quiet sun contribution to variations in the total solar irradiance

Solar Physics (2006) 235: 369–386

DOI: 10.1007/s11207-006-0070-0 C© Springer 2006

QUIET SUN CONTRIBUTION TO VARIATIONS IN THE TOTAL SOLARIRRADIANCE

GEORGE L. WITHBROEGeorge Mason University, Fairfax, Virginia, U.S.A.

(e-mail: [email protected])

(Received 19 September 2005; accepted 2 February 2006)

Abstract. An analysis of spatially-resolved measurements of the intensity of the photospheric con-

tinuum by the Michelson Doppler Imager (MDI) on the SOHO spacecraft indicates that these data can

be used to study variations of the Total Solar Irradiance (TSI). Since the techniques employed depend

upon ratios of intensities measured by MDI, they are independent of the absolute photometric calibra-

tion of the instrument. The results suggest that, while it is possible to account for short-term (weeks

to months) variation in TSI by variations in the irradiance contributions of regions with enhanced

magnetic fields (larger than ten G as measured by MDI), the longer-term variations are influenced

significantly by variations in the brightness of the quiet Sun, defined here as regions with magnetic

field magnitudes smaller than ten G. The latter regions cover a substantial fraction of the solar surface,

ranging from approximately 90% of the Sun near solar minimum to 70% near solar maximum. The

results provide evidence that a substantial fraction, 50% or more, of the longer term (≥one year)

variation in TSI is due to changes in the brightness of the quiet Sun.

1. Introduction

This paper discusses use of data from the Michelson Doppler Imager (MDI) on theSOHO spacecraft to study sources of the variation of the total solar irradiance (TSI).These variations appear to be due primarily to the competing effects of sunspots,faculae, and bright network that contribute increases (faculae and bright network)or decreases (sunspots) in total solar irradiance; see review by Frohlich and Lean(2004). This hypothesis suggests that the approximately 0.1% increase of TSI at so-lar maximum compared to the value at solar minimum results from the net positivecontribution of faculae and bright network at solar maximum being larger by 0.1%on average than the net negative contribution of sunspots. This hypothesis is sup-ported by results of several types of analyses. Models of TSI variations constructedfrom regression analyses of sunspot indices and facular indices applied to measure-ments of TSI provide good fits to TSI data (e.g., Frohlich and Lean, 2004). Whitelight images can provide good information on spot indices and their contributionsto decreases in TSI. However, determination of reliable facular indices from whitelight images is difficult due to the low contrast of the faculae and bright networkcompared to their surroundings. Facular indices are usually obtained from proxydata. Facular proxies include indices obtained from measurements of the intensity

370 GEORGE L. WITHBROE

(flux) in the core of the Ca K line, the core/wing intensity ratio of the Mg II line,and measurements of the photospheric magnetic field. Modeling and/or regressionanalyses are required to use these proxies to infer the contributions of these regionsto variations of TSI (e.g., Foukal and Lean, 1990; Chapman, Cookson, and Dobias,1996; Fligge et al., 1998, 2000; Lean et al., 1998; Fontenla et al., 1999; Fligge,Solanki, and Unruh, 2000; Foukal, 2002; Krivova et al., 2003; Solanki, Krivova, andWenzler, 2005). The success of proxy methods in accounting for measured changesin solar irradiance variations on time scales of days to years provides strong sup-port for the hypothesis that these irradiance variations are caused by changes in theamount and distribution of magnetic flux on the solar surface (cf. Krivova et al.,2003; Frohlich and Lean, 2004; Solanki et al., 2005). However, a disadvantage ofusing proxies is that they provide indirect determinations of the contribution offaculae and bright network to the total solar irradiance. This has posed difficultiesin determining quantitatively the relative importance of faculae and the network ingenerating solar cycle irradiance variations. For example, Walton, Preminger, andChapman (2003) concluded that faculae are responsible for 80% of the solar cyclevariation in TSI, while Ermolli, Berrille, and Florio (2003) estimated that duringthe ascending phase of the solar cycle (years 1996 – 2002) the network contributed40 – 50% of the solar cycle variation in TSI.

What has been lacking is a direct comparison of long-term measurements ofthe spatial distribution and intensity of solar photospheric radiation with measure-ments of TSI. The purpose of the present study is to address this lack through useof spatially-resolved measurements of variations of photospheric intensities andmagnetic fields over nearly a solar cycle that have been acquired using the MDIinstrument. MDI images the Sun on a 1024 × 1024 CCD camera through a seriesof narrow spectral filters (see Scherrer et al., 1995). The full disk images havefour arcsec resolution. Although the primary purpose of the instrument is probingthe solar interior using measurements of solar oscillations, MDI data can be used toaddress a variety of other problems. In addition to the extensive data acquired forstudy of solar oscillations, MDI obtains daily magnetograms and images of the con-tinuum intensity near the Ni I 676.8 nm spectral line. These data have been used bythe MDI team to construct synoptic maps of the continuum intensity and magneticfield from Carrington Rotation 1909 to the present. Figure 1 illustrates synopticmaps of intensity and magnetic field for Carrington Rotation 1911 ( June 28, 1996to July 25, 1996) near solar minimum and Carrington Rotation 1970 (November23, 2000 to December 21, 2000) near solar maximum. The daily images and syn-optic maps are potentially a rich source of information about the contributions ofdifferent types of solar regions to variations in the total solar irradiance. Of par-ticular interest in the present study is the contribution of the background quietSun.

Three types of MDI data were employed in this study: synoptic maps of themagnetic field, synoptic maps of the continuum intensity at 676.8 nm, and selectedindividual full disk images or filtergrams of the continuum intensity.

QUIET SUN CONTRIBUTION TO VARIATIONS IN THE TOTAL SOLAR IRRADIANCE 371

Figure 1. Synoptic maps of the continuum intensity at 676.8 nm (upper map for each Carring-ton Rotation) and the photospheric magnetic field (lower map for each Carrington Rotation) for

Carrington Rotations 1911 (near solar minimum) and 1970 (near solar maximum). The vertical axis

is sine latitude, the horizontal axis is Carrington longitude.

The magnetic fields in the published synoptic maps have projection correctionsfor the inclination of the solar equator (B angle) and longitude assuming that MDImakes line-of-sight measurements of a radial magnetic field; however no correctionwas made for the heliographic latitude projection. For this investigation we includeda correction for the latter so that the magnetic field magnitudes used here areassumed to represent the radial field. For the total solar irradiance, we used dailymeasurements in a total solar irradiance composite (Frohlich, 2000, 2006) that isbased primarily on data from the VIRGO instrument (Frohlich et al., 1995) onSOHO for the time interval studied in the present paper, with some small gapsfilled in by measurements from ACRIM II (Willson, 1994). VIRGO and ACRIM IImeasure TSI with high photometric precision. The data used in the analysis were


obtained from the MDI data library at http://soi.stanford.edu/ and theVIRGO data library at http://www.ias.u-psud.fr/virgo/.

2. A Comparison of MDI Irradiances and TSI Values

The objective of the first stage of the analysis was to determine whether or not itis possible to use MDI data to obtain irradiances at 676.8 nm that have sufficientlylow noise, equal to or less than a few times 0.01% over a month or more, toenable pursuit of the objectives of the study. In principle it is simple to obtainirradiances or fluxes from full disk intensity images or filtergrams. The difficulty inpractice is obtaining the required accuracy for an instrument (MDI) not designedspecifically for measuring irradiances with the required high photometric precision.Of particular importance is obtaining a reliable time-dependent flat field correctionfor MDI’s 1020 × 1024 detector.

Figure 2 contains plots of irradiances derived from MDI data using four differentmethods along with corresponding values of TSI measured by the VIRGO instru-ment. The individual curves have been shifted vertically to separate them. The MDIirradiances were determined from filtergrams acquired at approximately one weekintervals during Carrington Rotations 1973 – 1982 near solar maximum. Each ofthe four methods was used to determine the mean intensity 〈I〉 from each filtergram.These mean intensities, which are proportional to the irradiance at 687.8 nm, were

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Figure 2. Irradiances determined at intervals of approximately one week intervals during Carring-

ton Rotations 1973 – 1982 near solar maximum. The uppermost curve is the total solar irradiance

from VIRGO and the other four curves are irradiances derived from MDI continuum data by four

methods (see text). The long-dash curve, dot-dash curve, dotted curve, and heavy-solid-line curve are

respectively for irradiances determined using Methods 1, 2, 3, and 4 (see text).


then compared with values of TSI interpolated to the times that the filtergrams wereacquired. For two of the techniques (results plotted as dashed and dash-dot lines)the amplitude of the noise is much larger than the amplitude of the variations inTSI measured by VIRGO (uppermost curve). The third technique (dotted line) hasirradiance variations with amplitudes comparable to those measured by VIRGO,but poorly correlated with the latter, while the fourth technique (heavy solid line)has amplitudes and phases that correlate well with those measured by VIRGO.

What accounts for the success or failure of each method? All four methods makeuse of flat-field corrections employed by the MDI team in constructing synopticmaps such as those shown in Figure 1. These flat-field corrections were obtainedfrom http://soi.stanford.edu/magnetic/synoptic/carrot/Ic/. Theflat-field corrections for each Carrington rotation were derived from filtergramsused in constructing the synoptic map for that rotation. For example, in the MDIlibrary one finds a “med.1980.fits” file. The “med” image is the per pixelmedian of the intensities measured for each pixel and corrected for limb-darkeningfrom the set of filtergrams used in constructing the synoptic map for CarringtonRotation 1980. Intensities for individual pixels that lie outside the expected rangeare eliminated prior to the calculation of the median. This eliminates the effect ofsunspots and cosmic ray hits. The “med.1980.fits” array is assumed to providea flat-field correction ( fi ) for each pixel (i) in all of the filtergrams obtained duringCarrington Rotation 1980.

In the present study, the flat-field corrections ( fi ) based on the median intensitieshave been converted into corrections (Fi ) that are scaled to the intensity of the quietSun at each pixel (i), where the quiet Sun is assumed to have the mean intensity ofregions with radial magnetic field strengths less than ten G as measured by MDIafter correction for projection effects. This limit corresponds to having about 90%of the area of the polar regions at solar minimum defined as being representativeof the quiet Sun. An analysis of the synoptic maps indicates that quiet regionsdefined in this manner cover a substantial fraction of the solar surface, from about70% of the surface at solar maximum to 90% at solar minimum. The procedure fordetermining Fi will be given later.

Methods 1 and 2 are based on sums over the solar disk of the intensity fromeach pixel of the image,

Ii = kci

Fi, (1)

where k is a constant, which is assumed to be equal to unity in the initial calculations,ci is the count at pixel i in the filtergram, and Fi is the flat-field correction for pixeli. Method 1 is simply the average intensity 〈I〉 over all non-zero pixels in the image:

〈I 〉 = 1

Ni

∑Ii , (2)

where Ni is the number of non-zero pixels in the filtergram.


Method 2 takes the sum (∑

Ii ), which corresponds to the solar flux measuredby MDI (and which varies with the square of the distance R of SOHO from theSun), and obtains a determination for 〈I 〉 by the expression,

〈I 〉 = K

R2

∑Ii , (3)

where K is a constant that, in the absence of experimental errors, would equal R2/Ni .The primary source of the noise in the irradiances determined by Methods 1 and

2 appears to be the “ragged” edge of the Sun caused by having an image constructedwith finite-sized pixels where some pixels near the limb are filled by the solar diskand others are partially or nearly entirely empty (most of the area of the pixel isbeyond the solar limb).

Methods 3 and 4 eliminate the above source of noise (“ragged” limb) by usingonly those pixels that are fully on the disk. We used pixels whose centers werewithin 0.99 solar radii of the center of the solar disk. Method 3 is identical toMethod 1 except that the summation uses only pixels fully on the disk. Figure 2illustrates that this reduces the noise substantially as shown by the comparison ofthe dotted curve with the dashed and dash-dot curves. However, the variations inirradiance obtained using Method 3 (dotted) do not correlate very well with thosemeasured by VIRGO during the same time period near solar maximum.

We found that the problem is with the flat-field correction ( fi ) which, as dis-cussed above, is based on the median value of various intensities measured at pixeli from the filtergrams acquired during a given Carrington rotation. This techniquefor deriving flat-field corrections appears to be a good method for cross-calibratingthe sensitivity of pixels in the longitudinal (East – West) direction as solar rotationcarries different solar regions across the disk. However, the cross-calibration inthe latitudinal (North – South) direction is not as reliable, because intensities fromthe same regions are not used to cross-calibrate the sensitivities of pixels at dif-ferent latitudes (i.e., pixels at different latitudes that do not sample the same solarregions as they rotate across the disk). For Method 4 we added an additional pole-to-pole flat-field correction ( fp) calculated for each filtergram so that Equation (1)is modified to

Ii = kci

Fi fp

. (4)

The following procedure was used to determine fp. For the filtergram under analysiswe set k and fp equal to unity and corrected the intensity (Ii ) of each pixel for limb-darkening, thereby creating an array with limb-darkening corrected intensities.From the latter array we determined the mean intensity (Iθ ) and its standard deviationfor the pixels in a given latitude band θ ± �θ that are within 0.95 solar radii of thecenter of the disk. Pixels with low intensities due to sunspots were ignored. We thenrecalculated Iθ , after eliminating all pixels with intensities more than two standarddeviations above or below the mean, to eliminate bright faculae and low intensity


regions such as pixels adjacent to sunspots. We then assumed that the set of valuesdefining the variation of Iθ with latitude provides an estimate for the pole-to-poleflat-field corrections fp. That is,

fp(θ ) = Iθ = 1

Nθ

∑ ci

Fi Di=

⟨ci

Fi Di

⟩θ

(5)

where the summation includes only those pixels which have intensities within twostandard deviations of the mean limb-darkening-corrected intensity in the latitudeband, Nθ is the number of those pixels, and Di is the limb darkening calculated forpixel i. By dividing the intensities for each pixel (Ii ) by the value of fp correspondingto the latitude (θ ) of the pixel, we obtain a correction for time-dependent pole-to-pole variations in the sensitivity of the detector. This method of estimating fpassumes that the background quiet Sun for all filtergrams has the same pole-to-polevariation.

We found that the optimum agreement between the MDI irradiances and mea-sured TSI values was obtained when the mean 〈 f −1

p 〉 averaged over all pixels wasunity to better than a few times 0.001%. With the exception of filtergrams fromCarrington Rotations 1926 – 1966, nearly all filtergrams yielded values of 〈 f −1

p 〉equal to unity within this limit. For filtergrams with larger differences from unity,we adjusted the limb darkening correction as discussed below.

For the limb-darkening correction Di for each pixel i we used the same functionutilized by the MDI team in deriving the flat-field corrections fi for the synopticmaps,

Di = 0.42 + μ[0.81 − 0.23μ], (6)

where μ is the cosine of the angle between center of the disk and the center of pixeli and the distance from Sun center to pixel i is calculated under the assumption thatthe limb is at 0.995Rsun where Rsun is the solar radius as observed from SOHO. Useof this “artificial” value for the distance from disk center to the limb yields a betterestimate of the limb darkening and superior “flattening” of the image than the useof Rsun for this distance. For filtergrams with initial values of 〈 f −1

p 〉 with departuresfrom unity of more than a few times 0.001%, we made small adjustments (less thanor equal ±0.001 Rsun) to the assumed distance from disk center to the limb to forcethe value of 〈 f −1

p 〉 to unity within this limit.Figure 3 compares values of MDI irradiances derived by Method 4 with total

solar irradiances TSI (solid blue lines in the upper and lower set of curves) obtainedfrom VIRGO data. (Method 4 uses intensities corrected with the latitudinal flat-field correction fp.) The three sets of curves have been shifted vertically relative toeach other to separate them. The scale of the abscissa is not uniform because theMDI filtergrams employed in the analysis were acquired at irregular intervals oftime, typically at one to two week intervals. There were also several gaps in theMDI data. The MDI irradiances plotted in Figure 3 are 〈I 〉/k (green line in middleset of curves) and 〈I〉 (red line in lower set of curves). In the middle set of curves


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Figure 3. Irradiances determined at irregular intervals of time, typically at one to two week intervals.

The blue lines in the upper and lower sets of curves gives the variation of TSI for the times corre-

sponding to the MDI irradiances. The green solid line in the middle set of curves gives the variation

of 〈I〉/k derived from MDI data via Method 4 and the red curve in the lower set of curves gives the

corresponding variation of 〈I〉. The black dot-dash line in the upper set of curves gives the variation

of the parameter k (see text) obtained from the ratio of the smoothed values of TSI (blue dashedline in middle set of curves) and smoothed values of 〈I〉/k (dark green dotted line in middle set ofcurves). The three sets of curves have been shifted vertically relative to each other to separate them.

For reference, horizontal lines are drawn across the graph at values corresponding to the irradiance

at solar minimum for each set of curves.

the smoothed variations of TSI (dashed blue line) and 〈I〉/k (dark green dotted line)are also plotted. The parameter k (black dot-dash line in the upper set of curves)in Figure 3, was obtained from the ratio of the smoothed values of TSI and 〈I〉/kplotted in the middle set of curves). (The total solar irradiance is proportional to〈I〉, hence the ratio TSI/(〈I〉/k) is proportional to k.)

Figure 3 illustrates several things. First, the good correspondence between theshort-term MDI irradiances (solid green and solid red curves) and the total solarirradiances TSI (solid blue curves) demonstrate that it is possible to obtain relativeirradiances from MDI data that correlate well over time intervals of a several monthsto a year with corresponding values of TSI measured by VIRGO. Second, thereis a long-term (≥year) difference in the variations in TSI and the MDI relativeirradiances 〈I〉/k that suggests that the parameter k varies with time. The smoothedvariation of the MDI relative irradiance 〈I〉/k (green dotted line in middle set of


curves) is nearly constant over the solar cycle as compared to the significantlylarger variation in the smoothed variation of TSI (dashed blue line in middle setof curves). Since the variations in 〈I〉/k are determined by the variations in the netflux of radiation from sunspots and regions with radial magnetic field amplitudeslarger than ten G (as measured by MDI), this is evidence that the primary cause ofthe long-term (≥year) or solar-cycle variation in TSI is not the competing effects ofsunspots and faculae, but instead is due to variations in the intensity of the regionswith magnetic field amplitudes less than ten G as measured by MDI, defined hereas the quiet Sun. As we show below, the parameter k is proportional to the meanintensity of the quiet Sun. Thus our results suggest that short term (<year) variationsin solar irradiance are caused primarily by competing effects of sunspots and faculaewhile a substantial fraction of the longer-term variations is caused by changes inthe radiative output of the quiet Sun, presumably due to changes in the networkover the solar cycle.

3. Variation of the Quiet-Sun Irradiance

The intensities used in constructing Figures 2 and 3 are based on modified values(Fi ), of the original flat-field corrections ( fi ), employed by the MDI team in con-structing the synoptic maps. As indicated earlier, the latter flat-field corrections arebased on median intensities at each pixel (i) obtained from the set of filtergramsemployed in constructing each synoptic map. An analysis of the synoptic mapsprovides evidence that these flat-field corrections are “contaminated” by the effectsof solar activity, such that the values of fi in the activity belts are systematicallylarger than the equatorial and polar values during times of higher solar activity.

This is illustrated in Figure 4 where mean values of quiet-Sun intensities, Iq(θ ),at a given latitude (θ ) are plotted as a function of sin θ for two Carrington Rotations1910 (near solar minimum) and 1970 (near solar maximum). These intensitieswere determined from the synoptic maps, which utilize the original MDI flat-fieldcorrections ( fi ) rather than the modified values (Fi ) used elsewhere in our analysis.The parameter sin θ is positive in the northern hemisphere and negative in thesouthern hemisphere. The intensities Iq(θ ) were calculated from the expression

I q(θ ) = 1

Nθ

∑I qi =

⟨cqi

fi Di

⟩θ

, (7)

where Iqi = cqi /fi Di is the relative quiet-Sun intensity measured at pixel (i) in asynoptic map, cqi is the quiet-Sun count at that pixel, fi is the corresponding flat-field correction used in constructing the synoptic maps, Di is the limb darkening(see Equation (6)), the summation is over all pixels in a band of latitudes corre-sponding to sin θ ± � sin θ , and Nθ is the number of pixels in the band. A pixelis assumed to be from the quiet Sun if the magnitude of the radial magnetic fieldassociated with that pixel is less than ten G as obtained from the magnetic synoptic


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)

Figure 4. Values of the mean limb-darkening corrected quiet-Sun intensities (Iq(θ )) at latitude θ

plotted as a function of sin θ for Carrington Rotations 1910 (heavy line) and 1970 (light line).

map for the same Carrington Rotation as the intensity synoptic map. As might beexpected, the values of Iq(θ ) are nearly independent of latitude for data from nearsolar minimum (heavy line corresponding to Carrington Rotation 1910) while nearsolar maximum (light line corresponding to Carrington Rotation 1970) there is astrong dependence on latitude. (Note, that in a given latitude band, the ratio cqi /fi Di

is expected to be constant because (1) this ratio is independent of the instrumen-tal sensitivity at that pixel and (2) the procedure used to derive the values of fiat a given latitude yields values scaled by the same solar activity factor for thatlatitude.)

If one assumes that the absolute intensities of all quiet regions (i.e., regions withradial magnetic fields magnitudes less than ten G) are the same, then the variationof Iq(θ ) illustrated in Figure 4 is caused by a solar cycle dependent variation in theflat-field correction fi , where fi increases with increasing solar activity, especiallyat latitudes with the most activity. That is, the increased number of brighter pixels,due for example to faculae, shifts the median intensity at a given latitude to largervalues, particularly in the sunspot belts where a larger fraction of the area is coveredwith faculae.

We now define the flat-field correction used in Equation (1) as

Fi = cqi

Di(8)

where cqi is the estimated quiet-Sun count at pixel i which can be obtained fromEquation (7) via the relation,

cqi = fi Di

⟨cqi

fi Di

⟩θ

= fi Di I q(θ ), (9)


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Figure 5. The mean irradiance (〈Iq〉) of the quiet Sun (regions with magnetic fields less than ten G)

plotted as a function of Carrington Rotation. The heavy and light solid lines give, respectively, the

variation of the smoothed and unsmoothed determinations of 〈Iq〉 while the dotted curve gives the

corresponding of variation of the smoothed values of TSI. For reference, a horizontal line has been

drawn across the graph corresponding to a value on the ordinate of 1.0.

where, as discussed above, we have assumed that for pixels at latitude θ the ratiocqi /fi Di is constant and equal to 〈cqi /fi Di 〉θ . Thus we have from Equations (2), (4),(8), and (9):

〈I 〉 = k

Ni

∑ ci

fi fp I q(θ ). (10)

From Equations (9) and (10) we obtain an expression for the mean quiet Sunirradiance,

〈I q〉 = k

Ni

∑ Di

fp

= 0.849k, (11)

where ci in Equation (10) becomes cqi , and the factor 0.849 is determined fromthe data. Hence, the MDI relative irradiances 〈I〉/k plotted in the middle graph inFigure 3 are, except for the constant 0.849, equal to 〈I〉/〈Iq〉.

Since the total solar irradiance (TSI) is proportional to 〈I〉, the mean intensity orirradiance of the quiet Sun (〈I〉) is proportional to the ratio TSI/(〈I〉/〈Iq〉). Figure 5contains a plot of 〈Iq〉 (light solid curve) determined in this manner, a smoothedvalue for 〈Iq〉 (heavy solid curve), and the corresponding smoothed value of TSI(dotted curve), all normalized to unity at solar minimum. This result suggests thata substantial fraction, 50% or more on average, of the variation of the total solarirradiance over the solar cycle is due to a variation in the intensity of the quiet Sun,defined here as solar regions with radial magnetic field magnitudes less than ten G.

If the above interpretation were correct, one would expect that the mean intensity(〈Ip〉) of polar regions, which are dominated by quiet solar regions, would vary in amanner similar to 〈Iq〉, or conversely that 〈Ip〉/〈Iq〉 would be constant with time. We


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Figure 6. The smoothed (heavy solid line) and unsmoothed (light solid line) variations of the ratio

〈Ip〉/〈Iq〉 of the polar and quiet-Sun irradiances along with the smoothed variation of TSI (dotted line)

are plotted as a function of Carrington Rotation. For reference, a horizontal line has been drawn across

the graph corresponding to a value on the ordinate of 1.0.

determined 〈Ip〉/〈Iq〉 by application of Equation (10) to those pixels more than 0.7Rsun from the solar equator and obtained the result in Figure 6. The dashed curvein Figure 6 gives, for comparison, the smoothed variation of TSI. The smoothedvalues of 〈Ip〉/〈Iq〉 (heavy solid line) are consistent with a value of unity given theamount of noise in the unsmoothed determinations of 〈Ip〉/〈Iq〉 (light solid line).This differs from the behavior of 〈Iq〉 illustrated in Figure 5 where smoothed andunsmoothed determinations of the quiet-Sun intensity (〈Iq〉) depart significantlyfrom unity during times of high solar activity.

4. Discussion

We have searched for systematic factors or errors in the analysis that could affectour determination of the solar cycle behavior of the mean intensity of the quietSun. Some irradiances derived from MDI filtergrams contain some unexplainednoise, for which one possible cause is errors in the empirical flat-field correctionsfor latitudinal variations in the sensitivity of the MDI detector for those filtergrams.The importance of including flat-field corrections that include latitudinal effectsis demonstrated in Figure 7 which compares the ratio of irradiances derived byMethod 3, which does not include the pole-to-pole flat field correction ( fp), andirradiances derived by Method 4 which includes the latter correction. The smoothed(heavy solid line) and the noisier unsmoothed (light solid line) are consistent withthe ratio of the two types of irradiances being unity on average near solar minimum(Carrington Rotations < 1926) and during solar maximum and the declining phaseof the solar cycle (Carrington Rotations 1967 – 2026). However, the data between


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Figure 7. The smoothed (heavy solid line) and unsmoothed (light solid line) ratio of irradiances

derived by Methods 3 (which does not include the polar flat-field correction fp) and 4 (which includes

the polar flat-field correction) plotted as a function of Carrington Rotation. The heavy dotted line is

the smoothed variation of TSI and the lower light dotted line gives the variation of the unsmoothed

initial values of the ratio 〈Ip〉/〈Iq〉 (see text). The latter curve has been shifted downward by 0.004

units to separate it from the other curves. For reference, horizontal lines have been drawn across the

graph corresponding to values on the ordinate of 1.0 and 0.996.

Carrington Rotations 1926 and 1966 exhibit a significant departure from unity.The initial determinations of 〈Ip〉/〈Iq〉 also had values that departed systematicallyfrom unity in the same range of Carrington Rotations (dotted line in Figure 7,which has been shifted downward to separate it from the other curves). The initialdeterminations of 〈Ip〉/〈Iq〉 there had values of 〈 f −1

p 〉 that differed from unity bysignificantly more than a few times 0.001% while, as discussed earlier, the finalvalues (e.g., values plotted in Figure 6) were calculated by a procedure that forced〈 f −1

p 〉 to equal unity within a few times 0.001%. Since Method 3 for determiningirradiances does not include a flat-field correction that improves the accuracy of flatfield errors in the latitudinal direction, as is the case for Method 4, Method 3 is moresensitive to the effects of such errors as is demonstrated in Figures 2 and 7. Becauseof the findings shown in Figure 7, it is likely that the MDI irradiances in Figure 3and the quiet-Sun irradiances (〈Iq〉) in Figure 5 are less reliable for CarringtonRotations 1926 – 1966 that those for earlier and later Carrington Rotations.

The results plotted in Figure 7 are also relevant to one of the possible sources ofsystematic error that could vary with the solar cycle. If the magnitude of the pole-to-pole flat-field correction (fp) varied with the solar cycle, this could introducean error in the solar cycle variation of 〈Iq〉. However, as indicated above, with theexception of the period between Carrington Rotations 1926 – 1966, the value of〈 f −1

p 〉 was typically equal to unity to within a few times 0.001%. There was no


significant solar cycle variation. This point is illustrated graphically in Figure 7 bythe near unity values (except for the 1926 – 1966 period) of the smoothed ratio ofirradiances derived by Method 3 (which does not utilize the flat-field correction fp)and irradiances derived by Method 4 (which uses fp). The intervals with unit valuesof the ratio of the two types of irradiance cover times near the last solar minimum(Carrington Rotations 1909 – 1926) and times from solar maximum to near thenext solar minimum (Carrington Rotations 1966 – 2026).

Another possible source of systematic error in the derived quiet-Sun irradiancesis contamination of the quiet regions by bright areas from faculae. The most likelyregions where this could happen are in the active region or sunspot belts. If the valuesof 〈Iq〉 were contaminated by bright faculae in the activity belts, one would expectthe ratio 〈Ip〉/〈Iq〉 to be smaller near solar maximum than near solar minimum.As shown above, the amplitude of the variation in the smoothed values of the ratio〈Ip〉/〈Iq〉 is consistent with 〈Ip〉/〈Iq〉 being constant over the solar cycle, while 〈Iq〉 issignificantly larger at solar maximum than at solar minimum (cf. Figures 5 and 6).

We also explored the effect of placing a tighter constraint on which pixels are de-fined as quiet in the synoptic maps by using only those pixels for which the adjacentpixels up to two pixels away also had magnetic fields of less than ten G. There wasa small difference in the resulting curves used to determine the parameter Iq(θ ) =〈cqi /fi Di 〉θ (see Equation (7) and Figure 4) used to convert the MDI flat-field correc-tions ( fi ), which are relative to the median intensities at each pixel, to values Fi = fiIq(θ ) (see Equations (8) and (9)), which are relative to the quiet-Sun intensity at eachpixel. This results in only a small difference, 0.01%, between the relative values ofthe mean quiet-Sun intensity (〈I q〉) at minimum and maximum, an amount too smallto account for the difference, about 0.07%, between the corresponding differencebetween the values of 〈Iq〉 at maximum and minimum as illustrated in Figure 5.

5. Behavior of the Magnetic Field and Bright Pixels

We used the MDI synoptic maps of the photospheric magnetic fields to determineif there were significant differences in the mean magnetic fields in quiet regionsduring the time interval studied here (Carrington Rotations 1909 – 2026). Thesynoptic magnetic maps were used to determine the mean radial magnetic fieldmagnitudes (average of the absolute values magnetic field strengths) measured foreach Carrington rotation for the entire Sun and for quiet regions as defined earlier(regions with radial magnetic field magnitudes less than ten G). In defining theaverage magnitude of the magnetic field, each pixel is weighted by the fractionalarea of a sphere that it occupies when the synoptic map is mapped onto a sphere so asto avoid over representation of high-latitude pixels. Figure 8 illustrates the results.We see that the smoothed quiet-Sun magnetic field (solid line) is nearly constantover the solar cycle, as compared with the smoothed variation of the magnetic fieldaverage over the entire disk (dotted line). There is a small increase in the former


0

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Carrington Rotation

Mag

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Figure 8. Mean magnetic field strength in quiet regions (solid line) and for the entire disk (dottedline) obtained from the MDI synoptic maps plotted as a function of Carrington Rotation.

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Figure 9. Ratio (compared to solar minimum) of the fraction of quiet regions occupied by areas with

pixels having intensities greater than 1.005〈Iq〉 (heavy solid line), the standard deviation (dashed line)

of 〈Iq〉, and mean magnetic field (dot-dash line) in quiet regions plotted as a function of Carrington

Rotation. For comparison, renormalized values of 〈Iq〉 are also plotted (lower dotted line); renormal-

ized to artificially increase the amplitude of the variations in 〈Iq〉. For reference, horizontal lines have

been drawn across the graph corresponding to values on the ordinate of 1.0 and 0.5.

from about 2.8 G at solar minimum to 3.8 G at solar maximum. The variation ofthe magnetic field from the quiet Sun is illustrated with an expanded vertical scalein Figure 9 (dash-dot line).

What may be changing over the solar cycle is the relative proportion of brightpixels in the quiet regions due to the evolution of the network as the magnetic


elements in active region faculae break up and disperse over the disk (cf. Skumanichet al., 1984). We made a preliminary analysis of data from the synoptic maps andfound evidence for a solar cycle variation in the population of bright pixels, howeverthe variation does not behave as expected. This is illustrated in Figure 9 whichcontains a plot (dashed line) of the smoothed variation of the standard deviation(σ ) of the quiet-Sun intensities used to derive the mean intensity 〈Iq〉 as a functionof time (or Carrington Rotation) and a plot (heavy solid line) of the smoothedvariation in the fraction of the pixels in quiet regions that have intensities largerby 0.5% or more than 〈Iq〉 for the synoptic maps corresponding to the filtergramsused in obtaining the results in Figures 3 and 5 – 7. In Figure 9 the values of σandthe fractional area for the intensities larger than 1.005〈Iq〉 have been normalizedby dividing them by the mean values of these parameters, 0.00567 and 0.1955respectively, near solar minimum. Also plotted is the mean magnetic field (dot-dash line) where the values have been divided by 2.83. For comparison, the lowercurve (dotted line) is a plot of rescaled values of 〈Iq〉 obtained by renormalizingthe smoothed values of 〈Iq〉 illustrated in Figure 5 so as to artificially increase theamplitude of the variations for comparison with the other curves in Figure 9.

Although the number of bright elements in quiet regions increases from min-imum to maximum, as evidenced by the increase in σ (dashed line in Figure 9)and in the fractional area covered by brighter elements (heavy line), there is nocorresponding decline after maximum. Instead, these parameters remain at nearlythe same, or slightly larger, values as at maximum. The mean magnetic field in thequiet regions (dash-dot line) does appear to decline after solar maximum, howeverit does not return to the low values found near solar minimum prior to CarringtonRotation 1925. Hence, the results in Figure 9 are not particularly supportive of thehypothesis that the solar cycle variation in 〈Iq〉 stems primarily from variations inthe population of bright elements in quiet regions. However, it is important to notethat the three upper curves in Figure 9 were derived from the MDI synoptic maps.The synoptic maps are based primarily on intensity filtergrams and magnetogramsacquired near central meridian passage of regions on the solar disk. Hence, they donot take into account the influence of differences in the limb darkening of varioustypes of solar regions, such as faculae and other bright features, which typicallyare limb darkened by smaller amounts than the background quiet Sun (e.g., Ortizet al., 2002; Ortiz, 2005). More study is required to determine which features inquiet solar regions are the source of the solar cycle variation in 〈Iq〉 found here.The network is the most likely candidate given the correlation of solar irradiancevariations with magnetic activity.

6. Summary

This analysis of spatially resolved measurements of the intensity of the photo-spheric continuum at 676.8 nm by the Michelson Doppler Imager (MDI) indicates


that these data can be used to study variations of the total solar irradiance. Since thetechniques employed depend upon ratios of intensities measured by MDI, they areindependent of the absolute photometric calibration of the instrument. The resultssuggest that while it is possible to account for short term (weeks to months) vari-ation in TSI by variations in the irradiance contributions of regions with enhancedmagnetic field (larger than ten G as measured by MDI), the longer term variationsare influenced significantly by variations in the brightness of the quiet Sun, definedhere as regions with magnetic field magnitudes smaller than ten G as measured byMDI. The results provide evidence that a substantial fraction, 50% or more, of thelonger term variation (≥one year) in TSI is due to changes in the brightness of thequiet Sun.

Acknowledgements

We wish to express our appreciation to the MDI team for use of synoptic intensityand magnetic maps and intensity data obtained from their public library of data. Weespecially thank Philip Scherrer and Todd Hoeksema for their helpful commentsand suggestions. We also wish to express our appreciation the VIRGO team for useof their data and acknowledge receipt of the dataset, total solar irradiance compositeversion d41 61 0505, from PMOD/WRC, Davos, Switzerland and use of unpub-lished data from the VIRGO Experiment on the cooperative ESA/NASA MissionSoHO. This work was supported by NASA Cooperative Agreement NCC5-714.

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