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ISSN 0038 0946, Solar System Research, 2012, Vol. 46, No. 2, pp. 170­176. © Pleiades Publishing, Inc., 2012. Original Russian Text © V.M. Fedorov, 2012, published in Astronomicheskii Vestnik, 2012, Vol. 46, No. 2, pp. 184­189.

Interannual Variability of the Solar Constant
V. M. Fedorov
Moscow State University, Moscow, 119991 Russia
Received February 14, 2011

Abstract--It can be concluded from the calculations performed of interannual variations of the distance between the Sun and the Earth in the moments of the Earth's position in the equinoctial and solstitial points that the mean amplitude (approximately the same for all the equinoctial and solstitial points) is determined to be equal to 5700 km (at the maximum values being approximately equal to 15 000 km). The values of the solar constant have been calculated on the basis of the data of varying distances, and the values of its interan nual variability (for the period from 1900 up to 2050) have determined. Based on the analysis of the series, new periodic characteristics of a long term variation of the solar constant, related to the celestial mechanical process, namely, to the perturbed orbital motion of the Earth, are obtained. A three year cycle is distinguished in the interannual variability of the solar constant, which alternates with a two year cycle every eight and eleven years. The amplitude of the interannual variability in the series of equinoctial and solstitial points is on average about 0.1 W/m2 (about 0.008% of the solar constant value). This is comparable to the interannual variability of the solar constant in the eleven year cycle of the solar activity. The series obtained can be repre sented by alternation of eleven year and eight year cycles. The eleven year cycle is composed of three three year cycles and one two year cycle, and the eight year cycle is composed of two three year cycles and one two year cycle. DOI: 10.1134/S0038094612020049

INTRODUCTION The solar constant is the total solar radiant flux that passes through the unit area, oriented perpendicular to the flux at a distance of one astronomical unit (AU) from the Sun outside the Earth's atmosphere, per unit time (Eygenson, 1963; Kondratyev, 1965). According to the data of extraterrestrial measurements, the solar constant equals 1367 W/m2 (Kislov, 2001). The solar constant varies with time. Its variations are conven tionally related to the solar activity variation, i.e., to the physical processes on the Sun. Attention is paid to the fact that the variations of the solar constant are caused by the variation of the radiant flux with the variation of the number and total area of sunspots, with the relative variations of the radiant flux occur ring most intensively in the X ray and radio regions (Eygenson, 1963; Vitinskiy, 1983; Abdusamatov, 2009). The roles of facular regions and magnetic fields are also noticed as important factors. They contribute to the cyclic variations of the radiant flux (Foukal et al., 2006). The period of direct measurements of the solar constant is short (since 1978), and the amplitude of variation of the solar constant during the eleven year cycle of the solar activity (Schwabe­Wolf cycle) for the period of measurements is about 1 W/m2 (or approximately 0.07% of the solar constant (http://www.pmodwrc.ch)). The data for direct mea surements of the estimation of the solar constant vari

ation for longer solar cycles (Hale cycle, Gleisberg cycle, etc.) are absent (Makarova et al., 1991). The long term variations of the solar constant are related not only to the variation of solar activity (a great number of publications and monographs are devoted to the investigation of solar activity), but also to the celestial mechanical processes that change the distance between the Sun and the Earth. During the unperturbed (keplerian) motion of the Earth, the solar constant varies within an annual orbital motion of the Earth around the Sun with regular annual variation (the maximum is at perigee, the minimum is at aph elion). However, the real orbital motion of the Earth is a perturbed motion (Marov, 1981; Fedorov, 2000; 2002). In this connection, the Earth is from year to year at a different distance from the Sun, for example, while passing through the equinoctial and solstitial points. This is one more reason for the long term vari ations in the solar constant, which has not been stud ied yet. The purpose of this work is to estimate the influence of this factor on the long tem variations of the solar constant. In the last century, M. Milankovich carried out the calculations for solar radiation allowing for variations in the earth's orbit due to factors such as inclination of the axis, eccentricity, and longitude of perihelion. the periodicity of variation of these factors is tens of thou sands of years (Milankovich, 1939). In our calcula tions one astronomic parameter, namely, the distance

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from the Earth to the Sun, is used; and the variation of the solar constant is considered in other time scales (years, tens of years) that are comparable to the topical problems of modern civilization. TECHNIQUE OF THE INVESTIGATIONS The distances between the Earth and the Sun in the moments of the Earth's position in the equinoctial and solstitial points within the time interval from 1900 up to 2050 (http://www.ssd.jpl.nasa.gov; http://www. willbell.com) are determined according to the ephe merides data (JPL Planetary and Lunar Ephemerides (DE 406)). The accuracy of ephemerides in distance is 10­9 AU or 0.1496 km. The value of the solar con stant at a distance between the Earth and the Sun of 1 AU equals J0 = 1367 W/m2. It is known that if a is the average distance between the Sun and the Earth, which is equal to the major semiaxis of the ellipse of Earth's orbit (1 AU), then at distance l we have:

Jl = J0 a . l

()

2

(1)

The values of solar constant in the moments of the Earth's position in the equinoctial and solstitial points for each year within the time interval from 1900 up to 2050 have been calculated by Eq. (1). The series of val ues of the interannual variability of solar constant for the cardinal points of earth orbit (equinoctial and sol stitial points) are obtained through successive subtrac tion. As a result of the analysis of the calculated series, new periodical characteristics of the long term varia tion of solar constant, which are related to the per turbed orbital motion of the Earth, have been obtained. RESULTS AND DISCUSSION The distances between the Earth and the Sun in the moments of the Earth's position in the equinoctial and solstitial points within the time interval from 1900 up to 2050, obtained through the ephemerides, are pre sented in Fig. 1. The values of the interannual variability of the dis tance for the equinoctial and solstitial points have been calculated on the basis of the values of distance between the Sun and the Earth (it was performed by successive subtraction; therefore, the influence of the trend related to the precession was eliminated from further calculations and results). The average ampli tude of the interannual variations of the distance between the Earth and the Sun in the moments of the Earth's position in the equinoctial and solstitial points (which is approximately the same for all the equinoc tial and solstitial points) is determined to be equal to
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5738 km (at the maximum values being on average equal to 15042 km). The three year and two year peri odicities are observed in the distance variability (the maximum of spectral density occurs in a period of 2.7 years). The corresponding values of the solar con stant have been calculated on the basis of the values obtained for the distance between the Earth and the Sun. It may be seen from the graphs (Fig. 2) that the interannual variability of the solar constant for the equinoctial and solstitial points is periodical. A three year cycle is distinguished in the interan nual variability of the solar constant, which alternates with a two year cycle in eight and eleven years. The amplitude of the interannual variability in the series of the equinoctial and solstitial points is on average about 0.1 W/m2 (0.008% of the solar constant value). For a series of the interannual variability values, averaged over these four points, the amplitude decreases by 0.06 W/m2. The investigated period contains 38 com plete three year cycles and 16 complete two year cycles. In other words, the series are represented by alternating eleven year and eight year cycles. The eleven year cycle is composed of three three year cycles and one two year cycle, and the eight year cycle is composed of two three year cycles and one two year cycle (Figs. 3, 4). These characteristics are still beyond the limits of measurement accuracy of the solar con stant that are ±0.3­0.7 W/m2 or 0.02­0.05% of the solar constant value (Makarova et al., 1991). According to the data of the spectrum analysis, the increase in the spectral density within the approximate range from 2.3 up to 3 years, with the maximum being equal to a period of 2.7 years, is observed for all the series (Fig. 5). According to the data of the direct observations (Willson, 1997; Willson and Mordvinov, 2003; Abdusamatov, 2009; http://www.pmodwrc.ch; http:// www.acrim.com) the amplitudes of the eleven year cycles of the solar constant are 0.955 W/m2 in the 21st cycle, 0.919 W/m2 in the 22nd cycle, 1.039 W/m2 in the 23rd cycle. The data of the direct measurements of the solar radiant flux, corrected with respect to the Sun­Earth distance, reflect the physical processes that take place on the Sun. With allowance for the duration of these cycles (10.25, 9.75, and 12.50 years, respectively), the interannual variability is on average 0.415, 0.427, and 0.332 W/m2, respectively. These val ues are approximately 3­4 times greater than the values of the average interannual variability of the solar constant for the equinoctial and solstitial points (0.105 W/m2 and 0.008% of the solar constant), obtained in the work, that are related to the variation in the Sun­Earth dis tance. Thus, the solar constant interannual variations of a different physical nature have rather similar amplitude characteristics. However, an additional analysis of the results is required for the determination


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Fig. 1. Variability of the Earth­Sun distance for the period from 1900 up to 2050 at the Earth's position in the following points: 1, the vernal equinox, 2, the June solstice, 3, the autumnal equinox, 4, the winter solstice. SOLAR SYSTEM RESEARCH Vol. 46

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INTERANNUAL VARIABILITY OF THE SOLAR CONSTANT 0.3 0.2 0.1 W/m2 0 ­0.1 ­0.2 ­0.3 2034 2006 2027 2020 2048 2 2048 2048 4 2048 3 1985 1922 1943 1908 1978 2013 2041 1929 1936 1950 1957 1964 1992 1999 1901 1971 1915 ­0.4 1

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Years 0.3 0.2 0.1 W/m2 0 ­0.1 ­0.2 2034 2034 2034 2006 2027 2020 1922 1943 2013 2041 2041 2041 1929 1936 1950 1957 1964 1992 1999 1985 1915 1908 1978 1901 1971 ­0.3

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Fig. 2. Interannual variability of the solar constant for a period from 1900 to 2050 at the Earth's position ay the following points: 1, the vernal equinox, 2, the June solstice, 3, the autumnal equinox, 4, the winter solstice. SOLAR SYSTEM RESEARCH Vol. 46 No. 2 2012


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Fig. 3. Dynamics of the average (over four points, equinoxes and solstices) values of the interannual variability of the solar con stant for a period from 1900 to 2050.

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Fig. 4. Fragment of the dynamics of the average (over four points, equinoxes and solstices) values of the interannual variability of the solar constant for a period from 1981 to 2011.

of the correlation of the two year and three year orbital cycles with the solar activity cycles (Schwabe­ Wolf eleven year cycle) in the instrumentally obtained variations of the solar constant (satellite measure ments since 1978). In addition, the studies on the determination of the influence of the demonstrated cycles of long term

variation of the solar constant on climate variations are necessary. The two year cycles and three year cycles are multiple of the Earth's period of oscillation (one year) and are close to it (closer in value than the Schwabe­Wolf eleven year cycle); therefore, such investigations seem to be perspective due to a probable resonant response of the Earth climate system to the
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INTERANNUAL VARIABILITY OF THE SOLAR CONSTANT 3.0E­01 2.7 2.5E­01 4 Spectral density 2.0E­01 1.5E­01 1.0E­01 5.0E­02 6.4516 4.0816 4.6512 5.4054 28.5710 200.0000 15.3850 10.5260 8.0000 0.0E+00 1 2.5316 2.9851 2.3529 3.6364 2.7397 3.2787 2.1978 2.0619 3 5 2

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Fig. 5. Spectra of the interannual variability of the solar constant for the following points: 1, the vernal equinox, 2, the June sol stice, 3, the autumnal equinox, 4, the winter solstice, 5, the spectrum of the interannual variability of the solar constant averaged over the four points of equinox and solstice.

revealed characteristics of the solar constant variation. A distinct expression of the observed two year and three year cycles in contrast to the eleven year cycle of solar activity with less stable duration also points to this fact. From about the mid 18th century, the dura tion of separate cycles of solar activity, determined by the maxima of the number of sunspots, was within 7­ 17 years, and the same duration, determined by the min ima of the number of sunspots was within 9­14 years (Makarova et al., 1991). In addition, the three year periodicity is observed in a number of geophysical and hydrometeorological processes (Eigenson, 1963). CONCLUSIONS New orbital characteristics, related to the processes of celestial mechanics, namely, to the periodic varia tion in the distance between the Sun and the Earth, were found in the long term variation of the solar con stant for the equinoctial and solstitial points. The characteristics found are probably also related to other points of earth orbit. A dominant factor associated with a marked long term variation in the solar con stant is a well defined three year cycle with an ampli tude approximately equal to 0.1 W/m2. This cycle alternates, as a rule, every 8 or 11 years with the two year cycle. The value of the interannual variability of the solar constant in the three year cycle is compara ble (is three four times less) with the interannual vari ability of the solar constant in the eleven year cycle of solar activity.
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The data obtained are important for both under standing the nature of the solar constant variability and the clarification of ideas about solar activity varia tions and the solar climate of the Earth. ACKNOWLEDGMENTS The work was supported by the Russian Founda tion for Basic Research, grant no. 11 05 01134. REFERENCES
Abdusamatov, Kh.I., Solntse diktuet klimat Zemli (The Sun Dictates the Earth's Climate), St. Petersburg: Logos, 2009. Eigenson, M.S., Solntse, pogoda i klimat Zemli (Sun, Weather and the Earth's Climate), Leningrad: Gidro meteoizdat, 1963. Fedorov, V.M., Gravitatsionnye factory i astronomicheskaya khronologiya geosfernykh protsessov (Gravitation Factors and Astronomical Chronology of Geosphere Pro cesses), Moscow: MGU, 2000. Fedorov, V.M., Astronomicheskaya klimatologiya (Astro nomical Climatology), Moscow: MGU, 2002. Foukal, P., FrÆhlich, C., Spruit, H., and Wigley, T.M.L., Variations in Solar Luminosity and Their Effect on the Earth's Climate, Nature, 2006, vol. 443 (7108), pp. 161­166. Kislov, A.V., Klimat v proshlom, nastoyashchem i budush chem (Climate in the Past, Present and Future), Mos cow: Nauka, 2001. Kondrat'ev, K.Ya., Aktinometriya (Actinometry), Lenin grad: Gidrometeoizdat, 1965.


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FEDOROV Willson, R.C., Total Solar Irradiance Trend during Solar Cycles 21 and 22, Science, 1997, vol. 277, no. 5334, pp. 1963­1965. Willson, R.C. and Mordvinov, A.V., Secular Total Solar Irradiance Trend during Solar Cycles 21­23, Geophys. Res. Lett., 2003, vol. 30, no. 5, pp. 1199­1202. http://www.acrim.com http://www.ssd.jpl.nasa.gov http://www.pmodwrc.ch http://www.willbell.com

Makarova, E.A., Kharitonov, A.V., and Kazachevskaya, T.V., Potok solnechnogo izlucheniya (Solar Radiation Flux), Moscow: Nauka, 1991. Marov, M.Ya., Planety solnechnoi sistemy (Solar System Planets), Moscow: Nauka, 1981. Milanovich, M., Matematicheskaya klimatologiya i astro nomicheskaya teoriya kolebaniya klimata (Mathematical Climatology and Astronomical Theory of Climate Vari ation), Moscow­Leningrad: GONTI, 1939. Vitinskii, Yu.I., Solnechnaya aktivnost' (Solar Activity), Moscow: Nauka, 1983.

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