A&A 372, 1005-1018 (2001)
DOI: 10.1051/0004-6361:20010280
J. Pap 1,
- J. P. Rozelot 2
- S. Godier 2
- F. Varadi 3
1 - Department of Physics and Astronomy, University of
California, Los Angeles, 405 Hilgard Ave., Los Angeles, CA 90095, USA
2 - OCA/CERGA, Avenue Copernic, 06130 Grasse, France
3 - UCLA, Department of Physics, 90095-1562, Los Angeles, USA
Received 29 August 2000 / Accepted 18 January 2001
Abstract
We use Singular Spectrum Analysis (SSA) to analyze total solar irradiance
variations and CERGA radius measurements. Total solar irradiance has been
monitored from space for more than two decades, whilst ground-based radius
measurements are available as a coherent time series from 1975. We compare
these indicators to try to understand the origin of energy
production inside the Sun.
One of the main objectives was to assess the reality of the observed
variations of the Sun's radius by distinguishing the signal from the noise.
Two approaches were used: one using SSA on ground-based
data averaged over 90 days, in order to smooth the signal (especially over
periods when no data were obtained, mainly in winter time); the second
repeats the analysis on individual measurements corrected by
reporting data to the zenith. As expected, the level of noise is higher in
the first case and the reconstructed noise level, which is large,
indicates the difficulty in ascertaining the solar origin in the
apparent variability of the solar radius. It is shown from the
reconstructed components that the main variation in amplitude (over 930 days) is pronounced during the
first part of the measurements and seems to disappear after 1988. There is also
a variation with a periodicity of 1380 days, of
lower amplitude than that of the shorter component. In both cases,
these variations disappear during the rising portion of cycle 23. The first
reconstructed component shows
that total irradiance varies in parallel with the solar cycle, being
higher during maximum activity conditions. The reconstructed
radius trend indicates that the solar radius was higher during the minimum
of solar cycle 21, but its decrease with the rising activity of cycle
23 is less obvious. The observed value of the solar radius increased by
about 0.11 arcsec from the maximum of cycle 21 to the minimum
between cycles 21 and 22. Most importantly, we report a long-term
radius variation which increased from the maximum of cycle 21 to
minimum by about 0.015%, while a smaller decrease (around 0.01%) is
seen
from the minimum of cycle 21 to the maximum of cycle 22. This study
indicates need for measurements of the degree
of the radius changes taken from space, together with total irradiance
measurements to establish the phase
relation between these two quantities.
Key words: Sun: general - Sun: fundamental parameters
![]() |
Figure 1: Summary of total solar irradiance measurements (Fröhlich 1998). |
Open with DEXTER |
![]() |
Figure 2: Summary of radius measurements from the time of the Maunder Minimum. These radius measurements, covering a long time interval, suggests that the Sun may have been larger during its exceptionally low magnetic activity during the Maunder Minimum. Triangles represent the Fiala et al. (1994) determinations of the solar radius deduced from eclipse observations. |
Open with DEXTER |
Visual measurements of the solar radius have been made for more than twenty years at the Calern Observatory (South-East France) using a visual Danjon astrolabe (Laclare et al. 1983). Several technical improvements were made to optimize the observations of the semi-diameter measurements. The number of zenithal angle observations was increased from one possibility per day in 1975 to eleven nowadays; in the near future, quasi-continuous observations will be possible. In 1986 a CCD camera was installed to reduce biaises arising from visual observations (Laclare & Merlin 1991). Both visual and CCD observations have been performed daily since then, and comparison of these measurements shows that they are consistent (Laclare et al. 1996). Other measurements of the semi-diameter of the Sun, based on the same principle of the solar astrolabe, have been performed in other places, such as Brazil and Chile (Jilinski et al. 1999; Noël 1997, 1998). In this paper we rely only on the CERGA measurements, as the observations were made by a single observer over a long period of time. Comparison of the CERGA radius measurements with those obtained in other observatories and with different measuring methods will be the subject of another paper.
![]() |
Figure 3: Principle of the measurement of the Sun's radius by means of an astrolabe. The observed radius is the difference between the zenith distances of the upper and lower solar edges. |
Open with DEXTER |
In this study, we averaged the measurements made at several zenithal angles, for a given day, when such observations were available. A weighted running mean over 90-day periods was obtained, which smooths the periods of time where no data was obtained. To assign an error bar to a point which is a 90-days average, we can disregard as "noise" the variations present in the data, or we can assign to each point an error bar whose length is the standard deviation of a single observation divided by the square root of the number of added data. We took their inverse squares as weights in the above-mentionned "weighted running average". The general level of noise is slightly increased, as the quality of the results, mainly due to the seeing, is affected by the altitude of the observations (the lowest altitudes give poorest results). It would be better to correct each measurement by the zenithal function which can be determined only when sufficient data are available, which is just becoming the case now. This was done in the second step of this analysis: in that case, all solar data rather than mean values were used, each individual measurement being only corrected by reporting its value to the zenith.
![]() |
Figure 4: The CERGA radius measurements for November 16, 1978 to December 31, 1998. This plot is compliant in dates with the composite solar irradiance. Data has been corrected for the zenith distance effect, i.e. reset to Zenith. |
Open with DEXTER |
The radius measurements used in our analysis are plotted in Fig. 4. Total irradiance varies in parallel with the solar activity cycle, being higher during maximum solar activity. Despite the apparent noise in the semi-diameter data, the CERGA solar diameter seems to show a small variation over the solar cycle, being higher during the solar minimum. In addition to long-term variation in the semi-diameter, Delache et al. (1986) reported variations with periodicities of about 1.800, 1.000, 300 and 120 days. Their analysis was carried out from only 1975 to 1984. One major problem in analyzing observed time series is the presence of gaps which make the detection of periodicities very difficult. The choice of a detection threshold is fundamental and frequencies in the power spectrum may appear which are not real. The origin of these variations is not yet understood; the major question is whether these are real variations related to solar effects or to other effects, such as changes in the atmospheric transparency and/or random fluctuations associated with the noise in the data.
To separate the possible intrinsic solar radius variations from other effects, it is necessary to compare the radius changes to other solar parameters, such as total irradiance. Furthermore, to understand the effect and contribution of various solar features to the radius changes, it is necessary to decompose the examined time series to their major oscillatory components and to separate these components from the noise in the signal. We note that for long, statistically stationary time series with relatively small noise, the traditional methods of time series analysis, such as Fourier Transforms, work well. However, measurements rarely satisfy all these requirements; most time series representing solar variability can be regarded as nonstationary signals with variable spectral properties, e.g., change of periods, amplitudes, and phases (Fröhlich & Pap 1989; Vigouroux & Delache 1993; Vigouroux et al. 1997). Therefore, to separate the possible intrinsic solar radius variations from other effects, we have applied a relatively new technique, "Singular Spectrum Analysis" (SSA, Vautard et al. 1992). The advantage of using SSA is that SSA performs as a data adaptive filter instead of using fixed basis functions, as in the case of Fourier transform or wavelet techniques, therefore, it is highly capable of distinguishing between the dominant and possibly nonlinear oscillations of the system and clarifying the noise characteristics of the data.
Singular Spectrum Analysis (SSA) has been
developed and applied to study and understand nonlinear and chaotic
dynamical systems. SSA has been successfully used in nonlinear dynamics
and in analyzing paleoclimatic and
geophysical data sets and global temperature changes.
Pap & Varadi (1996) applied SSA to study
the effect of the evolution of active regions on total solar irradiance, and
Pap (1997a,b) used SSA to study variations in total solar and UV
irradiances.
SSA is based on Principal Component Analysis in the time domain. The
examined
time series is augmented into a number of lagged time series up to a fixed
value M, called the window length. The basis of SSA is the
eigenvalue-eigenvector decomposition of the lag-covariance matrix which
is composed of the covariances determined between the shifted time
series. The eigenvectors of the lag-covariance matrix,
the so-called temporal empirical orthogonal functions (T-EOFs), provide
moving average filters which extract uncorrelated parts of the signal
and whose contributions to the complete signal are given by the
corresponding eigenvalues.
Projection of the time series onto the T-EOFs yields the so-called principal
components (PCs); these are filtered versions of the original time series.
The T-EOFs are data adaptive analogs of sine and cosine functions and the
PCs
are the analogs of coefficients in Fourier analysis.
The actual computation proceeds as follows.
One takes the time series
![]() |
Figure 5: The Singular Spectra of total irradiance and radius are presented. The dots represent the composite total irradiance and the squares the radius. |
Open with DEXTER |
The original time series above the noise level and/or only parts of interest
can be reconstructed as a projection to certain T-EOFs. These reconstructed
components are usually associated with trends or with particular,
possibly nonsinusoidal, oscillations. The window length M determines
which components are resolved and which ones are reconstructed as part of the
trend. To reconstruct strange attractors, a larger M is better, but to
avoid statistical errors, the window length should be less than one third of
the length of the investigated times series (Vautard et al. 1992).
![]() |
Figure 6: The composite total solar irradiance and its first reconstructed component (upper panel) plotted together with the CERGA radius measurements (lower panel) and its first two SSA-reconstructed Components. The solid lines show the SSA reconstructed trend in total irradiance (RC1) and in the radius (RC's 1 and 2). By comparison the 5th degree polynomial fit is also indicated. |
Open with DEXTER |
The window length was chosen as 400, considering the yearly gaps in the radius measurements, and was applied with the same window length to the composite total irradiance.
The SSA spectra of the radius and the composite total irradiance are plotted in Fig. 5, where the squares show the composite eigenvalues and the dots show the radius eigenvalues. While there are many oscillatory components above the noise level in the SSA spectrum of total irradiance, in the case of the radius one can identify only 6 significant components; the rest of the eigenvalues fall to the so-called noise floor. This exercise indicates the difficulty of radius measurements from the ground where they are highly affected by the changing atmospheric conditions.
In the case of the solar radius, the solid line in Fig. 6 shows the
first two SSA-Reconstructed Components which represent the solar cycle
trends in the data. In the case of total irradiance, the solar cycle trend
is represented by the first reconstructed components (faint line). As can be seen, total irradiance varies in parallel with the solar cycle, being
higher during maximum activity conditions. In contrast, the reconstructed
radius trend indicates that the solar radius was higher during the minimum
of solar cycle 21, but its decrease with the rising activity of cycle
23 is less obvious. The value of the solar radius increased by about
0.11 (0.1) arcsec from the maximum of cycle 21 to the minimum between
cycles 21 and 22. The long-term percentage radius variation is given in
Fig. 7. As can be seen, the observed increase from the maximum of cycle 21
to minimum is about 0.015%, while a smaller decrease (around 0.01%) is seen
from the minimum of cycle 21 to the maximum of cycle 22.
In addition to the first two components in the radius, which represent
the solar cycle trends, we have reconstructed the 3rd, 4th, 5th and
6th
components, respectively. RCs 34 represent the variation of about 1.000 days, which is pronounced during the first part of the measurements but
seems to disappear after 1988. RCs 5
6 seem to represent a variation
with about a 1.500 day periodicity; however, the amplitude of this variation
is much smaller than that of the 1.000 day component. Note that in both
cases, these variations disappear during the rising portion of cycle 23. It
remains to be seen whether the lack of these periodicities during cycle
23 is related to real solar effects or additional effects (e.g.
instrumental effects).
Finally, we have reconstructed the so-called "noise" of the radius data, as shown in Fig. 8. The noise was calculated from the eigenvectors of the eigenvalues in the noise level in the SSA spectrum. As this plot shows, the noise accounts for a considerable portion of the radius signal. This indicates the difficulty in distinguishing between real solar variability and measurement uncertainties in the current ground-based radius measurements. This underscores the necessity of developing more advanced instrumentation to detect small changes in solar diameter and the need for observing these small variations from space (which is free from atmospheric blurring and weather conditions, see also Kuhn et al. 1998). The forthcoming PICARD experiment (a microsatellite mission which is scheduled to be launched in 2005), and other proposed space experiments, such as SPHERIS, will hopefully determine the degree of the radius changes and their phase relation to irradiance variations. Knowledge of these quantities will lead to a better understanding of the physical mechanisms within the convective zone or below, which is essential for an understanding of the role of solar variability in long-term climate changes.
![]() |
Figure 7:
The long-term radius variation in percentage from cycle 21 to rise of cycle 23
(2 first components, i.e. noise removed). From this figure, it can be deduced
that the peak-to-peak value of the solar radius is less than
0.11 arcsec (![]() |
Open with DEXTER |
![]() |
Figure 8: Reconstruction of the CERGA radius (RC's 1 to 6) and reconstructed noise of the radius data (RC's larger than 6). Even if the picture clearly depicts that the noise is the dominant effect, a weak signal appears, stronger in the data up to 1988. This figure indicates the necessity to measure from space the degree of the radius changes. |
Open with DEXTER |
A careful determination of the time-dependence of the solar shape and diameter, as well as distortion effects when they can be taken into account, is of importance. The subadiabatically stratified plasma in the convective zone provides both plasma motions (mainly differential rotation) and mechanical stability (mainly an intensification of the magnetic field). Solar cycles, for which the basic physics is still an enigma, are probably driven by convection in the solar interior. Apparent solar radius variations can provide useful information regarding the processes which may be occuring. Moreover, as pointed out by Ulrich & Bertello (1995), a change in the apparent solar radius could assist in the interpretation of frequency variations (as the most probable interpretation of the diameter variability is a change in the temperature profile of the solar limb). The magnitude of the radius fluctuations, compared to the irradiance change during a solar cycle, contains information that has not been previously completely used. Showing that a change in the irradiance leads to a change in the diameter will indicate a unique relationship between these two parameters (Sofia & Endal 1979). Such a law would be particularly useful to tackle problems relevant to our Earth's climatic system.
Several observations of the Sun's diameter exist obtained using different techniques. No convincing results have been obtained, due to the difficulty of detecting a hundred milliarcsecond point swamped by a fuzzy solar limb, worsened by the atmospheric seeing effects. The unavoidable question which arises is whether the results from ground can be considered sufficiently trustworthy to warrant more ground-based efforts. Even from space, the question is not trivial (see Emilio et al. 2001, from the MDI-SOHO limb observations). One of the major goals of this paper is thus to try to determine unambiguously the level of noise in at least one ground data set, in order to separate the contribution of the Sun's intrinsic variability from that originating from the instrument and the atmosphere. Ground-based measurements can be briefly report as follows (see also for instance Ribes et al. 1988)
1. Ulrich & Bertello (1995) showed that the apparent radius of the Sun, when viewed in the spectral line of neutral iron at 525 nm, varies in phase with solar activity. Observations were made at the Mount Wilson observatory and the method of measuring the Sun's apparent radius has been described by La Bonte & Howard (1991). The error on an annual average measurement (the mean derived from the daily observations for each year) is about 0.03''. It was concluded that the variations in the apparent radii of the Sun's disk (at all wavelengths) should contribute between 0.013 and 0.025 per cent variation in the total irradiance;
2. Between mid-1981 and 1987, Brown et al. (1982) observed the solar radius at the High Altitude Observatory by means of a horizontal instrument timing the interval required for solar meridian transits. They report no significant variation in the solar radius at all. This was based on a definition of the solar limb that minimizes the effect of the seeing. A reanalysis of the data carried out by means of a model of the solar limb intensity permitted the elimination of several uncertainties affecting earlier determinations, allowing an accurate measurement of the solar photospheric radius (Brown & Christensen-Dalsgaard 1998). The authors concluded that no significant variation could be found during the observation period (1981-1987);
3. Wittman (1997) studied possible time variations in the solar diameter,
using a method combining the advantage of both photoelectric and visual drift
techniques; the system was put into operation at Observatorio del Teide in
Tenerife. He concluded that, at least for his data between 1990 and 1992,
there was no strong indication of a variation in diameter with the 11-yr
activity cycle in excess of 0.05'';
4. All the above-mentioned measurements were performed using techniques other
than the solar astrolabe. In order to render the measurements more homogeneous
and to cover a larger set of heliographic latitudes, a network of solar
astrolabes has been set up in Brazil and in Chile; two other instruments are
under construction in Turkey and Algeria.
The measurements, performed from 1974 in Brazil (Leister & Benevides
1990) are from visual observations, which were followed by CCD measurements
since 1997 (Jilinski et al. 1999) and since 1990 in Chile (Noël 2000).
A recent comparison of these data has been made by Lefebvre (2000).
If all these observations lead to a variability in the solar diameter, the
peak-to-peak amplitude and the phase disagree from one dataset to another one.
In the case of the Santiago observations, the measurements, made at
and
zenith distances, show around a 1 arcsec
amplitude variation in the data over a solar cycle, the peak-to-peak
variability being taken from a sinusoidal fit of the data. The standard
deviation given by second order fits of the data is
at
(461 observations) and
at
(817 observations) zenith distances. Over the solar cycle, the monthly mean
values of solar radius measurements show a clear variation in phase with
solar magnetic activity.
In the case of the Brazilian measurements, the results show a variability out
of phase with the solar cycle, of peak-to-peak amplitude approximately 1.2 arcsec; the average error bar is 0''20 (4630 observations).
All published results
are similar to those of the CERGA observatory. Other results can be found, for example
Chapman et al. (1999) and Costa et al. (1999). What could
be the possible explanations for such discrepancies?
Brown and Christensen-Dalsgaard argued that the limb-position slope shows fairly substantial variations, variable in magnitude. This suggests that the long-term variation may result from localized activity-dependent features such as faculae.
Observations made by Ulrich and Bertello were not conducted at the same optical depth as observations made with the solar astrolabe. The iron line is strongly dependent on magnetic field, suggesting that the measurements may follow the solar activity cycle.
To explain discrepent values from Laclare and Noel, we note that they measure different heliographic latitudes on the same day (due to the different latitudes of observations). When the measurements are grouped by classes of latitude of at least 5 to 10 degrees, the results are more in accordance (Godier & Rozelot 1999). This can be explained, as part of the radius variation is due to solar deformation, linked on the one hand with the royal zones where the existence of spots diminishes the measured brightness, and as a consequence the measured radius; and on the other hand, with the high-latitude activity zones where the presence of faculae have the inverse effect (Pecker 1996).
Obviously, all these measurements are affected by the atmosphere. The degradations are directly related to the optical turbulence along the line of sight and evolve rapidly in time. Therefore, it is important to evaluate the level of noise in the data due to the combination of the instrument and the atmosphere.
The importance of noise in the two sets of data (irradiance and diameter) is well illustrated in Fig. 5. As the eigenvalues are arranged in a monotonically decreasing order, the cut-off at a certain order forms a "tail'' in the spectrum which is the noise floor of the data. The number of eigenvalues above the floor represents the degrees of freedom of the variability, or, in other words, the statistical dimension of the data, which is associated with the number of oscillatory components in the signal. The highest eigenvalues represent the fundamental oscillations in the data and are related to the trend. The two Singular Spectra are far from being identical as only six components can be recognized in the diameter data, the reminder creating the level of noise. By comparison, as many as 20 degrees of freedom in the irradiance data are established from the eigenvalues above the noise level. The reconstruction of these eigenvalues lead to a fairly good agreement with the observed irradiance data, indicating that these first 20 components represent the observed variability; the noise contributes to only a very small fraction of the observed irradiance signal. However, the reconstruction of the 6 first eigenvalues of the diameter data do not reproduce the observed data (Fig. 8, bottom), which are better described by all the components larger than 6. As one may expect, the noise contributes to a large part of the measured variable signal. Figure 5 shows that the level of noise inside the diameter data is far above that of the irradiance, a result not surprising as the previous data are ground-based measurements and the second from space observations.
Analyzing the long term trend (the 2 first components of the diameter
data), the peak-to-peak variation is only about 100 milliarcsecond. Is such
a value statiscally significant over the studied time range? The full
amplitude of the cosine function adjusting the semi-diameter data is
0''62, greater than five times the root mean square residuals (0''18). The
reduced for such a fit is 2.25. Under normal statistics one expect a good fit to have
a reduced
of about 1. A larger
can result from
systematic errors in the data sets
(including seeing effects), random variations, or underestimated
i. The trend deduced from the SSA shows a modulation
of
so that
drops to about unity, indicating a
confidence level of more than 95%. This does not demonstrate that the
radius variability is 100 milliarcsecond in amplitude, but indicates that
if such a variability exist, it must be no more than this value.
One of the most important implications concerns the changes in the irradiance
due to the changes in the radius. Sofia et al. (1979), calculated the
so-called w parameter which measures such changes. According to
different authors (Spruit 1982; Gilligand 1980; Deadborn & Blake 1980),
the value of w ranges from
to 0.075
and is strongly model dependent. Observational measurements are still very
poor. Assuming that w is constant in time, Ribes et al. (1991)
give w close to unity, while Rozelot (2001) gives a
w of about 0.85. This rather high value may certainly comes from the
fact that the data used in the study were not filtered. These present results
yield a value of w of 0.05, not too far
from the first theoretical value given by Sofia et al. i.e. 0.075.
However, space measurements lead to a value of w = 0.02, which is
claimed to be an upper limit (Emilio et al. 2001). How reliable are
these estimates? Our ability to make more precise measurements may
be expected to improve in an era of dedicated space missions (mainly
PICARD and SPHERIS). A direct application of a precise determination of
w would help in understanding how small perturbations in the solar
irradiance may induce changes in the terrestrial surface temperature of a
fraction of a degree centigrade (see for instance Eddy et al. 1982
or Sofia 1998).
1. From SSA, an amplitude in the solar diameter signal of no more than 110
milliarcsecond can be detected, the remainder being noise, coming jointly
from the atmospheric seeing and from the instrumental effects. A statistical test indicates that this level is significant
,
but as an upper limit;
2. The CERGA measurements show that the long-term variation of the radius over the solar cycle anticorrelates with both total irradiance and sunspot number. In contrast, measurements on other sites indicate a positive correlation between the solar radius and the activity cycle. Some explanations have been attempted: in the case of observations made by techniques other than the solar astrolabe, it can be argued that measurements probe different optical depths, or result either from variations in the limb-temperature profiles or from localized activity dependent features. In the case of measurements made by the solar astrolabe, but leading to different results, it can be suggested that the discrepancy comes from observations at different heliographic latitudes due to the different latitudes of the observation sites. Gathered by classes of heliographic latitudes, the discrepancies tend to disappear;
3. Considering the pure signal-to-noise ratio in the radius data, further measurements above the Earth's atmosphere are required to determine the phase between the radius and irradiance variations and to probe the solar interior. Such space experiments whose main objectives are to measure simultaneously the solar diameter and the solar irradiance would permit us to unambiguously confirm or disaprove the occurence of diameter variations without atmospheric effects. In turn, these space observations will also help to calibrate the ground-based radius measurements if the variations are real. Moreover, these measurements can be compared with theoretical models to identify the physical processes responsible for the observed changes of both diameter and irradiance. Such high-quality measurements and their analyses are absolutely necessary not only for astrophysics, but also for climate research. One of the main goals is to understand the origin of climate changes which occurred prior to the industrial era and to determine the degree of the contribution of solar variability to the current and future climate changes.
Acknowledgements
We gratefully acknowledge the referee for his helpful remarks which have enabled us to improve the paper.