WHY IS THE SOLAR CONSTANT NOT A CONSTANT?

The PMOD composite is an accurate measurement record of daily TSI during the last three solar cycles (Fröhlich 2006, 2009). Daily TSI from 1978 November 7 to 2010 September 20 in the PMOD composite is used here to investigate to what extent TSI varies on various timescales. The time series can be downloaded from the Web site.⁷ Figure 1 shows the PMOD composite of daily TSI. The two most striking features of the observed record of the daily TSI (for details, see Fröhlich 2009) are the inter-solar-cycle variations by about 0.1% in phase with the solar activity cycle and sharp dips with a comparable or even greater amplitude typically lasting 7–10 days (Krivova & Solanki 2008). The mean value of the composite is 1365.91 W m⁻² during the time interval considered.

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The EMD is a nonlinear time–frequency analysis method (Huang et al. 1998; Gao et al. 2011). It is an algorithm which decomposes an input signal into a finite set of oscillating functions, namely the so-called intrinsic mode functions (IMFs), which are the intrinsic periodicities of the original signal. These IMFs are extracted from the data themselves, and they are not restricted to have constant phases or amplitudes. Essentially EMD is an empirical algorithm which decomposes a signal, which can be non-stationary and nonlinear, into a finite set of IMFs (Barnhart & Eichinger 2011). These IMFs are defined to be functions which are symmetric about their local mean, and whose number of extrema and zero-crossings are equal or differ at most by one (Huang et al. 1998). These IMFs are extracted from a signal using a process called sifting. The sifting process essentially iteratively removes the local mean from a signal to extract the various cycles present. The sifting process is performed until the signal meets the definition of an IMF (for details, please see Barnhart & Eichinger 2011). Here, the PMOD composite is decomposed into 10 IMFs through the EMD analysis which are shown in Figure 2.

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The code of the wavelet transform analysis, which is provided by Torrence & Compo (1998), is utilized to study periodicity in the first nine IMFs of the PMOD composite. IMF 10 is excluded because it is the secular trend of the composite, and the limited length of the data used gives no period to the trend at present. Figure 3 shows their global wavelet power spectra and the corresponding 95% confidence level. Table 1 gives the periods in the first nine IMFs, which are significant at the 95% confidence level. Also given in the table are the period intervals of these period values.

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The periods 9.8, 14.5, 58.1, and 86.7 days are inferred to be the 1/3-, 1/2-, 2-, and 3 multiple harmonics of the period of about 29 days, which is approximately the solar rotation period. There are only those periods in IMFs 1–5, which are related to the rotation cycle, thus IMFs 1–5 are called the rotation-variation signal of TSI. The sum of IMFs 1–5, which is called here Component I of daily TSI, is shown in Figure 4. Component I is inferred to be mainly caused by magnetic structures, including sunspot groups, due to the following aspects. (1) Short-term changes of TSI on timescales of a few days to weeks are known to be dominated by magnetic structures (Chapman 1987; Solanki et al. 2003). (2) The figure shows that Component I fluctuates with much higher amplitude around the maximum times of the Schwable cycles than around the minimum times, and long-lived solar magnetic structures usually appear around the maximum times of solar cycles. And (3) as the figure displays, sharp dips appear only in this component and around the maximum times of solar cycles, and maximum variation amplitude can even exceed 3 W m⁻². Here, the variation amplitude is given relative to the mean value of the composite. These dips, lasting 7–10 days, are caused by the passage of sunspot groups across the visible disk as the Sun rotates. Toward activity maxima, when the number of sunspots grows considerably, the frequency and depth of the dips increase (Krivova & Solanki 2008). Figure 5 shows daily sunspot area and daily TSI from 2003 September 10 to 2003 November 17. The figure displays the very well-known fact that when large sunspot groups pass across the solar visible disk, a sharp dip appears in the daily TSI with its amplitude decreasing from about 1366 W m⁻² to about 1362 W m⁻². These sharp dips of TSI are caused by the passage of sunspot groups across the visible disk as the Sun rotates.

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The periods 781.1 and 1570 days are considered to be the 2 and 4 multiple harmonics of the period of 390.6 days, respectively, and these three periods show a broad peak in their power spectra. Thus, IMFs 6–8 show periodical annual variations, and they are called the annual-variation signal of TSI.

The periodical annual-variation signal had not been determined by the helioseismic probing of the solar interior (Howe et al. 2000). The one-year periodicity is found in several solar-activity indices, but its origin is doubtful. That is, it is difficult to rule out the possibility that this periodicity is not due to the influence of seasonal effects (Javaraiah et al. 2009). Of course, it must be pointed out that so far there has been no quantitative analysis about the effect of Earth's helio-latitude on the measurement of the Sun, and the origin of the annual periodical signal of the Sun is an open issue. Here, we speculate that IMFs 6–8 are possibly caused by Earth's orbital revolution. However, this needs to be independently confirmed through the analysis of Earth's/spacecraft orbital data along with the TSI time series. The sum of IMFs 6–8, which is called here Component II of the daily TSI, is shown in Figure 4, and almost all variation amplitudes are found to be less than 0.5 W m⁻².

We also calculate the correlation coefficient (cc) between daily TSI of the PMOD composite and daily sunspot number, which is available from the Solar Influences Data Analysis Center's (SIDC) Web site, and cc = 0.4456, which is statistically significant at the 99.9% confidential level. When the annual-variation signal of daily TSI, namely Component II is deduced from the original daily TSI, cc obviously increases to be 0.4659. Based on the method used to test the statistical difference of two correlation coefficients by Li et al. (2002), the difference between these two cc values is found significant with a probability of about 91%, that is to say, the difference is not caused by randomness, and the two values are statistically different from each other.

The period of 3880.2 days (≈10.63 years) corresponds to the so-called Schwabe cycle, and the period of 1104.7 days is inferred to be the 3 multiple harmonic of the approximately annual period of about 376.0 days. IMF 9, probably plus IMF 10, is therefore related to magnetic activity of the Schwabe cycle.

Jin et al. (2011) have divided the full Sun's magnetograms into active regions (ARs) and quiet regions (QRs) and calculated the monthly average magnetic flux F_AR and F_QR, respectively, in the time interval of 1996 September to 2010 February with the MDI/Solar and Heliospheric Observatory (SOHO) data used. They found that the flux of network magnetic elements in QR could be further divided into four components: (1) those elements whose fluxes are in the range of (1.5–2.9) × 10¹⁸ Mx are basically independent of the sunspot cycle, and thus called by them no-correlation elements (F_no); (2) the elements in the flux range of (2.9–35.9) × 10¹⁸ Mx show an in-phase correlation with the sunspot cycle, and thus they are anti-phase elements (F_anti); (3) those in the flux range of (35.9–42.7) × 10¹⁸ Mx are called transition elements (F_tran), which represents a transition from anti-phase to in-phase with the sunspot cycle; and (4) the so-called in-phase elements (F_in), in the range of (4.27–38.01) × 10¹⁹ Mx, which is in-phase with the sunspot cycle. Based on IMFs 9 and 10, we calculate the monthly average value ($\overline{{\rm IMF}_{9}}$) of IMF 9 and that ($\overline{{\rm IMF}_{9+10}}$) of IMF 9 plus IMF 10 in the time interval of 1996 September to 2010 February. Then we calculate the correlation coefficient of $\overline{{\rm IMF}_{9}}$ and $\overline{{\rm IMF}_{9+10}}$, respectively, with F_AR, F_QR, F_no, F_anti, F_tran, and F_in, and the results obtained are given in Table 2.

Table 2. Correlation Coefficients of TSI Components with Magnetic Activity

	FAR	FQR	Fno	Fanti	Ftran	Fin
$\overline{{\rm IMF}_{9}}$	0.8319	0.6737	−0.4891	−0.6991	−0.3991	0.7198
$\overline{{\rm IMF}_{9+10}}$	0.9671	0.9518	−0.0046	−0.5804	0.0824	0.9818

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The relation of $\overline{{\rm IMF}_{9+10}}$ respectively to F_in and F_AR gives the maximum two correlation coefficients among these coefficients in the table, which are correspondingly much larger than those given by the relations of $\overline{{\rm IMF}_{9}}$ respectively to F_in and F_AR, and the maximum correlation coefficient is given for the relation of $\overline{{\rm IMF}_{9+10}}$ to F_in. Thus, it is seemingly $\overline{{\rm IMF}_{9+10}}$, not $\overline{{\rm IMF}_{9}}$, that is most probably related with magnetic activity, and the magnetic activity is referred to the magnetic flux of F_in. That is to say, IMF 9 plus IMF 10 should be related to magnetic activity of the Schwabe cycle. Figure 6 plots $\overline{{\rm IMF}_{9}}$ and $\overline{{\rm IMF}_{9+10}}$ together with the monthly average magnetic flux values of F_in and F_AR, in order to illustrate the relations of $\overline{{\rm IMF}_{9}}$ and $\overline{{\rm IMF}_{9+10}}$, respectively, to F_in and F_AR. The sum of IMFs 9 and 10, which is called here Component III of the daily TSI, is shown in Figure 4. Variation amplitude of Component III can reach 0.6 W m⁻², and the difference of its maximum and minimum values can still match up to TSI variations of about 0.1%.

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In order to examine the significance in the difference of the maximum two coefficients in the above table, a statistical test is carried out following Li et al (2002), and the difference is found significant with a probability of about 97%. Thus, Component III is inferred to be caused by the network magnetic elements, whose magnetic fluxes are of (4.27–38.01) × 10¹⁹ Mx. The above significant difference somewhat confirms that magnetic fields of different strengths could even act on TSI in reverse ways: intense magnetic fields, as thermal "plugs" to divert heat flow from solar deep layers, decrease TSI, but small-scale magnetic fields, as local thermal "leaks," increase TSI (Domingo et al. 2009).

The complex Morlet wavelet transform is utilized to study the periodicity respectively in Components I and II. Figure 7 shows their global wavelet power spectra and corresponding 95% confidence levels. For Component I the periods of significance are 14.2 ± 1.1 and 31.7 ± 2.8 days, and for Component II the periods of significance are 366 ± 15.0 and 726.3 ± 38.4 days, which are all significant at the 95% confidence level. Thus, Component I is indeed the rotation signal of TSI, and Component II is the annual-variation signal.

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Finally, we determine the contribution of each of the three components to the daily TSI. We calculate the sum of Components I, II, and III over the whole time interval, respectively. The sum of Component I counts up to 42.31% of the total sum of Components I–III, namely daily TSI related to its mean value (TSI minus its mean value). The sum of Component II counts up to 15.17%, and the sum of Component III up to 42.52%.