Advances in Space Research 35 (2005) 725–738 www.elsevier.com/locate/asr
Solar cycles: A tutorial X. Moussas a
a,*
, J.M. Polygiannakis
a,z
, P. Preka-Papadema a, G. Exarhos
a,b
Department of Astrophysics, Astronomy and Mechanics, National and Kapodistrian University of Athens, Panepistimiopolis, GR 15783 Zographos, Athens, Greece b Siemens A.E., Artemidos 8, GR 151 25 Marousi, Athens, Greece Received 2 July 2004; received in revised form 11 March 2005; accepted 11 March 2005
Abstract The Sun is the nearest stellar and astrophysical laboratory, available for detailed studies in several fields of physics and astronomy. It is a sphere of hot gas with a complex and highly variable magnetic field which plays a very important role. The Sun shows an unprecedented wealth of phenomena that can be studied extensively and to the greatest detail, in a way we will never be in a position to study in other stars. Humans have studied the Sun for millennia and after the discovery of the telescope they realized that the Sun varies with time, i.e., solar activity is highly variable, in tune scales of millennia to seconds. The study of these variabilities helps us to understand how the Sun works and how it affects the interplanetary medium, Earth and the other planets. Solar power varies substantially and greatly affects the Earth and humans. Solar activity has several important periodicities, and quasi-periodicities. Knowledge of these periodicities helps us to forecast, to an extent, solar events that affect our planet. The most prominent periodicity of solar activity is the one of 11 years. The actual period is in fact 22 years because the magnetic field polarity of the Sun has to be taken into account. The Sun can be considered as a non-linear RLC electric circuit with a period of 22 years. The RLC equivalent circuit of the Sun is a van der Pol oscillator and such a model can explain many solar phenomena, including the variability of solar energy with time. Other quasi-periodicities such as the ones of 154 days, the 1.3, 1.7 to 2 years, etc., some of which might be harmonics of the 22 year cycle are also present in solar activity, and their study is very interesting and important since they affect the Earth and human activities. The period of 27 days related to solar rotation plays also a very important role in geophysical phenomena. It is noticeable that almost all periodicities are highly variable with time as wavelet analysis reveals. It is very important for humans to be in a position to forecast solar activity during the next hour, day, year, decade and century, because solar phenomena affect life on Earth and such predictions will help politicians and policy makers to better serve their countries and our planet. 2005 COSPAR. Published by Elsevier Ltd. All rights reserved. Keywords: The Sun; Solar cycles; Solar activity
1. Introduction The Sun is the nearest star and the only one that affects our lives.1 It supplies almost all the energy of our
*
Corresponding author. E-mail address:
[email protected] (X. Moussas). z Deceased. 1 We can assume astronomers are the only humans that their life is affected by the stars since they make their living with them, although people believing in astrology are also affected by the stars.
planet, regulates the light and heat on the Earth, the day and night, the seasons and the years. Humans understood the importance of the Sun and for millennia adored the nearest star as a significant god. Humans prefer stability, especially those who have power. When science started in antiquity, humans for centuries believed, and preferred to believe, that the Sun is a perfect celestial body and that its power does not change with time. Primitive human societies have a fear of solar eclipses and even had the habit to fight (firing arrows towards the Sun) for its reappearance
0273-1177/$30 2005 COSPAR. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.asr.2005.03.148
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during this natural phenomenon that takes place when the Moon is in front the Sun and shadows a small part of the Earth. Observing the daily motion of the Sun2 and the stars and the planets, humans realized that the Universe has rhythms, regularities, normalities, steadiness, symmetry, periodicities, and that laws of nature govern the cosmos.3 Humans started studying the Sun from the beginning of astronomy and have greatly benefited from these studies. The motion of the Sun was the main interest of ancient astronomers, but they soon started their inquiries about its nature and the nature of the stars. In antiquity some astronomers observed sunspots,4 but perhaps science under the pressure of philosophy refused to accept the facts. Since the time of Galileo, astronomy and solar physics in particular changed drastically and astronomers began to have good quality images and continuous time coverage of the solar ÔsurfaceÕ (photosphere). They measured the sunspots and other indices of solar activity, which enable us to study the variability of this life-giving star. The Sun is a common star (G2V), one of those that we call dwarf stars. It belongs to the main sequence of stars, i.e., stars that still burn their hydrogen. The Sun started its life almost 4.6 billion years ago, has an age approximately equal to 1/3 of the life of the Universe and we expect it will live another 5 billion years or so. It is a second generation star, i.e., it is made out of a huge cloud with material that came partly from stars of the first generation, which died before the Sun was born. Its mass is 1.9891 · 1030 kg or 333,000 times the Earth mass, and its average radius is 696,000 km or 109.2 times EarthÕs radius. Solar power is 3.846 · 1026 J/s. This power requires that every second a mass of 4.3 · 109 kg is transformed into energy through thermonuclear reactions DE = Dmc2 in the solar core. The power emitted by every square meter is P ¼ rT 4 ¼ 6; 329 106 J=m2 s or 63 GW/m2, and if we assume that the Sun is a black body its temperature is 5700 K (effective temperature). Knowing our star we are able 2 Recently Prof. Dr. Francois Bertemes, archaeologist at the MartinLuther-Universitat Halle-Wittenberg discovered a large circle in a field in Goseck, Germany, which is the oldest observatory in Europe (c. 5000 b.C). Bronze Age people had astronomers and some 200 other prehistoric astronomical observatories (circles) exist. I have recently discovered several bronze celestial spheres (which look like the symbol ¯, c. 750 b.C) at the museum of Volos coming from a temple of Itonia Athena in Thessaly, Greece. 3 During the second millennium b.C. in Orphic verses we read that the stars are governed by the astrothetis nomos, i.e., the law that governs the stars and puts them in order, Papathanasiou (1978). 4 Sunspots are dark regions on the Sun. They are cooler than the surrounding surface of the Sun. Their temperature is approximately 2000 K lower than the photosphere. The number of sunspots vary with time (Fig. 1). Sunspots are manifestations of very strong local magnetic fields. Solar activity is the variation of the appearance or of the energy output of the Sun, which is associated with the number of sunspots and solar magnetic fields.
to predict its behaviour, with its short and long term variations. Solar activity causes direct and indirect effects in our every day life and it is a matter of importance for humanity to understand these effects. Studying solar variability is important because it permits us to better design our future, to plan it more carefully, by informing politicians and policy makers on the future of our planet. Statesmanship is to foresee, to plan and act accordingly and for this politicians need to know all the possible scenarios for the climate and the Sun.
2. Sunspots and solar cycles Theophrastus (AristotleÕs student) and Aratus in Greece and Chinese astronomers at the same era (4th and 3rd century b.C.) observed first the presence of big sunspots on the Sun by naked eye, probably when the Sun was near a cloudy horizon and it had some large spots for long time. Thomas Harriot was one of the first to use a telescope to observe the universe after Galileo and observed the first sunspots on the 8th of December 1610, followed by David Fabricius and his son Johannes and Scheiner and naturally Galileo. Observing sunspots gradually became an interesting and every day duty for many astronomers, even amateurs, and eventually led to solar patrol for several astronomical observatories. These frequent solar observations have been accompanied by drawings from the very beginning. Sunspots appear mainly near the equator (5–40 in latitude) and their frequency of occurrence varies with the years. Comparison of the consecutive solar images lead to the discovery of the rotation of the Sun and subsequently to the discovery of the 11 year periodicity of solar activity (solar cycle) by Schwabe (1841, 1844, 1876). He was initially a pharmacist who become an astronomer in the pursuit for a new planet near the Sun in 1827 and finally discovered the main periodicity of solar activity (see Fig. 1). Since then, many scientists have been working on solar cycles and solar activity. Astronomers analyzed old and new data, and today a complete series of modern
Fig. 1. The monthly values of sunspot number vary periodically (11 years) and is a very good index of solar activity.
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Fig. 2. The rising time measured from the minimum to the maximum of sunspot numbers varies significantly from one solar cycle to the next. The same applies for falling time. The actual period is not constant and it varies substantially. In this figure the distributions of rising time, falling time, their ratio and the length of solar cycle are presented.
and ancient sunspot time series is available. Data of sunspot observations from the time of Galileo enabled us to have a reliable time series for several centuries. The record of the sunspots constitutes the longest time series of actual observations made by humans (Usoskin et al., 2002; Usoskin and Mursula, 2003). Analyses of all the data showed that the actual period varies between 8 and 15 years (cf. Fig. 2, second panel). The rising time is usually shorter than the falling time of the solar cycle (Fig. 3, 2nd and 3rd panels). The shape of the variation from minimum to maximum and to minimum again is usually very asymmetric. The average falling time is double than the rising time, like an asymmetric pulse, but there are several solar cycles with larger rising time than falling time. The rising time varies between 2.8 and 9 years, the falling time between 3.5 and 10.2 years. The sunspot number varies between almost zero to 250. The number of sunspots at the maxima varies between 49 and 250, while at the minimum from 0 to 12 (Fig. 4). On the average, big solar cycles, i.e., solar cycles with large maxima, have short rising times and long falling times. The solar power and the solar constant5 vary with time following the number of sunspots. Several characteristics of the solar activity have been revealed over the years, starting with Carrington (1860) who observed flares,6 with a telescope in his garden. Humans know that prediction is power and knowledge of sunspot activity in advanced gives politicians
5
Solar constant is the power per square meter received from the Sun at 1 AU. 6 A flare is a sudden energy release during a short time period, usually a few minutes, which takes place in a small area near sunspots (active regions).
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and humanity the opportunity to design well the future. Scientists have tried to predict the average long term sunspot activity several years in advance. They have tried to relate the values of various parameters of one cycle with the values of the parameters (maximum, duration, rising time, falling time, etc.), of the previous solar cycle. From Fig. 5 it is evident that the parameters of one solar cycle are not related with the values of the previous one. Flares and CMEs (Coronal Mass Ejections)7 are the most important manifestations of solar activity and consequently space weather which directly affects our every day lives (Chian, 2003, 2005). Solar activity affects the EarthÕs climate, but to what extent? Are quasi-biennial variations of climatic parameters (e.g., Kayano et al., 2005) related to the quasi-biennial solar activity period? Sun-Earth relations are an intriguing and hot issue, still open (Kane, 2005). Solar activity is the driving force for all large scale phenomena on this planet. This becomes evident from the long term data base which was constructed out of a unique 2500 year long rings record from South America (Nordemann et al., 2005). This is an exceptionally long and reliable climatic record suitable for the study of our planetÕs climate, that is directly related to the solar variability (e.g., solar irradiance, Bouwer, 1992; Mendoza, 2005). Sun–Earth relations are greatly affected by the solar wind8 plasma dynamics. The interplanetary disturbances (Gonzalez, 2005) are caused by solar disturbances (e.g., Raulin and Pacini, 2005), which affect the magnetosphere (Mendes, 2005; Valdivia, 2005), the ionosphere (Kintner and Ledvina, 2005), cosmic rays (Valdes-Galicia, 2005) and the heliosphere (Exarhos and Moussas, 1999a,b, 2000a,b).
3. Differential rotation The Sun rotates around its axis with an average period of approximately 25 days with respect to the position of the stars or 27 days as observed from the Earth, which moves around the Sun once per year.
7 Coronal Mass Ejection (CME) is an eruptive solar effect in which a magnetic loop is ejected in the interplanetary medium with a velocity of the order of 1000 km/s (and up to 3500 km/s). Coronal mass ejections are solar eruptions which inject billions of tons of solar magnetized plasma into the interplanetary space. Coronal mass ejections come from magnetic loops which become unstable, erupt and throw material in the corona and the interplanetary space. 8 The solar wind is a fast magnetized plasma which blows continuously almost radially away from the Sun. It is supersonic with speeds of the order of 400 km/s (slow wind) to 800 km/s (fast streams), temperature of the order of a few tens of thousands Kelvin to a few millions. It is the extension of the solar corona into the interplanetary medium. Fast solar wind comes from the coronal holes, regions with open magnetic filed lines, which are near the poles of the Sun.
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Fig. 3. The rising times and falling times as well as their ratio as a function of the number of sunspots at maxima and minima for all solar cycles. The falling time (tfall) and the rising time (trise) seem to be related to the strength of the solar cycle (maximum of sunspot numbers). Perhaps this simply reflects the non-linearity of the phenomenon, which is still an open and intriguing issue.
tude. It is faster near the equator and slower near the poles (Fig. 6.). The period is 24.56 days at the equator and increases to 35 days near the poles. The angular velocity varies with heliolatitude following the relation X ¼ 14.37 2.33 sin2 ðlatitudeÞ 1.563 sin4 ðlatitudeÞ degrees per day.
Fig. 4. The distribution of sunspot maxima varies from 40 to 220, and the minima varies from 0 to 12.
The rotation axis makes an angle of 7.175 with the normal to the ecliptic plane which is the plane of the EarthÕs trajectory around the Sun. The angular velocity of the Sun around its axis is differential, i.e., varies with lati-
Sporer and Carrington discovered in the 1860s the differential rotation of the Sun. They also established the fact that sunspots appear at the beginning of the 11 year cycle at high heliolatitudes and gradually drift towards the solar equator (cf. Fig. 9, which shows the heliolatitude of flares during the last three solar cycles). This drift of sunspot appearence is called SporerÕs Law. The extension of the solar atmosphere is the corona which expands radially in the interplanetary space and
Fig. 5. Scatter plots of sunspot maxima with parameters of the previous solar cycle and for all solar cycles. The plots show that there is no correlation of maximum sunspot number with parameters of the previous solar cycle (maximum, duration, rising time, falling time, etc.).
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Fig. 7. The duration of solar flares as a function of time follows the solar cycle of 11 years. Fig. 6. The rotational velocity of the Sun (measured by the rotation of sunspots and Doppler effect) varies with latitude.
forms the solar wind.9 This differential rotation of the Sun plays a key role in the development and evolution of the magnetic field of the Sun. The rotational velocity varies also with the depth inside the Sun and height in the corona of the Sun.
4. Solar magnetic field and solar cycles The development of solar magnetographs based on the Zeemann effect offered additional information on the solar magnetic fields and the nature of sunspots and other features of the Sun. Hale (1908) first discovered and measured the magnetic field in sunspots using very effectively the Zeemann effect and spectroscopy from a distance. Magnetograms showed that the general magnetic field of the Sun is dipolar and changes polarity every 11 years. Hence, if we take into account the reversal of the general dipolar field of the Sun, the period of solar activity has in fact a period of 22 years and not 11 years. Sunspots usually appear in pairs, the preceding and following spot. The two spots of every pair have opposite magnetic polarities and form bipolar or multipolar active regions. The magnetic field of the sunspots is over a thousand times stronger (usually 2000–3000 G) than the general bipolar magnetic field of the Sun (1–2 G). Most preceding spots in every hemisphere have the polarity of the polar magnetic field. In every hemisphere almost all pairs of sunspots have their bipolar filed oriented in the same sense and the opposite happens on the other hemisphere. The polarity of the pairs of sunspots reverses every solar cycle. At the beginning of the solar
9 Al Birouni (864–930 a.D.), observed the corona around the Sun during solar eclipses and concluded that a rarefied substance continuously leaves the Sun, i.e., he foresaw the solar wind and Beniamin of Lesbos (late 18th century) stated that all celestial bodies continuously emit a substance pan-tachy-kineton (=fast omni directional) which causes gravitational forces (gravitons?) and the aurora.
cycle the bipolar regions (pair of sunspots) appear at high heliolatitudes (SporerÕs law) with opposite polarity with respect to the polarity of the bipolar regions of the same hemisphere of the previous solar cycle. It is evident that the Sun is a physical system with highly variable behaviour. Observations of the magnetic field based on the Zeemann effect and images in X-rays which capture radiation emitted by energetic electrons trapped in quasi bipolar or multipolar magnetic bottles show that the phenomena on the Sun change continuously and almost all the local magnetic fields change within hours. Several large scale characteristics, such as the field in sunspots and coronal holes can remain almost the same for a few months and perhaps for over one year, respectively (Figs. 2–7).
5. Ground based and space observatories During the last decades we have continuous observations of good quality of the Sun in various wavelengths, from ground based telescopes, radio spectrographs (e.g., the Solar Submillimeter Telescope (SST), Kaufmann et al., 2001, the ARTEMIS IV solar radio spectrograph, Caroubalos et al., 2001a), and satellites (SOHO, Ulysses, WIND, TRACE, STEREO). All these are extremely valuable to solar physics but also to astrophysics in general and also to space physics, geophysics, and plasma physics, to name a few other disciplines. Solar radio observations with high resolution in time and frequency led to the discovery of new periodic phenomena (pulsations, fine structure, fiber bursts) in explosive solar events (see for example Caroubalos et al., 2001b; Kaufmann et al., 2004; Raulin et al., 2003). Based on all these data we know what is happening (almost every moment) on the surface and the interior of the Sun, and we can also see the back side of the Sun. Additionally we have very important observations of the interplanetary medium in 3D (Ulysses, STEREO). Realistic modelling based on all the detailed observations will permit, very soon, accurate predictions of solar eruptive phenomena and space weather.
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6. Helioseismology Combinations of the magnetic field measurements together with measurements of the velocity fields, with the aid of the Doppler effect, in sunspots, active regions, granules, and all the surface of the Sun disclosed numerous, previously almost inconceivable facets of solar physics (Goode, 2001). Helioseismology has been established by Leighton and his co-workers (Leighton et al., 1961; Noyes and Leighton, 1963; Simon and Leighton, 1964) who pioneered in this field taking measurements of the first velocity fields and opening a new chapter in solar physics and the solar oscillations. It gave new, almost unexpected, previously unimaginable and unrevealed knowledge of the Sun, its internal structure, its time variability with short (5 min oscillations) and longer period variations in the interior of the Sun, where the flow patterns change with the phase of the solar cycle. Helioseismic analyses showed the 1.3-year periodicity in solar rotation at the base of the solar convection zone (for example see Krivova and Solanki, 2002). The opposite side of the Sun has also been studied with similar methods, which also combined a new type of solar holography (Braun and Lindsey, 2000). In a similar way the regions under sunspots have been studied.
7. Periodicities of solar activity Solar activity, as we measure it with the number of sunspots and other solar parameters (see the duration of solar flares versus time in Fig. 7) that we observe over the years, varies periodically with a dominant periodicity of 11 years, which in fact is 22 years, if we take into account the time of the overall photospheric magnetic field reversal. The dipolar magnetic field of the Sun varies with the 22-year period and inverses every 11 years (Fig. 8). The frequency of flare occurrences (Fig. 3) as well as the one of coronal mass ejections (CMEs), which are explosions of huge magnetic loops in the interplanetary medium, and their-average velocity, or their: heliographic distribution of occurrence follow the dominant 11-year periodicity (Fig. 9). The position of appearance of other features on the Sun, such as the prominences or filaments, varies with the 11-year solar cycle. The solar X-ray, UV and solar radio emission vary substantially over the 11 years. The coronal holes, which are extensive regions on the Sun with open magnetic field lines, and from which the fast solar wind blows, change their-area and position from the minimum to the maximum of the 11 year solar cycle. The overall solar irradiance varies also with a relative amplitude of about 0.001 over the 11-year cycle. Frohlich et al. (1994) examines the solar irradiance variations with respect to the solar cycle. Solanki and Unruh (1998) provide a model of the wavelength dependence of solar irradiance variations using a Kurucz
Fig. 8. The inclination of the magnetic dipole of the Sun changes with a period of 22 years (Hale cycle).
(Kurucz, 1979, 1991; Kurucz and Avrett, 1981) model for star spectra with an effective temperature of 5777 K. The solar activity (coronal mass ejections, solar flares, fast solar wind streams, etc.), causes numerous effects on Earth, such as climatic variability, geomagnetic storms, ionospheric changes, cosmic ray modulation and others, which statistically also follow the general solar periodicities. Beyond the main 11-year periodicity others have been observed, among them periodicities of 28, 24, 19,16, 13, 8.8, 5.6, 5.5, 3.8, and 2.8 years. The 2-year period (quasibiennial oscillation, QBO) and many more (1.4, 1.07, 0.97, 0.68, 0.49, 0.42, 0.34, 0.26, 0.2 years) were previously detected in several solar, geo and climatic indices, which might be harmonics of the QBO (Polygiannakis et al., 2003). All these suggest that it is possible to have a double-cycle solar magnetic dynamo model with two main periods, the 22 years and the 2 year period (Benevolenskaya, 1998, 2000). The 22 year period can be explained in terms of a non linear RLC oscillator (van der Pol type, Polygiannakis et al., 1996). One can also assume that the Sun, which is a highly variable electromagnetic object can be represented by an equivalent electric circuit. The van der Pol non-linear oscillator was developed in 1929 to explain the behaviour of electronic valves working in non-linear state. This classic non-linear oscillator can be considered
Fig. 9. The positions of solar flare appearance changes dramatically during the solar cycle. Flares appear at high latitudes at the beginning of solar cycle and later drift towards the equator, following the generation of sunspots (SporerÕs law). This is also called the Butterfly diagram.
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also as the electric equivalent of the human heart. If we construct the phase space ofpthe the sunspot ffiffiffiffiffiffiffiffisquare ffi pof ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi number R(t) and we plot RðtÞ vs Rðt þ sÞ, with s ¼ 66 months, we see that the phase space represents the very familiar non linear RLC oscillator of van der Pol type (see Fig. 4 of Polygiannakis et al., 1996). Then it is very natural to assume that the Sun, with its magnetic fields and emissions has an equivalent electric circuit. One can additionally assume that a part of the solar power (very small if compared with the power of thermonuclear reactions), which is emitted by this electromagnetic object, is proportional to the square of its equivalent electric current. So the square root of an emission (lets say in radio frequencies) of the Sun is proportional to the electric current of the equivalent circuit. The number of sunspots is known to be proportional to the solar radio emission (Hoyt et al., 1992). Consequently one can use the square root of the sunspot number as a proxy of the electric current of the Sun. Taking this electric current and plotting it against the current values several months earlier we have the phase space reconstruction. The van der Pol model reproduces well the effects observed in actual solar cycles such as the Waldmeire effect (unequality of rising to falling time, cycles with larger amplitude are more asymmetric and reach the maximum in shorter times), it allows for the existence of a Maunder minimum (as a non-linear RLC circuit) and explains the characteristic of solar activity near a Maunder type minimum, which occurred in the middle ages and probably caused the little ice age. It can also explain the variation of the solar power during the 11 year solar cycle which can be due to the thermal emission of the electric circuit of the Sun, which has a current, impendence, capacitor and inductance that vary with time.
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Wavelet analysis is a very useful tool for time series analysis and other types of data, (cf. Domingues, 2005). The continuous wavelet transform (Polygiannakis et al., 2003; Addison, 2004; Arneodo et al., 1988, 1995, 1998) gives a unique analysis of a time series in to signal-like and noise-like components. From the overall wavelet spectrum two mutually independent skeleton spectra can be extracted and hence enable us to separate time evolving data and detect not only stable but also transient, highly variable periodicities and scale-invariant structures, and also signal and noise (Fig. 10). The method enables us to estimate the most significant wavelet components (with the largest amplitude) at any given time or scale. This gives the instantly maximal wavelet skeleton spectrum and scale maximal wavelet skeleton spectrum. The scale maximal spectrum was proposed by Arneodo et al. (1988) for studying multifractal scaling properties of the time series. We have analysed sunspot series with this method and observed the well known 11-year period together with several harmonics such and the 2-year periodicity (quasi-biennial oscillation, Benevolenskaya, 1998, 2000) which is the most noticeable, as well as other periods as the one of 1.7 years, first observed by Valdes-Galicia et al. (1996) in cosmic ray modulation studies, and in solar wind parameters. Another important periodicity is the 154 days one and other related periodicities found in sunspot and flare data (Rieger et al., 1984; Bai, 1987, 1992, 1994; Bai and Sturrock, 1987, 1991; Bai and Cliver, 1990; Ozguc and Atac, 1989; Verma and Joshi, 1987; Verma et al., 1992; Oliver and Ballester, 1995). Wavelet analysis is an extremely valuable method for analyzing high resolution data. A good example of such an analysis is made using the data of the Solar Submillimeter Telescope (Makhmutov et al., 2003; Kaufmann
Fig. 10. Wavelet analysis of the sunspot series. This analysis reveals all periodicities, including ephemeral periodicities, which characterise a dynamic time series and a non-linear system such as the Sun.
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et al., 2004) which permits us to study highly variable quasi periodic phenomena which appear in various ranges of frequencies during a solar flare and enables us to see the heart-beat of a solar flare at the time of a maximum. Many solar wind parameters such as velocity, density, magnetic field, temperature, pressure, energetic particles, cosmic rays the geomagnetic index Kp are presented in Figs. 11 and 12. It is evident that the 11 year periodicity is dominant in the modulation of all interplanetary parameters.
8. Long-term variability of the Sun, periods of low solar activity, maunder minimum and the little ice age Long time periods of high solar activity or low solar activity are of great importance for the energy balance of our planet. The average solar power increases in the presence of a huge number of sunspots and decreases in their absence. The appearance of a large sunspot on Fig. 12. Sunspot numbers, solar wind pressure, T, the Kp geomagnetic index, together with the magnetic pressure, (B2 /2/l), which does not vary with the solar cycle, as well as the solar wind flux nVsw and the main component of the solar wind pressure nV 2sw which shows substantial variability with the solar cycle of 11 years and makes the heliosphere to vary significantly with the same period. We note that the temperature of the solar wind somehow follows the solar cycle, showing a maximum after the solar maximum and the solar wind pressure follows the temperature variations. The 27 day averages of solar wind, geomagnetic and sunspot numbers are used to smooth out variations related to the rotation of the Sun.
Fig. 11. Sunspot number, interplanetary magnetic field, B, solar wind velocity, Vsw, the geomagnetic index Kp, and the intensities of cosmic rays at two energies during four solar cycles. The low energy cosmic rays (E P 1MeV/nucleon) follow the solar activity. These are particles partly of solar and interplanetary origin and hence follow the solar activity. Higher energy particles (E P 60 MeV/nudeon) show a twocomponent variability. Their background is anticorrelated to solar activity, since these particles are modulated by the changes in the heliosphere (see Valdes-Galicia, 2005). A second component is superimposed on it and follows the solar activity as the lower energy one. The magnetic field does not show co-variation with the solar cycle. The solar wind velocity varies at times following the solar cycle periodicity. The geomagnetic index Kp, follows the cycle of solar activity.
the Sun temporarily decreases the irradiance of the Sun. This is caused by the presence of the large area of a dark and cool sunspot on the Sun, which causes the decrease of power. But the average power (irradiance) of the Sun increases, when many sunspots are present. This power variation can be explained in terms of the van der Pol non-linear RLC oscillator. The absence of sunspots during the second half of the 17th century and near the end of the 18th century (Maunder minimum from 1645 to 1700, Dalton minimum near the end of 18th century) is associated with a prolonged period of low temperature at the Earth. This cold period is named Little Ice Age and it is believed that it has been caused by a substantial lowering of the solar output associated with the absence of sunspots, and perhaps absence of strong magnetic fields on the Sun. E.W. Maunder, between 1890 and 1920, continued the work of Gustav Sporer who, around 1870, using data compiled by Wolf (between 1850 and 1870) and others, studied the lack of solar activity (sunspots) and established their absence during the second half of the seventeenth century. This coincided with the, well recorded in
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Europe, exceptionally cold winters. The Little Ice Age, as this period has been called, followed the Medieval Warm Period, well known for warm weather which permitted growth of grapes in England. Other very cold or hot eras are also well known.10 Periodicities with periods of several decades are very important but difficult to study and accept beyond any doubt. The limitations come from the small extent of the time series. Such long term periodicities have been found in various solar indices. Yoshimura (1979) studies the asymmetries of the solar cycle and discusses the effects of a non-linear magnetic oscillation as well as a 55 year cycle. These long term periodicities, such as the 55 year one, the 80–100 years, 200 years or 400– 500 years can be studied better with the several centuries long time series of tree rings (Nordemann et al., 2005), while other longer ones (100,000 years) are based on geological data (see also Rigozo et al., 2001). Long time periods of solar activity or inactivity can be explained in terms of low RLC currents according to the non-linear van der Pol model mentioned above. The Sun has exhibited a very high activity during the last 5 solar cycles and it is very interesting and important for humans to understand and predict what to expect in the next few decades. It is also very important to explain and with appropriate models, in terms of physics and mathematics, the empirical rules of Gnevyshev-Ohl and the Waldmeier relations (see Usoskin and Mursula, 2003 for a review and Bracewell, 1988). The nature of the solar cycle is related to the intrinsic non linearity of the Sun (Veselovsky and Tarsina, 2002) and it is a difficult, but not impossible, task to understand it. It is very important to study carefully the energy production using actual numbers for the physical parameters of the Sun and to study the mechanisms and components of energy production, because this will help us to understand the nature of the variation of solar power (Veselovsky, 2002).
9. Coronal mass ejections and solar cycle Coronal mass ejections follow the general trend of the solar cycle. Their maximum occurs near the maximum of sunspots and the same goes for the minimum (Fig. 13). Near the minimum of solar activity, CMEs appear near the solar equator while later they spread to all heliolatitudes (Moussas et al., 2002), probably following 10 As Aeschylus describes in The Persians (MessengerÕs Speech 353– 516) ‘‘During this night a god caused unseasonal weather and froze all the stream of holy Strymon [river in Macedonia, northern Greece] Whoever before had dismissed the gods now called upon them with prayers and worshipped Earth and Heaven. When the long calling on the gods was over for the army, we set out across the frozen stream.’’, Translated by Niall Mc Closkey by Dr. Niall McCloskey, Univ. of Saskatchewan.
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Fig. 13. Coronal mass ejection velocities (CMEs) show a large diversity from a few km/s to over 3000 km/s. In this plot, all the measurements of CME velocities estimated at a constant distance of 20 solar radii are presented. It is evident that the CME velocities are increasing from solar minimum to solar maximum.
Fig. 14. The average CME velocities [ellipses] change substantially (almost doubles) from the minimum of the solar cycle to the maximum, while at the same time the solar wind velocity [rectangles] is almost constant.
the distributions of prominences on the solar disc. Additionally the extend of CMEs appearance follows the maximum mean tilt of the heliospheric current sheet which represents the magnetic equator of the Sun, and its inclination varies greatly with the phase of the solar cycle. It is interesting to notice that the velocity of CMEs does not seem to be related to the velocity of the solar wind at the same time period (Fig. 14). Also the position of appearance of CMEs is not at all related to the position of flares at the same time period.
10. Solar rotation, interplanetary medium, the Earth Another important periodicity arises due to the solar rotation with respect to the stars and the Earth. The nominal period is approximately 27 days for an observer on the Earth. This period appears in all solar indices,
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although it exhibits small fluctuations of the period from one index to the other (Kane, 2002) and for various time intervals. Solar rotation greatly affects the interplanetary medium because the sources of the solar wind rotate together with the Sun and hence the solar wind carries in the interplanetary medium all the inhomogeneities which exist on the solar surface. The fast solar wind sources, which are the coronal holes, are relatively stable on the Sun for several rotations and for as long as a year. Each solar wind source (regardless if it is a large coronal hole or a small region on the Sun) carries its own magnetic field with a certain polarity. The magnetic field of the source is carried away by the solar wind because it is a plasma fluid kept frozen-in in the magnetic field due to its high conductivity. Since the sources of the solar wind are stable, the values of the physical quantities of the fast solar wind streams are repeated several times with the period of solar rotation (Neugebauer and Snyder, 1962; Snyder and Neugebauer, 1964; Wilcox and Ness, 1965). Using the first observations of the solar wind by the Mariner II spacecraft they noticed a quasi-stationary corotating structure in the interplanetary medium which repeated itself four times every solar rotation during the period of measurements with the exception of an interplanetary disturbance. They measured recurrent solar wind streams with well-defined dominant magnetic filed polarity. The field changes drastically direction in the heliospheric current sheet, which separates the two fast streams of solar wind. The heliospheric current sheer separates the two fast streams of solar wind that come from the northern coronal hole and the southern coronal hole, respectively. Consequently, we expect to observe two (at times four) streams of fast solar wind per solar rotation. There are two fast streams of solar wind coming from the two dominant coronal holes, which exist around the poles for most of the time around a solar minimum. The appearance of two or four steams near the solar equator depends upon the inclination and shape of the magnetic equator of the Sun. If the magnetic equator is inclined with respect to the plane of the observer (i.e., the ecliptic plane) and if the magnetic equator is a plane (ideally a great circle of the Sun), we expect two sectors to develop. This happens because the magnetic equator and the ecliptic plane have a section along their common diameter. In reality the magnetic equator is not a plane. At times, it diverges substantially from a plane. In this case four sections of the heliospheric current sheet with the ecliptic plane might appear. The magnetic field is frozen in the plasma of the solar wind and it is carried away to the limits of the heliosphere. The magnetic field is stronger near shock waves, which are formed when one fast stream interacts with a slower, proceeding solar wind plasma and cause distur-
bances in the interplanetary medium. Magnetic clouds are large structures, with diameters of the order of 1/ 4 AU which propagate in the solar wind with large velocities. They have a strong magnetic field which rotates characteristically and are associated with low temperature of protons in their interior. Several magnetic clouds are caused by CMEs. Most CMEs have large velocities. They are associated with strong magnetic fields and they also cause disturbances in the interplanetary medium. Some of them, which have supersonic velocity, develop a shock wave in front of them. If these shock waves hit the Earth, they can cause geomagnetic variations (substorms). A magnetic substorm is a series of magnetic disturbances at the Earths magnetic field, which are caused by solar wind disturbances, usually due to a fast solar wind stream with a southern interplanetary magnetic field. A magnetic storm is a much larger and longer in duration disturbance of the geomagnetic field. Both are associated with intense auroras. When the interplanetary magnetic field orientation is such that it has a component antiparallel to some geomagnetic field lines, substorms usually occur as a result of magnetic recombination which can take place. This mainly happens when the interplanetary magnetic field has a southward orientation (Valdivia, 2005 and Gonzalez, 2005). The cosmic ray intensity usually decreases substantially (Forbush effect) at the Earth or at a spacecraft, when it is shielded by the strongly magnetized plasma (such as the CMEs) which reflects charged particles, and therefore cosmic rays from the Galaxy can not easily enter in the magnetized region (Figs. 15 and 16).
11. Solar cycles in the heliosphere The heliosphere is defined as a large volume around the Sun, which is formed by the solar wind that blows in all directions. The limits of the heliosphere are formed by the balance of pressures of the solar wind inside the heliosphere and the pressure of the interplanetary medium outside, i.e., the pressure of the interstellar medium. As the solar wind varies with the solar cycle so does the solar wind pressure and, as a result, the heliosphere varies in extent as well substantially (see for example Exarhos and Moussas, 1999a,b, 2000a,b, 2003). The interplanetary magnetic field also varies with the solar cycle. During the first decade of interplanetary measurements (1960–1970) the interplanetary magnetic field was found to be almost constant, but since then the magnetic field of the solar wind follows the solar cycle (Veselovsky et al., 2000). The heliospheric termination shock radius (RTs), shown in Fig. 15, can be calculated using the Rankine–Hugoniot conditions (Parker, 1961; Axford, 1985; Exarhos and Moussas, 1999a) based on in-ecliptic measurements since 1961 at 1 AU (OMNI database) and it
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Fig. 15. Shape (latitudinal variation) of the Heliospheric Termination Shock based on Ulysses/SWOOPS measurements during four passes over the solar poles. The results from the first pass are shown with large squares, while progressively smaller squares denote the next ones. Rst is the radius of the heliospheric termination shock. The x-axis is towards Apex, the Sun is moving towards Apex with a speed of 25 km/ s, the z-axis is perpendicular to the ecliptic. Units are in AU.
shows an 11-year periodicity. The calculated magnetic field on the termination shock is anticorrelated with the cosmic ray intensity (Exarhos and Moussas, 1999b). The velocity of the termination shock is also estimated (with values as high as 350 km/s) since it plays an important role in heliospheric turbulence apart from the ion driven one. The heliospheric termination shock radius is calculated from the c equation 2 h i ðcþ1Þ2 c1 2 Kq0 sw RrTS0 u2sw ¼ pstag , where K ¼ cþ1 , assum4c r 2 0 ing qsw ¼ q0 sw r , where p0 sw the solar wind mass density at the reference distance r0 (here 1 AU) and pstag the interstellar pressure at the stagnation point of the heliopause. The solar wind velocity usw is assumed to be constant with radial distance but varying with time (Exarhos and Moussas, 1999a). The (RTS) minimum and maximum are 65 and 125 AU. The mean velocity of the termination shock towards the Sun is 25 km/s and outwards 45 km/s. The magnetosonic velocity inside the termination shock is of the same magnitude (40 km/s) based on the most frequently observed solar wind parameters, T 8000 K, n 0.001/cm3, B 0.06 nT. We suggest that Voyager 1 was very close to the termination shock at 1999 and probably crossed it a few times. These crossings have been observed by Voyager 1 LECP (Krimigis et al., 2003) in 2002 at 85 AU and at 87 AU (Fig. 17). Our estimations suggest that there is also a possibility of multiple encounters of Voyager with the termination shock in 2005–2006. The radius of the termination shock at different heliolatitudes has been calculated using the following equation 2 q ðcþ1Þðu2 þ u2s Þ 0 (Exarhos and Moussas, 2000a) rr0s ¼ 2cðp þ1q1 uc1 2 Þ 1
2 1 1
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which can be solved numerically. Measurements of the Ulysses spacecraft for all pole-to-pole transitions have been used for the calculations. The results show that the termination shock is polar, caudal elongated and time variable (Fig. 15). The shock radius is 85 AU at the nose, 135 AU at the tail and 115 AU at the poles. A time variable model of the heliosphere permits the estimation of cosmic ray modulation (Exarhos and Moussas, 2001, 2003) (Fig. 16). The model consists of concentric solar wind shells with the magnetic field (measured by spacecraft) frozen-in in the solar wind. This magnetized plasma, which fills up all the heliosphere is responsible for the cosmic ray modulation (Forbush, 1954, 1958; Parker, 1958, 1965; Caballero and Valdes-Galicia, 2003a,b). The cosmic ray variations at the Earth, or any other place (Ulysses, Voyager) can be estimated (Exarhos and Moussas, 2001) with a diffusion–convection model. Without calculation of the drift effect, the difference of cosmic ray modulation in even and odd solar cycles cannot be reproduced. Estimations of the radius and shape of the solar wind sonic and Alfve´n critical surfaces, can be made. These surfaces are important for the solar wind acceleration
Fig. 16. Cosmic ray measurements during a period of almost three solar cycles together with a model calculation of cosmic rays at the Earth based on a variable heliosphere which follows the solar cycle. In this model the radius of the heliosphere varies with the solar wind pressure nV 2sw , the heliosphere consists of numerous concentric shells with various magnetic fields, B, as measured by satellites, and the cosmic rays are calculated with a diffusion convection adiabatic declaration model (Exarhos and Moussas, 1999a,b, 2003).
Fig. 17. Temporal variations of the Heliospheric Termination Shock radius (RTs). Krimigis et al. (2003) using Voyager 1 data showed that for two time intervals the s/c probably exited the solar wind, August 2002 at a distance of 85 AU (heliolatitude 34N), and re-entered the solar wind about six months later at 87 AU.
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and the development of the heliospheric magnetic field (deviations from dipole). All critical surfaces are equatorial elongated. The sonic surface is the most asymmetric (polar radius 1.5Rx, equatorial 2Rx). The Alfve´n polar and equatorial radii are 14Rx and 17Rx. The Alfve´n radius follows the 11-year solar cycle while the sonic radius is constant. For the estimation of the radius of the critical surfaces, we need to know the dependence of the solar wind parameters on the radial distance. For this purpose we use the model of Lima and Priest (1993) modified for the case of the solar wind (Exarhos and Moussas, 2000b). The sonic surface is the surface where the solar wind velocity Vr equals the speed of sound cs, Vr(Rs) = cs(Rs), Rs being the sonic surface radius and Cq2YðRðRssÞRÞ 2 ð1 þ lsin2e hÞ ¼ 1. The sonic surface 0 s has an equatorial elongation (polar radius 1.5Rx and equatorial 2Rx) because the solar wind emanating from the polar coronal reaches the sonic speed faster than at the equator, and it is very stable to solar wind changes during both pole-to-pole journeys, while the equatorial radius is almost invariant. Acknowledgements We express out thanks to WDCA, Dr J. King, all the principal investigators and co-investigators of all spacecraft used of OMNI data base (1AU data since 1962), Ulysses spacecraft experiments P.I. and co-I, Voyager 1 and 2 spacecraft. We continue to be grateful to the University of Athens research committee for supporting us with the grant for Space Physics and ARTEMIS IV, as well as PLATON (Franco-Hellenic research programme). We express our thanks to all the organizers of ASSE, WISE INPE/MCT, IAG/USP, CAPES, CNPq, FAPESP, FINEP for their hospitality and the financial support. We express our gratitude especially Dr. A. Chian. Thanks are also due to Mr S. Dimitrakoudis and Mr P. Papaspyrou for critical reading of the manuscript. References Addison, P. The little wave with the big future. Physics World (March), 35–39, 2004. Arneodo, A., Grasseau, G., Holschneider, M. Wavelet transform of multifractals. Phys. Rev. Lett. 20, 2281–2284, 1988. Arneodo, A., Bacry, E., Muzy, J.F. The thermodynamics of fractal revisited with wavelets. Physica A 213, 232–275, 1995. Arneodo, A., Manneville, S., Muzy, J.F. Towards log-normal statistics in high Reynolds number turbulence. Eur. Phys. J. B. 1, 129–140, 1998. Axford, W.I. The solar wind. Sol. Phys. 100, 575–586, 1985. Bai, T., Sturrock, P.A. The 152d periodicity of the solar flare occurrence rate. Nature 327, 601–604, 1987. Bai, T., Sturrock, P.A. The 154d and related periodicities of solar activity as subharmonics of a fundamental period. Nature 350, 141–143, 1991.
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