Modelling solar irradiance variability on time scales from minutes to months
aa r X i v : . [ a s t r o - ph . S R ] M a r Astronomy&Astrophysicsmanuscript no. aa2008-11138 c (cid:13)
ESO 2018September 8, 2018
Modelling solar irradiance variability on time scales fromminutes to months
Andrey D. Seleznyov , Sami K. Solanki , , and Natalie A. Krivova Max-Planck-Institut f¨ur Sonnensystemforschung, 37191, Katlenburg-Lindau, Germany School of Space Research, Kyung Hee University, Yongin, Gyeonggi 446-701, KoreaReceived October 14, 2008; accepted
ABSTRACT
We analyze and model total solar irradiance variability on time scales from minutes to months, excluding variations due to p-modeoscillations, using a combination of convective and magnetic components. These include granulation, the magnetic network, faculaeand sunspots. Analysis of VIRGO data shows that on periods of a day or longer solar variability depends on magnetic activity, butis nearly independent at shorter periods. We assume that only granulation affects the solar irradiance variability on time scales fromminutes to hours. Granulation is described as a large sample of bright cells and dark lanes that evolve according to rules deduced fromobservations and radiation hydrodynamic simulations. Comparison of this model combined with a high time resolution magnetic-fieldbased irradiance reconstruction, with solar data reveals a good correspondence except at periods of 10 to 30 hours. This suggests thatthe model is missing some power at these periods, which may be due to the absence of supergranulation or insufficient sensitivityof MDI magnetograms used for the reconstruction of the magnetic field-based irradiance reconstructions. Our model also shows thateven for spatially unresolved data (such as those available for stars) the Fourier or wavelet transform of time series sampled at highcadence may allow properties of stellar granulation, in particular granule lifetimes to be determined.
Key words.
Sun: activity – Sun: granulation – Sun: magnetic fields – Sun: photosphere
1. Introduction
Models of solar irradiance on time scales of days to the so-lar cycle have reached a certain maturity. They reproducethe observations with high accuracy (e.g., Solanki et al. 2005;Solanki & Krivova 2006; Krivova et al. 2011, and referencestherein). Shorter time scales have been dealt with much moresummarily. Traditionally, interest in time scales of minutes todays has derived from helioseismology (and more recently aster-oseismology) since the Sun’s ‘noise’ background produced byconvection and magnetism is a limiting factor in detecting os-cillation modes (Harvey & Duvall 1984; Andersen et al. 1994;Rabello-Soares et al. 1997). In recent years variability on hoursto days time scales has become important in connection with ex-trasolar planet transit detection programs. Stellar noise at thesetime scales is the factor finally limiting the size of the planetsthat can be detected with this technique (Carpano et al. 2003;Aigrain et al. 2004).Here we model solar total irradiance on time scales of min-utes to months. However, we concentrate on what helioseismol-ogists call ‘solar noise’ and explicitly do not consider irradiancefluctuations caused by p-modes. Traditionally, the influence ofgranulation, mesogranulation and supergranulation is consideredtogether in highly parameterized models, while the magneticfield is not taken into account explicitly (Harvey & Duvall 1984;Andersen et al. 1994; Rabello-Soares et al. 1997; Aigrain et al.2004). Here we take a different approach, assuming that all so-lar variability not due to oscillations is produced by magnetismand granulation. We compute the variability and compare it withobservations, mainly recorded by VIRGO on SoHO and TIMon SORCE. This test should then show if the assumption is jus-
Send offprint requests to : S. K. Solanki tified. If not then it suggests that some convective or magneticcomponent is missing. Such modelling also provides the basis ofusing stellar power spectra to infer the convective and magneticproperties of stars, since it allows us to explore the diagnosticcapabilities of such spectra.We model the influence of the granulation in an explicit man-ner, although using an empirical model. Main granular character-istics such as size, lifetime, contrast are based on observationaldata. Modelled power spectra of the irradiance caused by thegranulation are compared with the corresponding spectra of theVIRGO data. Not surprisingly, it is found that total irradiancevariability on time scales longer than a few hours is not wellreproduced by the granulation alone. We therefore also apply amodel of solar irradiance variations based on the evolution of thesolar surface magnetic field, which reproduces solar total andspectral irradiance changes on time-scales from days to years.This model is extended to shorter time scales down to an hour(the shortest time scale on which MDI magnetograms are avail-able for an uninterrupted interval of multiple months). Finallywe combine irradiance variations caused by solar magnetic fieldchanges with those caused by granulation.
2. Sources of irradiance variations
We first analyze the observed irradiance variations on time scalesof minutes to a month. For this we mainly employ data obtainedby the Variability of Irradiance and Gravity Oscillations instru-ment (VIRGO; Fr¨ohlich et al. 1995, 1997) on SoHO. These datahave high relative accuracy and are recorded at a 1-minute ca-dence. They are thus best suited for our purpose. We considerthe total irradiance (version tsi d v4 90) as well as the measure-ments in the 3 VIRGO spectral channels: red, green and blue
Seleznyov et al.: Modelling of solar irradiance variability
Fig. 1.
Fourier (grey line) and global wavelet (black dotted line)power spectra (in ppm / µ Hz) of the VIRGO data set for theyear 2002 sampled at a 1 minute cadence. Black solid line showsthe global wavelet spectrum of the SORCE TIM data for the year2003 sampled every 6 hours.centered at 862 nm, 500 nm and 402 nm, respectively (versionspma level2 d 2002; Fr¨ohlich 2003). Detrended data are used,in order to remove the large trends introduced by degradation ofthe photometers in the color channels. This mainly affects verylow frequencies, which are not of interest here. We also fill inthe numerous gaps in the 1-minute sampled data by linearly in-terpolating across them. Most gaps are only a few minutes longand the interpolation should influence the results mainly at thehighest frequencies.We also consider data from the Total Irradiance Monitor(TIM; Kopp & Lawrence 2005; Kopp et al. 2005) onboard theSolar Radiation and Climate Experiment (SORCE; Woods et al.2000; Rottman 2005) satellite. SORCE data for the year 2003sampled each 6 hours were taken from the LASP (Laboratoryfor Atmospheric and Space Physics, Boulder, Colorado) DataProduct web page: http://lasp.colorado.edu/sorce/tsi data.html.Following the procedure for the VIRGO data, gaps were filledin using linear interpolation.We have applied Fourier and Morlet wavelet transforms tothe data. Both gave essentially the same results (see Fig. 1), ex-cept that the global wavelet power spectrum shows smaller fluc-tuations due to the smoothing introduced by Morlet wavelets.The peak in the power at about 5 minutes is due to low degree p -modes, eigen oscillations of the Sun. VIRGO data sampled everyminute or every hour (not shown in the figure) display enhancedpower at frequencies above 5 µ Hz relative to SORCE/TIM data.This enhancement of power in the VIRGO data is due to noisein the electronic calibration. This noise is roughly constant at alevel of 50 ppm at low frequency, but drops at higher frequen-cies, having its 3-db point at around 10 µ Hz, which makes itappear as a bump in the solar spectrum (C. Fr¨ohlich 2008, priv.comm.). Differences between VIRGO and TIM data at frequen-cies below 1 µ Hz are due to the different times they refer to(2002 for VIRGO and 2003 for TIM).In order to estimate the time scale at which the relative con-tribution of magnetic field evolution (which is solar cycle phasedependent) and of convection (which is almost independent) be-comes equal, we have analyzed VIRGO records for quiet (1996–1997) and active (1999–2000) Sun periods individually. For the
Fig. 2.
Ratio of wavelet global power spectra of activity maxi-mum to those at activity minimum for the TSI (solid line) andthree VIRGO channels: red (dot-dashed line), green (dashed)and blue (dotted).component related to active region magnetic fields we expectstronger variations at activity maximum than at minimum.The ratio of the wavelet power spectrum of the 1999–2000period to that of 1996–1997 is shown in Fig. 2. The ratio exceedsunity at periods longer than a day. Obviously at these periodsmagnetic fields dominate the variability. At shorter periods, theratio is essentially 1 in the total irradiance. Values below 1 areseen in the 3 color channels. They may be an instrumental ar-tifact, although the reason has not been identified (C. Fr¨ohlich2003, priv. comm.). Fr¨ohlich & Lean (2004) employed the databetween 2000.8 and 2002.8 for the active period to arrive at avalue somewhat higher than 1. Since the Sun was more activeduring this latter period than during 1999–2000, this may partlyexplain the higher value. They have also apparently used a laterversion of the SPM data. From this diagram alone it is not pos-sible to say whether there is a non-magnetic source of the ir-radiance variations above 10 µ Hz, e.g. convection, or a mag-netic source, e.g. the network, which does not change stronglyin strength over the solar cycle (Harvey 1994). At frequenciesbetween approximately 10 and 100 µ Hz (which are of particularinterest for planetary transits and g-mode search) both magneticfield and convection may contribute to the power.
3. Short term variability: Granulation model
To extrapolate from the Sun to other stars it is necessary to con-struct models that are easily scalable to other stars. Here weconcentrate on convection. Of the main scales of solar convec-tion, there is no evidence that the larger scales (meso- and su-pergranulation) show any intrinsic brightness contrast after thecontribution from magnetic fields is eliminated. Mesogranularstructure is best visible when following ‘corks’ over an houror two (November & Simon 1988; November 1989) or whenconsidering the spatial distribution of, e.g., exploding gran-ules (Roudier & Muller 1987; Ploner et al. 2000). There is noevidence that they show any intrinsic brightness contrast (cf.Straus & Bonaccini 1997; Hathaway et al. 2000; Rieutord et al.2000; Roudier et al. 2009). Supergranulation is mainly evidentin intensity images that sample wavelengths at which small-scale eleznyov et al.: Modelling of solar irradiance variability 3 a b Fig. 3.
Normalized distributions: a) granule lifetimes; b) granulesizes. Plotted is a snapshot at a moment of the evolution. Dashedlines give observed distributions, solid lines the distributions inthe ‘standard’ model.magnetic elements are bright (e.g., Solanki 1993). Rast (2003)has set tight limits on the visible contrast between supergranulecell interior and boundaries if the magnetic features are maskedout. If correct, any significant contribution to irradiance varia-tions produced by a supergranule must come from the evolutionof the magnetic field at its boundary. We therefore concentratehere on modelling the granulation and deal separately with themagnetic field (Sect. 4).To model the influence of the roughly granules on the so-lar disk, we use a simple parameterized description of individualgranules and a statistical approach to the evolution of the wholeensemble. Each granule is treated as a brightness structure, witha time-dependent bright part (granule) that is assigned a bright-ness value and area and a dark part (intergranular lane), whosebrightness is fixed and does not change with time. The granuledistribution and evolution are determined by the following freeparameters: the granule expansion or contraction rate, the gran-ule lifetime, rate of brightness increase or decrease, ratio of thesplitting granules to ones that dissolve.The key features of our model are: 1) The main birthand death mechanisms of granules are fragmentation (birthand death) and emergence from (birth), or dissolution into(death) the background (e.g., Mehltretter 1978; Kawaguchi1980; Hirzberger et al. 1999). 2) The brightness value of eachgranule is assigned randomly within a given range. 3) The life-time distribution of all granules follows an exponential law(Fig. 3a). For the models designated as the ‘standard’ model the lifetime distribution follows the one derived empirically byHirzberger et al. (1999) with the decay time of τ = 4 . min-utes. 4) The initial distribution of granule sizes in the standardmodel is described by the best fit exponential function to the ob-served distribution published by Roudier & Muller (1987). Theobserved distribution is represented by the dashed line in Fig. 3b.While granules evolve their size changes such that largergranules expand and smaller ones shrink. Due to the excess pres-sure large granules build up in the photosphere relative to theirneighbors (e.g., Ploner et al. 1999). At the end of its life, depend-ing on its size, each granule can either split into two parts, thusforming 2 new granules, or dissolve into the background. Theend of a granule’s life is reached either when the preassignedtime (lifetime) has run out or when its size reaches the limits ofthe size distribution. The rate at which the granule’s size changesgrows linearly from 0 in the middle of the size distribution to themaximum of 0.1 arcsecond per minute at the lowest and high-est diameter distribution edges. This effectively means that thesmaller the granule the faster it is squeezed out of existence byits neighbors, and the bigger the granule the faster it reaches theupper size limit imposed in this model and splits into two parts.The individual areas of the child granules lie randomly between1/4 and 3/4 of the area of the parent granule.We attempt to reduce sudden jumps in the brightness at thetime of death or birth of a granule (which produce artificial high-frequency power). Thus the brightness of granules born out ofthe splitting of a larger granule slightly increases, in order tocompensate for the increased area of the intergranular network,which after splitting surrounds 2 granules instead of one beforesplitting. Small granules that are close to disappearing or havejust been born have brightness levels close to the background in-tergranular lanes. Such transitions are linear and limited in time:brightness drops or rises are completed in 3 minutes. The to-tal number of granules is kept constant by maintaining a bal-ance between appearing and dying granules. Also, the size dis-tribution is roughly maintained by allowing fresh granules withthe appropriate size distribution to emerge from the intergran-ular lanes. In Fig. 3b we plot the empirical size distribution ofgranules (dashed curve), which is very close to the initial sizedistribution. The solid line represents the size distribution of thesynthetic granules at a typical later time. The output irradianceof the model is normalized by the total area of the sun at eachtime step.In order to check the importance of model parameters,we have varied each of them individually, while keeping therest fixed. The results of this parameter study are presented inSect. 3.2. Two other parameters, the total number of granules, N tot , and the average brightness contrast between granules andintergranular lanes move the irradiance power spectrum up ordown, but do not influence its slope. They are therefore not dis-cussed further here. Power spectra from a parameter study carried out with our modelare shown in Figs. 4 and 5. Except for the parameter explic-itly mentioned in each case, all others remain unchanged at theirstandard values: mean granule lifetime is approximately 5 min-utes, mean granule diameter is approximately 0.9 arcseconds.At low frequencies all model power spectra are relatively in-dependent of frequency (but with increasing fluctuations towardslower frequency due to the fewer periods of this length sampledby the simulation). Above a certain frequency the power dropsapproximately as a power law. The flattening of the power spec-
Seleznyov et al.: Modelling of solar irradiance variability
Fig. 4.
Wavelet power spectra (in ppm / µ Hz) for differentgranule lifetimes. Solid line shows the result based on the solarlifetime distribution with a mean lifetime of ∼ ∼ ∼ Fig. 5.
Wavelet power spectra (in ppm / µ Hz) for differentgranule sizes (solid line: maximum diameter of a granule is 3arcsec, dotted line: maximum diameter is 9 arcsec, dashed line:maximum diameter 2 arcsec).trum at higher frequencies is due to aliasing introduced by thefact that granules evolve also on periods shorter than the em-ployed time step (1 minute); imposed by the typically 6 monthssolar time that a model has to be run in order to obtain reliableresults also at lower frequencies.The curves in Fig. 4 refer to different average granule life-times. As expected, the frequency at which the power startsto decrease (i.e. the ‘knee’ in the wavelet power spectrum) in-creases with decreasing lifetime of granules. The correspondingperiod roughly doubles as the average granule lifetime is dou-bled, so that this parameter is potentially a diagnostic of stellargranule lifetimes. Note, however, that the period of the ‘knee’ isroughly 10 times longer than the average granule lifetime.In Fig. 5 the diameter of granules has been varied. The num-ber of granules is kept constant, so that the total area coveredis larger in the case of larger granules (corresponding to a big-ger star). Obviously small granules produce less noise than big ones with a shallower slope of the power below several hours.This is caused by the fact that the ratio of the granule area tothat of the surrounding intergranular lanes is higher for big gran-ules; the ratio grows with granular diameter because the widthof the intergranular lanes is fixed. Hence the ratio grows almostlike the ratio of area to circumference, i.e. nearly linearly withgranule diameter. Changing the width of the intergranular laneswhile keeping the size distribution fixed gives basically the sameresult.Note that changing a single parameter can lead to subtle in-direct changes in the properties of the ensemble of granules andhence of the wavelet power spectrum. As an example, a changein the size distribution of granules also leads to a change oftheir effective lifetime distribution. Thus for a very small av-erage granule size more granules die early by reaching the lim-its of the size distribution. For a star of constant size the num-ber of granules is reduced if the average granule size is bigger.Therefore the curves in Fig. 5 would lie even further apart for afixed stellar surface area. The dotted line would lie higher by afactor of approximately √ , while the dashed line would movedown by the same factor. We also found that splitting granules onthe stellar surface produce higher levels of noise than dissolvinggranules. This is partly due to the fact that splitting granules areon average bigger, so that a similar picture emerges as shown inFig. 5 (with some differences, since the shape of the lifetime andsize distributions change significantly with the ratio of splittingto dissolving granules).
4. Magnetic field contribution to irradiancevariations
Here we employ the SATIRE-S (Spectral And Total IrradianceREconstruction for the Satellite era; Fligge et al. 2000;Krivova et al. 2003; Solanki et al. 2005; Krivova et al. 2011)model, which is based on the assumption that all irradiancechanges on timescales of days to years are entirely due to theevolution of the magnetic flux on the solar surface and includesfour components of the solar photosphere: quiet Sun (solar sur-face free of magnetic fields), umbra and penumbra of sunspots,as well as bright magnetic features forming faculae and the net-work (described as a single component). Calculation of solar ir-radiance requires two input data sets. The first one is the inten-sities of each atmospheric component as a function of the wave-length and of the angle between the line of sight and the nor-mal to the solar surface (i.e. the center-to-limb variation). Thesetime-independent spectra are taken from Unruh et al. (1999).The second data set is composed of the maps describing thedistribution of the magnetic features (umbrae, penumbrae, plageand faculae) on the solar surface at a given time. They are pro-duced from magnetograms and continuum images (see Krivovaet al. 2003 for details) recorded by MDI on board SOHO(Scherrer et al. 1995). The changing area coverage and distribu-tion of the different components introduces the time variabilityof the brightness.Usually, MDI does not record continuum images and full-disk magnetograms with the necessary low noise level (5-minintegration, corresponding to a noise level of 9 G) at the cadencerequired for the present investigation. However, we found a 2.2-month period from 15th of March to 19th of May 1999 whenMDI magnetograms were recorded at a rate of at least one 5-min sequence every half an hour practically without interrup-tions. Unfortunately, the continuum images, required to deter-mine umbral and penumbral area, were recorded with a cadence eleznyov et al.: Modelling of solar irradiance variability 5
Fig. 6.
Total solar irradiance observed by VIRGO betweenMarch and May 1999 (upper black curve) and reconstructed us-ing MDI data (lower black curve). Also plotted (in gray) is thecombination of magnetic reconstruction with granulation model.The sampling rate of the VIRGO data is 1 minute, while themagnetic reconstruction alone is sampled each 30 minutes. Themagnetic reconstruction and the combined reconstruction plusgranulation model are shifted down by 500 ppm to facilitate thevisual comparison.of only 96 minutes over the same period of time. Therefore, ev-ery continuum image is used for 3 points in time by rotating itto the times of the corresponding magnetograms. Since sunspotscan evolve over this interval, the obtained power at periods of afew hours is probably somewhat underestimated.SATIRE-S has a single free parameter, B sat , which entersin how the magnetogram signal in facular and network regionsis converted into brightness. It is described in detail by, e.g.,Fligge et al. (2000); Krivova et al. (2003); Wenzler et al. (2006).Here we adopt the value of B sat =280 G following Krivova et al.(2003).The time series produced in this manner is plotted in Fig. 6(lower black curve). Compared with the irradiance fluctuationsmeasured by VIRGO (top curve in Fig. 6) the irradiance varia-tions induced by the magnetic field follow the longer term vari-ations very well, but are too smooth, i.e. they obviously lackpower at very short periods. Some artifacts (small brighteningspikes) are seen at selected times between days 5 and 11. Theyare due to artifacts in some of the images recorded on thesedays. The power spectrum resulting from this reconstruction isrepresented by the dotted line in Fig. 7 (between about 1 and250 µ Hz).
5. Combined short and long time scale variability
Now we have 2 parts of the solar irradiance — one is recon-structed using only the magnetic activity of the Sun, the other isbased exclusively on the convective ‘noise’ of the solar surface,i.e. caused only by granulation. In order to compare modelledvariability and observed one, we combine both magnetic andgranulation parts. The granulation model used here is the ‘stan-dard’ model with granule parameters based on observed solarvalues (see Sect. 3.1 for a description). It corresponds to the solidcurves in Figs. 4 and 5. Both models were run for the same lengthof time here, namely 2.2 months. To combine magnetic recon-structions with granulation model results we re-sample magnetic
Fig. 7.
Wavelet power spectra (in ppm / µ Hz). Thick solidline: VIRGO data set (same as dotted line in Fig. 1). Thin solidline: power spectrum of the granulation model, dotted line: mag-netic reconstruction with 30 minutes sampling, dashed line in theleft part of the plot: daily sampled magnetic reconstruction.reconstructions from 30 minutes down to a 1 minute samplingrate. As seen in Fig. 6, the combined model (gray curve) followsvery closely the observed one.Power spectra of the modelled and observed irradiance al-low a more detailed comparison. Figure 7 shows the waveletpower spectra of the observed data (thick solid line), of theirradiance produced by the granulation model (high frequen-cies; thin solid line) and the magnetic field model (frequencies ≈ − µ Hz; dotted line). Since half-hourly sampled mag-netograms are available only for 2.2 months, while the diagramcovers 4 months (and VIRGO data allow even longer periods tobe considered), we have also employed the reconstruction car-ried out by Krivova et al. (2003) which is sampled at a cadenceof 1 day and represented by the dashed line in the left part ofFig. 7. The low frequency part, caused exclusively by the mag-netic activity, is in good agreement with the observed spectrum.The high frequency part obtained using the solar surface granu-lation model (now run for a larger number of granules) recreatesreal power rather well at frequencies above 100 µ Hz and gives anegligible contribution to the combined irradiance at frequenciesbelow 10 µ Hz. Figure 7 confirms that the solar total irradiancenoise background is equally determined by convection (granula-tion) and magnetism at periods of 10 hours (cf. Fig. 2).In order to compare the observed and modelled power spec-tra in more detail, we combine all 3 components of the model.Figure 8 shows wavelet power spectra of the observed VIRGO(thin solid line) and SORCE data (thick solid line) and of thereconstructed irradiance (dotted) produced by combining thevarious reconstructions plotted in Fig. 7. SORCE data are forthe year 2003 and sampled every 6 hours, while the employedVIRGO data are sampled each minute and refer to 2002. Thecomparison with the SORCE spectrum confirms that the VIRGOpower enhancement in the band from 7 to 30 µ Hz is an arti-fact (C. Fr¨ohlich, priv. comm.). The different years to which theSORCE TIM and VIRGO data refer to should not affect this re-sult, since variability at 10–100 µ Hz is hardly affected by solarcycle phase (see Fig. 2). Note, however, that the modelled vari-ability lies below both VIRGO and SORCE TIM measurementsin the frequency range 8–70 µ Hz.
Seleznyov et al.: Modelling of solar irradiance variability
Fig. 8.
Wavelet power spectra (in ppm / µ Hz). Thin solidline: VIRGO data set. Dotted line: power spectrum of the com-bined magnetic reconstruction and granulation model. Thicksolid line: SORCE wavelet power spectrum.
6. Conclusions
We have analyzed solar irradiance variations on time scales be-tween minutes and months. Whereas on time scales of a dayand longer, the main mechanism of variations is the evolutionof the solar magnetic field, on time scales shorter than roughlyhalf a day granular convection becomes dominant. The crossoverbetween magnetic and convective signatures coincides with thefrequency band of most interest for planetary transit observa-tions.If we neglect the 5-min band, then results of combined mod-elling of irradiance variations due to granular convection andsurface magnetism suggest that we are able to reproduce solarirradiance variability using only data of magnetic activity on thesurface of the Sun and ‘noise’ produced by granulation, exceptfor periods between 5 and 30 hours. In this range of periods thesolar variability is underestimated by the model by up to a fac-tor of 5 when comparing with VIRGO TSI and up to a factorof 2 when comparing with SORCE TIM. Due to the influenceof electronic noise on the VIRGO data, we believe the latterratio is the more reliable one. The discrepancy may have oneof the following causes: 1. The lack of larger scales of convec-tion, such as supergranulation (and possibly mesogranulation)in the model. 2. Insufficient sensitivity of MDI magnetogramsto weak fields (i.e. to quiet Sun fields) and therefore also totheir variability. 3. Insufficient spatial resolution of the magne-tograms, so that flux with mixed polarities at small spatial scales,as is typical of the quiet Sun, can be missed (Krivova & Solanki2004). 4. The breakdown of one of the assumptions underlyingthe magnetic field-based reconstructions. For example, the sim-plifying assumption that all magnetic features at a certain limbdistance and a given magnetic flux have the same brightness isexpected to break down at some level. 5. The high frequencycomponent of the evolution of sunspots and pores is inaccuratelyrepresented by the interpolation between the continuum imagesseparated by 96 minutes. The Nyquist frequency of this data setlies at 45 µ Hz, making this explanation less likely. Note that thetime-scale on which the quiet Sun network flux is replaced byephemeral regions, 14 hours according to Hagenaar (2001), fallsin the critical time range and supports causes 2 and 3 above. The present investigation also provides an evaluation of thediagnostic potential of power spectra of radiative flux time se-ries for determining the properties of stellargranulation. Granulecontrasts, the number of granules, granule lifetimes and diame-ters, as well as contrast and thickness of intergranular lanes areimportant factors determining the amplitude and the shape ofthe power spectrum at periods shorter than several hours. In par-ticular, power spectra, such as those that can be obtained fromCOROT (Baglin et al. 2002) and Kepler (Borucki et al. 2003)data, allow granule lifetimes to be determined in a relativelyunique manner. This is of some interest, since traditional tech-niques for diagnosing stellar granulation, such as line bisectors,do not provide any information on granule lifetimes. Note thatthis diagnostic is not affected by any uncertainties in the modelbelow 100 µ Hz.Important next steps are to identify the source of the miss-ing power around periods of 10–20 hours and to extend such ananalysis to other stars with different effective temperatures andgravities, as well as with different rotation rates and magneticactivity levels. Magnetograms and continuum images obtainedat high cadence by the HMI instrument on SDO could play aprominent role in the former exercise.
Acknowledgements.
We thank C. Fr¨ohlich for providing data and clarificationsregarding the SoHO/VIRGO instrument. We thank the SoHO/MDI team for pro-viding access to magnetograms and continuum images. SoHO is a project ofinternational cooperation between ESA and NASA. Also we thank the SORCEteam and in particular G. Kopp for providing data for comparison. This workwas supported by the
Deutsche Forschungsgemeinschaft, DFG project num-ber SO 711/1-1/2 and by WCU grant (No. R31-10016) funded by the KoreanMinistry of Education, Science and Technology.
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