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The Inﬂuence of Solar System Oscillation on the

Variability of the Total Solar Irradiance

Harald Yndestad∗

Norwegian University of Science and Technology Aalesund, 6025 Aalesund, Norway

Jan-Erik Solheim1,∗

Department of Physics and Technology UiT The Artic University of Norway, 9037 Tromsø,

Norway

Abstract

The total solar irradiation (TSI) is the primary quantity of energy that is

provided to the Earth. The properties of the TSI variability are critical for

understanding the cause of the irradiation variability and its expected inﬂuence

on climate variations. A deterministic property of TSI variability can provide

information about future irradiation variability and expected long-term climate

variation, whereas the non-deterministic variability can only explain the past.

This study of solar variability is based on an analysis of the TSI data series

from 1700 and 1000 A.D., a sunspot data series from 1611, and a solar orbit

data series from 1000. The study is based on a wavelet spectrum analysis.

First the TSI data series are transformed into a wavelet spectrum. Then the

wavelet spectrum is transformed into an autocorrelation spectrum, to identify

stationary, subharmonic and coincidence periods in the TSI variability.

The results indicate that the TSI and sunspot data series have periodic

cycles that is correlated to the solar position oscillation and controlled by gravity

variations from the large planets Jupiter, Saturn, Uranus and Neptune and the

solar dynamo. A possible explanation is forced oscillation gravity between the

∗Corresponding author

Email address: Harald.Yndestad@ntnu.no (Harald Yndestad)

1Retired, Address now: Wilh. Wilhemsen v 71, 1362 Hosle, Norway

Preprint submitted to New Astronomy May 12, 2016

large planets and the solar dynamo.The major solar variability is controlled by

the 12-year Jupiter period and the 84-year Uranus period. The TSI data series

from 1700 has a direct relation to the 84-year Uranus period with subharmonics.

The phase lag between the solar position oscillation and this TSI oscillation is

estimated to about 0.15π(rad/year) for the dominating 84-year period, and is

phase locked to the perihel state of Uranus.

The long TSI data series from 1000 has stationary periods of approximately

125 years and 210 years, which are controlled by the same stationary period of

84 year. The minimum of the 125 year period coincide with the time Uranus

perihel. The 125-year and the 210-year period subsequently produce a new set

of subharmonic periods. The sunspot data series from 1611 has a stationary 12-

year Jupiter period and a stationary period of approximately 210 years, which

are controlled by a 5/2 resonance to the 84-year Uranus period. The study

conﬁrms that the 12-year Jupiter period and the 210-year de Vries/Suess period

have coincidence periods in TSI and sunspot variability. The phase lag between

the solar position oscillation and TSI and sunspot oscillation is estimated to

about 0.7π(rad/year) for the dominating 210-year period.

A model of the stationary periods in TSI and sunspot variability conﬁrms

the results by a close relation to known long solar minimum periods since 1000

and a modern maximum period from 1940 to 2015. The model computes a new

Dalton sunspot minimum from approximately 2025 to 2050 and a new Dalton

period TSI minimum from approximately 2040 to 2065.

Keywords:

solar oscillation, solar irradiation oscillation, wavelet analysis, grand minima

1. Introduction

The total solar irradiation (TSI) is the primary source of energy that is

provided to the Earth’s climate system. A variation in the TSI irradiation

will contribute to a natural climate variation on the Earth. The variability of

the irradiation from the Sun was approximately 0.3% over the last 300 years5

2

(Scafetta & Willson, 2014). A better understanding of the TSI variability prop-

erties is critical for understanding the cause of the irradiation variability from

the Sun. A TSI data series has information that reﬂects the cause of the TSI

variability. If the TSI variability has deterministic oscillating periods, we can

forecast expected TSI variation, whereas a random TSI variability can only ex-10

plain the past. The intermittency of the solar variation is preferably explained

as stochastic noise (Charbonneau, 2010). In this investigation, we introduce a

simple hypothesis: if the TSI variability has a periodic oscillation, the variabil-

ity oscillation must have an oscillation source that inﬂuences the solar energy

oscillation. A possible oscillation source is the oscillating gravity between the15

Sun and the large planets.

1.1. Solar variability

The concept of a perfect and constant Sun, as postulated by Aristotle, was

undisputed for many centuries. Although some transient changes of the Sun

were observed with the naked eye, the introduction of the telescope in approxi-20

mately 1600 demonstrated that the Sun had spots that varied in number and lo-

cation. From 1610 systematic observations were reported. A pattern of sunspot

variations was established when Heinrich Schwabe began regular observations

of sunspots in 1826. He reported a possible period of approximately ten years

(Schwabe , 1844). Wolf (1859) ) presented the opinion that the planets Venus,25

Earth, Jupiter and Saturn modulate the solar variability.

The solar activity cycle (Hathaway, 2015) consists of dark sunspots and

bright regions (faculae) in addition to active regions that display sudden energy

releases (ﬂares). The average cycle length is 11.1 years. During a cycle, the

number of spots increases to a maximum number and then decreases. The30

average lifetime of a sunspot is slightly longer than the solar rotation period.

They are bipolar, with the same magnetic polarity that leads with respect to

the direction of the solar rotation. When the next cycle starts, spots appear

with opposite polarity at high latitudes in both hemispheres, and as the cycle

progress, they appear closer to the Equator.35

3

The 11.1-year sunspot period is referred to as the Schwabe cycle, and is

proposed to be created by the tidal torque from the planets Venus, Earth and

Jupiter (Wilson, 2013). The 22-year magnetic reversal period is referred to as

the Hale period. Scafetta (2012) showed that the 11-year Schwabe sunspot cycle

consists of three periods of 9.98, 10.90 and 11.86 years, which are close to the40

Jupiter/Saturn spring period of 9.93 years, a tidal pattern of Venus, Earth and

Jupiter of 11.07 years and the Jupiter orbital period of 11.86 years. A relation

between the planets periods and sunspot periods indicates the possibility of a

deterministic long-term relation between planet periods and hidden periods in

sunspot data series.45

1.1.1. Sunspot data series

The sunspot number time series is a measure of the long-term evolution of

the solar cycle and the long-term inﬂuence of the Sun on the Earth’s climate.

The relative sunspot number (R) as deﬁned by Wolf (1861) is based on the total

number of individual sunspots nand the number of sunspot groups g, according50

to the formula R=k(10g+n), where kis a correction factor for the observer.

It was introduced to correct for the use of diﬀerent telescopes and observers. R

is referred to as the Z¨urich, Wolf or International Sunspot Number. Today SN

is used for the International Sunspot Number (Clette et al., 2014).

Rudolf Wolf started systematic observations of sunspot numbers in 1849.55

He also collected previous observations to construct daily sunspot numbers to

1750 and a yearly series to 1700. The cycle that started in 1755 became sunspot

cycle 1. The sunspot numbers had to be scaled upwards several times due to

missing spots. By approximately doubling the number of recovered observations

and cleverly interpolating between sparse observations (Hoyt et al., 1994), gaps60

were reduced and the series was extended to the ﬁrst recording of sunspots

by telescope in 1611. The history of the sunspot series and the last extensive

corrections are described by Clette et al. (2014). The revised yearly series, which

is available from the World Data Center SILSO from July 2015, was employed

in our analysis.65

4

Because the standard sunspot series is a composite time series based on sin-

gle spots and groups, the accuracy signiﬁcantly decreases the possibility of going

back in time. Similar to poorer telescopes and locations, smaller spots were diﬃ-

cult to see and frequently lost. To correct for this situation, Hoyt and Schatten

(1998a,b) constructed a new group sunspot number RGthat was normalized70

to the Z¨urich sunspot number. Their series covered the period 1610-1995 and

was based on a larger and more reﬁned observational database. Although the

group sunspot number corresponded to the relative sunspot number in the 20th

century, the maximum group number was 40% lower in the 19th century and

previous centuries (Clette et al., 2014). The group sunspot numbers were re-75

cently revised, and the diﬀerence between the series may now be considered as

random noise. However, during the last two sunspot cycles (nos. 23 and 24),

30% fewer sunspots per group were observed, which may be a sign of changes

in the solar dynamo (Clette et al., 2014).

1.1.2. Solar activity periods – grand maxima and minima80

In the 1890s, G. Sp¨orer and E. W. Maunder (Maunder, 1890) reported that

the solar activity was strongly reduced over a period of 70 years from 1645 to

1715 (Eddy, 1976, 1983). Based on naked-eye observations of sunspots, records

of aurora activity, and a relation between 14C variations and solar activity, a

grand maximum (1100-1250) and the Sp¨orer minimum (1460-1550) were also85

identiﬁed (Eddy, 1976).

The distribution of the solar activity can be interpreted as bi-modal, which

implies distinct modes of activity. The main (regular) mode corresponds to

moderate activity, which has a maxima of the 10-yr average spot number be-

tween 20 spots and 67 spots. In addition, we obtain grand maxima and grand90

minima that are above this range and below this range, respectively (Usoskin

et al., 2014). Studies that employ cosmogenic isotope data and sunspot data

indicate that we are currently leaving a grand activity maximum, which started

in approximately 1940 and is now declining (Usoskin et al., 2003; Solanki et al.,

2004; Abreu et al., 2008).95

5

Because grand maxima and minima occur on centennial or millennial timescales,

they can only be investigated using proxy data, i.e., solar activity reconstructed

from 10Be and 14 C time-calibrated data. The conclusion is that the activity

level of the Modern Maximum (1940-2000) is a relatively rare event, with the

previous similarly high levels of solar activity observed 4 and 8 millennia ago100

(Usoskin et al., 2003). Nineteen grand maxima have been identiﬁed by Usoskin

et al. (2007) in an 11,000-yr series.

Grand minimum modes with reduced activity cannot be explained by only

random ﬂuctuations of the regular mode (Usoskin et al., 2014). They can be

characterized as two ﬂavors: short minima in the length range of 50-80 years105

(Maunder-type) and longer minima (Sp¨orer-type). Twenty-seven grand minima

are identiﬁed with a total duration of 1900 years, or approximately 17% of the

time during the last 11,500 years (Usoskin et al., 2007). An adjustment-free

reconstruction of the solar activity over the last three millennia conﬁrms four

grand minima since the year 1000: Maunder (1640-1720), Sp¨orer (1390-1550),110

Wolf (1270-1340) and Oort (1010-1070) (Usoskin et al., 2007). The Dalton

minimum (1790-1820) does not ﬁt the deﬁnition of a grand minimum; it is more

likely a regular deep minimum that is observed once per century or an immediate

state between the grand minimum and normal activity (Usoskin et al., 2013).

Temperature reconstructions for the last millennium for the northern hemi-115

sphere (Ljungquist, 2010) show a medieval maximum temperature at approx-

imately the year 1000 and a cooling period starting at approximately 1350,

immediately after the Wolf minimum and lasting nearly 500 years, with the

coldest period in what is referred to as the Little Ice Age (LIA) at the time of

the Maunder minimum. A cold period was also observed during the time of the120

Dalton minimum. The Maunder and the Dalton minima are associated with less

solar activity and colder climate periods. In this investigation, minimum solar

activity periods may serve as a reference for the identiﬁed minimum irradiations

in the TSI oscillations.

6

1.2. Total Solar Irradiance125

The total solar irradiance (TSI) represents a direct index for the luminosity

of the Sun measured at the Earths average distance from the Sun. The solar

luminosity was previously considered to be constant, and the TSI was also

named the solar constant. Since satellite observations started in 1979, the total

solar intensity (TSI) has increased by approximately 0.1% from the solar minima130

to the solar maxima in the three observed sunspot periods. The variation in

the TSI level does not adequately explain the observed variations in the global

temperature. In addition to the direct eﬀect, however, many indirect eﬀects

exist, such as UV energy changes that aﬀect the production of ozone, solar

wind modulation of the galactic cosmic ray ﬂux that may aﬀect the formation135

of clouds, and local and regional eﬀects on temperature, pressure, precipitation

(monsoons) and ocean currents. The Paciﬁc Decadal Oscillation (PDO) and

the North Atlantic Oscillation also show variations that are related to the phase

of the TSI (Velasco & Mendoza, 2008). A signiﬁcant relation between sunspots

and ENSO data has also been observed (Hassan et al., 2016).140

Composite TSI records have been constructed from a database of seven in-

dependent measurement series that cover diﬀerent periods since 1979. Diﬀerent

approaches to the selection of results and cross-calibration have produced com-

posites with diﬀerent characteristics: the Active Cavity Radiometer Irradiance

Monitor (ACRIM) and the Physikalisch-Meteorologisches Observatorium Davos145

(PMOD) series. The ACRIM composite uses the TSI measurements that were

published by the experimental teams (Willson, 2014), whereas the PMOD com-

posite uses a proxy model that is based on the linear regression of sunspot

blocking and faculae brightening against satellite TSI observations (Fr¨olich &

Lean, 1998). To construct a TSI from a previous time period, two diﬀerent ap-150

proaches are employeda reconstruction that is based on several diﬀerent proxies

for the suns irradiance (ACRIM) or a statistical approach (PMOD). Proxies

for the Suns irradiance include the equatorial solar rotation rate, the sunspot

structure, the decay rate of individual sunspots, the number of sunspots without

umbra, the length and decay rate of the sunspot cycle, and the solar activity155

7

level.

Hoyt & Schatten (1993) constructed an irradiance model that was based on

the solar cycle length, cycle decay rate, and mean level of solar activity for the

period 1700-1874. From 1875-1992, a maximum of ﬁve solar indices were em-

ployed. The correlation between these indices and the phase coherence indicated160

that they have the same origin. Hoyt & Schatten (1993) interpret this ﬁnding

as a response to convection changes near the top of the convection zone in the

Sun. All solar indices have maxima between 1920 and 1940; the majority of

the maxima occur in the 1930s. The Hoyt-Schatten irradiance model has been

calibrated and extended with the newest version of ACRIM TSI observations165

(e.g. Scafetta & Willson, 2014, Fig. 16); it is employed in this analysis. In the

following section, this reconstruction is referred to as TSI-HS. A mostly rural

Northern Hemisphere composite temperature series 1880 -2013 show strong cor-

relation with the TSI-HS reconstruction, which indicates a strong solar inﬂuence

on Northern Hemisphere temperature (Soon et al., 2015).170

The TSI-HS series covers the period from 1700-2013. To investigate longer

periods to search for minimum periods, we have employed a statistical TSI index

that was estimated by Velasco Hererra et al. (2015) from 1000 to 2100. The

index, which is referred to as TSI-LS, is estimated by the least squares support

vector machine (LS-SVM) method, which is applied for the ﬁrst time for this175

purpose. The method is nonlinear and nonparametric. The starting point is a

probability density function (PDF) that was constructed from the PMOD and

ACRIM composites. The function describes how many times a certain level

of TSI has been observed. From this normalized annual power, anomalies are

constructed. The TSI between 1610 and 1978 was determined by the LS-SVM180

method using the group sunspot number as an input after calibration between

1979 and 2013 with the ACRIM or PMOD composites. To estimate the TSI

from 1000 to 1510 and from 2013 to 2100, the LS-SVM method and a nonlinear

autoregressive exogenous model (NARX) were employed. In this study, we

have employed the TSI reconstruction that was calibrated by the ACRIM TSI185

composite (Velasco Hererra et al., 2015).

8

1.3. Solar energy oscillation

An oscillation TSI variability is produced by irradiation from an oscillating

energy source. This oscillation energy source may be the solar inertial motion,

processes in the interior of the Sun, solar tide and/or solar orbit oscillation190

around the solar system barycenter (SSB). The energy source for the solar ac-

tivity is the deceleration of the rotation of the Sun by magnetic ﬁeld lines that

are connected to interplanetary space. The solar wind carries mass away from

the Sun; this magnetic braking causes a spin down of the solar rotation. Part

of the decrease in rotational energy is the energy source for the solar dynamo,195

which converts kinetic energy to electromagnetic energy.

The classical interpretation of the solar dynamo is that it is placed in the

transition zone between convection and radiation near the solar surface: the

tachocline, approximately 200,000 km below the surface. Strong electric cur-

rents originate by the interaction between the convection and the diﬀerential200

solar rotation, which causes the formation of strong magnetic ﬁelds, which rise

to the surface and display various aspects of solar activity, such as spots, facu-

lar ﬁelds, ﬂares, coronal mass emissions, coronal holes, polar bright points, and

polar faculae, after having detached, as described by De Jager & Duhau (2011).

They explain the 22-year Hale cycle as attributed to magneto-hydrodynamic205

oscillations of the tachocline. This period is not constant and persisted for ap-

proximately 23 years prior to the Maunder Minimum, during which it increased

to 26 years. During the maximum of the last century, this period was as brief as

21 years. Gleissberg (1958, 1965) discovered a cycle of approximately 80 years

in the amplitude of the sunspot numbers. It is interpreted as the average of210

two frequency bandsone band from 50-80 years and one band from 90-140 years

(Ogurtsov et al., 2002). An examination of the longest detailed cosmogenic

isotope record (INTCAL98) of 14C abundance, with a length of 12,000 years,

reveals an average Gleissberg cycle period of 87.8 years. It is resolved in two

combination periods of 91.5±0.1 and 84.6±0.1 yr (Peristykh & Damon, 2003).215

Proxies that describe the magnetic ﬁelds in the equatorial and polar regions

can describe the variability of the tachocline. A proxy for the equatorial (or

9

toroidal) magnetic ﬁeld is Rmax (the maximum number of sunspots in two suc-

cessive Schwabe cycles), and a proxy for the maximum poloidal magnetic ﬁeld

strength is aamin (the minimum value of the measured terrestrial magnetic ﬁeld220

diﬀerence). In a phase diagram based on theRmax and aamin values, two Gleiss-

berg cycles (1630-1724) and (1787-1880) are shown (Duhau & De Jager, 2008).

The years 1630 and 1787 represent transition points, where phase transitions

to the grand episodes (Maunder and Dalton minima) occurred. The lengths

of the two Gleissberg cycles were 157 years and 93 years. The next Gleissberg225

cycle lasted 129 years until 2009 with an expected phase transition to a high

state in 1924. Duhau & De Jager (2008) predicted that the transition in 2009

indicates a transition to a Maunder-type minimum that will start with cycle 25

in approximately 2020.

In addition to the variable Gleissberg period, a de Vries period from 170-260230

years is observed in the 14C and 38Cl records. This period is fairly sharp with

little or no variability (Ogurtsov et al., 2002). Almost no existing models for

the solar activity predicted the current weak cycle 24. A principal component

analysis of full disc magnetograms during solar cycles 21-23 revealed two mag-

netic waves that travel from opposite hemispheres with similar frequencies and235

increasing phase shifts (Shepherd et al., 2014; Zharkova et al., 2015). To under-

stand this phase shift they introduce a non-linear dynamo model in a two-layer

medium with opposite meridional circulation. One dynamo is located in the

surface layer and the other dipole deeply in the solar convection zone. The solar

poloidal ﬁeld is generated by these two dynamos in diﬀerent cells with oppo-240

site meridional circulation. The observed poloidal-toroidal ﬁelds have similar

periods of oscillation with opposite polarities that are in an anti-phase every

11 years, which explains the Schwabe period. The double-cell meridional cir-

culation ﬂow is also detected with helioseismology by HMI/SDO observations

(Zhao et al., 2013). Extrapolations backward of these two components revealed245

two 350-year grand cycles that were superimposed on a 22-year cycle. The beat

between the two waves shows a remarkable resemblance to the sunspot activity,

including the Maunder and Dalton minima, and forecasts a deep minimum in

10

this century. The low frequency wave has a variable period length from 320 year

(in 18-20 centuries) to 400 year predicted for the next millennium.250

Another model is based on the observation that the thermal relaxation time

in the convection zone is on the order of 105years (Foukal et al., 2009), which

is too long to explain the rapid decay of the magnetic ﬁeld during one solar

cycle. A simple and elegant solution is to place the dynamo in small bubbles

in the solar core, which change polarity every cycle due to interaction with255

the interplanetary magnetic ﬁeld (Granpierre, 2015). The liberated rotational

energy then forms buoyant hot bubbles that move toward the solar surface.

These bubbles are observed on the solar surface as precursors for large ﬂares.

The largest ﬂares have a high probability of appearing near the closest position

of one or more of the tide-producing planetsMars, Venus, Earth and Jupiter260

(Hung, 2007; M¨orner et al., 2015). The energy of the hot bubbles is boosted

by thermonuclear runaway processes in the bubbles, which appear at the solar

surface as hot areas with a frozen magnetic ﬁeld. In this process, planetary

eﬀects serve an important role (Granpierre, 1990, 1996; Wolf & O’Donovan,

2007; Scafetta, 2012).265

1.4. External forcing generated by the planets

Although the various dynamo models can explain the variations to some

extent, few or no constraints on the periods exist. The majority of the expla-

nations operate with a range of possible periods. The models do not explicitly

determine whether the observed periods are random and stochastic or if some270

period-forcing from external or internal sources occurs. In the following section,

we investigate the external forcing that is generated by the planets in the solar

system.

1.4.1. Solar inertial motion

Charv´atov´a & Heida (2014) have classiﬁed the solar inertial motion (SIM)275

in an ordered (trefoil) pattern with a length of approximately 50 years and dis-

ordered intervals. Exceptionally long (approximately 370 years) trefoil patterns

11

appear with a 2402-year period (Hallstadt period). They determined that the

deepest and longest solar activity minima (of Sp¨orer and Maunder types) ap-

peared in the second half of the 2402-year cycle, in accordance with the most280

disordered type of SIM. The Dalton minimum appeared during a mildly disor-

dered SIM (1787–1843), which repeats from 1985-2040. The solar orbit in the

period 1940-2040, which is shown in Figure 1, demonstrates this phase.

1.4.2. Interior of the Sun as a rotating star

Wolf & Patrone (2010) have investigated how the interior of a rotating star285

can be perturbed when the star is accelerated by orbiting objects, as in the solar

system. They present a simple model in which ﬂuid elements of equal mass

exchange positions. This exchange releases potential energy (PE) that is only

available in the hemisphere that faces the barycenter of the planetary system,

with a minor exception. This eﬀect can raise the PE for a few well-positioned290

elements in the Suns envelope by a factor of 7, which indicates that a star with

planets will burn nuclear fuel more eﬀectively and have a shorter lifetime than

identical stars without planets. However, occasional mass exchanges occur near

the solar center, which activate a mixed shell situated at 0.16rswhere rsis the

solar radius. For this reason, the close passages of the barycenter are important295

because they can cause negative pulses in the PE. The energy is a result of the

roto-translational dynamics of the cell around the solar system barycenter. An

analysis of the variation of the PE storage reveals that the maximum variations

correspond to the documented grand minima of the last 1000 years because

the PE minima are connected to periods in which the Sun moves close to the300

barycenter. Large reductions in the PE values occur when the giant planets are

quasi-aligned, which occurred in approximately 1632, 1811 and 1990, separated

by 179 years (Jose cycle). Because the planetary positions never exactly repeat,

the PE variations show a complex pattern that creates diﬀerent minima (Cionco

& Soon, 2015).305

12

1.4.3. Solar inertial oscillations

The complex planetary synchronization structure of the solar system has

been known since the time of Pythagoras of Samos (ca. 570-495 BC). Jose

(1965) showed that the solar center moves in loops around the solar system

barycenter (SSB). The average orbital period of 19.86 years corresponds to310

the heliocentric synodic period of Jupiter and Saturn. The modulation of the

orbit by the outer planets Uranus and Neptune produces asymmetries in the

orbital shape and period variations between 15.3 and 23.4 years (Fairbridge &

Shirley, 1987). The solar motion diﬀers from the Keplerian motion of planets

and satellites in important ways. For instance, the velocity is some time highest315

when the distance from the Sun to the SSB is largest, and the solar angular

momentum may vary by more than one order of magnitude over a period of ten

years (Blizzard, 1981). An analysis of solar orbits from A.D. 816 – 2054 covered

seven complete Jose cycles of 179 years and indicated that prolonged minima

can be identiﬁed by two parameters: the ﬁrst parameter is the diﬀerence in320

axial symmetry of the orbit, and the second parameter is the change in angular

momentum (torque) about SSB. Based on these criteria, a new minimum should

begin between 1990 and 2013 and end in 2091 (Fairbridge & Shirley, 1987).

The distance of the Sun from the barycenter, the velocity, and the angular

momentum show the same periodic behavior. The motion of the solar center325

around the SSB is typically prograde; however, in 1811 and 1990, the Sun

occasionally passes near the SSB in a retrograde motion. Because the 1811

event occurred at the time of the Dalton minimum, a new minimum may occur

in approximately 1990 (Cionco & Soon, 2015).

Scafetta (2014)reviews the investigation of the patterns that are described by330

the Sun and planets. He concludes that modern research shows that the plane-

tary orbits can be approximated by a simple system of resonant frequencies and

that the solar system oscillates with a speciﬁc set of gravitational frequencies,

many of which range between three and 100 years, that can be constructed as

harmonics of a base period of ∼178.78 years.335

13

1.4.4. Solar tidal oscillation

The tidal elongation at the solar surface is on the order of 1-2 mm from

the planets Venus and Jupiter with less tides from the other planets. Scafetta

(2012) proposed that tidal forces, torques and jerk shocks act on and inside the

Sun and that the continuous tidal massaging of the Sun should involve heating340

the core and periodically increasing the nuclear fusion rate. This action would

amplify weak signals from the planets with a factor ∼4×106. Even if the

amplitude is small in the direction of a planet, it creates a wave that propagates

with the velocity of the planet. If the planet has an elliptical orbit, the variation

in distance creates a disturbance that will aﬀect the nuclear energy production345

in the center of the Sun. Since more planets participate, the eﬀect will be a

combination of phases and periods, which can be highly nonlinear.

Our hypothesis is that the solar position oscillation (SPO) represents an in-

dicator of the tidal and inertial interaction between the giant planets Jupiter,

Saturn, Neptune, Uranus and the Sun. The SPO can be calculated from plane-350

tary Ephemeris as the movement of the Sun around the Solar System Barycenter

(SSB). In section 2, we describe the methods and data sets used to demonstrate

a connection between SPO and TSI and SN variations. In section 3, we present

the results; in section 4, we discuss the results and relate them to other investi-

gations. We conclude the paper in section 5. Because the solar system and its355

planets has a long lifetime, we can expect forces in the same direction over long

periods of time that may have a strong eﬀect on long periods.

2. Materials and methods

2.1. Data

The motive of the study is to identify possible stationary periods in TSI360

variability. In this study possible stationary periods are represented by ﬁrst

stationary periods, subharmonic periods and coincidence periods. First sta-

tionary periods have a period Tin the data series. Subharmonic periods have

periods n∗Tfor n= 2,3,4. . . Coincidence periods have a coincidence between

14

two ore more periods and may be represented by n∗T1=m∗T2. Coincidence365

periods are stationary periods and introduce a new set of subharmonic periods.

The study compares the identiﬁed stationary periods and period phase in two

TSI data series, a sunspot data series and a SPO data series.

The SPO data series represents an indicator of the oscillating tidal and in-

ertial interaction between the Sun and the large planets. The large planets370

have the following periods (in years): Jupiter P(J, 11.862), Saturn P(S, 29.447)

, Uranus P(U, 84.02) and Neptune P(N, 164.79). The SPO covers the pe-

riod from 1000 to 2100, where S P Oxrepresents the x-direction of the xyz-

vector. The source of the SPO data series is the JPL Horizon web interface

(http://ssd.jpl.nasa.gov/horizons.cgi#top),which is based on the Re-375

vised July 31, 2013 ephemeris with the ICRF/J2000 reference frame, down-

loaded 30.09.14 and at subsequent dates.

The total solar irradiance (TSI-HS) data series (e.g. Scafetta & Willson,

2014, Fig. 16) covers the period from 1700 to 2013. The source of the data series

is Scafetta (personal communication. Dec. 2013). The total solar irradiation380

(TSI-LS) is based on the LS-SVM ACRIM data series (Velasco Hererra et al.,

2015) and covers the period from 1000 to 2100. The source of the TSI-LS data

series is Velasco Herrera (21.09.14. Personal communication). The sunspot

data series is the group sunspot numbers from 1610 to 2015. The source of

the sunspot data series is SILSO (The World Data Center for the production,385

preservation and dissemination of the international sunspot number).

2.2. Methods

Possible stationary periods in the data series are identiﬁed in two steps. First

a wavelet transform of the data series separates all periods in into a wavelet

spectrum. The autocorrelation for wavelet spectrum then identiﬁes dominant390

ﬁrst stationary periods, subharmonic periods and the coincidence periods. Prior

to the wavelet analysis, all data series are scaled by

x(t)=(y(t)−E[y(t)])/var(y(t)) (1)

15

where y(t) is the data series, E[y(t)] is the mean value, var(y(t)) is the

variance and x(t) is the scaled data series. The data series are scaled to compare

the amplitudes from the oscillation periods and reduce side eﬀects in the wavelet395

analysis.

A wavelet transform of a data series x(t) has the ability to separate periods

in the data series into a wavelet spectrum. The wavelet spectrum is computed

by the transformation

Wa,b(t)=1

√aZ

R

x(t)ψt−b

adt (2)

where x(t) is the analyzed time series, ψ() is a coif3 wavelet impulse function400

(Daubechies, 1992; Matlab, 2015); which is chosen for its symmetrical perfor-

mance and its ability to identify symmetrical periods in data series; W(a, b(t))

represents the computed wavelet spectrum, the parameter arepresents a time-

scaling parameter, and the parameter brepresents a translation in time in the

wavelet transformation. When translating b= 0 and s= 1/a, the wavelet405

spectrum W(s, t) represents a set of moving correlations between x(t) and the

impulse function ψ() over the entire time series x(t). The relationship between

the wavelet sand a sinus period Tis approximately T∼1.2swhen using the

coif3 wavelet function. In this investigation, the wavelet spectrum W(s, t) has

the spectrum range s= 0,1,2. . . 0.6N, where Nis the number of samples in410

the data series.

An autocorrelation transformation of the wavelet spectrum W(s, t) identiﬁes

ﬁrst periods, subharmonic periods and coincidence periods as maximum values

in the computed set of autocorrelation functions. The set of autocorrelation

functions are estimated by the transformation415

R(s, m) = E[W(s, t)W(s, t +m)] (3)

where R(s, m) represents the correlations between samples, at a distance m

years, for a wavelet sin the wavelet spectrum W(s, t).

16

3. Results

3.1. Sun Position Oscillation

The Sun moves in a closed orbit around the barycenter of the solar system.420

Figure 1 shows the SPO in the ecliptical plane from 1940 to 2040. The solar

system oscillation (SSO) is caused by the mutual gravity dynamics between

the planet system oscillation (PSO) and the solar position oscillation (SPO).

The solar position oscillation has oscillations in the (x, y, z ) directions; they are

represented by the data series SP Ox, S P Oy and SP O z. The movement looks425

rather chaotic, as shown in Figure 1, because it mirrors the movements of the

planets in their orbits. A ﬁrst step in this investigation is to identify stationary

periods and phase relations in the solar position between year 1000 and year

2100.

A wavelet spectrum represents a set of moving correlations between a data430

series and a scalable wavelet pulse. When the data series in the y-direction

-SP O y - is transformed to the wavelet spectrum W spoy(s, t), the spectrum

represents a collection of dominant periods in the SPOy data series. A visual

inspection of the wavelet spectrum W spoy(s, t) shows a long-term dominant

period of approximately P spoy(164) years. This period has a coincidence to the435

Neptune period P(N, 164.79). The data series SP Ox and SP Oy have the same

periods; however, SP Oy has a 90-degree phase delay.

The wavelet spectrum W spoy (s, t) is transformed to a set of autocorrelation

functions Rspoy(s, m), as shown on Figure 2, where each colored function rep-

resents a single autocorrelation. The set of autocorrelations Rspoy(s, t) shows440

the identiﬁed stationary periods in the wavelet spectrum W spoy(s, t). The ﬁrst

maximum represents the correlation to a ﬁrst stationary period. Subharmonic

periods have a maximum correlation at a distance ﬁrst period∗nwhere n=

1,2,3. . . .Rspoy(s, m) identiﬁes stationary periods P(spoy , 12) for Rspoy(12) =

0.98, P (spoy, 29) for Rspoy(29) = 0.95, P (spoy, 84) for Rspoy(84) = 0.9 and445

P(spoy, 164) for Rspoy(164) = 0.9. The same periods are associated with the

PSO periods P(J, 11.862), P (S, 29.447), P (U, 84.02) and P(N, 164.79), which in-

17

dicates that the planets Jupiter, Saturn, Uranus and Neptune in the planetary

system are controlling the SPO.

A coincidence between subharmonic periods will amplify the coincidence pe-450

riod and introduce a new sett of stationary periods. The autocorrelation spec-

trum Rspoy(s, m) of Figure 2 shows a set of subharmonic periods - P(spoy , n ∗

12), P (spoy, n ∗29) and P(spoy, n ∗84) - where n= 1,2,3. . ... The identiﬁed

coincidence periods have mean values of

(P(spoy, 5∗12) + P(spoy, 2∗29))/2 = P(spox, 59) for R W (spoy, 59) = 0.95,455

P(spoy, 7∗12)+P(spoy, 3∗29)+P(spoy , 84))/3 = P(spoy, 85) for RW (spoy, 85) =

0.9,

(P(spoy, 10 ∗12) + P(spoy, 4∗29))/2 = P(spoy , 118) for Rspoy(118) = 0.9

and (P(spoy, 2∗84) + P(spoy, 164))/2 = P(spoy , 166) for Rspoy(166) = 0.9.

Figure 2 reveals that the majority of the SPO periods are mutually related by460

resonance. The new modulated periods are P(spoy, 59) and P(spoy, 118).

The stationary long wavelet periods W spoy(84, t) and W spoy(164, t) have

maxima in approximately 1820, and W spoy(29, t) has a maximum in approx-

imately 1812. The identiﬁed stationary periods may be transformed to the

model:465

P(spoyc, 29, t) = RW (spoy, 29) cos(2π(t−1812))/29.447) (4)

P(spoyc, 84, t) = RW (spoy, 85) cos(2π(t−1820))/84.02) (5)

P(spoyc, 164, t) = RW (spoy, 164) cos(2π(t−1820))/164.97) (6)

By this model, the year 1820 may serve as a phase reference for the SP Oy

periods, TSI variability and solar variability (SN). The data series S P Ox in has

the same stationary periods but a diﬀerent phase. P(spoxc, 84, t) has a max-

imum at approximately 1797, which represents a phase shift of approximately

π/2.P (spoxc, 164, t) has a maximum at approximately 1779. The maxima in470

SP oy and S P Ox corresponds to minima in S P O ¨yand SP O ¨x. The determin-

istic model has the sum P(spoyc, 29, t) + P(spoy c, 84, t) + P(spoyc, 164, t) and

a maximum in approximately 1812.

18

3.2. TSI-HS variability

The total solar irradiation (TSI) represents the measured irradiation Wm−2

475

from the Sun to the Earth. Figure 3 shows an annual mean total solar irradiance

(TSI-HS) data series (Scafetta & Willson, 2014) that covers the period from 1700

to 2013. A simple visual inspection of this data series shows some variability

properties. The TSI-HS data series irradiation has ﬂuctuations of approximately

3-4 Wm−2. The TSI ﬂuctuations have minima in approximately 1700 (or480

before), 1800, 1890, and 1960, with gaps of approximately 100, 90, and 70

years, or a mean minimum period of approximately 86 years. The TSI-HS data

series has maxima in 1770, 1830, and 1950, with gaps of approximately 60 and

120 years. The mean maximum ﬂuctuation period in the TSI-HS data series

is approximately 75 years or 11 years less than the mean minimum period.485

Transformation of the TSI data series into a wavelet spectrum may identify

stationary periods.

The transformed wavelet spectrum W hs(s, t) represents a set of separated

wavelet periods from the TSI-HS data series. Figure 4 shows the computed

wavelet spectrum of the TSI-HS data series from 1700 to 2013. In this presen-490

tation, the wavelet scaling range is s= 1 . . . 0.6N, and the data series contains

N= 313 data points. A visual inspection of the TSI wavelet spectrum shows

the dominant periods in the TSI data series in the time window between 1700

and 2013. The long wavelet period has a maximum in 1760, 1840, 1930, and

2000, with a mean gap of approximately 80 years.495

The autocorrelation spectrum Rhs(s, m) of the wavelet spectrum W hs(s, t)

identiﬁes hidden stationary periods in the wavelet spectrum. The maximum

values in the autocorrelation spectrum Rhs(s, m) represent a correlation to sta-

tionary periods in the TSI-HS wavelet spectrum. Figure 5 shows the autocor-

relation spectrum Rhs(s, m) of the wavelet spectrum W hs(s, t) of the TSI-HS500

data series.

A study of the autocorrelation spectrumRhs(s, m) shows a set of stationary

periods in the W hs(s, t) wavelet spectrum. The identiﬁed ﬁrst cause station-

ary periods comprise the period set P(hs, 11) for Rhs(11) = 0.55, P (hs, 49) for

19

Rhs(49) = 0.55, P (hs, 86) for Rhs(86) = 0.65 and P(hs, 164) for RW hs(164) =505

0.7. The identiﬁed stationary periods are associated with the PSO periods

P(J, 11.862), P (S, 29.447), P (U, 84.02) and P(N , 164.79). The identiﬁed sta-

tionary period P(hs, 49) is explained by a modulation between the Saturn os-

cillation and the Neptune oscillation: 2/(1/P (S, 29.447) + 1/P (N, 164.79)) =

P(S, N, 49.96). This ﬁnding indicates that the TSI-HS variability is related510

to the solar position oscillation, which is controlled by the planet oscillation

from the large planets Jupiter, Saturn, Uranus and Neptune. Additional analy-

sis indicates that the dominant wavelet periods W hs(11, t) and W hs(49, t) are

mean estimates. W hs(11, t) has phase disturbance and W hs(49, t) has a phase-

reversal, as shown in Figure 6. They do not have a stable phase and represent515

mean periods.

Figure 6 shows the identiﬁed dominant stationary wavelet periods W hs(49, t),

W hs(86, t) and W hs(165, t) from the autocorrelation functions in Rhs(s, m). It

shows that the wavelet period P(hs, 49) has a time-variant phase and is not a

stable period. The TSI-HS wavelet periods W hs(49, t) and W hs(86, t) have a520

negative value coincidence in the period from 1786 to 1820. W spoy(84, t) has an

estimated maximum velocity and W spox(84, t) has maximum state at approx-

imately 1797. The dominant wavelet period W hs(84, t) has a minimum state

at approximately 1803, or a phase delay of approximately 0.15πbetween the

W spox(84, t) maximum and the minimum W hs(84, t). Uranus was in perihelion525

in 1798. This indicates a relation between a minimum Uranus distance to the

Sun and a minimum in TSI-HS.

The correlation between the TSI-HS data series and the identiﬁed domi-

nant wavelet periods W hs(49, t) + W hs(86, t) + W hs(164, t) is estimated to be

R= 0.93, Q = 46.6 (Pearson correlation coeﬃcient) in N= 312 samples. The530

correlation R= 0.93 reveals a close relation between the TSI-HS variability and

the solar position oscillation, which is controlled by Jupiter, Saturn, Uranus and

Neptune.

20

3.2.1. Deterministic model

The identiﬁed stationary periods W hs(86, t) and W hs(164, t) may be repre-535

sented by a deterministic model from the sum of the stationary cosine functions:

P(hsc, 84, t) = −Rhs(86) cos(2π(t−1803)/84.02) (7)

P(hsc, 164, t) = −Rhs(164) cos(2π(t−1860)/164.97) (8)

P(hsc, t) = P(hsc, 84, t) + P(hsc, 164, t) (9)

where R(hs, 86) and R(hs, 164) represent estimated correlations in the autocor-

relation. The phase relation between the maximum value of P spox(84, t) in 1797

and the minimum value P(hsc, 84, t) in 1803 is approximately 0.15π(rad/year).

The year 1797 is also the year of Uranus in perigel. We also notice that the min-540

imum value of P(hs, 164, t) (Eq. 8) is close to the time of Neptune in perihel.

This indicates a delayed response from the SPO periods on the TSI-HS vari-

ability. The correlation between the 84-year wavelet periods P(hsc, 84, t) and

W(hsc, 84, t) is estimated to be Rhs, hsc(84) = 0.83. The correlation between

the 164-year periods is Rhs, hsc(164) = 0.88. The correlation between the data545

series TSI-HS and the sum P(hsc, t) is estimated to be R= 0.55. These results

indicate that the dominant wavelet periods W hs(84, t) and W hs(164, t) have

stable phases from 1700 to 2013.

From the deterministic model (Eq. 9) of the data series TSI-HS, we estimate

grand minimum periods when P(hsc, t)≤ −1. These minima, which are com-550

pared with named solar minima, are shown in Table 2. The next deep minimum

is estimated at approximately 2050. The TSH-HS data series can estimate time

period up to a maximum of (2013-1700)/2=156 years and supports reasonable

good estimates of periods of approximately one hundred years. Longer time

period estimates require longer data series.555

3.3. TSI-LS variability

The TSI-LS data series (Figure 7) covers a period of 1100 years from 1000 to

2100, where the time period from the present to 2100 is forecasted. A realistic

21

hundred-year forecast or hindcast has to be based on possible hidden determin-

istic periods in the data series. A coherence analysis of the wavelet spectra560

W hs(s, t) and W ls(s, t) shows a coherency Chs, ls = 0.8−0.95 for periods be-

tween 48 years and 86 years, which indicates that the TSI-HS data series and

the TSI-LS data series have the same periods from 48 - 86 years from 1700 to

2013.

The data series TSI-LS is analyzed by computing the wavelet spectrum565

W ls(s, t) and the autocorrelation spectrum Rls(s, m), the latter shown in Figure

8. The identiﬁed stationary periods in the autocorrelation spectrum Rls(s, m)

are P(ls, 11) for Rls(11) = 0.8, P(ls, 18) for Rls(18) = 0.3, P(ls, 29) for

Rls(29) = 0.2, P(ls, 83) for Rls(83) = 0.17, P(ls, 125) for Rls(125) = 0.6,

P(ls, 210) for Rls(210) = 0.35 and P(ls, 373) for Rls(373) = 0.5, the last not570

shown in Fig. 8. These periods are associated with the identiﬁed stationary

periods in the TSI-HS data series, the SPO data series periods and the PSO pe-

riods P(J, 11.862), P(S, 29.447) and P(U, 84.02). The diﬀerence is the smaller

correlation value in the autocorrelation Rls(s, m). Smaller correlation values

may be explained by phase errors in this long data series.575

The autocorrelation spectrumRls(s, m) (Figure 8) shows coincidence peri-

ods between P(ls, 3∗11) = P(ls, 33) and P(ls, 2∗18) = P(ls, 36), between

P(ls, 5∗11) = P(ls, 55) and P(ls, 3∗18) = P(ls, 54), and between P(ls, 8∗11) =

P(ls, 88) and the ﬁrst period P(ls, 83). The coincidence period P(ls, 55) in-

troduces the subharmonic periods P(ls, n ∗55) for n= 1,2,3. . .. The new580

information in Rls(s, m) is an identiﬁcation of the dominant ﬁrst cause pe-

riods P(ls, 18), P(ls, 125) and P(ls, 210). These periods have a combination

resonance that is created by a 2/3 resonance and a 5/2 resonance. The sta-

tionary model has a perfect relation to the Jupiter period and the Uranus

period when P(ls, 18) = P(ls, 3∗11/2) is related to P(ls, 3∗11.862/2 =585

17.793), P(ls, 126) = P(ls, 3∗84/2) is related to P(ls, 3∗84.02/2 = 126.03) and

P(ls, 210) = P(ls, 5∗84/2) is related to the period P(ls, 5∗84.02/2 = 210.05).

The period P(ls, 125) introduces a set of subharmonic periods P(ls, n ∗125),

where n= 1,2,3. . . . In this investigation, we have only selected the third sub-

22

harmonic period P(ls, 3∗126.03 = 378.09), which is the most dominant.590

The autocorrelation spectrum Rls(s, m) shows that the period W ls(125, t)

represents the dominant amplitude variability in the TSI-LS data series. Fig-

ure 9 shows the identiﬁed long-term stationary periodsW ls(124, t), W ls(210, t),

W ls(373, t) and the mean of the periods. The correlation between TSI-LS

and the mean is estimated to be R= 0.7 for N=1100 samples and the qual-595

ity Q=27.4. The mean of the identiﬁed wavelets W ls(125, t), W ls(210, t),

W ls(373, t) has a negative state in the periods (1000-1100), (1275-1314), (1383-

1527), (1634-1729), (1802-1846) and (2002-2083). The mean has a minimum

state in the years 1050, 1293, 1428, 1679, 1820, and 2040.

3.3.1. Deterministic model600

The identiﬁed dominant periods W ls(125, t), W ls(210, t), and W ls(373, t)

may be represented by the deterministic stationary model from the sum of the

cosine functions

P(lsc, 126, t) = Rls(125) cos(2π(t−1857)/(3 ∗84.02/2) (10)

P(lsc, 210, t) = Rls(210) cos(2π(t−1769)/(5 ∗84.02/2) (11)

P(lsc, 378, t) = Rls(373) cos(2π(t−1580)/(9 ∗84.02/2) (12)

P(lsc, t) = P(lsc, 126, t) + P(lsc, 210, t) + P(lsc, 378, t) (13)

where Rls(125), Rls(210) and RW l s(373) represent the maximum period corre-

lations in the autocorrelation Rls(s, m). The correlation between the 125-year605

wavelet period W(ls, 125, t) and the stationary period P(l sc, 126, t) is estimated

to be Rls, lsc(125) = 0.9 for N=1040 samples and Q=53.7, Rls, lsc(210) = 0.67

for N=1000 and Q=28.9, and Rls, lsc(378) = 0.68 for N=1000 and Q=28.8.

The period P l sc(375, t) has the correlation Rls, lsc(378) = 0.67 to the identi-

ﬁed wavelet period W ls(373, t) for N=1000 samples and Q=2813. The domi-610

nant wavelet periods W l s(125, t) and W ls(375, t) have a stationary period and

an approximately stable phase in the period from 1000 to 2100. A correlation

of long data series is sensitive to phase noise. The sum of the stationary periods

P(lsc, t) represents a mean TSI-LS variability. The correlation to the TSI-LS

23

data series is estimated to be Rlsc, ls(126 + 210 + 378) = 0.55 for N=1100 and615

Q=21.5. This analysis indicates that the TSI-LS variability has been inﬂuenced

by stationary periods that are controlled by the Uranus period P(U, 84.02). A

minimum of P(lsc, 126, t) is in 1794, which is close to the time of Uranus perihel

position, while P(lsc, 2010, t) has a minimum in 1874 which is 0.7πafter the

P spox(max). This indicates that the phase of these periods are synchronized620

with Uranus perihel position.

The deterministic model (Eq.11) of the data series TSI-LS may represent an

index of minimum irradiation periods as shown in Table 2. By this index, the

chosen data series references a TSI minimum when the state is P(lsc, t)≤ −0.5,

a Dalton-type minimum when P(lsc, t)≤ −0.7 and a grand minimum when625

P(lsc, t)≤ −1.0. The identiﬁed minima from this model are P(lsc, t)≤ −1.0 for

the time period (1014-1056); P(lsc, t)≤ −0.5) for (1276-1301); P(lsc, t)≤ −1.0

for (1404-1435), which has a minimum -1.215 in the year 1419; P(lsc, t)≤ −0.5

for (1662-1695) which has a minimum -0.91 in the year 1672; and P(lsc, t)≤

−0.5 for (1775-1819), which has a minimum -0.81 in the year 1796. The com-630

puted subsequent minimum time period is P(lsc, t)≤ −0.5 for (2035-2079),

which has a minimum -0.79 in the year 2057. In this model, a Dalton-type min-

imum has a minimum at approximately -0.7. The Maunder minimum is between

-0.7 and -1.0, as shown in Table 2. The computed minimum -0.79 in the year

2057 indicates an expected Dalton-Maunder-type minimum. The determinis-635

tic model state has a state P(lsc, t)≥+0.5 index for the periods (1093-1134),

(1198-1241) and (1351-1357); P(lsc, t)≥+1.0 index for the period (1582-1610);

P(lsc, t)≥+0.5 for (1945-2013); and P(lsc, t)≥+1.0 for (1959-2001), which

has a maximum 1.4 in 1981.

3.4. Sunspot variability640

The sunspot data series SN(t) is an indicator of the solar variability. Figure

10 shows the group sunspot number data series that covers a period of approx-

imately 400 years from 1610 to 2015. From this 400-year data series, we can

estimate periods of approximately up to 200 years. Periods with few sunspots

24

are associated with low solar activity and cold climate periods. Periods with645

many sunspots are associated with high solar activity and warm climate periods.

If a relation exists between solar periods and climate periods, we may expect a

relation between the hidden periods in the TSI variability and solar variability.

Figure 11 shows the computed wavelet spectrum W sn(s, t) of the SN(t) data

series from 1610 to 2015, with the wavelet scaling parameter s= 1 . . . 6N. A650

visual inspection of the wavelet spectrum shows a maximum at the approximate

years (1750, 1860, 1970), which represents periods of approximately 110 years.

The time from 1750 to 1970 represents a period of 220 years. Temporary periods

of approximately 50 years from approximately 1725 and 1930 may be conﬁrmed

by computing the autocorrelation wavelet spectrum Rsn(s, t).655

The computed set of autocorrelations Rsn(s, m) of the wavelet spectrum

W sn(s, t) is shown in Figure 12. The wavelet spectrum W(sn, t) has the sta-

tionary periods P(sn, 11) for Rsn(11) = 0.73, P(sn, 22) for Rsn(22) = 0.35

and P(sn, 86) for Rsn(86) = 0.35. The identiﬁed period P(sn, 11) repre-

sents the Schwabe cycle and corresponds to the TSI P(tsi, 11), the SPO period660

P(spox, 11) and the Jupiter period P(J, 11.862).

The period P(sn, 11) introduces the subharmonic period P(sn, 5∗11) =

P(sn, 55) for Rsn(55) = 0.43, which introduces the subharmonic periods P(sn, 110)

for Rsn(110) = 0.40 and P(sn, 210) for Rsn(210) = 0.36. The period P(sn, 55)

is a temporary stationary period from 1610 only when P(sn, 110) has a posi-665

tive state. An inspection of P(sn, 55) shows that the period is stationary when

P(sn, 210) has a positive state from 1726-1831 and from 1935. The period

P(sn, 55) shifted to P(sn, 2∗55) when P(sn, 220) has a negative state from 1831-

1935. A possible explanation is an 5/2 relation between the periods P(U, 84.04)

and P(sn, 210) (Eq. 16).670

Figure 12 shows that the period P(sn, 55) has combination resonance peri-

ods with a 3/2 relation P(sn, 3∗55/2 = 84) to the Uranus period P(U, 84.02).

The 3/2 correlation to the P(ls, 84) period and the Uranus period P(U, 84.02)

explains the synchronization between the SN variability and the TSI-LS vari-

ability. The dominant period P(sn, 110) is a coincidence period in the subhar-675

25

monic period P(sn, 2∗55 = 110), which has a combination resonance to the

Neptune period by P(sn, 2∗164,79/3 = 109,86). The long stationary iden-

tiﬁed period P(sn, 210) is related to a 5/2 combination resonance to Uranus

by P(U, 5∗84.02/2 = 210.05). The period P(sn, 210) corresponds to the TSI-

LS period P(ls, 210). The identiﬁed periods have a subharmonic resonance in680

the Jupiter period P(J, 11.862). The correlation between the data series SN(t)

and the dominant wavelet periods W(sn, 55, t) + W(sn, 110, t) + W(sn, 210, t)

is estimated to be R= 0.51 for N=404 and Q=11.8.

3.4.1. Deterministic model

The identiﬁed temporary stationary periods W sn(55, t), W sn(110, t) and685

W ls(210, t) may be represented by a deterministic model

P(snc, 56, t) = Rsn(55) cos(2π(t−1782)/(2 ∗84.02/3) (14)

P(snc, 112, t) = Rsn(110) cos(2π(t−1751)/(4 ∗84.02/3) (15)

P(snc, 210, t) = Rsn(210) cos(2π(t−1770)/(5 ∗84.02/2) (16)

P(snc, t) = P(snc, 56, t) + P(snc, 112, t) + P(snc, 210, t) (17)

where Rsn(56), Rsn(112) and Rsn(210) represent the maximum correlation

in the autocorrelation Rsn(s, m). This model is, however, a simpliﬁed lin-

ear model. Figure 11 shows that the Rsn(55) amplitude is controlled by the

Rsn(110) amplitude, which indicates that the period P(sn, 55) is temporarily690

stable. The correlation between the 55-year wavelet periods W(sn, 55, t) and the

stationary period P(snc, 55, t) is estimated to be Rsn, snc(55) = 0.66 for N=354

samples and Q=16.6. The correlations are Rsn, snc(110) = 0.9 for N=304 and

Q=36 and Rsn, snc(210) = 0.9 for N=304 and Q=36. The correlation between

the sum W(sn, 55, t)+W(sn, 110, t)+ W(sn, 210, t) and the deterministic model695

from (Eq. 17) is estimated to be R= 0.84 for N=304 and Q=29.8. Minimum

states that correspond to negative values of the stationary model correspond

to the observed minima, as shown in Table 2. The model indicates a future

minimum in the period 2018 - 2055 with an extreme value in 2035.

26

This analysis indicates that the sunspot variations is controlled by the Uranus700

period P(U, 84.02), which introduces a 2/3 resonance to the period P(sn, 55, t)

and a 5/2 super-resonance to the P(sn, 210) period. The TSI-LS data series and

the sunspots data series have stationary coincidence periods with P(ls, 11) and

P(sn, 11), P(ls, 125) and P(sn, 110) and with P(ls, 210) and P(sn, 210). The

diﬀerence between the stationary periods P(ls, 125) and P(sn, 110) indicates a705

limited direct relation between the data series.

3.5. Stationary dominant periods and minima

The relations between the identiﬁed dominant periods in SN(t), TSI-HS and

TSI-LS are shown in Table 1, where Ris the autocorrelation of the wavelet

spectrum..710

In Table 2 we compare values of the stationary models P(hsc, t), P (lsc, t)

and P(snc, t) at minima corresponding to the solar activity minima determined

by Usoskin et al. (2007). The grand minimum periods are calculated from the

stationary models in Equations 9, 13 and 17, and compared with Spox and Spoy

maxima. The model P(snc, t) computes a new Dalton sunspot minimum from715

approximately 2025 to 2050; the model P(hsc, t) computes a new Dalton TSI

minimum period (2035-2065), and the model P(lsc, t) computes a new Dalton

TSI minimum period (2045-2070).

The SN model in Eq. 17 is a simpliﬁed linear model. It has a minimum

P(snc, t)≤ −0.5 in 1907-1931, which is not shown in the table. The HS-model720

from Eq. 9 has grand minima in 1200-120 and 1876-1887, which are not shown

in the table. For this model the Dalton minimum is less deep. The LS-model

from Eq. 13 has the maximum index P(lsc, t)≥0.5 for the periods (1093-

1134), (1198-1241), (1351-1357), and (1945-2013) and the grand maximum index

P(lsc, t)≥+1.0 for the periods (1582-1610) and (1959-2001).725

4. Discussion

The study of the TSI variability is based on the TSI-HS data series from

1700-2013, the TSI-LS data series 1000-2100, sunspots data series 1610-2015

27

and a Solar Barycenter orbit data series from 1000-2100. The results are, how-

ever, limited by how well they represent the solar physics and how well the730

methods are able to identify the periods in the data series. The investiga-

tion is based on a new method. The data series are transformed to a wavelet

spectrum to separate periods, and the wavelets are transformed into a set of

autocorrelations to identify the ﬁrst periods, subharmonic periods and coinci-

dence periods. The identiﬁed stationary periods in the TSI and SN series are735

supported by the close relations with the well-known solar position periods and

documented solar minimum periods. The solar orbit data will then provide a

stable and computable reference. We have used the Dalton minimum (1790-

1820) as a reference period, since our two TSI-series and the SN-series cover

this minimum. We notice that maxima in SP Ox and SP Oy corresponds to740

minima in SP O ¨xand S P O ¨ywhich means maximum negative acceleration. We

use P(spoxc, 84, max) = 1797 as a time of reference. This is close to the tim of

Uranus in perihelion (1798) and Neptune in aphelion (1804), which indicates a

possible relation between the distance to these planets and the minimum.

4.1. TSI-HS variability745

The hidden dominant periods in the TSI-HS variability are related to the

large planets, as shown in Table 1. The correlation between the TSI-HS data

series and the identiﬁed dominant wavelet periods W hs(49, t) + W hs(86, t) +

W hs(164, t) is estimated to be R= 0.93. The dominant periods P(hs, 11) and

P(hs, 49) have a time-variant phase and represent mean estimates. A possi-750

ble source of the P(hs, 49) period is the interference between the Saturn pe-

riod P(hs, 29) and the Neptune period P(hs, 164). The periods P(hs, 84) and

P(hs, 164) have a stable period and phase in the time period from 1700 to

2013. The TSI-HS data series from 1700 is too short for a reasonable estimate

of P(hs, 164). A possible alternative is a coincidence resonance between the755

subharmonic period P(hs, 2∗84 = 168) and the 164 year Neptune period.

The model P(hsc, t) computes the deterministic oscillations in the TSI-HS

variability. Table 2 shows that P(hsc, t) computes a minimum in the period

28

from 1796-1830. In the same time period, P(spox, 84) and P(spoy , 84)have

maxima in 1797 and 1820. A close relation between the minimum of the period760

P(hs, 84) and the maximum states of P(spox, 84) and P(spoy, 84) is observed.

The identiﬁed wavelet stationary periods W hs(84, t) and W hs(164, t) are

transformed to a simpliﬁed model in Eq. 9, which produces a simpliﬁed deter-

ministic TSI-HS data series from 1000 to 2100. The computed results in Table

2 show a close relation between the P(spox, 84) maxima periods and minimum765

sunspots periods. The stationary model predicts minima in 1880 and 1960,

which is seen in the TSI-HS reconstruction (Figure 3). The Eq. 9 model iden-

tiﬁes three additional P(hsc, t)≤0 minimum periods, which are not shown in

Table 2. These periods are (1296-1313), (1629-1656) and (1962-2002). The last

period had a P(hsc, t) = −0.40 state in 1979. The model estimates a minimum770

P(hsc, t)≤ −0.5 in the period (2030-2065), a grand minimum P(hsc, t)≤ −1.0

period (2044-2054) and a local minimum irradiation state in approximately 2050.

These estimates support the identiﬁcation of Uranus, in resonance with Nep-

tune, as the major cause of TSI-HS variability.

The implication of this result is a chain of events between the solar inertial775

motion due to the large planets and the TSI-HS variability. The SPO period

P(spoy, 84), controlled by the 84-year Uranus period may serve as a reference

for the TSI-HS variability.

4.2. TSI-LS variability

The TSI-LS variability is inﬂuenced by the periods from the large planets,780

as shown in Table 1. The major variability is, however, inﬂuenced by the long

stationary periods P(ls, 125) and P(ls, 210). A stationary period is dependent

on a stationary source. The autocorrelations in Figure 8 indicate a 3/2 and

5/2 combination resonance to P(ls, 84), which produce the stationary periods

P(ls, 3∗84/2 = 165) and P(ls, 5∗84/2 = 210). The same stationary deter-785

ministic periods produce a new set of subharmonic periods P(ls, n ∗126) and

P(ls, n ∗210) for n= 1,2,3. When P(ls, 125) and P(ls, 210) are related to

the stationary Uranus period P(U, 84.02), they will produce a set of subhar-

29

monic stationary periods. The period P(ls, 3∗84/2 = 126.03) will produce

the subharmonic periods P(ls, 2∗126.03 = 252.06), P(ls, 3∗126.03 = 378.09),790

P(ls, 4∗126.03 = 504.12), P(ls, 5∗126.03 = 630.15) and P(ls, 6∗126.03 =

756.18). The period P(ls, 5∗84/2 = 210.05) will produce the subharmonic pe-

riods P(ls, 2∗210.05 = 420.1) and P(ls, 3∗210.05 = 630.15), which indicate

that P(ls, 126.03) and P(ls, 210.05) have a coincidence resonance in periods of

approximately 630 years (Nayfeh & Mook, 2004; Ghilea, 2014).795

The TSI-LS data series is reconstructed by Velasco Hererra et al. (2015), who

performed a wavelet analysis of their TSI-PMOD and TSI-ACCRIM reconstruc-

tions for the years 1000-2100 and discovered periods of 11 ±3,60 ±20,120 ±30

and 240 ±40 years. They interpret the 11-year period as the Schwabe cycle

and the 60-year period as the Yoshimura-Gleissberg cycle, which is associated800

with solar barycentric motion. The 120 years period they associated with solar

magnetic activity (Velasco Hererra, 2013), and the 240-year period was asso-

ciated with barycentric motion as discovered by Jose (1965). They concluded

that the negative phase of the 120-year period coincides with the grand minima,

the positive phase of the 120-year period coincides with the grand maxima. The805

next minimum should appear between 2010 and 2070 according to this inter-

pretation. Long periods were also identiﬁed byMcCracken et al. (2014), who

analyzed 10Be from cosmic ray variation over the past 9400 years by a Fourier

spectrum analysis and identiﬁed three related periods (65 and 130), (75 and

150), and (104 and 208) and the periods 350, 510 and 708 years. The identiﬁed810

210-year period is known as the 210-year de Vries/Suess period. Suess (1980)

identiﬁed a stationary period of approximately 210 years in the radiocarbon

dating of pine tree rings of the last 8000 years.

The identiﬁed periods in this investigation support the study by Suess (1980);

Velasco Hererra et al. (2015) and McCracken et al. (2014). The new informa-815

tion reveals that all long periods in the TSI-LS variability are traced to the

deterministic Uranus period P(U ranus, 84.02). This study conﬁrms that the

TSI-LS variability is dominated by deterministic periods and explains why the

identiﬁed periods from Suess (1980) and McCracken et al. (2014) are found in

30

series of 8000 and 9400 years long.820

An identiﬁcation stationary periods in TSI variability can provide informa-

tion about future irradiation variability and expected long-term climate varia-

tion. The computed minima from the deterministic model (Eq. 13) show a close

relation between the solar grand minimum periods and the computed minimum

periods from the model (Table 2). From this deterministic model, we may ex-825

pect a new TSI minimum P(lsc, t)≤ −0.5 for the period 2040 – 2080, a Dalton

state level P(lsc, t)≤ −0.7 in the time-period 2048 – 2068 and a minimum state

P(lsc, t) = −0.9 at approximately 2060.

4.3. Sunspot variability

A study of the sunspot data series from 1611 identiﬁed the ﬁrst dominant pe-830

riods P(sn, 11), P (sn, 55), P (sn, 110) and P(sn, 210), as shown in Table 1. The

period P(sn, 11) is a mean estimate from a time-variant phase. The wavelet

spectrum in Figure 11 shows that the period P(sn, 55) has a time variant am-

plitude that is controlled by the period P(sn, 110). The stationary ﬁrst pe-

riod P(sn, 11) is related to the Jupiter period P(J, 11.862) and produces the835

subharmonic periods P(sn, 5∗11.862 = 59.31), P (sn, 2∗59.31 = 118.62) and

P(sn, 4∗59.31 = 237.24).

The periods have a combination resonance to the Uranus period P(U, 84.02).

The autocorrelation spectrum in Figure 12 shows that the period P(sn, 55) has

a 2/3 combination resonance to the period P(sn, 84). This ﬁnding indicates that840

the identiﬁed period P(sn, 55) is a stationary period that is controlled by the

Uranus period P(U, 84.02) from the 2/3 relation 2 ∗84.02/3 = 56.01 years. This

ﬁrst stationary period is expected to produce a new set of subharmonic periods

of approximately 2 ∗56 = 112 and 4 ∗56 = 224 years. The identiﬁed period

P(sn, 210) has a 5/2 combination resonance to the Uranus period P(U, 84.02) by845

the relation 5 ∗84.02/2 = 210.05 years. Table 1 shows a close relation between

the identiﬁed TSI period P(hs, 11), P (ls, 11), the sunspots period P(sn, 11) and

the Jupiter period P(J, 11.862). This study has demonstrated that the Uranus

period P(U, 84.02) introduces a deterministic TSI period of approximately 5 ∗

31

84/2 = 210.05 years, a deterministic sunspots period of approximately 4(2 ∗850

84.02/3) = 224.05 years and a mean coincidence period of 217 years.

The sunspot data series has been investigated for decades. Schwabe (1844)

proposed Jupiter as a source for P(sn, 11), and Ljungman (1879) presented the

theory that the long-term herring biomass ﬂuctuation was related to a 111-

year sunspot cycle. The 210-year de Vries/Suess period is related to a climate855

cycle (Suess, 1980). The new information from this study is that the 210-

year de Vries/Suess period, which is identiﬁed as deterministic period in the

TSI-LS and the sunspots data series, has its minimum at a phase diﬀerence of

0.7π/2(rad/year) from the SPOx maximum in 1797, which coincide with Uranus

perigel. This shows that they are controlled by the same 84-year Uranus period,860

This study shows that solar variability and TSI variability have deterministic

coincidence periods of approximately 11 and 210-220 years. The deterministic

model of the solar variability indicates that we may expect a new sunspot solar

variability minimum P(snc, t)≤ −0.5 in the period from approximately 2025

to 2050, a Dalton level minimum P(snc, t)≤ −0.7 in the period from approxi-865

mately 2030 to 2040 and a minimum state P(snc, t) = −0.84 approximately at

the year 2035.

4.4. Possible explanation

This study of long solar variable data series has identiﬁed a deterministic

relation among TSI variability, sunspot variability, the solar position oscillation870

and the periods from the four large planets. In this chain of events, we may

understand the solar dynamo oscillation as a coupled oscillator, forced by the

oscillating gravity between the Sun and the large planets.

The study of the solar position oscillation shows that the 84-year Uranus

period P(spox, 84) may serve as a reference for the forced gravity oscillation875

inﬂuence on the solar dynamo. The real SPO gravity inﬂuence on the solar

dynamo is more complex. A mutual gravity oscillation exists between four large

planets and the solar position oscillation that controls the angular momentum

on the solar dynamo (Sharp, 2013).

32

Since the direct gravitational eﬀect is small (Scafetta, 2012), an ampliﬁcation880

mechanism is necessary to produce the TSI variations. Proposed mechanisms

are a nonspherical shape of the tacholine (Abreu et al., 2008): the two meridional

circulating magnetic waves (Shepherd et al., 2014; Zharkova et al., 2015); the

tidal massage of the solar center resulting in greater nuclear energy production

(Scafetta, 2012); movement of elements near the center of the Sun (Wolf &885

Patrone, 2010; Cionco & Soon, 2015) or reconnection of magnetic ﬁeld lines

which create magnetic bubbles (Granpierre, 2015). A sudden loss of angular

momentum from solar rotation to solar and planetary orbit may cause variation

in diﬀerential rotation that modulates the dynamo, which generates a magnetic

ﬁeld and sunspot variations (Blizzard, 1981). Transfer of angular momentum890

between the rotation of the Sun and the orbit of the planets is possible because

of the wobble of the Sun. The axis of rotation is tilted with respect to the axes

of the orbital plane, and the shape of the Sun is elliptical in the polar directions.

Since the Earth also moves inside the solar wind, modulation of the solar wind

by the four large planets may also be directly felt by the Earth, in addition895

to exchange of angular momentum resulting in faster or slower rotation, which

modulates the Earths climate (M¨orner, 2010).

Other studies have identiﬁed stationary periods in the solar dynamo. Duhau

& De Jager (2008) analyzed the variation of the solar-dynamo magnetic-ﬁeld

since 800 and identiﬁed periods of approximately 11, 22, 88 and 208 years.900

Shepherd et al. (2014) and Zharkova et al. (2015) have identiﬁed two dynamo

waves that show periods of 320 - 400 years, with an amplitude modulation in

the range of 20 - 24 years. These periods are similar to some of the identiﬁed

periods in Table 1. The new information from this study is that the identiﬁed

solar dynamo periods have a deterministic relation to the stationary periods905

from the four large planets, the TSI variability and the sunspot variability.

The stationary solar dynamic periods explains why the 125-year TSI-LS period

produces a subharmonic period of approximately 3 ∗125 or 375 years.

33

5. Conclusions

A better understanding of the deterministic properties of the TSI variability910

is critical for understanding the cause of irradiation variability and how the TSI

irradiation will contribute to the natural climate variation on the Earth. In this

study, we have identiﬁed stationary periods in the TSI-HS data series from 1700-

2013, in the TSI-LS data series from 1000-2100 and in the sunspots data series

from 1610-2015. The identiﬁed stationary periods are related to the SPO and the915

periods from the four large planets. The results show that the TSI and sunspot

data series variability have stationary oscillating periods that is controlled by

the gravity from the large planets Jupiter, Saturn, Uranus and Neptune. The

identiﬁed periodic relation between the solar system oscillation and the TSI

variability, indicates a chain of events between the solar system oscillation and920

the TSI variability. A possible chain of events is that the oscillating gravity

between the Sun and the large planets inﬂuences the solar dynamo oscillation,

which produces the TSI variability and the sunspot variability.

The study demonstrates that the major TSI variability and sunspot variabil-

ity are controlled by the 11-year Jupiter period and the 84-year Uranus period.925

The TSI data series from 1700 has a variability that is controlled by the 11-year

Jupiter period and the 84-year Uranus period. The TSI data series from 1000

has a stationary dominant period of approximately 125 years, which is con-

trolled by a 3/2 resonance to the 84-year Uranus period, and a 210-year period

by a 5/2 resonances to the 84-year Uranus period. The stationary periods of930

approximately 125 and 210 years introduce a new set of deterministic subhar-

monic periods. The study conﬁrms the deterministic relation between 210-year

variability and TSI variability, which is known as the 210-year de Vries/Suess

period (Suess, 1980).

The identiﬁed stationary periods in TSI variability and sunspot variability935

are transformed to deterministic models of TSI oscillation and sunspot oscilla-

tion. The close relation between the computed minima and the known mini-

mum periods since 1000 conﬁrms the identiﬁed periods from this study. The

34

deterministic model of sunspots and TSI computes a new Dalton-type sunspot

minimum from 2025 to 2050 and a new Dalton-period-type TSI minimum from940

approximately 2040 to 2065.

Acknowledgements

We thank N. Scafetta and V. M. Velasco Hererra for providing the data sets

for TSI we have used in this investigation. We also thank an anonymous referee

for valuable suggestions, helping us to improve the manuscript.945

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41

-2.5

-2.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

2.5

-2.5

-2.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

2.5

-2.5 -2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0 2.5

-2.5

-2.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

2.5

y(solarradius)

-2.5

-2.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

2.5

-2.5 -2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0 2.5

x(solarradius)

1940

2040

Figure 1: Orbit of the solar center with respect to the solar system barycenter (SSB) (+)

for the period 1940–2040 in the ecliptic plane that is deﬁned in the direction of

the Earth vernal equinox (Υ). The outer yellow circle represents the diameter

of the Sun, and the inner circle at radius 0.65rsun represents a shell where the

potential energy (PE) of the solar radiative zone can be aﬀected if the solar

center moves closer to the SSB (Cionco & Soon, 2015).

42

0 50 100 150 200 250

Year

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

R

Figure 2: Autocorrelation spectrum Rspoy (s, m) of Sun position oscillation (S P Oy) wavelet

spectrum. Each colored line represents a single autocorrelation.

43

1700 1750 1800 1850 1900 1950 2000 2050

Year

-15

-10

-5

0

5

10

W(s,t)

Figure 4: Wavelet spectrum W hs(s, t) of the TSI-HS data series, for s= 1 ...0.6N.

45

0 20 40 60 80 100 120 140 160

Year

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

R

Figure 5: Autocorrelation spectrum Rhs(s, m) of the TSI-HS wavelet spectrum W hs(s, t) for

s= 1 ...190 and m= 0 ...160.

46

1700 1750 1800 1850 1900 1950 2000 2050

Year

-10

-8

-6

-4

-2

0

2

4

6

8

10

W(s,t)

Wx(29,t)

Wx(84,t)

Wx(165,t)

Figure 6: The identiﬁed stationary wavelet periods W hs(49, t), W hs(86, t) and W hs(165, t)

from the TSI-HS wavelet spectrum Whs(s, t).

47

0 50 100 150 200 250

Year

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

R

Figure 8: Computed autocorrelation Rls(s, m) of the TSI-LS wavelet spectrum W ls(s, t).

49

1000 1200 1400 1600 1800 2000 2200

Year

-10

-8

-6

-4

-2

0

2

4

6

8

10

W(s,t)

W(125,t)

W(210)

W(373,t)

Mean

Figure 9: The identiﬁed long stationary wavelet periods W ls(125, t), W l s(210, t)W ls(373, t)

and the period mean value

50

1600 1650 1700 1750 1800 1850 1900 1950 2000 2050

Year

0

2

4

6

8

10

12

14

SN(nr/yr)

Figure 10: Solar variability represented by the yearly average group sunspot number series

SN (t), estimated from 1610 to 2015 (SILSO data/image, Royal Observatory of Belgium,

Brussels)

51

1600 1650 1700 1750 1800 1850 1900 1950 2000 2050

Year

-8

-6

-4

-2

0

2

4

6

8

W(s,t)

Figure 11: Wavelet spectrum W sn(s, t) of the sunspot data series SN (t).

52

0 50 100 150 200 250 300 350 400 450

Year

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

R

Figure 12: Computed set of autocorrelations Rsn(s, m) of the sunspot wavelet spectrum

W sn(s, t).

53