Activity-brightness Correlations for the Sun and Sun-like Stars
ABSTRACT We analyze the effect of solar features on the variability of the solar irradiance in three different spectral ranges. Our study is based on two solar-cycles' worth of full-disk photometric images from the San Fernando Observatory, obtained with red, blue, and Ca II K-line filters. For each image we measure the photometric sum, Σ, which is the relative contribution of solar features to the disk-integrated intensity of the image. The photometric sums in the red and blue continuum, Σr and Σb, exhibit similar temporal patterns: they are negatively correlated with solar activity, with strong short-term variability, and weak solar-cycle variability. However, the Ca II K-line photometric sum, ΣK, is positively correlated with solar activity and has strong variations on solar-cycle timescales. We show that we can model the variability of the Sun's bolometric flux as a linear combination of Σr and ΣK. We infer that, over solar-cycle timescales, the variability of the Sun's bolometric irradiance is directly correlated with spectral line variability, but inversely correlated with continuum variability. Our blue and red continuum filters are quite similar to the Strömgren b and y filters used to measure stellar photometric variability. We conclude that active stars whose visible continuum brightness varies inversely with activity, as measured by the Ca HK index, are displaying a pattern that is similar to that of the Sun, i.e., radiative variability in the visible continuum that is spot-dominated.
The Astrophysical Journal Letters, 739:L45 (6pp), 2011 October 1doi:10.1088/2041-8205/739/2/L45
C ?2011. The American Astronomical Society. All rights reserved. Printed in the U.S.A.
ACTIVITY–BRIGHTNESS CORRELATIONS FOR THE SUN AND SUN-LIKE STARS
D. G. Preminger, G. A. Chapman, and A. M. Cookson
San Fernando Observatory, Department of Physics and Astronomy, California State University Northridge, Northridge, CA, USA
Received 2010 December 15; accepted 2011 July 27; published 2011 September 7
We analyze the effect of solar features on the variability of the solar irradiance in three different spectral ranges.
Our study is based on two solar-cycles’ worth of full-disk photometric images from the San Fernando Observatory,
obtained with red, blue, and Caii K-line filters. For each image we measure the photometric sum, Σ, which is
the relative contribution of solar features to the disk-integrated intensity of the image. The photometric sums in
the red and blue continuum, Σrand Σb, exhibit similar temporal patterns: they are negatively correlated with solar
activity, with strong short-term variability, and weak solar-cycle variability. However, the Caii K-line photometric
sum, ΣK, is positively correlated with solar activity and has strong variations on solar-cycle timescales. We show
that we can model the variability of the Sun’s bolometric flux as a linear combination of Σrand ΣK. We infer that,
over solar-cycle timescales, the variability of the Sun’s bolometric irradiance is directly correlated with spectral
line variability, but inversely correlated with continuum variability. Our blue and red continuum filters are quite
similar to the Str¨ omgren b and y filters used to measure stellar photometric variability. We conclude that active stars
whose visible continuum brightness varies inversely with activity, as measured by the Ca HK index, are displaying
a pattern that is similar to that of the Sun, i.e., radiative variability in the visible continuum that is spot-dominated.
Key words: stars: activity – stars: solar-type – Sun: activity – Sun: chromosphere – Sun: photosphere – Sun:
Total solar irradiance (TSI) is the bolometric solar energy
flux received at the Earth at its average distance of 1 AU,
measured since 1978 by a succession of spacecraft instru-
ments. Figure 1 shows the composite TSI record, compiled by
Fr¨ ohlich (2000, 2006) at the World Radiation Center at
Sunspot cycles 21, 22, and 23 are clearly visible. Sunspots are
manifestations of magnetic active regions, and their properties
can be measured directly on solar images and magnetograms.
Solar activity modulates TSI in a complex way: at solar max-
imum, when there are a large number of active regions, TSI
is high on average, but TSI drops sharply every time a large
sunspot group transits the solar disk.
Several Sun-like stars also show evidence of magnetic ac-
tivity, and their variability has been studied extensively with
photometry and spectroscopy (Radick et al. 1998; Lockwood
et al. 2007; Hall et al. 2009). For stars, magnetic activity cannot
be observed directly; instead, Ca HK emission index is taken
as a proxy for magnetic activity, and Str¨ omgren b+y bright-
ness is taken as a measure of bolometric brightness. One of
the fundamental questions asked regarding these solar analogs
is: do brightness and activity for these stars vary in a Sun-like
way? Observations by Lockwood et al. (2007) and Hall et al.
(2009) have shown that Sun-like stars seem to fall into two
groups, according to the correlation between their b+y bright-
ness and Ca HK emission: highly active stars display an inverse
correlation, while less active stars a direct one. They concluded
that there is some critical stellar activity level at which the
activity–brightness relation reverses. The Sun was put into the
less active group because we observe a direct correlation be-
tween time-averaged TSI and solar activity level.
In this work, we examine the relationship between TSI and
photometric solar data from the San Fernando Observatory
(SFO) and discuss what it reveals about the activity–brightness
correlations for the Sun-as-a-star. We then compare our results
with the activity–brightness relations that have been measured
for Sun-like stars.
2. SOLAR PHOTOMETRY AT THE SAN
2.1. Observing Program
At the SFO, full-disk photometric solar images are recorded
daily with two purpose-built telescopes, CFDT1 (5??pixels) and
CFDT2 (2.??5 pixels), described in Chapman et al. (1997). The
SFO observational record now includes about two solar-cycles’
worth of solar images at different wavelengths. The images
pertinent to this study are those at 672.3 nm with a 10 nm
bandpass (red), 472.3 nm with a 10 nm bandpass (blue), and
393.4 nm with a 1 nm bandpass (Caii K). Using the Kurucz
solar flux atlas (Kurucz 2005), we determined that photospheric
spectral lines block ∼2% and ∼11% of the continuum flux in
the passbands of the red and blue filters, respectively. Thus, our
red and blue images show primarily continuum emission, from
of the Caii K line, hence our Ca K images show emission in this
strong spectral line, which is formed in the chromosphere and
upper photosphere. Sample images appear in Preminger et al.
(2001). All images exhibit bright and dark features on a “quiet”
background. These features are manifestations of magnetic
active regions at different heights in the solar atmosphere. Dark
sunspots are the most prominent features on the red and blue
continuum images, while both sunspots and bright plages are
distinct in the Caii K-line images.
2.2. Data Analysis
SFO data reduction and analysis procedures are described
in detail in Walton et al. (1998) and Preminger et al. (2002).
Presented here is a brief overview. SFO images are suited to
The Astrophysical Journal Letters, 739:L45 (6pp), 2011 October 1Preminger, Chapman, & Cookson
TSI (PMOD composite)
Figure 1. Total solar irradiance (PMOD Composite). TSI is a bolometric measurement.
curve is computed, then used to produce a contrast map. For
image pixel i, we define its contrast, Ci, as
where Iiis the observed intensity of pixel i, μiis the cosine of
the heliocentric angle at pixel i, and I(μ) is the quiet-Sun limb
darkening curve for that image. On the contrast map, quiet-Sun
pixels have a contrast of zero, and solar features that are bright
or dark relative to the quiet Sun have positive and negative
contrasts, respectively. The photometric sum Σ is given by
where the sum extends over all image pixels, Ciis the contrast
of pixel i, and φ(μi) is the limb darkening curve for that image,
normalized to unit integral over the hemisphere. Thus for each
image, Σ is the relative change (in parts per million) in the
disk-integrated intensity of the image, due to the presence of
solar features. We measure Σ for all of our images, and we use
subscripts r, b, and K to refer to the measurement on continuum
red, continuum blue, and Caii K-line images, respectively. We
note that the photometric sum includes the contributions of
all solar features, bright and dark; noise pixels cancel in the
features that are difficult to isolate with threshold techniques.
However, the accuracy of the photometric sum does depend
on the correct identification of the quiet Sun. Preminger et al.
(2002) show that a slight bias of the quiet Sun is present in the
contrast maps, due to the presence of numerous low-contrast
bright features at solar maximum, but the effect of this bias can
be assessed and corrected for. We revisit the issue of bias in
Sections 2.3 and 3 of this manuscript.
2.3. SFO Results
In what follows we use the term “brightness” to mean disk-
integrated intensity. The photometric sum measures the change
were present. Σr and Σb measure the relative change, due
to active regions, in visible continuum brightness, while ΣK
measures the relative change, due to active regions, in the Caii
Figure 2 shows the photometric sums Σr, Σb, and ΣK,
computed from SFO images. All three sums show large short-
term variations that correspond to rotational modulation by
bright and dark active regions, but short-term variability is most
large dips in brightness as they transit the solar disk. In other
studies we examined the short-term effects of active regions on
regions over timescales of months to years. In Figure 2, the 81
day running mean is superimposed on the data, to highlight
As Figure 2 shows, ΣK is highest at solar maximum, i.e.,
the average effect of solar active regions is to enhance Caii K-
line brightness. We conclude from this plot and from Figure 1
that the Sun’s relative brightness in the Caii K line is directly
correlated with solar activity and TSI over timescales of months
to years. This is expected: magnetic activity has been shown to
brightness is directly correlated with the average photospheric
magnetic flux density, which is greatest at solar maximum
(Schrijver & Harvey 1989; Harvey & White 1999; Ortiz & Rast
2005; Preminger et al. 2010).
In Figure 2, Σr shows the effect of solar activity on the
Sun’s red continuum brightness. Contrary to ΣK,Σrshows no
enhancement at all at solar maximum: it seems almost flat.
However, the running mean of Σris slightly anti-correlated with
solar activity. We conclude that time-averaged red continuum
brightness is diminished at solar maximum. The pattern of
Σbis similar to that of Σr: blue continuum variability is also
anti-correlated with solar activity, on timescales of months to
years. We infer from the red and blue photometric sums that the
Sun’s visible continuum brightness is slightly anti-correlated
with solar activity, over solar-cycle timescales.
The Astrophysical Journal Letters, 739:L45 (6pp), 2011 October 1 Preminger, Chapman, & Cookson
198519901995 20002005 2010
19851990 1995 20002005 2010
Figure 2. Top: the Caii K-line photometric sum, ΣK, from Caii K-line images. Middle: the red photometric sum, Σr, from red continuum images. Bottom: the blue
photometric sum, Σb, from blue continuum images. Σ measures the relative change (in parts per million) in the disk-integrated intensity of the Sun due to the presence
of solar features. The bold curves are the 81 day running means of the data.
We now address the issue of bias in the determination of
the photometric sums. As stated, an accurate measurement of Σ
requires accurate identification of the quiet Sun. When the level
of solar activity is high, there are small, low-contrast bright
features present on the disk, tending to bias the intensity level
determined for the quiet Sun. In Preminger et al. (2002) we
assessed the effect of this bias: it causes a small, positive,
cycle-dependent trend in the data to be removed. To correct
for this bias, we must reduce the long-term trend of Σr by
∼50%, and increase the long-term trend of ΣKby ∼7%. This
correction does not affect our qualitative assessment of the
spectral variability: ΣKremains almost the same, while Σris
still negatively correlated with the solar cycle, albeit less so.
3. BRIGHTNESS VERSUS ACTIVITY FOR THE SUN
Studies show it is reasonable to conclude that solar magnetic
activity accounts for most, if not all, solar radiative variabil-
ity on timescales of days to decades, while the “quiet-Sun”
background remains constant (Domingo et al. 2009 have an
extensive review; Livingston et al. 2005). Now assuming this
is the case, the photometric sum measures all variability in a
given spectral band, and we can use this quantity to evaluate
activity–brightness correlations for the Sun-as-a-star.
From Figures 1 and 2, the relationship between solar bright-
ness and activity is clearly complex. The activity-related bright-
ness variations in different parts of the spectrum are not simi-
lar to each other, and it is not obvious how changes in specific
ever, we can use the photometric quantities, Σrand ΣK, to model
changes in total solar irradiance (ΔTSI) according to
ΔTSI(t) = aΣr(t) + bΣK(t).
reconstruct TSI variations during cycle 22. Figure 3 compares
this simple model to TSI variations from 1988 to 2010. The
fit is very good: the model, based on ∼8700 data points, has
a regression coefficient R2= 0.85. This is comparable to the
goodness of fit achieved by other empirical and semi-empirical
models of TSI (Domingo et al. 2009). We interpret the physical
significance of Equation (3) as follows: to a first approximation,
the net change in the Sun’s bolometric flux is the linear sum
of two components: the change in photospheric continuum flux
(represented by aΣr) and the change in spectral line blanketing
from the chromosphere and upper photosphere (represented
by bΣK). All changes are induced by magnetic active regions.
Equation (3) is a rather simplified model, but it can reconstruct
85% of TSI variability over a period of two solar cycles, which
indicates that it incorporates the primary components of the
Sun’s irradiance variability in terms of energy.
Figure 4 shows the relative variability of ΔTSI(t) and the two
contributing components from Equation (3). The pronounced
short-term dips in TSI corresponds to negative changes in the
continuum component, aΣr(t), that occur whenever a sunspot
group transits the solar disk. But the continuum component is
clearly not the primary source of longer-term TSI variations.
The long-term variability is in the spectral lines, represented by
out in Preminger et al. (2002).
Long-term activity–brightness correlations become clear if
we plot the time average of the different components. For our
best quantitative assessment, we first correct the photomet-
ric sums for quiet-Sun bias, as discussed in Section 2. Then
we re-compute the components of the fit from Equation (3).
Figure 5 shows the 81 day running mean of ΔTSI and the new
and spectral line brightness are strongly enhanced by solar ac-
tivity, on the order of 0.1% at solar maximum, while continuum
brightness is slightly diminished, on the order of –0.02%.
4. THE SUN VERSUS SOLAR ANALOGS
Many Sun-like stars appear to have magnetic activity cycles.
Generally, observations of Ca HK emission are used to infer the
level of stellar magnetic activity, and stellar brightness is mea-
sured using the Str¨ omgren b and y photometric system (Hall
The Astrophysical Journal Letters, 739:L45 (6pp), 2011 October 1 Preminger, Chapman, & Cookson
19851990 1995 2000
Σr, ΣK fit
Figure 3. TSI and model fit using SFO data, R2= 0.85 (1988–2010).
Relative Change (%)
change in TSI
Figure 4. Relative change in TSI and the two primary contributing components, computed from Equation (3). We infer that the component aΣrrepresents the change
in continuum irradiance and the component bΣKrepresents the change in spectral lines.
et al. 2009; Radick et al. 1998; Lockwood et al. 2007). Obser-
vations of Sun-like stars are carried out during an observing
season, and the seasonal mean is studied. An observing season
lasts about three months, therefore we can compare the results
for stellar variability with the 81 day average variability of the
Observations show that solar analogs appear to fall into two
HK emission) tend to have an inverse correlation between b+y
brightness and HK emission, while less active stars tend to have
a positive correlation. Let us compare these stellar observations
to those of the Sun.
The connection between magnetic active regions and en-
hanced emission in chromospheric lines such as the Caii H
and K lines is already well established (e.g., Leighton 1959;
Schrijver et al. 1989). Disk-integrated Ca K-line variability as
quantified by ΣKcan be considered to be directly related to the
Ca HK brightness index commonly used to measure chromo-
FilterStr¨ omgren bStr¨ omgren ySFO blue SFO red
spheric activity for stars. Both quantities are good measures of
Table 1 compares filters used for stellar Str¨ omgren b and y
photometry with the blue and red filters used at SFO for solar
For the Str¨ omgren filters, photospheric spectral lines block
∼8% and ∼11% of the continuum flux in the passbands of
the y and b filters, respectively (Conti & Deutsch 1966). Thus,
SFO blue and red filters are comparable to the Str¨ omgren b
and y filters: all measure continuum brightness because they are
broad and located above the point (about 450 nm) where line-
The Astrophysical Journal Letters, 739:L45 (6pp), 2011 October 1Preminger, Chapman, & Cookson
1985 1990 19952000
Relative change (%)
change in TSI
Figure 5. 81 day running mean of the relative change in TSI and the two primary contributing components. We infer that aΣrrepresents the change in continuum
irradiance and bΣKrepresents the change in spectral lines. Here, the components have been corrected for the effects of quiet-Sun bias.
blanketing becomes important (Crawford 1987). Stellar b+y
brightness is a measure of visible continuum brightness.
The most active Sun-like stars, those observed to have an
inverse correlation between b+y brightness and HK emission
between visible continuum and spectral line flux over these
timescales. This is the same pattern we see for the Sun. Thus,
if solar analogs fall into two groups, we must conclude that the
Sun belongs with the more active group.
This conclusion is opposite to that reached by Lockwood
et al. (2007). The problem is that, for Sun-like stars, bolometric
brightness is inferred, not measured. It is assumed that the
star radiates like a blackbody, and variations in flux arise from
changes in its effective temperature:
?b + y
For active Sun-like stars, b+y flux is low when activity is
high. Applying Equation (4) leads to the conclusion that the
bolometric brightness of these stars is reduced by activity.
However, our analysis of solar variability shows that
Equation (4) is invalid for the Sun, and likely also invalid for
5. SUMMARY AND CONCLUSIONS
The photometric quantity Σ measures the effect of all solar
features (bright and dark) on the relative brightness of the solar
disk. We examine Σ for two solar-cycles’ worth of red, blue,
and Caii K-line photometric images from the San Fernando
Observatory. Σrand Σbmeasure visible continuum variability,
due to solar activity, while ΣKmeasures variability in the Caii
K spectral line, caused by solar activity. When we average over
timescales of a few months, we find that the net effect of solar
features is to enhance the brightness of the solar disk in the Caii
K line and to diminish the brightness of the disk in the visible
continuum spectral range. We originally reported this result
based on SFO data for solar cycle 22 (Preminger et al. 2002);
our data for solar cycle 23 now confirm the result. New results
from the Spectral Irradiance Monitor (SIM) experiment on the
We construct a model of the total solar irradiance, TSI, of the
form ΔTSI = aΣr+ bΣK. This model successfully reproduces
ically, the model implies that two primary components of bolo-
metric variability exist: variations in continuum, photospheric
irradiance (represented by aΣr), and variations in spectral lines
(represented by bΣK). Over solar-cycle timescales, the Sun’s
bolometric irradiance is directly correlated with spectral line
irradiance, but inversely correlated with continuum irradiance.
The quiet-Sun irradiance is assumed to be constant.
We compare the activity–brightness relations determined for
the Sun to those observed for Sun-like stars. For those stars,
activity is measured using the Caii HK index while brightness
is measured using Str¨ omgren b+y photometry. These two mea-
sures are anti-correlated for very active stars (those with large
periodic changes in the Ca HK index), and positively correlated
for less active stars. The b+y brightness was assumed to be di-
rectly related to bolometric brightness, making the behavior of
Thus, Sun-like stars for which b+y brightness is inversely cor-
related with activity are behaving in a very Sun-like way.
If solar analogs are indeed Sun-like, we might expect the
magnetic active regions to be similar in nature, and to affect the
radiative properties of the stellar atmosphere in a similar way.
And, indeed, the relationship between photospheric magnetic
flux and radiative flux from the chromosphere or corona is
similar for the Sun and solar analogs (Schrijver et al. 1989;
Pevtsov et al. 2003). Therefore, we should not be surprised to
find Sun-like stars for which activity and photospheric radiative
flux are anti-correlated, as they are for the Sun. How, then,
can we explain that some Sun-like stars exhibit activity directly
correlated with photospheric brightness? It could be due to the
angle from which we view these stars: we know that solar active
regions appear at mid-to-low latitudes only and that sunspots
decay into facular regions which appear bright in continuum
images when they are on the solar limb. In a polar view of
the Sun, the active regions would circle the limb; the average
with activity, even for visible continuum wavelengths. The stars
with small HK variability, making this explanation plausible.