A CALIBRATION OF EK CEPHEI
1,2 ,
J. P. Marques
1
2
3
4
2,3
J. Fernandes
1,4
and M. J. P. F. G. Monteiro
´
Centro de Astrofsica da Universidade do Porto, Porto, Portugal
Observatório Astronómico da Universidade de Coimbra, Coimbra, Portugal
Departamento de Matemática da FCTUC, Coimbra, Portugal
Departamento de Matemática Aplicada da FCUP, Porto, Portugal
Abstract
EK Cephei (HD 206821) is a unique candidate to test predictions based on stellar
evolutionary models.
It is a double-lined detached eclipsing binary system with
accurate absolute dimensions available, good determination of the metallicity and
in particular of the lithium abundance.
mass (1.12
M¯
Most importantly, for our work, its low
) component appears to be in the pre-main sequence (PMS) phase,
being therefore the rst known solar-type PMS star with a directly determined
mass.
We have produced detailed evolutionary models of the binary EK Cep using
the CESAM stellar evolution code (Morel 1997).
Models were constructed using
the OPAL equation of state, OPAL opacities and NACRE thermonuclear reaction
rates.
A
χ2
-minimization was performed in order to derive the most reliable set
of modelling parameters (age,
αA αB
,
and
Yi
).
We have found that an evolutionary age of about 26.7 Myrs ts both components in the same isochrone.
The positions of EK Cep A and B in the HR diagram
are consistent (within the observational uncertainties) with our results.
In doing
this we have found that a very small time step for the evolution of the models is
able to t the observations much better.
This indicates that an adequate denition
of the time step is of great importance for modelling the PMS phase.
Our
results
show
clearly
that
EK
Cep
A
is
sequence, while EK Cep B is indeed a PMS star.
by
modeling
the
chemical
evolution
of
the
two
in
the
beginning
of
the
main
We will complement this study
components
given
the
special
relevance of their properties and age for studying this aspect of the evolution.
1.
Introduction
Modeling a single star is not a closed problem because the number of parameters
to
be
determined
is
larger
than
the
observational
parameters of a model of a star are the stellar mass
rameter
α
, the initial helium abundance
model's age
t?
.
Yi
M?
constraints.
The
, the mixing length pa-
i
, the initial metallicity [Fe/H]
and the
In most cases, there are only two reliable observables, namely
the effective temperature and the luminosity.
1
2
Good determinations of stellar masses are in general only possible for the
components of binary systems.
Based on the reasonable assumption of a com-
mon origin for both components (which yelds the same initial chemical composition
and
age),
the
problem
of
calibrating
a
binary
system
determination of ve so-called modeling parameters, namely,
αA
and
αB
.
reduces
Yi
to
i
the
, [Fe/H] ,
t?
,
If the metallicity of the stars is also known, the problem is closed,
since the number of observables (the effective temperatures and luminosities
for both stars) is equal to the number of modeling parameters.
A
good
candidate
binary
system
to
calibrate
must
have
well
determined
luminosities, effective temperatures, metallicities and dynamical masses.
Pre-
main-sequence (PMS) binaries that have all the required characteristics are very
rare at present (Palla and Stahler 2001).
The double-lined eclipsing binary system EK Cep has these unique characteristics.
Accurate absolute dimensions are available (Andersen 1991, Popper
1987) and a good determination of the surface lithium abundance and metal-
´
licity was made (Martn & Rebolo 1993).
M¯
And its secondary component (1.12
) seems to be in the pre-main sequence phase (Popper 1987), being there-
fore the rst solar-type PMS star with a well determined mass.
All this makes
EK Cep a perfect candidate to test theoretical models of pre-ZAMS solar-type
stars.
A summary of the physical characteristics of this binary is given in table
1.
Table 1.
Properties of EK Cep (HD 206821).
Masses, radii, effective and luminosities are from
Andersen (1991) and Popper (1987); spectral type from Popper (1987).
are
from
Mart
n
&
Rebolo
(1993).
expected given their mass ratio.
Note
that
the
radii
of
the
two
Chemical compositions
stars
are
much
closer
than
This indicates that EK Cep B is still contracting towards the
main-sequence.
EK Cep A
M? /M¯
Teff
L? /L¯
R? /R¯
Sp.
±
±
±
±
2.019
9000
0.016
200 K
14.8
1.58
Type
...
[Li]
...
±
±
±
±
1.124
5700
0.012
200 K
1.4
1.62
0.24
0.15
1.32
0.15
A 1.5 V
[Fe/H]
2.
EK Cep B
...
+0.07
3.11
±
0.05
±
0.3 dex
Physical Ingredients
Models
1997).
were
computed
using
the
CESAM
stellar
evolution
code
(Morel
Typically, each model is described by about 600 shells, and an evolution
by about 400–600 models.
We restricted the maximum time step of the evolu-
3
A Calibration of EK Cephei
tion:
for EK Cep A, the maximum time step was kept at 0.025 Myrs, while for
EK Cep B we allowed the time step to reach 0.1 Myrs.
Each evolution is initialized with an homogeneous, fully convective model
in quasi-static contraction,
temperature
of
the
with a central temperature inferior to the ignition
deuteron.
elapsed since initialization.
We
shall
call
the
“age”
of
the
model
the
time
This initial model for PMS evolution is not a re-
alistic assumption, since stars do not form by the homologous contraction of a
pre-stellar cloud.
Instead, a hydrostatic core rst forms, which accretes mass
from
cloud
the
parental
until
it
reaches
the
nal
stellar
mass;
then,
since
no
more surrounding matter obscures the star, it becomes optically visible for the
rst time:
the star is “born” (see Stahler 1983, Palla and Stahler 1991 & 1992).
If we want to model very young stars, the problem of the initial conditions is
very important; for EK Cep, however, the initial conditions are irrelevant since
it is much older than 1 Myr.
Our “age” of the model, although not a real age of
the star, is negligibly close to it.
The model of zero-age main-sequence is dened as the rst model where
nuclear reactions account for more than 99% of the energy generation.
We used the OPAL equation of state (Rogers et al.
1996) and the opacities
of Iglesias & Rogers (1996 and 2001) complemented, at low temperatures, by
the Alexander & Fergusson (1994) opacities.
The temperature gradient in convection zones is computed using the standard
mixing-length theory (Böhm-Vitense 1958).
l = αHP HP
,
The mixing length is dened as
being the local pressure scale height,
HP = −dr/d ln P
4 He, 7 Li, 7 Be, 12 C, 13 C, 14 N, 15 N, 16 O, 17 O, 9 Be
.
1 H, 2 H, 3 He,
The nuclear network we used contains the following species:
and a “extra” ctitious non-
CNO heavy element which complements the mixture; this element has atomic
mass 28 and charge 13.
Deuterium and lithium burning are taken into account,
as well as the most important reactions of the PP+CNO cycles.
The nuclear
reaction rates are taken from the NACRE compilation (Angulo et al.
1999).
We
don't consider diffusion processes due to the young age of the binary; however,
these effects must be taken into account if we want to follow the evolution of
the surface chemical composition.
3.
The Calibration Method
The calibration of a binary system consists in adjusting the stellar modeling
parameters so that the models reproduce the observational data of the stars.
used the
χ2
Morel et al.
The
tting, which was developed by Lastennet et al.
(1999;
We
see also
2000).
effective
temperature
and
luminosity
of
a
model
depend,
for
a
given
mass and xed input physics (EOS, opacity, nuclear energy generation rates)
on the modeling parameters: the age
t?
, initial helium abundance
Yi
and mixing
4
length parameter
α?
. We assume that the initial metallicity is equal to the present
surface metallicity because we didn't consider diffusion processes.
obtain observables as close as possible to the observations,
In order to
we minimize the
following functional:
χ
2
"µ
X
=
?=A,B
µ
+
log Teff (?)mod − log Teff (?)
σ(log Teff (?))
log L(?)mod − log L(?)
σ(log L(?))
¶2
+
¶2 #
log Teff (?)mod
This functional depends on the modeling parameters through
and
log L(?)mod
.
For a
grid
of
modeling
parameters
we
have
computed
the
evolution of models with the masses of EK Cep A and B. The functional
was then computed for each of these models using the above equation.
parameters that minimized
4.
χ2
χ2
The
were selected as the solution.
Results
χ2
The modeling parameters that minimize
are shown in table 2a, together
with the same parameters for a calibrated solar model using the same physics.
The uncertainties are derived by changing one parameter at a time around the
calibrated value, keeping all the others constant.
tainty associated with
αA
We do not present the uncer-
because the luminosity and effective temperature of
EK Cep A depend very weakly on the mixing length parameter.
This is due to
the disappearance of the convective envelope of the star during the late-PMS
phase (see gure 2).
Table 2b shows the characteristics of the calibrated EK Cep A and B models.
As shown in gure 2, both components of EK Cep have a convective core; EK
Cep
B
just
acquired
a
convective
core
due
to
the
start
of
12 C
burning.
That
convective core will be lost shortly after the arrival on the ZAMS.
The
ZAMS
tzams,B = 51.3
occurs
at
an
evolutionary
age
of
tzams,A = 10.8
Myrs for EK Cep A and B, respectively.
Myrs
and
Given our estimated
age of around 27 Myrs, EK Cep A is already in the main-sequence, while EK
Cep B is indeed a PMS star (see gures 1 and 2).
5.
The Role of the Time Step
Figure
3
shows
again
the
evolutionary
components of the EK Cep binary.
tracks
in
the
HR
diagram
for
both
The effects of an inappropriate choice of the
time step are seen; these effects only affect the PMS phase because in this phase
the star gets its energy from gravitational contraction; the rate of gravitational
contraction is miscalculated if the time step between two consecutive models
is too big.
5
A Calibration of EK Cephei
Table 2a.
Cep
Calibration parameters of a EK
model
boxes
(see
lying
text).
within
The
the
uncertainty
parameters
for
the
Table 2b.
models.
Characteristics of the calibrated
Tc
and
ρc
are
the
central
ties and temperatures, given in
−3
Sun are for a calibrated solar model using
cm
the same physics.
convective
, respectively.
core
and
Rco
Rce
106
is the radius of the
the
radius
base of the convective envelope.
EK Cep
Sun
densi-
K and g
of
the
The same
quantities are also given for a calibrated solar model using the same physics.
αA
1.340
αB
1.375+0.295
−0.185
Yi
t?
Figure 1.
0.261+0.005
−0.006
26.7+1.7
−1.8
Myrs
1.632
...
0.265
4.52 Gyrs
Tc
ρc
Teff
L? /L¯
R? /R¯
Rco /R?
Rce /R?
EK Cep A
EK Cep B
Sun
21.11
13.44
15.45
64.28
77.13
149
8974 K
5689 K
5777 K
15.74
1.497
1.000
1.644
1.260
1.000
0.120
0.0637
...
0.993
0.758
0.729
Evolutionary tracks of EK Cep A and B on the HR diagram.
The uncertainty boxes
of both stars are also shown.
This can be seen in the left panel of gure 3. The locus of the calibrated model
of EK Cep A in the HR diagram doesn't change if we don't control the time
6
Figure
2.
Convective
zones
and
radii
of
represent the extension of convective zones;
the calibrated model,
the
two
with an age of 26.7 Myrs.
observational radii of the two stars.
components
of
EK
Cep.
Vertical
lines
the vertical dashed line in both panels represents
The dashed zone represents the error in the
Our models t the radii of both stars within observational
uncertainties.
step carefully because the star derives most of its energy from nuclear reactions.
On
the
other
hand,
the
locus
of
the
calibrated
model
of
EK
Cep
B
changes
considerably; the luminosity of the model calculated without the restriction on
the time step is appreciably lower.
In fact, we can't t the observations within
their error boxes for both stars if we don't restrict the time step in this way;
the luminosity (and radius) of the low mass component is too low.
This might
´
explain the results cited in Martn & Rebolo (1993), that the luminosity of EK
Cep B was slightly higher than predicted by pre-main-sequence models.
The
radius of our model of EK Cep B (with a large time step) is below the error box
in radius, a result also obtained by Claret et al.
(1995).
Finally, the evolution of the models is faster if we don't restrict the time step.
The calibrated models of EK Cep have just
6.
' 23
Myrs.
Perspectives for Future Work
We plan to continue our study of this binary by analyzing the surface chemical composition evolution.
In order to do this,
we need to include diffusion
and convective overshooting in our physical ingredients.
This is particularly
important for EK Cep since we have at our disposal the surface chemical com-
´
position of EK Cep B, and particularly the lithium abundance (Martn & Rebolo
1993).
This element is very important for testing models of PMS stars since it
traces the evolution of the temperature of the base of the convective envelope
(Piau & Turck-Chièze 2002).
If the temperature of the base of the convective
envelope gets higher than the lithium ignition temperature (about
that region starts to exhaust its lithium.
2.5 × 106
K),
Only if the temperature of the base of
the convective zone dips bellow that value does the exhaustion of lithium stops.
7
A Calibration of EK Cephei
Figure 3.
The role of the time step.
Left panel:
EK Cep A; in full, the evolutionary track with
a maximum time step of 0.025 Myrs; dashed, the evolutionary track without such a strict control.
Right panel:
the same thing for EK Cep B. In full, the evolutionary track with a maximum time
step of 0.5 Myrs.
Acknowledgments
This
work
was
supported
by
the
Fundação
para
a
Ciência
e
Tecnologia
through the projects PESO/P/PRO/1528/99 and ESO/FNU/43656/2001.
JPM
was supported by grant SFRH/BD/9228/2002 from Fundação para a Ciência e
Tecnologia and would like to thank the local organizing committee for funds to
support the registration fee.
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