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Mon. Not. R. Astron. Soc. 000, 000{000 (0000) Printed 31 July 1999 (MN L A T E X style le v1.4)
Photometric modelling of starspots. II. The
FORTRAN code SPOTPIC
P.J. Amado 1? , J. G. Doyle 1 and P.B. Byrne y
1 Armagh Observatory, College Hill, Armagh BT61 9DG, N. Ireland
31 July 1999
ABSTRACT
A FORTRAN code which computes synthetic light and colour curves of
active, spotted stars has been developed. The main feature of this code is its
ability to model simultaneously the V light curve and the (V {R) c , (V {I) c and
(V {K) colour data. It also uses new eective temperature-colour and Barnes-
Evans-like calibrations, temperature and gravity-dependent limb-darkening
coeôcients and dierent eective surface gravities for the spotted and unspot-
ted photosphere. The code allows for two-component spots, i.e., spots with
umbral and penumbral components. Various problematic spot congurations
were investigated, taking us to the conclusion that, in order to be able to dif-
ferentiate spots with various thermal structures (umbrae, penumbrae, faculae)
or polar spots from equatorial bands, the modelling of the infrared colours,
especially (V {I) c and (V {K), is needed.
Key words: starspots { stars: late-type
1 INTRODUCTION
Photometric spot models have been used to model the light and colour curves of active
stars, including RS CVn, BY Dra, FK Com and T Tau type stars. Several authors have
employed much eort in developing formulations and computational codes to deal with this
problem (Budding 1977; Budding & Zeilik 1987; Dorren 1987; Eker 1994). Most of them
? Present address: Instituto de Astrofsica de Andaluca-CSIC. Apartado 3004. 18080 Granada. Spain. E-mail: pja@iaa.es
y Deceased
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2 P.J. Amado, J. G. Doyle and P.B. Byrne
have also studied the impact on these theoretical light curves of changing the spot and the
star parameters (especially the inclination of the star, i) (Strassmeier 1987; Eker 1994) or
of the intrinsic scatter of the observed data (K}ovari & Bartus 1997). However, very little
has been said on the eects of using one or another spectral ux distribution to compute
theoretical light and colour curves of spotted stars.
It is generally accepted that the lack of accuracy of the photometric data and the diô-
culties at determining important stellar parameters like the inclination of the star and the
unspotted magnitudes limit the goodness of the spot solutions (at least in the determination
of the latitudes of the spots) and their uniqueness. The inversion technique of recovering
spot positions by tting temporal series of observations of some photospheric lines of rapidly
rotating stars known as Doppler imaging is much better at determining the spatial distri-
bution of spots but temperatures must be xed beforehand. It has been showed, however,
that unique solutions can be recovered if VRI light curves and line proles are observed and
modelled simultaneously (see, for instance, Strassmeier 1987).
In this work we present a new FORTRAN code whose main ability lies on its being more
sensitive to temperature-area determinations of starspots, since it models the ux ratios
between the spot and the unspotted photosphere more reliably.
2 THE FORTRAN CODE
spotpic is a FORTRAN code developed to compute synthetic light and colour curves of
active late-type stars with temperature inhomogeneities (starspots) on their surfaces. It uses
the eective temperature{colour relationships derived by Amado (1997) and the calibration
of the surface brightness parameter FV against the (I c {K) colour index from Amado, Butler
& Byrne (1999a) (hereafter Paper I) to compute the colours and ux ratios, respectively,
of the spotted and unspotted photosphere of the star. The code calls a subroutine named
geom which calculates by direct integration on the visible surface of the star the projected
area of up to ve circular spots, although this number can be easily modied in order to
accommodate any number of spots. The formulation utilized for these integrations is the one
developed by Dorren (1987) and the input parameters for the code are, namely, the eective
temperature of the spot (T sp ) and unspotted photosphere (T ph ), the surface gravity of the
star (log g ph ) and the eective gravity of the spot (log g sp ), position of the spot on the star's
photosphere, i.e., latitude () and phase (), and its radius r sp , the inclination angle of the
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Photometric modelling of starspots. II. The FORTRAN code SPOTPIC 3
star i, its unspotted magnitudes and the contribution to each magnitude from a companion
if the star was a binary. The code allows for two-component spots, i.e., spots with concentric
umbra and penumbra, in which case, the ratio of the umbral radius to the total spot radius
(r u =r sp ) can be varied. Limb-darkening coeôcients are interpolated from the tables from
Daz Cordoves, Claret & Gimenez (1995) and Claret, Daz Cordoves & Gimenez (1995).
Finally, the code calculates and nds the best t to the observed light and colour curves
simultaneously in V , (V {R) c , (V {I) c and (V {K) by means of the Levenberg-Marquardt
least-square minimization procedure.
The fundamental dierences between this and other codes already in existence are the
use and modelling of infrared colours like (V {K), the use of a new Barnes-Evans-like FV {
(I c {K) relation to determine the ux ratios between the spot and unspotted photosphere
and the possibility of assigning dierent surface gravities and limb darkenings for the spotted
and unspotted photosphere. We call eective surface gravity of the spot to be the \gravity"
of the spot once the magnetic pressure is taken into account. Due to the presence of the
strong magnetic elds inside the spot, the gas pressure is lowered in order to keep the
pressure balance, leading to a dierent value of the surface gravity with respect to that of
the unspotted photosphere. This eect can be modelled with this code.
3 LIMB-DARKENING COEFFICIENTS
Many authors have studied the variations produced in the light and colour curves of spotted
stars by either ignoring the limb-darkening or using linear or non-linear limb-darkening laws
(see, for instance, the analysis by Eker 1994). They all arrived at the conclusion that a limb-
darkening law must be introduced in the calculations if reliable light and colour curves are to
be obtained and that a linear law is suôcient, considering the accuracy of the photometry,
if the U and B passbands are not used (Eker 1994).
The rst explicit expression to compute the eect of limb-darkening as a linear approx-
imation was derived by Milne (1921), who introduced the well-known linear limb-darkening
law
I() = I(1) (1 u (1 ))
where  = cos , and is the angle subtended between the line of sight and the direction of
the emerging ux. I(1) represents the specic intensity emitted at the centre of the stellar
disc and u is the limb-darkening coeôcient.
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4 P.J. Amado, J. G. Doyle and P.B. Byrne
For the study of the light curves of spotted stars, the linear approximation is easy to
introduce in the equations which compute the ux emitted by the star. For this reason and
because the eect of using dierent laws to obtain the light and colour curves of spotted
stars is very small, as showed by Eker (1994), only the linear law was considered here.
The linear limb-darkening coeôcients adopted for use with the code are those by Daz Cor-
doves et al. (1995) for the UBV passbands and by Claret et al. (1995) for the RIJHK pass-
bands. They tted the linear coeôcients computed from the model atmospheres by Kurucz
(1991) by means of the straight least-squares method. They followed this procedure because
some of the other methods produced unrealistic values of I()=I(1) for the coolest models
in which convective envelopes were used (see, for instance, Daz Cordoves et al. 1995).
The coeôcients given by these authors are tabulated versus T e and log g, the T e ranging
from 50000 K down to 3500 K and the log g between 0.0 and 5.0. In Fig. 1, the linear limb-
darkening coeôcients for the VRIK passbands calculated from the Kurucz models (upper
panel) are plotted. For T e 's less than 3500 K, which is the lowest temperature the Kurucz
models reach, the code assumes constant values for the limb-darkening coeôcient equal to
those at 3500 K. The code assigns dierent limb darkenings to parts of the photosphere
with dierent eective temperature in order to reproduce with more accuracy this eect on
the light and colour curves. It is clear that, although small, the darkening of the unspotted
photosphere will not be the same as that of a spot or facula.
Recently, Claret (1998) calculated linear limb{darkening coeôcients using model atmo-
spheres from the phoenix code (Allard et al. 1994; Allard & Hauschildt 1995a; Allard &
Hauschildt 1995b) for very cool stars, and his results are plotted in the lower panel of Fig. 1.
The coeôcients become larger than 1 for the coolest temperatures, which could indicate that
the linear law is not suitable in this region. The current approximation is accurate down to
3000 K, as evident by Fig. 1. For eective temperatures lower than 3000 K, new coeôcients
are required or perhaps non-linear limb darkening laws.
4 EFFECTIVE TEMPERATURE-COLOUR
AND SURFACE BRIGHTNESS-COLOUR RELATIONSHIPS
Input parameters of the code are the photospheric temperature and the spot temperatures,
for one-component spots, or the umbral and penumbral temperatures for two-component
spots. The code transforms these temperatures into colours via the calibrations taken from
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Photometric modelling of starspots. II. The FORTRAN code SPOTPIC 5
Figure 1. Linear limb-darkening coeôcients for VRI and K passbands from Kurucz models (upper panel) and from phoenix
models (lower panel). The family of curves for dierent log g for each passband has the same symbol, i.e., circles for V , squares
for R, triangles for I and crosses for K. Within each family, the surface gravity changes as follows: solid line for log g = 5:0,
dotted line for log g = 4:5, dashed line for log g = 4:0, long-dashed line for log g = 3:5 and dashed-dotted line for log g = 3:0.
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6 P.J. Amado, J. G. Doyle and P.B. Byrne
Figure 2. Colour-eective temperature relationships employed in the FORTRAN code spotpic. a) (V {R)c , b) (V {I)c , c)
(V {K) and d) (Ic {K) colours against eective temperature, T e . The open triangles are the data for the giant stars from
Amado & Byrne (1996) and the solid symbols for the dwarfs from Leggett et al. (1996) and Kirkpatrick et al. (1993) (triangles)
and Berriman & Reid (1987) and Veeder (1974) (squares). The lines represent the ts to these data and to the theoretical
colours from the phoenix model atmospheres for log g = 5:0 (solid), log g = 4:0 (dashed) and log g = 3:0 (dotted).
Amado (1997), given in Fig. 2, where the lines are ts to synthetic colours computed from the
phoenix model atmospheres and to data with eective temperatures derived from empirical
and semi-empirical methods (see Amado (1997) for details), trying to match the rst with
the second where they overlap. These ts were either expressed as polynomial functions
and introduced directly in the code or written in a tabular form from which the code could
interpolate using a cubic spline t.
The eective temperatures for dwarfs are computed using theoretical methods as was
explained in Paper I and, therefore, they might be aected by the deciencies of the models
and not be as reliable as those obtained from empirical methods. However, it is interesting
to note from Fig. 2 that the colours are able to better delineate stars of dierent luminosity-
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class as one goes towards longer wavelength baselines. However, for higher temperatures
most of the colour indexes are insensitive to gravity (Amado 1997; Bessell, Castelli & Plez
1998).
One of the approaches to evaluating uxes emitted by both the unspotted photosphere
and spots is by using the calibration of the surface brightness parameter against a colour
index. In spotpic, the FV {(I c {K) relationship derived in Paper I was used for this purpose,
i.e., the code calculates surface brightnesses once the colours for the spot and unspotted
photosphere have been computed via the colour{T e relations.
5 EFFECTIVE GRAVITY OF SPOTS
The maximum eld strength that the gas pressure can conne at the photospheric layers of
late-type stars can be determined by
B 0 =
q
8p e
The way B 0 depends on T e and log g was determined by Bunte & Saar (1993) from Kurucz
(1991) models with 3500K  T e  7000 K and 0:5  log g  5:0 and is shown in Fig. 1 in
their paper.
This dependence of B 0 with log g and T e can be interpreted as follows (Solanki 1996),
(a) dierences in density scale height: as the gravity decreases, one can see less deep into the
atmosphere; as the density scale height increases, both the geometrical and optical paths the
light must traverse from a given pressure level to the observer increase. This implies that the
gas pressure and, thereby, B 0 at  = 1 at 5000  A, where  is the continuum optical depth,
decreases with decreasing g, (b) continuum H opacity: as T e is lowered, the main source
of continuum opacity (due to the H ion) decreases. Consequently, deeper layers with larger
pressure can be seen, so that B 0 increases with decreasing temperature.
For stars of luminosity class I-III, however, the kinetic energy density derived from mea-
sured macroturbulence velocities can become comparable to the internal energy density of
the gas (Gray 1992). Thus the maximum achievable eld strength is larger for such stars.
Half or more of the magnetic energy density in ux tubes on giants and supergiants may be
due to the conning eect of ows.
In order to estimate the gas pressure inside the ux tube (spot) and, thereby a value
to the eective gravity of the spot, the following expression for the gravity of a star as a
function of the non-magnetic gas pressure (Gray 1992) will be used
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8 P.J. Amado, J. G. Doyle and P.B. Byrne
Table 1. Photospheric surface ( = 1 at 5000  A) gravity and gas pressure for dwarf stars of various spectral types (Allen
1973) and the corresponding equipartition values for the internal pressure and eective gravity for a spot with a magnetic eld
strength of B = 1000 G.
Sp.Type log gext P ext
gas P int
gas log g int
G5V 4.50 7:94 10 4 3:96 10 4 3.65
K0V 4.50 1:00 10 5 6:02 10 4 3.96
K5V 4.50 1:26 10 5 8:62 10 4 4.22
M0V 4.60 1:58 10 5 1:18 10 5 4.44
M5V 4.80 2:51 10 5 2:11 10 5 4.86
g  =
"
g 0:6

p 
p 
# 1
0:6
where p  is the stellar pressure, p  is the solar photospheric pressure and g  is the value for
the solar gravity at the surface ( = 1 at 5000  A), which is given by
g() = G M
R 2

= 2:7387 10 4 cm s 2
where M and R are the solar mass and radius and G is the Gravitational constant.
In sunspots, the main trend is for the eld strength to decrease from its maximum value
in the darkest part of the umbra (2000-3600 G) to the outer edge of the penumbra, where it
lies between 700-1000 G (e.g., Solanki 1992; Lites & Skumanich 1990; Lites et al. 1993). The
maximum umbral eld strength increases with increasing umbral size (Kopp & Rabin 1992),
but the eld strength averaged over the whole sunspot, i.e., over the whole ux tube, is
relatively independent of size, being roughly 1200-1700 G. Assuming a conservative value of
B = 1000 G for the magnetic strength in starspots, then p mag  B 2 =8 = 3:98 10 4 dyn cm 2 .
Varying the surface gravity of the star and taking the photospheric pressure values from Allen
(1973), the values for the eective gravity of the spot given in Table 1 are obtained. It can
be seen that the spot eective gravity is always lower than the star's surface gravity except
for the coolest dwarfs. However, the maximum magnetic eld strength increases for these
stars due to the increase in their photospheric pressure and to the decrease in T e (Bunte
& Saar 1993). Therefore, it is not too unrealistic to presume that the average magnetic
eld strength is going to increase as well, keeping the eective gravity of the spot below the
photospheric surface gravity.
6 MINIMIZATION PROCEDURE
The problem assessed here is the minimization of a function which, in our case, is the  2
of the t of the theoretical light and colour curves to the data points of spotted stars. In
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Photometric modelling of starspots. II. The FORTRAN code SPOTPIC 9
general, the problem of nding a global minimum (truly the lowest function value) instead of
a local one (the lowest in a nite neighbourhood) is a very diôcult one. This can be achieved
by using various dierent algorithms, among which, one can choose depending on the nature
of the function, the number of dimensions, the possibility of computing the derivatives and,
of course, our restrictions in computing memory and speed. Since our particular problem is
that of, given a set of observations, tting a particular model and since we can also compute
the derivatives of the equations of our model we chose the Levenberg-Marquardt method as
our minimization procedure. For a more detailed explanation on the workings and equations
of the Levenberg-Marquardt method see Press et al. (1992).
The parameters to be extracted from the ts to the observed data are the eective
temperature, position, i.e., latitude and phase and radius of the spots. The stellar parameters,
like the inclination angle of the star and its unspotted magnitudes, and the eective gravity
of the spots are xed. Since the code allows for two-component spots, the ratio of the umbral
radius to the total spot radius can also be determined.
The code nds the best t to the observed light and colour curves simultaneously in V ,
(V {R) c , (V {I) c and (V {K). Basically, what the code does is, given a set of initial values for
the spot parameters (trial parameters), it computes the theoretical V , (V {R), (V {I) and
(V {K) curves and compares them with the observed data. If this rst approximation of the
spot parameters is a good one, it knows how to jump from the current trial parameters to
the real ones by using the Hessian (second derivatives) matrix. On the other hand, if it is a
poor local approximation it takes a step down the gradient towards a better solution, as in
the steepest descent method (see also Press et al. 1992). By iterating the process, a solution
which minimizes the  2 can be found.
7 SOME THEORETICAL RESULTS
One of the problems we will try to shed some light on is the use of two-component spots to
model observed light and colour curves. This problem has been addressed by a few authors
like, for instance, Lodenquai & McTavish (1988), Kjurkchieva (1990) and Dorren (1987).
The last author investigated the question of whether the temperatures derived from tting
the light and colour curves produced by a two-component spot with a spot of uniform
temperature were more likely to represent the umbral or the penumbral temperature. He
found, against expectations expressed in the literature given the much higher ux emitted
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10 P.J. Amado, J. G. Doyle and P.B. Byrne
by the penumbra, that umbral temperatures were obtained for the one-component t, with
only temperatures derived from two-component spots with very small umbrae departing
signicantly from the umbral temperature.
Another longstanding controversy is about the existence of spots on the rotational poles
of active stars. Such spots distributions are used to explain at-bottomed photospheric line
proles and they have been found in Doppler maps of rapidly-rotating BY Dra, RS CVn
and FK Com stars by, for instance, Vogt & Penrod (1983) on HR 1099, Strassmeier (1994)
on HU Vir, or Kurster et al. (1992) on YY Gem. Researchers have put forward many
reasons to try to explain why polar spots are not artifacts of the Doppler imaging technique
and why they must be considered real. Hatzes et al. (1996) studied possible hypotheses
for the appearance of polar spots in Doppler images and found that dierential rotation
and facular bands succeeded in producing box-shaped spectral line proles, although they
did not reproduce the seen dependence with stellar inclination. However, it must be said
that this relation is based on a very small sample of stars. Theoretical interpretation of the
distribution of spots on rapidly rotating stars agree with the scenario of spots appearing at
high latitudes or even straddling the poles (Schussler & Solanki 1992; Schussler et al. 1996).
On the other hand, Byrne (1995) and Unruh & Cameron (1995), for instance, recom-
mended some caution since there are still many uncertainties when it comes to modelling
the observed line proles that the technique uses to extract the spot parameters. Byrne
(1995) enumerated possible physical conditions of the atmospheres of rapidly rotating ac-
tive stars which could simulate the conditions for the existence of polar spots. However, some
of them have been shown not to aect the line proles as much as to generate polar spots
in the Doppler maps. For instance, Unruh & Collier Cameron (1997) and Bruls, Solanki &
Schussler (1998) have demonstrated that chromospheric activity is not the main cause of
the at-bottomed cores of the line proles.
7.1 Two-component spots: umbra and penumbra
On the Sun, large spots have an umbral/spot radius ratio (r u =r sp ) of, typically, 0.42 and a ux
ratio such that 98% of the spot ux is contributed by the penumbra (Allen 1973). Therefore,
based on these gures, expectations among researchers were that temperatures derived for
starspots, if they had similar structures to those on the Sun, would be closer to penumbral
temperatures. However, ratios of derived spot temperature to the photospheric temperature
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for most RS CVn stars were closer to the solar umbral-to-photospheric temperature ratio,
suggesting that spot temperatures from the ts re ected those of the starspot umbrae (Vogt
& Penrod 1983). It was, therefore, suggested that the structure of starspots might be quite
dierent from that of sunspots, either with a relatively larger umbral component or even
of a uniform temperature (Vogt & Penrod 1983; Poe & Eaton 1985). Nevertheless, a direct
comparison of the umbral-to-spot temperature ratio between the Sun, a G2 dwarf star, and
RS CVn's is not very meaningful due to the very dierent nature of these two types of stars.
In his study of this problem, Dorren (1987) found that the eect of introducing an
umbral component to the spot was approximately wavelength-independent and that the
important factor in determining the light-curve amplitude was the loss of light due to the
presence of the spot, rather than simply the uxes from the umbra and the penumbra. In
one example of his work, he tted the light curves produced by a concentric two-component
30 ô radius spot with a one-component spot. The resulting temperature for the solid spot T s
was much closer to the umbral temperature (T s T u = 260 K) than to the penumbral one
(T p T s = 1180 K), with the spot decreasing to a radius of 19:5 ô . The simulation adopted
the Sun's photospheric temperature, i.e., T ph = 6050 K, a spot penumbral temperature of
T p = 5680 K (
T p  T ph T p = 370 K) and an umbral temperature of T u = 4240 K
(
T u  T ph T u = 1810 K), which are typical values for large sunspots.
That study was partly repeated here but for two cooler photospheric temperature, viz.,
5200 K and 4700 K, since the code spotpic was developed especially for stars cooler than
the Sun. The results do not agree with those found by Dorren (1987), i.e., the modelled
solid spot temperatures were nearer to those of the penumbral component of the concen-
tric two-component spot, for the hotter photospheric temperature, and roughly in the um-
bral/penumbral average temperature for the cooler photospheric temperature. The adopted
values for the radius, the ratio r u =r sp ,
T p and
T u were the same as used by Dorren (1987).
In order to see the eect of two-component spots on the colour curves, a comparison
between three dierent types of spots will be considered next. These spots will be on the
equator and will have a radius of  sp = 30 ô and an average eective temperature of T sp =
4300 K on a photosphere with an eective temperature of T ph = 4700 K.
The maximum projected area for a spot is reached when the spot is at the centre of
the visible disk, and is A sp =  sin 2 . Let us consider the three following models for two-
component spots,
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Table 2. V -band ux ratios, eective temperatures and radii for the umbral and penumbral
regions of three dierent type of spots.
Model Fu=F Fp=F 1
 Fsp=F Tu T 1
p Tsp ru=rsp rsp
Solid spot { { 0.667 { { 4300 0.00 30 ô
Umbra/Penumbra Spot 0.412 0.719 0.667 3527 4400 4300 0.40 30 ô
Umbra/Facula Spot 0.455 1.305 0.667 3604 4900 4300 0.85 30 ô
1 For the Umbra/Facula Spot Model, these values represent those of the facular compo-
nent.
(i) Solid Spot Model, computed with a one-component spot of eective temperature T sp =
4300 K and radius  sp = 30 ô
(ii) Umbra/Penumbra Spot Model, calculated with a concentric two-component spot of
total radius  sp = 30 ô and an umbral radius 0.4 times that of the spot radius, which is
a typical value for large sunspots (Allen 1973), i.e.,  u = 12 ô . So, then, for the umbral
component, we can compute an area factor as
sin 2  u
sin 2  sp
= sin 2 12 ô
sin 2 30 ô = 0:1729
and for the penumbral one,
sin 2  sp sin 2  u
sin 2  sp
= 0:8271
then,
0:1729

F u
F 

T u + 0:8271

F p
F 

T p = 1:0000

F sp
F 

T sp
(iii) Umbra/Facula Spot Model, represented by a one-component spot and a concentric
facular region surrounding it. This `active region' (spot + facula) will have a total radius of
 tot = 30 ô , the radius of the dark component being 0.85 times that of the spot radius (Allen
1973), i.e.,  u = 0:85
 tot = 25:5 ô and of the bright component  fac = 4:5 ô . So, then, for
the dark (umbral) component,
sin 2  u
sin 2  tot
= sin 2 25:5 ô
sin 2 30 ô = 0:7413
and for the bright (facular) one,
sin 2  tot sin 2  u
sin 2  tot
= 0:2587
then,
0:7413

F u
F 

T u + 0:2587

F fac
F 

T fac = 1:0000

F sp
F 

T sp
In the equations above, F u =F  , F p =F  , F fac =F  and F sp =F  are the ux ratios in the V
band of the umbral, penumbral, facular and solid spot regions, respectively, with respect
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to the photosphere, and their values are listed in Table 2 along with the various eective
temperatures.
This procedure, viz., multiplying the ux ratios by the area factors and temperatures,
leads to identical V light curves for the three spot models and allows the eects on the colours
to be investigated. They are shown in Fig. 3 and it is evident that the three models are
indistinguishable in V , having been constructed in this way. They are also similar in (V {R) c
due to the low amplitudes produced in this colour index. If an observer had to distinguish
between them, he would have to obtain (V {I) c and (V {K) in order to determine uniquely the
eective temperature of the spot. Moreover, the eects on the (V {K) colours are the largest
when these type of spots with very dierent temperature structure are considered. If the
umbra of the third model (umbra + facula) was actually an average of two other components
(umbra and penumbra), then, the meaning of tting these spots with a single temperature
takes another dimension. The third model is not so far from what is actually believed to
occur on the surface of active stars. Catalano et al. (1995) showed for some active binaries a
close spatial association of a bright, chromospheric H emitting region to photospheric spots
and suggested that these two regions should be considered as a single, evolving, spatially
connected entity. In this case, (U{B) and (B{V ) colour curves in anti-phase with the V and
infrared bands might be an indication for the existence of that bright component. Although
this situation is rarely found in the literature, it could be more common than previously
thought, since the spatial and/or temperature distribution could disguise their presence.
Notwithstanding its rarity, some examples can be found, for instance, in Catalano et al.
(1995) or in Amado et al. (1999b) for the BY Dra are star YZ CMi.
Although, at the present, spotpic does not produce theoretical (U{B) and (B{V ) colour
curves, we intend to implement it in this way in order to study the eect mentioned above.
7.2 Polar versus equatorial belt spots
If light and colour curves of stars having only a non-modulating distribution of spots (polar
spots or belts) on their photospheres were observed, they would show only a dimming of the
maximum brightness of the star with, in principle, no way of dierentiating between which
of these two congurations would be producing it. This is not actually the case because of
the eect of limb-darkening. For instance, assuming a value for the inclination angle of a
star of i = 90 ô , a spot placed at the pole and a belt of spots situated at the equator will be
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14 P.J. Amado, J. G. Doyle and P.B. Byrne
Figure 3. Light and colour curves for the three two-component spot models given in Table 2. The solid spot model is represented
by the solid line, the umbra/penumbra spot model by the dotted line and the umbra/facula spot model by the dashed line.
positioned on dierent darkened parts of the photosphere and, therefore, will be contributing
dierently to the observed ux at dierent wavelengths.
In this Section, the eect of a polar spot and an equatorial band on the light and colour
curves of the star and their distinctive features, if any, will be studied. In order to do this,
the projected area on the visible stellar disk for both distributions must be the same. This
area must be purely geometric, i.e., without accounting for limb darkening so that any
dierences between the two spot distributions arise entirely from their interaction with the
limb-darkened parts of the disk. The geometrical area of a spot, i.e., without accounting
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Photometric modelling of starspots. II. The FORTRAN code SPOTPIC 15
Figure 4. Diagram of the spot distributions used to determine the eect of limb-darkening on the zero-points of light and
colour curves.
for limb darkening, is given by the coeôcient A from the integration procedure of Dorren
(1987),
A =  + ( ô) cos  sin 2  sin  sin  cos 
where  is the spot radius,  is the angle subtended by the arc of a great circle connecting
the centre of the spot and the centre of the visible disk and ô and  are angles that occur
when the circumference of the visible disk intersect the spot (see Fig. 1 in Dorren 1987). If
the spot is placed on the pole then  = , ô = 90 ô and  = 90 ô , reducing A to
A =  sin  cos 
Since the projected area of the polar spot and the equatorial band must be the same,
then
A pc = A eq
where A pc and A eq are the respective projected areas.
If a polar cap with a radius of 45 ô were to be assumed, the equatorial band with the same
projected area would cover an area around the equator of 4:093 degrees. Such examples
are represented in Fig. 4 and their eects on the zero-points of the light and colour curves
are shown in Fig. 5, where the polar spot eect is depicted by the solid line and that of
the equatorial band by the dashed line. These spots are placed on the photosphere of a star
with eective temperature T ph = 4700 K and log g ph = 5:0 (thick line) and log g ph = 4:0
(thin line), the spots having the same eective gravity as the unspotted photosphere. The
curves show the dimming of the V light and colour curves that these two spot distributions
produce when their temperatures are varied between 3000 K and 4700 K. Obviously, when
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16 P.J. Amado, J. G. Doyle and P.B. Byrne
Figure 5. Eect of limb-darkening on the zero-points of the light and colour curves produced by a polar spot of radius  = 45 ô
(solid line) and by an equatorial belt of the same projected area (dashed line). The spots have been placed on a star with a
photospheric temperature of 4700 K and a surface gravity of log g = 5:0 (thick line) and log g = 4:0 (thin line), the eective
gravity of the spots being the same as the photospheric surface gravity.
the spot reaches the temperature of 4700 K, which, in this case, is the same as the unspotted
photospheric temperature, the spots have a null eect on the curves.
It is readily seen that the equatorial band produces a much more eective dimming of
the light and colour curves than the polar cap due to its position on the least darkened part
of the disk. This eect is  0:04 mag fainter in (V {K) for the equatorial band than for
the polar cap for the coolest spots, but it becomes smaller as the temperature of the spot
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Photometric modelling of starspots. II. The FORTRAN code SPOTPIC 17
increases. It can also be deducted from Fig. 5 that the eect is more pronounced in stars
with lower surface gravity.
One of the problems arising when such congurations are to be modelled is the knowledge
of the inclination of the rotational axis of the star. If there is not a good determination of
its value from an independent method, the problem can be partially solved if Doppler maps
can be constructed simultaneously since they achieve a minimal solution for the correct
setting of the inclination. Another problem arises when setting the unspotted levels, or zero
points, for the magnitudes. This is a more complicated obstacle since these are known to
be ambiguous from the changes of season-to-season maxima and one of the most important
sources of errors when it comes to modelling the observational data (K}ovari & Bartus 1997).
For instance, the star II Peg showed a historical maximum brightness in 1974 (Chugainov
1976) that has not been reached since. However, the results of Ne, O'Neal & Saar (1995)
using the TiO band technique suggest that the true unspotted magnitude for this star is
even brighter than that observed by Chuiganov. Therefore, in order to be able to model
such spot distributions, the unspotted magnitudes of a star must be known at least with a
precision better than  0:04 in the (V {K) colour.
8 CONCLUSIONS
We have developed the FORTRAN code spotpic to model the photometry of active, spot-
ted stars, cooler than the Sun. It uses eective temperature-colour and Barnes-Evans-like
calibrations especially sensitive to the cool temperatures of the photosphere of these stars
and to the even cooler temperatures of starspots (Amado et al. 1999a; Amado 1997). The
main dierences between this and other codes are the use and modelling of the infrared
colour (V {K), the use of temperature and gravity-dependent limb-darkening coeôcients and
the possibility of assigning dierent eective surface gravities for the spotted and unspot-
ted photosphere. The code also allows for two-component spots, i.e., spots with umbral
and penumbral components and computes and nds a best t to the observed V , (V {R) c ,
(V {I) c and (V {K) simultaneously via the Levenberg-Marquardt least-squares minimization
procedure.
Various investigations were conducted to determine the code's ability to discern between
dierent problematic spot congurations and the results suggested that:
(i) temperatures derived by modelling two-component spots with solid spots were, as
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18 P.J. Amado, J. G. Doyle and P.B. Byrne
expected, closer to the penumbral temperatures for higher photospheric temperatures and
roughly in the umbral-penumbral average temperature for cooler photospheric temperatures,
(ii) spots with various thermal structures (umbrae, penumbrae, faculae) can be dieren-
tiated and modelled if the infrared colours (V {I) c and (V {K) are utilised, with the largest
eect being produced in this last colour index, and
(iii) polar spots and equatorial bands may be dierentiated in (V {K) if high-accuracy
photometry is obtained and if the inclination and unspotted magnitudes of the star are well
known. This dierentiating eect, due to the equatorial band been positioned on the less
darkened parts of the disk, can reach a value of up to 0.04 mag fainter in this colour index
than for the polar spot.
Although diôcult, the accuracy mentioned above is not impossible to achieve and the
better the detectors become and the longer the stars are monitored in UBV(RI) c K the more
probable it will be to nd with high precision the inmaculate colours of the star and to,
therefore, dierentiate between these spot congurations. Furthermore, the combined use of
the Doppler imaging technique, which can determine quite unambiguously the modulating
spot distribution and can help in establishing stellar parameters like the inclination of the
star, and the modelling of the V , (V {R) c , (V {I) c and especially the (V {K) photometric
curves for determining the temperatures will undoubtly improve our understanding of the
temperature and brightness distributions of spots on cool active stars.
ACKNOWLEDGMENTS
Research at Armagh Observatory is grant-aided by the Dept. of Education for Northern
Ireland. This work used computer hardware and software provided by the UK Starlink
Project which is funded by the UK PPARC. PJA acknowledges nancial support from
Armagh Observatory and computing and technical support from the Instituto de Astrofsica
de Andaluca-CSIC, Spain. This research has made use of the SIMBAD database, operated at
CDS, Strasbourg, France. The authors thank the anonymous referee for his constructive
comments on this manuscript which helped to improve it signicantly.
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APPENDIX A:
The surface brightness parameter can be dened for any passband in the same way as it was
done for V by Barnes & Evans (1976), i.e, using the equation
F V = 4:2207 0:1V 0 0:5 log  0 (A1)
for a generic passband 
F  = 4:2207 0:1m  0:5 log  0
So, adding and subtracting V 0 , this Equation can be expressed as
F  = 4:2207 0:1V 0 0:1(V 0 m  ) 0:5 log  0
which, making use of Eq. (A1), yields
F  = F V 0:1(V 0 m  ) (A2)
So, the surface brightness parameter for an specic passband can be expressed as a function
of that for another passband plus a colour factor.
The surface brightness is not a ux parameter but a magnitude parameter, so that,
changing Eq. (A2) into a ux scale, gives
F V F  = 0:1

2:5 log F V
F 

(A3)
where F V and F  are the surface uxes in the V and  passbands.
Using Eq. (A2) to represent the surface brightness parameters for the unspotted (F V p )
and spotted (F V s ) photosphere, yields
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F p = F V p 0:1(V m  ) p
F s = F V s 0:1(V m  ) s
which, combined, give
F p F s = F V p F V s + 0:1[(V m  ) p (V m  ) s ] (A4)
Using Eq. (A3), Eq. (A4) can be transformed into a ux scale
0:1

2:5 log F p
F s

= 0:1

2:5 log F Vp
F Vs

+ (A5)
+0:1[(V m  ) p (V m  ) s ]
which, in turn, gives
F p
F s
= F Vp
F Vs
+ 10 +0:4[(V m )p (V m ) s ] (A6)
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