Vol.3, No.8, 641-645 (2011) Natural Science
http://dx.doi.org/10.4236/ns.2011.38087
On chromospheric variations modeling for
main-sequence stars of G and K spectral classes
Bruevich Elena Alecsandrovna
Sternberg Astronomical Institute, Moscow, Russia;
Received 2 March 2011; revised 13 April 2011; accepted 21 April 2011.
ABSTRACT
We present a method of 13 late-type main-se-
quence stars chromospheric flux observation
data calculations. These Sun-like stars have
well-determined cyclic flux variations similar to
the 11-year solar activity cycle. Our flux predic-
tion is based on chromospheric HK emission
time series measurements from the Mount Wil-
son Observatory and comparable solar data. We
show that solar three-component modeling ex-
plains well the stellar observations. We find that
the 10% - 20% of K-stars disc’s surfaces are
occupied by bright active regions.
Keywords: The Sun; Late-Type Stars;
Chromospheric Emission
1. INTRODUCTION
This paper continues the study of variability among
Sun-like stars. Here the purpose is to obtain the possibil-
ity of modeling the behavior of the star’s chromospheric
emission in future or for periods of time without meas-
urements.
Observations of chromospheric variability requires at
least a decade to reveal variations with timescales to the
11-year solar cycle.
We use the data from the observation program that
was initiated by Wilson who discovered the widespread
occurrence of activity cycles by monitoring Ca II H and
K variations in 91 stars on or near the lower main se-
quence over 12 year. Two sets of measurements (named
“HK-project”) have been combined to make more than
30 years records of stellar chromospheric activity. Wil-
son made observations from 1966 to 1977 at monthly
intervals on 2.5 m telescope at Mount Wilson Observa-
tory. The survey moved in 1977 to 1.5 m telescope with
instrument whose measurements can be compared to
those of Wilson’s system. Some new stars were added to
91 Wilson’s stars to bring the total in the monitoring
program to 111 stars [1,2]. Ca II H (396.8 nm) and K
(393.4 nm) emission is observed in stars with spectral
class later than approximately F2 V, i.e. less massive
than about 1.5
M
. Areas of concentrated magnetic
fields on the Sun and Sun-like stars emit Ca II H and K
more intensely than areas with less magnetic field pre-
sent. So the contrast of Active Regions (AR) emission
(where the local magnetic fields are more then some
orders higher than average global magnetic field) in
these Ca II lines changes from 1.2 to 1.5 with changing
of chromospheric activity cycle phase.
Comparing of variability of H and K emission in
main-sequence stars should provide important validation
for theories of magnetic activity, as well as place of solar
activity in a general perspective.
The influence of photospheric flux in the total solar or
stars irradiance we can interpretate as the cyclic flux
variations caused by slight imbalance between the flux
deficit produced by dark sunspots and the excess flux
produced by bright faculae. average emission (that varies
with the so-called the 11-year chromospheric cycle).
Besides of such structures as AR in solar and stars
chromosphere there is another regular structure—“chro-
mospheric network” (connecting with the supegranula-
tion). It also varies its own relative brightness with
chromospheric activity cycle.
We can note that the maximum amplitude of photo-
spheric flux variability in the 11-year solar cycle may be
as much as 1% - 3% of the average photospheric flux
level but the maximum amplitude of Ca II chromos-
pheric flux may be as much as 20% of the average level.
These values are our estimations for the maximum am-
plitudes of the 11-year variations of Sun-like stars pho-
tospheric and chromospheric fluxes.
2. THE THREE-COMPONENT MODEL
OF STELLAR CHROMOSPHERIC
EMISSION
The processes in solar atmosphere caused the emis-
sion in different spectral intervals and lines are studied

B. E. Alecsandrovna / Natural Science 3 (2011) 641-645 642
well enough. But it’s very difficult to take into account
the contribution of all different structures that emitted
from the solar surface. As a successful example of solar
flux model calculations in spectral intervals of 40 - 140
nm (that are in agreement with SKYLAB’s observations)
we can point out the [3].
These calculations (made in [3]) take into account the
influence of 6 main different components on the solar
surface and their contributions to the total emission in
this spectral interval. These components are: the dark
areas inside the chromospheric network cells, the centers
of networks, the areas of quiet Sun, the average level of
network emission region, the bright areas of network, the
most bright areas of network. When the observations are
not made with high accuracy all this structures we see as
quiet Sun
These structures contribute significantly to the full
flux emitted from the quiet Sun chromospheric average
emission.
The next most important source of solar chromos-
phere emission is the Active Regions (AR) emission.
The SKYLAB’s observations show the brightness of AR
are 1.5 - 2.5 times greater the average quiet surface
brightness [4]. This AR brightness contrast is depend of
wavelengths. They note also that the AR surface bright-
ness depends of the AR area and the number of spots the
AR consists of.
The model [5] (that’s made for 40 - 140 nm spectral
interval) based on NIMBUS 7 observations assumed that
the full flux from chromosphere is determined by three
main components.
These components are: 1) the constant component
with uniform distributed sources on solar surface, 2) the
“active” network component (uniformly distributed too
but also connected with destroyed parts of previous AR
and so is proportional to total AR areas), 3) the AR
component.
So one can use formulae from [5] for calculation of
the flux in chromospheric lines:



  
=1 1
2π11
QNN
Qiiipi
II fC
FARCW




 
(1)
where I is the full flux of chromospheric emission, Q I

is the contribution of the constant component (BASAL),
p
C

is the values of AR contrasts and they are similar
to contrasts from [6],
N
C

is the value of “active net-
work” contrast: they are equal to 0. 5
p
C

 for contin-
uum and 13
p
C

 for lines,
N
f
is part of disk (with-
out AR) that is occupied by the “active network”.
The second member in the right part of (1) describes
emission from all AR on the disk; Ai are values of their
squares, μi describes the AR position: =cos cos
iii


(where i

and i

are the coordinates of AR number i).


i
R


describes the relative change of the surface
brightness


Qi
F


with moving from center to edge
of disk. The relative adding AR contribution to full flux
from the different AR is determined by the factor i W
that is linearly changed from the value 0.76 to 1.6 de-
pending of the brightness ball of flocculae (according to
ball flocculae changes from 1 to 5).
So the “active” network part in all the surface without
AR is determined by the AR decay, the next relationship
between
N
f
in time moment t and average values Ai in
earlier time is right:


5
= 13.31027
Ni
ft At

 (2)
where the time-averaging is taken for 7 previous rotation
periods, Ai is measured in one million parts of the disk.
To analyzed the H and K Ca II flux long-time varia-
tions in case of Sun-like stars we assume that full flux
()
CaII
St is consists of three main components:
1) the “constant part” (so-called BASAL in solar
physics—we call this component min P),
2) the “low-changed background” (we call this com-
ponent ()
CaII
Pt) and
3) AR on the disk of star (we call this component
()
AR
St).
So the full flux will be


=
CaIICaII AR
StPtSt
The component (b) ()
CaII
St consists of constant
BASAL component min
P and low-changed pseudo-
sinusitis component which we can see from the Sun ob-
servations and will describe it's approximation later.
It’s evident (from solar observations and their inter-
pretations) that between the values ()
CaII
St and ()
CaII
Pt
there is close connection.
According to [7] the average amplitude of flux varia-
tions may be 20% in maximum phase of chromospheric
cycle.
This point of view is according well enough with
Lean’s model [5] for solar L

line (in case of solar L

line flux the maximum amplitude of this flux variation in
different 11 yr cycles reached the value of 20%).
Than we determine the analog coefficient k for star’s
chromospheric cycle as equal to ratio of maximum am-
plitude of so called “background” component to maxi-
mum amplitude of full flux in long-term activity cycle:

 
max max
min min
=CaII CaII kP PSP (3)
We consider that k is constant ratio between ()
CaII
Pt
and ()
CaII
St for all moments during star’s cycle.
We also assume that max
min
=1.2
CaII
PP .
It’s evident from our previous consideration that min P
is a constant value during all long-term cycles but differs

B. E. Alecsandrovna / Natural Science 3 (2011) 641-645
643643
for different stars and the Sun. Most likely that the value
min
P characterizes the average level of outer atmosphere
activity of stars and may correlate with ROSAT observa-
tions of their X-ray fluxes (X-ray luminosity are ob-
served on ROSAT for 65% “HK-project” stars only).
According to these we connect the full flux value
()
CaII
St and “background” flux value ()
CaII
Pt by ana-
log coefficient k (3):
 
=
CaII CaII
PtkS t (4)
The ()
CaII
St values one can take from observations [2].
In Figure 1 we can see records of relative Ca II H + K
emission fluxes (CaII
S) for the Sun and HD 81809 star of
“EXCELLENT” class as determined in [1] and [2].
It’s evident (from solar observations in different spec-
tral intervals) that ()
CaII
Pt and ()
CaII
St have similar
behavior in chromospheric cycle.
We’ve calculated the regression coefficients (see Ta-
ble 1) a and b for the linear regression relation:
 
=
CaII CaII
StaPtb (5)
To make “background” flux prediction we use the
method from [8] for solar “background” flux variations
in the 11-year cycle. Using considerations from [8]
we’ve obtained the next approximation for “back-
ground” flux ()
CaII
Pt:


π
4
min
=1 π
sin tT
CaII
PtP tTe

  (6)
where min
P is the minimum value of “background” flux
is equal to BASAL level flux. It corresponds to the
minimum “background” flux value for the star and it’s
constant for all observed long-term cycles, see Figure 2.
T is the period of long-term chromospheric cycle cal-
culated by [2], t is the time expressed in parts of period
T:
= 0.1,0.2,tTT


So we have two methods of “background” flux calcu-
lations: 1) from observations records using (3) and (4)
(see Figure 2), 2) from analytic approximation (6) using
min
P only. Note that the both methods give us very iden-
tifiable values of ()
CaII
Pt which differ some percents
only.
So if we want the chromospheric flux to predict we
may calculate ()
CaII
St with help of (5) using a and b
coefficients which are calculated earlier with help of
standard regression methods and presented in the Table
1.
In the Table 1 we present also the relative full flux
variation in activity cycle maximum: (max
min
CaII
SP) and
relative AR adding flux in activity cycle maximum:
(max
min
AR
SP).
These values of flux variations are presented in % of
min
P. The value min P - that is equal to BASAL emission
for different stars which we can determine from observa-
tions [2], see Figure 1.
Figure 1. Records of relative CaII emission fluxes (S-index) from the Mount Wilson observations
[1,2] for the Sun and HD 81809 star of “EXCELLENT” class. (B - V) values are presented.

B. E. Alecsandrovna / Natural Science 3 (2011) 641-645 644
Figure 2. For the Sun and HD 81809 we show BASAL level and other components for (3) - (5).
Table 1. Observed parameters of 13 “EXCELLENT” class stars and regression coefficients a and b
calculated from (5).
Object B - V
cyc
T, yr min P a b max
minCaII
SPmax
min ,%
AR
SP
Sun 0.66 10 0.162 1.19 –0.031 23.4 3.4
HD 81809 0.64 8.2 0.155 1.13 –0.020 22.6 2.6
HD 152391 0.76 10.9 0.32 1.56 –0.180 31.6 11.3
HD 103095 0.75 7.3 0.17 1.23 –0.040 24.7 4.7
HD 184144 0.80 7.0 0.19 1.45 –0.085 28.9 8.9
HD 26965 0.82 10.1 0.18 1.39 –0.07 27.8 7.8
HD 10476 0.84 9.4 0.17 1.61 –0.104 32.4 12.3
HD 166620 0.87 15.8 0.175 1.43 –0.075 28.6 8.6
HD 160346 0.96 7.0 0.24 1.88 –0.21 37.5 17.5
HD 4628 0.88 8.4 0.19 1.96 –0.183 39.4 19.4
HD 16160 0.98 13.2 0.19 1.61 –0.116 32.6 12.6
HD 219834B 0.91 10.0 0.17 1.92 –0.157 38.2 18.2
HD 201091 1.18 7.3 0.51 1.85 –0.434 37.2 17.2
HD 32147 1.06 11.1 0.22 1.67 –0.147 45.4 25.4

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The Table 1 data we may employed in our full flux
chromospheric predictions: for the certain moment t (t is
the time expressed in parts of period value T) we can
calculated the value ()
CaII
Pt from (6). Then from (5)
for moments t and for min
P star’s values we can calcu-
late the predicted flux ()
CaII
St.
3. SUMMARY AND CONCLUSIONS
When we analyze results of our predictions (in the Ta-
ble 1 we presented the observed values that we’re dis-
cussed in this issue and our estimations as max
minCaII
SP
and max
minAR
SP) some conclusions can be made:
K-stars of “EXCELLENT” class (as was for stars with
the most evident determination of chromospheric activ-
ity cycles [1,2]) have enough number of AR at stars sur-
faces and these AR can emit the addition flux near 10% -
20% of full flux in chromospheric cycle maximum, see
Table. So we can see that in K-stars the most bright floc-
culae (its flux is about two times brighter than the aver-
age chromosphere flux) may occupy almost 10% - 20%
of star’s disk.
This 10% - 20% of AR additional flux (“background”
flux evaluation) we show in Figure 2. Also we see the
star’s “background” flux ()
CaII
Pt (smoothly changed in
chromospheric cycle) and BASAL component (constant
in chromospheric cycle).
Note that all our three components we can see in Fig-
ure 1 (“HK-project” observations of the Sun and HD
81809 star) similar to solar case, described in [5].
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