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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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Zharkova, V.V., Shepherd, S.J., Popova, E. and Zharkov, S.I.,2015, Heartbeat
of the Sun from Principal Component Analysis and prediction of solar activity1095
on a millennium timescale, www.nature.com/Scientific reports, 5:15689, DOI:
10.1038/srep1569.
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
