calibrate their methods.
Unfortunately, the few stars with empirical values of
Teff are too bright to have accurate 2MASS photometry and
they are of no use to calibrate the SEDF method. As an alter-
native, we have used the list of photometric solar analogues
compiled by Cayrel de Strobel (1996). We assume that, as an
ensemble, the average of the effective temperatures of these
photometric solar analogues should be equal to the effective
temperature of the Sun (i.e., 5777 K).
After selecting a subsample of 50 unreddened stars
with non-saturated 2MASS photometry from table 1 of
Cayrel de Strobel (1996), we computed their temperatures us-
ing the SEDF method. We obtained an average temperature of
5832 ± 14 K, i.e., 55 K (or ∼1%) higher than the solar effec-
tive temperature. Exactly the same value is obtained if we use
the subset of solar “effective temperature analogues” from ta-
ble 5 of Cayrel de Strobel. Without a profound analysis of all
the ingredients involved – from the stellar atmosphere model to
the absolute flux calibration, – it is very difficult to assess the
reasons for such difference. However, it seems clear that the
temperature scale as obtained from the synthetic photometry
alone needs a correction to agree with the average of the solar
analogues. From a formal point of view, this correction can be
computed from the synthetic photometry that results from forc-
ing a value of Teff = 5777 K to the entire sample. After doing so,
we replaced the zero points given by Cohen et al. (2003b) (see
Sect. 2.1) by the average difference (for each band) between the
observed and synthetic photometry computed for the solar ana-
logues. Assuming that there is no offset on the V band, the off-
sets for the other bands are: 0.027±0.003 mag (J); 0.075±0.005
mag (H); 0.022±0.005 mag (K). It is interesting to note that
both in the case of Cohen et al. (2003a) and in our case, the
value of the offset in the H band differs significantly from the
offsets in J and K. It should be stressed that the effective tem-
peratures given by Cayrel de Strobel (1996) have not been used
here. We have only used the property of the stars in being clas-
sified as solar analogues, and, consequently, we assumed their
average temperature to be equal to the solar effective tempera-
ture.
In our procedure, we are implicitly assuming that the cor-
rection in our temperature scale is just a zero point offset and
that no dependence on temperature or metallicity is present.
These assumptions are justified a posteriori in Sect. 3, where
several comparisons of SEDF temperatures with other photo-
metric and spectroscopic determinations are shown.
The angular semi-diameters computed from Eq. (4) were
used to check the consistency of the new zero points in our
temperature scale. These angular semi-diameters were com-
pared with the direct values compiled in the CHARM2 cata-
logue (Richichi & Percheron 2005). We restricted the compar-
ison to stars with accurate VLBI or indirect (spectrophotome-
try) measurements of the semi-diameter. Only 10 of these stars
fulfill the conditions for applicability of the SEDF method.
Figure 1 shows the comparison of the semi-diameters for these
10 stars. The agreement is excellent, with an average differ-
ence (θdir − θSEDF), weighted with the inverse of the square
of the error, equal to −0.3% with a s.d. of 4.6% (see Table
1). All the direct values used in the comparison correspond
to an uniform stellar disk. A crude comparison of both uni-
form disk and limb darkened values for about 1600 F, G and
K stars in the CHARM2 catalogue indicates a ∼4% positive

E. Masana et al.: Effective temperature scale and bolometric corrections 5
3.1. Methods based on IR photometry: Alonso et al.
(1996a) and Ramı´rez & Mele´ndez (2005)
As mentioned above, the IRFM is the most popular method to
compute effective temperatures from IR photometry. The work
by Alonso et al. (1996a) is undoubtedly the widest applica-
tion of the IRFM to FGK stars. The authors computed effective
temperatures for 462 stars with known interstellar absorption,
surface gravity and metallicity. After selecting the stars in the
Alonso et al. sample in the range 4000 < Teff < 8000 K and
with errors in the 2MASS photometry below 0.05 mag, we ob-
tained effective temperatures from the SEDF method for a sub-
set of 315 stars. The comparison between both determinations
of Teff is shown in Fig. 2. The average difference ∆Teff (IRFM
− SEDF) was found to be −67 K, with a standard deviation of
81 K. The dependence of this difference on the temperature is
not significant: T IRFM
eff = 1.030 T
SEDF
eff − 239 K. As shown in the
bottom panel of Fig. 2, there is no dependence of the tempera-
ture difference with metallicity.
In a recent work, Ramı´rez & Mele´ndez (2005) have re-
computed the IRFM temperatures of almost all the stars in
Alonso et al. (1996a) using updated input data. According to
the authors, the difference between the old and new temper-
ature scales is not significant. They also compare their effec-
tive temperatures with some direct determinations. The authors
conclude that there is a systematic difference of about 40 K at
solar temperature (in the sense IRFMAlonso – their values). The
comparison between the temperatures of Ramı´rez & Mele´ndez
and our determinations is shown in Fig. 3. For 385 stars in com-
mon we find ∆Teff (IRFM −SEDF) equal to −58 K (σTeff=67
K), and T IRFM
eff = 1.061 T
SEDF
eff − 403 K. Unlike in the case
of Alonso et al. (1996a), the dependence of ∆Teff in [m/H] is
relevant (Fig. 3, bottom panel). For [m/H] < −2.0 the tem-
peratures from Ramı´rez & Mele´ndez are clearly hotter than
our temperatures. The same trend was found by Charbonnel &
Primas (2005) when comparing their temperatures of 32 halo
dwarfs (−3.5 < [Fe/H] < −1.0) with the values of Ramı´rez &
Mele´ndez.
3.2. Other methods
Besides the IRFM, which uses IR photometry, we have also
compared the effective temperatures obtained using the SEDF
method with other determinations. Two of these (Fuhrmann
1998 and Santos et al. 2003) are spectroscopic works, while
in another case (Edvardsson et al. 1993) the temperatures are
based on uvby−β photometry. The chief problem in the case of
spectroscopic determinations is that, in general, they are mostly
applied to bright stars, which have poor 2MASS photometry
(the 2MASS detectors saturate for stars brighter than K ≈ 4
mag). This fact reduces the number of stars in the Fuhrmann
(1998) and Santos et al. (2003) samples that can be compared
with SEDF method.
3.2.1. Fuhrmann (1998)
This sample is composed of about 50 F and G nearby stars,
both main sequence and subgiants, of the Galactic disk and
4000
5000
6000
7000
8000
T e
ff
(IR
FM
) (
K)
4000 5000 6000 7000 8000
T
eff (SEDF) (K)
-300
-150
0
150
300
T e
ffI
R
M
F
-

T e
ffS
ED
F (K
)
-4 -3 -2 -1 0
[m/H]
-300
-150
0
150
300
T
ef
f I
R
M
F

-

T
ef
f S
ED
F
(K
)
Fig. 2. Comparison of the effective temperatures from the IRFM and
the SEDF method for 315 stars in the sample of Alonso et al. (1996a).
The bottom panel shows the temperature difference as a function of
the metallicity.
halo. Effective temperatures were determined from fits to the
wings of the Balmer lines. Of those stars, 24 have accurate
2MASS photometry so that reliable SEDF temperatures can be
derived. The comparison is shown in Fig. 4. The mean aver-
age difference ∆Teff (Fuhrmann − SEDF) is 12 K, (σTeff=45
K), with a slight dependence on the temperature: T Fuhrmann
eff =
0.895 T SEDF
eff +618 K. No dependence was found between ∆Teff
and [m/H] (Fig. 4, bottom panel).
3.2.2. Santos et al. (2003)
To study the correlation between the metallicity and the prob-
ability of a star to host a planet, Santos et al. (2003) obtained
spectroscopic temperatures for 139 stars based on the analy-
sis of several iron lines. Effective temperatures for a total of
101 stars in the sample of Santos et al. can be obtained us-
ing the SEDF method. In this case, ∆Teff (Santos − SEDF) is
28 K, with σTeff = 68 K, and practically independent of the
temperature: T Santos
eff = 1.053 T
SEDF
eff − 270 K (Fig. 5). There is
no dependence of ∆Teff with [m/H] (Fig. 5, bottom panel).

6 E. Masana et al.: Effective temperature scale and bolometric corrections
4000
5000
6000
7000
8000
T e
ff
(IR
M
F)
(K
)
4000 5000 6000 7000 8000
T
eff (SEDF) (K)
-300
-150
0
150
300
T
ef
f I
R
M
F
-

T
ef
f S
ED
F
(K
)
-4 -3 -2 -1 0 1
[m/H]
-300
-150
0
150
300
T
ef
f I
R
M
F

-

T
ef
f S
ED
F
(K
)
Fig. 3. Comparison of the effective temperatures from the IRFM and
the SEDF method for 386 stars in the sample of Ramı´rez & Mele´ndez
(2005). The bottom panel shows the temperature difference as a func-
tion of the metallicity.
3.2.3. Edvardsson et al. (1993)
The sample of Edvardsson et al. is composed by 189 nearby F
and G type stars. In contrast with the previous two, the effective
temperature is not derived from spectroscopy but from uvby−β
photometry. To do so, the authors built a grid of synthetic
photometry using the atmosphere models of Gustafsson et al.
(1975) and further improved it by adding several new atomic
and molecular lines. Effective temperatures for 115 stars in
their sample could be derived using the SEDF method. The
average difference ∆Teff (Edvardsson − SEDF) is 10 K, with
a dispersion of 70 K and no dependence on the temperature:
T Edvardsson
eff = 1.006 T
SEDF
eff − 27 K (Fig. 6). As in the two previ-
ous cases, the bottom panel of Fig. 6 shows that the temperature
difference is not correlated with [m/H].
4. Parametric calibrations
The practical use of the SEDF method as it has been described
in Sect. 2 is not straightforward since it requires the calcula-
tion of synthetic photometry from stellar atmosphere models
and then use a numerical algorithm to minimize the χ2 func-
tion. Parametric calibrations (as a function of one or more pa-
rameters) may offer a suitable means to estimate reliable ef-
fective temperatures in cases where simplicity and speed are
5000
5250
5500
5750
6000
6250
6500
T e
ff
(S
pe
c.)
(K
)
5000 5250 5500 5750 6000 6250 6500
T
eff(SEDF) (K)
-200
-100
0
100
T e
ff
Sp
ec
. -

T e
ff
SE
D
F (K
)
-2.5 -2 -1.5 -1 -0.5 0 0.5
[m/H]
-200
-100
0
100
200
T
ef
f S
pe
c.
-

T
ef
f S
ED
F
(K
)
Fig. 4. Comparison of the effective temperatures from fits to Balmer
lines and the SEDF method for 24 stars in common with the sample of
Fuhrmann (1998). The bottom panel shows the temperature difference
as a function of the metallicity.
to be preferred over the best possible accuracy. In this section
we present calibrations for both Teff and BC as a function of
(V − K)0, [m/H]and log g. To calculate the calibrations, the
SEDF method was applied to a sample of stars in the Hipparcos
catalogue, as described below. Note that these calibrations are
subject to two limitations with respect to the full SEDF method:
First, they are simplifications since not all the available infor-
mation is used, and second, individual uncertainties cannot be
determined.
4.1. The stellar sample
We collected a sample of FGK dwarfs and subdwarfs in the
Hipparcos catalogue, and therefore with measured trigono-
metric parallaxes. Their V magnitudes come mainly from the
Hauck & Mermilliod (1998) catalogue, except for those stars
with less than two observations, where we used the Hipparcos
catalogue. The entire sample has complete and non-saturated
JHK photometry in the 2MASS catalogue. The metallicity
was extracted from the compilation of Cayrel de Strobel et al.
(2001) or computed from uvby − β photometry – either mea-
sured from our own observations or obtained from the Hauck
& Mermilliod (1998) catalogue –, using a slightly revised ver-
sion of the Schuster & Nissen (1989) calibration. The range of

E. Masana et al.: Effective temperature scale and bolometric corrections 7
4500
5000
5500
6000
6500
T e
ff
(S
pe
c.)
(K
)
4500 5000 5500 6000 6500
T
eff(SEDF) (K)
-300
-150
0
150
T e
ff
Sp
ec
. -

T e
ff
SE
D
F
(K
)
-1 -0.5 0 0.5
[m/H]
-300
-150
0
150
300
T
ef
f S
pe
c.
-

T
ef
f S
ED
F
(K
)
Fig. 5. Comparison of the effective temperatures from iron line fits and
the SEDF method for 101 stars in common with the sample of Santos
et al. (2003). The bottom panel shows the temperature difference as a
function of the metallicity.
metallicities covered by the sample is −3.0 . [m/H] . 0.5.
Values of log g were computed from uvby − β photometry
(Masana 1994; Jordi et al. 1996). Originally, the sample was
built to study the structure and kinematics of the disk and halo
of the Galaxy (Masana (2004)) and a full description including
the photometry and a complete set of physical parameters will
be provided in a forthcoming paper (Masana et al. 2006).
In spite of the proximity of the stars (90% of them are closer
than 200 pc), we computed individual interstellar absorptions
from uvby − β photometry and corrected the observed magni-
tudes. As discussed below, interstellar absorption is one of the
most important sources of uncertainty in the Teff determination.
4.1.1. Errors
For our sample, the errors in the magnitudes, metallicity and
surface gravity were estimated in the following manner:
– Errors in the VJHK magnitudes: The total error in each
magnitude was computed as the quadratic sum of the ob-
servational error, the error in the absolute flux calibration
and the error in the determination of the interstellar ex-
tinction. The first one comes from the photometric cata-
logues. However, to prevent the underestimation of the er-
ror in the V band, usually computed from the average of
5500
6000
6500
7000
T e
ff
(P
ho
t.)
(K
)
5500 6000 6500 7000
T
eff (SEDF) (K)
-300
-150
0
150
300
T
ef
fP
ho
t.
-

T
ef
fS
ED
F

(K
)
-1.5 -1.25 -1 -0.75 -0.5 -0.25 0 0.25 0.5
[m/H]
-200
-100
0
100
200
T
ef
fP
ho
t.

-

T
ef
f S
ED
F
(K
)
Fig. 6. Comparison of the effective temperatures from photometry
and the SEDF method for 115 stars in common with the sample of
Edvardsson et al. (1993). The bottom panel shows the temperature
difference as a function of the metallicity. The standard deviation for
a single star in Edvardsson et al. (1993) is 81 K.
a few measurements (and with no evaluation of systemat-
ics), we have set a minimum error in V equal to 0.015 mag.
The uncertainties in the absolute flux calibration are given
by Cohen et al. (2003a,b) and are in the range 1.5–1.7%
(0.016–0.019 mag), depending on the band. For those stars
affected by interstellar reddening, the uncertainty in AV as
derived from photometric calibrations based on uvby − β
photometry (Jordi et al. 1996) is expected to be of about
0.05 mag, or ∼1.5% in Teff.
– Errors in [m/H] and log g: As mentioned above,
[m/H] was obtained, whenever possible, from spectro-
scopic measurements, and otherwise we used photometric
calibrations, with assigned uncertainties of 0.10 dex and
0.15 dex, respectively. We assigned uncertainties of 0.18
dex to log g values determined from photometric calibra-
tions. The effect on the final effective temperatures due to
the uncertainties of both [m/H] and log g is very small:
an error of 0.5 dex in [m/H] has an effect in Teff of less
than 0.5%, whereas the same error in log g has an effect
that ranges between 0% and 1%, depending on the value of
Teff and log g.
No error was attributed to the flux in the stellar atmosphere
models. Comparisons carried out by using other stellar atmo-

8 E. Masana et al.: Effective temperature scale and bolometric corrections
Fig. 7. Relative error (%) in effective temperature assuming σVJHK = 0.015 mag, σ[m/H] = 0.2, σlog g = 0.2 and the error of the absolute flux
calibration. Top left: [m/H] = 0.0. Top right: log g = 4.5. Bottom left: [m/H] = −2.0. Bottom right: log g = 2.0.
sphere models such as those by Castelli et al. (1997) and the
NextGen models by Hauschildt et al. (1999) show resulting dif-
ferences in temperature below ∼ 0.3% in all cases (Ribas et al.
2003).
An estimation of the final errors in Teff as function of Teff,
[m/H] and log g is shown in Fig. 7. As can be seen, the fi-
nal error is almost independent of [m/H] and log g, but not of
Teff. Hotter stars have greater uncertainties (slightly >1% for
Teff = 7500 K) than cooler stars (0.6% for Teff = 5000 K).
It is important to note that, in the case of reddened stars, an
uncertainty of 0.05 mag in AV can double the error in Teff com-
pared to the values in Fig. 7. For the angular semi-diameter
the behaviour of the errors is very similar to those of the ef-
fective temperature, with values for unreddened stars of about
1.0–2.5%. This means that for Hipparcos stars with good paral-
laxes, we are able to determine the stellar radii with remarkable
uncertainties of about 1.5–5.0%.
Figure 8 shows the cumulative histograms of the relative
errors in effectiVe temperature, angular semi-diameter and ra-
dius for the 10999 stars of the sample. As can be seen, about
85% of the stars have determinations of Teff better than 1.1%.
The relative error in the angular semi-diameter is also better
than 1.5% for about 85% of the stars. In the case of the radii,
the main contributor to the error is the uncertainty in the paral-
lax. Even so, 50% of the stars have radius determinations better
than 10%, and 85% of the stars better than 25%. It should be
kept in mind that most of the stars in our sample are essentially
unreddened, thus yielding the best possible accuracy.
Table 3 lists effective temperatures, angular semi-
diameters, radii and bolometric corrections in the V and K
(2MASS) bands with the corresponding uncertainties for the
entire sample. Using these values, we calculated simple para-
metric calibrations of effective temperature and bolometric cor-
rection as described below.
4.2. Effective temperature calibration
Although the effective temperature for FGK type stars is
strongly correlated with the (V − K)0 index (see for instance
Alonso et al. (1996a)), it also depends weakly on the metallicity
and surface gravity, as we mentioned in Sect. 2. Therefore, an
empirical calibration of Teff should include terms in all (V−K)0,
[m/H] and log g. Furthermore, in our case the calibrations were
constructed separately in two (V−K)0 intervals. Stars departing
more than 3σ from the fit were rejected. The resulting expres-
sions are:
– 0.35 < (V − K)0 < 1.15 (4954 stars):
θeff = 0.5961 + 0.1567(V − K)0 + 0.0309(V − K)20 +
+ 0.009[m/H] + 0.0022[m/H]2
+ 0.0021(V − K)0[m/H] − 0.0067 log g

E. Masana et al.: Effective temperature scale and bolometric corrections 9
0 0.5 1 1.5 2 2.5 3
Relative error in T
eff (%)
0
0.25
0.5
0.75
1
St
ar
’s
fr
ac
tio
n
0.25 0.75 1.25 1.75 2.25 2.75
St
ar
’s
fr
ac
tio
n
0 0.5 1 1.5 2 2.5 3
Relative error in θ (%)
0
0.25
0.5
0.75
1
St
ar
’s
fr
ac
tio
n
0.25 0.75 1.25 1.75 2.25 2.75
St
ar
’s
fr
ac
tio
n
0 5 10 15 20 25 30 35 40 45 50
Relative error R (%)
0
0.25
0.5
0.75
1
St
ar
’s
fr
ac
tio
n
St
ar
’s
fr
ac
tio
n
Fig. 8. Cumulative histograms of the relative error in effective tem-
perature, angular semi-diameter and radius for the 10999 stars in the
sample.
σθeff = 0.0028 (8)
– 1.15 ≤ (V − K)0 < 3.0 (5820 stars):
θeff = 0.5135 + 0.2687(V − K)0 − 0.0174(V − K)20 +
+ 0.0298[m/H]− 0.0009[m/H]2
− 0.0184(V − K)0[m/H] − 0.0028 log g
σθeff = 0.0026 (9)
where θeff = 5040Teff . The standard deviation of Eqs. (8) and (9) is
about 20 K and 25 K, respectively. As shown in Fig. 9, there
is no residual trend as a function of (V − K)0, [m/H] or log g.
Equation (8) is aplicable in the range 3.25 . log g . 4.75 and
eq. 9 in the range 3.75 . log g . 4.75. Furthermore the calibra-
tions are valid in the ranges of colours and metallicities of the
sample:
−3.0 < [m/H] < −1.5 for 1.0 < (V − K)0 < 2.9
−1.5 ≤ [m/H] < −0.5 for 0.5 < (V − K)0 < 2.9
−0.5 ≤ [m/H] < 0.0 for 0.4 < (V − K)0 < 3.0
0.5 ≤ [m/H] < 0.5 for 0.35 < (V − K)0 < 2.8 (10)
While (V − K)0 is an observational quantity and [m/H] can
be obtained from photometric and/or spectroscopic measure-
ments, a good determination of log g is usually unavailable
for the most of the stars. This could severely restrict the ap-
plicability of the above calibrations. However, some photo-
metric indexes, as the Stro¨mgren δc1 (see Crawford (1975) or
Olsen (1988)), are good surface gravity indicators and, if avail-
able, can help to estimate log g . On the other hand, catalogues
of spectroscopic metallicities usually provide an estimation of
the surface gravity. Finally, a crude estimation of log g can be
Fig. 9. Residuals of the Teff fit as function of effective temperature,
metallicity and surface gravity.
get from MK classification. The error in effective temperature
caused by an error in log g will be:
∆Teff =
a
5040T
2
eff∆ log g (11)
where a is the coefficient of the log g terms in Eqs. (8) and (9).
In the worst case, that of the hotter stars, ∆Teff = 85 ∆ log g.
Thus, even if the uncertainty in log g is as much as 0.5 dex, the
error induced in Teff is just 40 K.
The fits for four different metallicities and log g = 4.5 to-
gether with the stellar sample are shown in Fig. 10. Figure 11
shows the empirical Teff -(V − K)0 relationships as a function
of the metallicity.
4.3. Bolometric correction calibration
Since the SEDF method provides both effective temperature
and angular semi-diameter, it also naturally allows for the de-
termination of the bolometric correction in a specific band.
From this, if the distance is known, one can compute the lu-
minosity of the star. The bolometric correction in a given band
is defined as the difference between the bolometric magnitude
and the magnitude in that band:
BCi = Mbol − Mi = mbol − mi (12)
where mbol and mi are assumed to be corrected of interstellar
reddening. Mbol can be easily expressed as a function of the
radius and effective temperature:
Mbol = −5 log
R
R⊙
− 10 log Teff
Teff ⊙
+ 4.74 (13)

10 E. Masana et al.: Effective temperature scale and bolometric corrections
Fig. 10. Teff -(V − K)0 fits for four groups of stars with different
metallicities. The empirical relationships correspond to log g =4.5 and
[m/H] =−2.0, −1.0, −0.25 and +0.25.
0 1 2 3 4
(V-K)0
4000
5000
6000
7000
8000
T e
ff
(K
)
[m/H] = -3.0
[m/H] = -2.0
[m/H] = -1.0
[m/H] = 0.0
Fig. 11. Teff -(V − K)0 relationships for log g =4.5 and four different
metallicities.
where R⊙=6.95508 108 m and Teff ⊙ = 5777 K. For the Sun we
adopt V(⊙) = −26.75 mag and mbol(⊙) = −26.83 mag, and
therefore BCV (⊙) = −0.08 (Cox 2000).
Using the definition of the absolute magnitude at a given
band (Mx = mx + 5 logπ + 5) and expressing the radius as
function of the parallax (π) and the angular semi-diameter
(R = θ/π), we obtain the following formula for the bolomet-
ric correction:
BCx = Mbol − Mx =
= −5 log
(
K θ
R⊙
)
− 10 log T
T⊙
− 0.26 − mx (14)
where K is the factor corresponding to the transformation of
units. Once the bolometric correction for a band i is known, the
bolometric correction for any band j can be determined from:
BC j = (mi − m j) + BCi (15)
The error in the bolometric correction can be expressed as a
function of the uncertainties in Teff, θ (or A) and mi, as in Sect.
2.4:
(σBCi )2 =
(
5
ln 10
σθ
θ
)2
+
(
10
ln 10
σTeff
Teff
)2
+ (σmi )2 =
= (σA)2 +
(
10
ln 10
σTeff
Teff
)2
+ (σmi )2 (16)
The procedure described here was used to compute the
bolometric correction in the K (2MASS) band for the stars
in our sample. In the same way as for the effective tempera-
ture, we calibrated BC as a function of (V − K)0, [m/H] and
log g with the following results:
– 0.35 < (V − K)0 < 1.15 (4906 stars):
BCK = 0.1275 + 0.9907(V − K)0 − 0.0395(V − K)20 +
+0.0693[m/H] + 0.0140[m/H]2
+0.0120(V − K)0[m/H] − 0.0253 log g
σBC = 0.007 mag (17)
– 1.15 ≤ (V − K)0 < 3.0 (5783 stars):
BCK = −0.1041 + 1.2600(V − K)0 − 0.1570(V − K)20 +
+0.1460[m/H] + 0.0010[m/H]2
−0.0631(V − K)0[m/H] − 0.0079 log g
σBC = 0.005 mag (18)
The range of validity of these calibrations is the same as in the
case of the effective temperature. The bolometric correction in
any band can be obtained from BCK via Eq. (15).
Figure 12 shows the fits for four different metallicities, to-
gether with the stars in the sample used to obtain the calibra-
tions. The BCK − (V − K)0 relationships as a function of the
metallicity are shown in Fig. 13. The calibration is tabulated
in Table 2 and compared with the calibrations by Alonso et al.
(1995) and Flower (1996) in Fig. 14, showing good agreement.
5. Discussion
The procedure described in this paper yields three basic stellar
parameters: the best-fitting effective temperature and angular
semi-diameter and, from them, the bolometric correction. If the
distance is known, θ can be transformed into the true stellar
radius. The accuracies of the parameters for the stars in our
sample are 0.5–1.3% in Teff, 1.0–2.5% in θ and 0.04–0.08 mag
for the BC.

E. Masana et al.: Effective temperature scale and bolometric corrections 11
40005000600070008000
T
eff(K)
−1.25
−1
−0.75
−0.5
−0.25
0
0.25
B
C V
40005000600070008000
T
eff(K)
−1.25
−1
−0.75
−0.5
−0.25
0
0.25
B
C V
40005000600070008000
T
eff(K)
−1.25
−1
−0.75
−0.5
−0.25
0
0.25
B
C V
40005000600070008000
T
eff(K)
−1.25
−1
−0.75
−0.5
−0.25
0
0.25
B
C V
[m/H] = −1.0 [m/H] = −2.0
[m/H] = 0.0[m/H] = +0.5
Fig. 14. Comparison between BCV values for log g =4.5 given in Table 2 (solid line) and the values given by Alonso et al. (1995) for log g = 4
(dashed line) and log g = 5 (long-dashed line). In the panel corresponding to [M/H] = 0.0, the calibration of Flower (1996) is also shown
(dotted line).
Comparisons with other determinations described in Sect.
3 show general good agreement, with differences below 0.5σ,
except for Alonso et al. (1996a) and Ramı´rez & Mele´ndez
(2005), where the difference is about 0.8σ. The use of differ-
ent atmosphere models and the intrinsic nature of the meth-
ods (photometric for Edvardsson et al., Alonso et al., Ramı´rez
& Mele´ndez and ours; spectroscopic for Santos et al. and
Fuhrmann) can explain in part the small differences. In the
case of the IRFM, the main difference between the implemen-
tation of both Alonso et al. (1996a) and Ramı´rez & Mele´ndez
(2005), and the SEDF method is the absolute flux calibration:
Alonso et al. (1994) for the IRFM and Cohen et al. (2003b) for
the SEDF. This, together with the use of different versions of
the ATLAS9 atmosphere models, is probably the reason for the
∼60 K differential between both implementations of the IRFM
and our determination. For Ramı´rez & Mele´ndez (2005) there
is a dependence of ∆Teff with [m/H] in such a way that the
temperature difference (Ramı´rez & Mele´ndez- SEDF) increase
abruptly for [m/H] . 2.0. In all the other cases, the temperature
differences are not correlated with [m/H].
The most important factor to explain the systematics among
the effective temperatures computed from different methods is
the absolute flux calibration affecting photometric determina-
tions and inaccuracies of model atmospheres (non-LTE effects,
3D effects, treatment on convection ,...) affecting both photo-
metric and spectroscopic determinations. Bohlin & Gilliland
(2004) pointed out a probable 2% overestimation of the IR flux
in the Vega model used by Cohen et al. (2003a). A 2% shift in
absolute flux calibration is equivalent to a difference of about
40 K in temperature and to a zero point offset in the synthetic
photometry of 0.022 mag. Such value would be compatible
with our magnitude zero points in Sect. 2.3.
Beyond the internal errors, which in the case of the SEDF
take into account the uncertainty in the flux calibration and all
other error sources, the comparison with other methods shows
that, at present, the systematic errors involved in the determi-
nation of effective temperature are of about 20–30 K, equiva-
lent to the 2% uncertainty in the IR fluxes of Vega claimed by
Bohlin & Gilliland (2004) to be a realistic value.
6. Conclusions
We have presented a method (called SEDF) to compute ef-
fective temperatures, angular semi-diameters and bolometric
corrections from 2MASS photometry. We have adopted an ap-
proach based on the fit of the observed VJHK magnitudes us-

12 E. Masana et al.: Effective temperature scale and bolometric corrections
Table 2. Bolometric correction as a function of effective temperature for log g =4.5 and different metallicities (valid for dwarf and subdwarf
stars). Tables for log g =3.5 and log g =4.0 are available in electronic form.
[m/H] = +0.5 [m/H] = 0.0 [m/H] = −1.0 [m/H] = −2.0 [m/H] = −3.0
Teff BC(V) BC(K) BC(V) BC(K) BC(V) BC(K) BC(V) BC(K) BC(V) BC(K)
4000 −1.536 2.336 −1.344 2.375 −1.067 2.417 — — — —
4100 −1.310 2.324 −1.156 2.348 −0.933 2.371 — — — —
4200 −1.122 2.298 −0.999 2.311 −0.818 2.319 — — — —
4300 −0.964 2.262 −0.865 2.266 −0.719 2.263 — — — —
4400 −0.830 2.219 −0.751 2.217 −0.635 2.204 −0.559 2.186 −0.509 2.166
4500 −0.716 2.171 −0.653 2.164 −0.562 2.144 −0.504 2.121 −0.468 2.097
4600 −0.618 2.119 −0.568 2.108 −0.499 2.082 −0.457 2.055 −0.433 2.029
4700 −0.532 2.064 −0.495 2.050 −0.444 2.020 −0.416 1.990 −0.403 1.962
4800 −0.459 2.008 −0.431 1.991 −0.396 1.958 −0.380 1.926 −0.377 1.896
4900 −0.395 1.950 −0.376 1.932 −0.355 1.896 −0.350 1.862 −0.356 1.831
5000 −0.339 1.892 −0.327 1.872 −0.319 1.834 −0.324 1.799 −0.337 1.767
5100 −0.290 1.833 −0.285 1.813 −0.288 1.773 −0.301 1.737 −0.321 1.704
5200 −0.247 1.775 −0.248 1.753 −0.261 1.713 −0.282 1.676 −0.308 1.643
5300 −0.209 1.717 −0.216 1.695 −0.237 1.653 −0.265 1.616 −0.298 1.582
5400 −0.177 1.659 −0.188 1.636 −0.217 1.595 −0.251 1.557 −0.289 1.523
5500 −0.148 1.601 −0.163 1.579 −0.199 1.537 −0.239 1.499 −0.282 1.466
5600 −0.123 1.545 −0.142 1.522 −0.184 1.480 −0.229 1.443 −0.276 1.409
5700 −0.101 1.489 −0.123 1.467 −0.171 1.425 −0.221 1.387 −0.272 1.354
5800 −0.081 1.434 −0.107 1.412 −0.160 1.370 −0.214 1.333 −0.269 1.301
5900 −0.065 1.380 −0.093 1.358 −0.151 1.317 −0.209 1.280 −0.267 1.248
6000 −0.050 1.327 −0.081 1.305 −0.143 1.265 −0.205 1.228 −0.266 1.197
6100 −0.038 1.275 −0.071 1.253 −0.137 1.213 −0.202 1.178 −0.266 1.147
6200 −0.027 1.224 −0.062 1.202 −0.132 1.163 −0.200 1.128 −0.267 1.098
6300 −0.018 1.173 −0.055 1.152 −0.128 1.114 −0.199 1.080 −0.268 1.050
6400 −0.002 1.123 −0.049 1.104 −0.125 1.066 −0.199 1.032 −0.270 1.003
6500 0.003 1.074 −0.045 1.054 −0.118 1.022 −0.199 0.986 −0.272 0.957
6600 0.007 1.026 −0.040 1.007 −0.112 0.975 — — — —
6700 0.011 0.978 −0.035 0.959 −0.106 0.929 — — — —
6800 0.015 0.931 −0.031 0.913 −0.101 0.883 — — — —
6900 0.019 0.884 −0.026 0.866 −0.096 0.838 — — — —
7000 0.023 0.837 −0.023 0.821 −0.092 0.793 — — — —
7100 0.026 0.792 −0.019 0.775 −0.087 0.748 — — — —
7200 0.029 0.746 −0.016 0.730 −0.083 0.704 — — — —
7300 0.032 0.701 −0.012 0.686 −0.079 0.660 — — — —
7400 0.034 0.656 −0.009 0.642 −0.076 0.617 — — — —
7500 0.037 0.612 −0.007 0.598 −0.072 0.574 — — — —
7600 0.039 0.568 −0.004 0.555 −0.069 0.532 — — — —
7700 0.041 0.525 −0.002 0.512 −0.066 0.490 — — — —
7800 0.043 0.481 0.000 0.469 — — — — — —
7900 0.044 0.438 0.002 0.427 — — — — — —
8000 0.046 0.396 0.004 0.385 — — — — — —
ing synthetic photometry, and it yields accuracies around 1%
in Teff, 2% in θ, and 0.05 mag in BC, in the temperature range
4000–8000 K. A zero point offset was added to the synthetic
photometry computed from the Kurucz atmosphere models to
tie in our temperature scale with the Sun’s temperature through
a sample of solar analogues. From the application to a large
sample of FGK Hipparcos dwarfs and subdwarfs we provide
parametric calibrations for both effective temperature and bolo-
metric correction as a function of (V − K)0, [m/H] and log g.
Note that the method presented here has been selected as one of
the main sources of effective temperatures to characterize the
primary and secondary targets of the COROT space mission
(Baglin et al. 2000). Also, it is being currently implemented
as one of the tools offered by the Spanish Virtual Observatory
(Solano et al. 2005).
The resulting temperatures have been compared with sev-
eral photometric and spectroscopic determinations. Although
we obtained remarkably good agreement, slight systematic dif-
ferences with other semi-empirical methods, such as the IRFM,
are present. This is probably due to the uncertainties in the ab-
solute flux calibration used by different techniques. It is possi-
ble that, in spite of the great effort carried out by Cohen et al.
(2003a) and others to construct a consistent absolute flux cal-
ibration in both the optical and the IR regions, some prob-
lems still remain, which introduce small systematic effects in
the temperatures. However, these effects seem to be as small

E. Masana et al.: Effective temperature scale and bolometric corrections 13
0 1 2 3 4
(V−K)0
0
0.5
1
1.5
2
2.5
3
B
C K
−0.5 < [m/H] < 0.0
0 1 2 3 4
(V−K)0
0
0.5
1
1.5
2
2.5
3
B
C K
−3.5 < [m/H] < −1.5
0 1 2 3 4
(V−K)0
0
0.5
1
1.5
2
2.5
3
B
C K
0.0 < [m/H] < +0.5
0 1 2 3 4
(V−K)0
0
0.5
1
1.5
2
2.5
3
B
C K
−1.5 < [m/H] < −0.5
Fig. 12. BCK − (V − K)0 fits for four groups of stars with different
metallicities. The empirical relationships correspond to log g =4.5 and
[m/H] =−2.0, −1.0, −0.25 and +0.25.
0 0.5 1 1.5 2 2.5 3 3.5
(V−K)0
0
0.5
1
1.5
2
2.5
3
B
C K
[m/H] = −3.0
[m/H] = −2.0
[m/H] = −1.0
[m/H] = 0.0
Fig. 13. BCK − (V − K)0 relationships for log g =4.5 and four different
metallicities.
as 20–30 K and could be explained through uncertainties in
the IR fluxes of about 2%. In conclusion, the results presented
here strongly suggest that, given the small differences found
between methods, the effective temperature scale of FGK stars
(4000–8000 K) is currently established with a net accuracy bet-
ter than 0.5–1.0%.
Acknowledgements. We are grateful to Dr. Angel Alonso for the
suggestion of using solar analogues to calibrate our method. We
thank P. Nissen and the anonymous referee for their remarks that
helped to improve the paper. We also acknowledge support from
the Spanish MCyT through grant AyA2003-07736. I. R. acknowl-
edges support from the Spanish Ministerio de Ciencia y Tecnologı´a
through a Ramo´n y Cajal fellowship. This publication makes use of
data products from the Two Micron All Sky Survey, which is a joint
project of the University of Massachusetts and the Infrared Processing
and Analysis Center/California Institute of Technology, funded by
the National Aeronautics and Space Administration and the National
Science Foundation.
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14 E. Masana et al.: Effective temperature scale and bolometric corrections
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