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Can origin of the 2400-year cycle of solar activity be caused
by solar inertial motion?
I. Charva
Âtova
Â
Geophysical Institute AS CR, Boc
Ïnõ
ÂII, 141 31 Praha 4, Czech Republic
Received: 30 September 1999 / Revised: 14 January 2000 / Accepted: 17 January 2000
Abstract. A solar activity cycle of about 2400 years has
until now been of uncertain origin. Recent results
indicate it is caused by solar inertial motion. First we
describe the 178.7-year basic cycle of solar motion. The
longer cycle, over an 8000 year interval, is found to
average 2402.2 years. This corresponds to the Jupiter/
Heliocentre/Barycentre alignments (9.8855 ´243). With-
in each cycle an exceptional segment of 370 years has
been found characterized by a looping pattern by a
trefoil or quasitrefoil geometry. Solar activity, evidenced
by
14
C tree-ring proxies, shows the same pattern. Solar
motion is computable in advance, so this provides a
basis for future predictive assessments. The next 370-
year segment will occur between
AD
2240 and 2610.
Key words: Solar physics (celestial mechanics)
1 Introduction
To ®nd a cause of solar variability is a key task for solar
physics. It is an important subject also for geophysics
due to the in¯uence of this variability on the Earth's
climate. Over recent centuries, enormous eorts have
been given to this problem. In recent decades satellites
have provided much data and in particular the obser-
vations of the satellite SOHO provides a more reliable
view of the solar interior.
It is now generally accepted that the two most
prominent long-term cycles are permanently present in
solar activity: a cycle of about 200 (160±210) years and
a cycle of about 2400 (2200±2600) years, see e.g. Suess
(1980), Sonett and Finney (1991), Damon and Linick
(1986), Damon et al. (1989), Damon and Sonett (1990),
Hood and Jirikowic (1991), etc. These studies have been
made on the basis of indirect, proxy
14
C record in tree
rings. A scatter of the cycle lengths observed is probably
due to the mixed quality of the records, the dierent
parts of the record being processed, and the various
methods of spectral analysis used, etc. It is dicult to
observe this periodicity of 200 years because the record
of sunspot numbers covers only 300 years. It is easier to
observe a periodicity of about 100 years (Gleissberg
cycle) connected with the amplitude modulation of the
sunspot number series. An origin of the prominent, and
permanently present, period of about 2400 years in solar
activity has still not been settled (i.e., is it solar,
geomagnetic, or extraheliospheric) or even enigmatic;
the underlying `forcing' being so far unknown. ``The
2400-year period appears to be stationary'' according to
Damon and Linick (1986).
The solar inertial motion (i.e., the motion of the Sun
around the centre of mass of the solar system) is the
central phenomenon of the solar system, caused by
varying positions, predominantly, of the giant planets
(Fairbridge and Sanders, 1987). The contribution by the
inner planets is minute. The varying positions of the
giant planets [Jupiter (J), Saturn (S), Uranus (U),
Neptune (N)] force the Sun to move inside a circular
area which has a diameter of 0.02 AU (astronomical
unit) or 3 á10
6
km, see Fig. 1. This is negligible in
comparison with the size of the solar system, but it is
very signi®cant with respect to the size of the Sun. The
diameter of the area in which the Sun moves represents
4.4 solar radii. The Sun moves with a velocity between
9 and 16 m ás
)1
, i.e. 30 and 60 km áh
-1
. This solar
motion is computable in advance, a great advantage that
opens up the possibility of establishing predictive
assessments of solar activity.
To understand the 2402-year cycle, it is necessary to
describe ®rst and explain the basic cycle of solar motion,
the cycle of 178.7 years and its relations to solar activity
behaviour over recent centuries. Both cycle values have
been found to correspond to the intervals between the
exceptional patterns in solar motion. The values repre-
sent, in both cases, the mean of the dispersion intervals of
the prominent cycles found in solar activity mentioned.
Ann. Geophysicae 18, 399±405 (2000) ÓEGS ± Springer-Verlag 2000
2 The ®rst basic cycle of 178.7 years in solar motion
and its response in terms of solar activity
The ®rst basic cycle of solar inertial motion, the cycle of
178.7 years, was found by Jose (1965) in a repetition of
solar motion characteristics computed between 1653 and
2060, and most important the time derivative of the
Sun's angular momentum was found. The cycle was
con®rmed by Fairbridge and Shirley (1987) since 760
AD
, and by Fairbridge and Sanders (1987) since 777
AD
.
Jakubcova
Âand Pick (1987) veri®ed the 178.7 year cycle
as the basic period of solar motion periodicities
(see their Fig. 1). These periods have been found as
its higher harmonics [also Fairbridge, 1997, e.g.
28 ´178.7=5004.5 (252 ´JS; 24 ´208.5 year VJU)]
and mostly correspond to the orbital periods of the giant
planets [i.e. the periods of 80±90 (U), 60, 45 (SN), 35
(SU), 30 (S), 13.8 (JU), 12.8 (JN), 11.9 (J), 10 (JS/2)].
The Sun moves in the plane of the solar system,
so this action is essentially planar (Z-coordinates have
been neglected for initial studies). Therefore, besides the
motion characteristics, the geometry of the solar orbit
should be most carefully considered. Eventually, this
geometrical approach did provide a key for the solution
of the solar motion ± solar activity relationship.
A key to solution consists in the separation of solar
motion into two basic orbital types (Charva
Âtova
Â, 1988,
1990a, b), see Fig. 2: the ordered (according to JS
motion order, 117.3°, 19.86 years in a trefoil) and the
disordered. The Sun enters into the orbital trefoils with
a spacing, on average, of 178.7 years and moves along
a trefoil (in one loop) in about 50 years (10 years),
respectively. While the trefoil orbits are, after a rotation,
nearly the same, the disordered orbits dier from one to
another. After separation, the solar motion itself was
discernible and ®xed in time as a precise and homoge-
neous basis suitable for solar-terrestrial studies. Then,
it has been possible to observe separated solar motion
from a particular view point. The trefoil intervals which
recur in regular steps of 178.7 years can be taken and
employed as the exceptional and stable pattern of solar
motion.
A response to solar motion has been observed in solar
activity. In fact, it was noticed that the intervals of
disordered solar motion coincide with particular
prolonged minima in solar activity, such as the Wolf,
Spo
Èrer, Maunder and Dalton minima of this millennium.
The results obtained for mutual relationships be-
tween solar motion and solar activity during the latest,
basic 178.7 year cycle are summarized in Fig. 3a±d. If
solar motion is really a cause of solar variability, then
the motion along the same (trefoil) orbit should create
the same series of sunspot cycles. In fact, the Sun
moving along the same orbits during the trefoil intervals
of the eighteenth century (1734±1785) and the twentieth
century (1913±1964) (taken from minimum to minimum
of the sunspot cycles), created nearly the same sets of
®ve sunspot cycles (Charva
Âtova
Â, 1990b, 1995a, 1997a, b):
cycles )1 to 3 and cycles 15 to 19. Small deviations can
be mainly ascribed to lower quality of sunspot numbers
in the eighteenth century. Unfortunately, the Wolf
number series is not homogeneous. It can be taken as
reliable only since about 1850. Earlier, it is of lower
quality and before 1749 not even the monthly data are
available. This is not sucient to con®rm an exact
response of the solar trefoil motion in the Wolf series.
The highest and the only signi®cant coecient of
correlation between the successive groups of ®ve sunspot
cycles since 1700 and the series from the trefoil intervals
is that (r= 0.81) between the two series of ®ve cycles
from both trefoil intervals. The basic cycle of 179 years
in those two phenomena, the mutual relation between
solar motion and solar activity thus became evident
(Fig. 3d).
It was shown (Charva
Âtova
Â, 1995b, c, 1997a) that the
lengths of ®ve sunspot cycles (15±19) created by the Sun
moving along the trefoil orbit where motion along one
motion loop lasts 10 years (see Fig. 2, top), is stable and
equal to 10.1 (~JS/2) years. The more precise length of
the cycles )1 to 3 cannot be calculated, mainly due to
the lack of monthly data before 1749. However, the
mean length of the respective ®ve cycles is also about
10 years. In the surrounding and intermediate intervals,
the lengths of sunspot cycles are variable and generally
greater (Fig. 3a). The dominant period of 10.1 (JS/2)
years was detected in sunspot numbers from the both
trefoil intervals (Charva
Âtova
Â, 1990b, Fig. 4), in contrast
to the longer dominant periods found for the other
intervals, up to 11.9 (J) years detected for the Dalton
minimum in solar activity (1790±1843) (Charva
Âtova
Âand
Str
Ïes
Ïtõ
Âk, 1994). Rabin et al. (1986) and Wilson (1988)
found that solar behaviour is bimodal, with modes being
10.1 and 11.9 years.
Fig. 1. The orbit of the centre of the Sun around the centre of
mass of the solar system (in units of 10
)3
AU, astronomical
unit = 149 á10
6
km), for further details see Fig. 2. The dashed
horizontal abscissa in the upper part of the ®gure represents the
diameter of the Sun
400 I. Charva
Âtova
Â: Can origin of the 2400-year cycle of solar activity be caused by solar inertial motion?
3 The second basic cycle in solar motion: 2402 years,
and its response in the proxy solar activity record
(
14
C in tree rings)
On the very long (millennial) time scale only indirect,
proxy
14
C records of solar activity (radiocarbon in tree-
rings) are available. Radiocarbon production in the
atmosphere and consequently in tree-rings, is owing to a
cosmic ray ¯ux which has signi®cantly varied, being in
inverse relationship with solar activity. The radiocarbon
record in tree-rings thus provides a reliable archive of
past solar activity.
The regularity of the 178.7-year cycle is sometimes
disturbed: e.g. between 158
BC
and 208
AD
, the ®rst
178.7-year basic cycle (the interval between the two
consecutive orbital trefoils) was twice shortened to 159
years and during the intermediate intervals the Sun
moved along an orbit that is not too far from a trefoil.
These 370-year segments of exceptional and nearly
stable motion of the Sun (along a trefoil to quasitrefoil
orbit) have been found to recur in steps of 2402 years,
the mean value being 2402.2 years.
Figure 4 displays the last three such segments: the
®rst from 158
BC
to 208
AD
, the second from 2561
BC
to 2193
BC
and the third from 4964
BC
to 4596
BC
. The
solar orbits are the same in all three cases, if we
imagine them after a rotation. Further back in the
past, the Sun moved along the same orbital pattern
between the years 7366
BC
and 6998
BC
. The next such
segment will occur between the years 2242
AD
and
2610
AD
.
The strongest variations in solar activity occurred
during each of the second millenia, approximately,
where the longest on prolonged solar minima of Spo
Èrer
(S) and Maunder (M) type occurred. The patterns of the
S- and M- minima have always been observed to be
dierent. Hood and Jirikowic (1991) note that the 2400-
year cycle modulates the amplitude of the cycle at about
200 (160±210) years. Charva
Âtova
Â(1995b, c) pointed out
the coincidence of S (M) solar orbital patterns with the
respective S (M) types of solar minima occurrence,
(see Damon and Linick, 1986).
Note should be made here of the symmetry between
the moments of the Sun's entrances into the 50-year
trefoil orbits during the ®rst millennium (158, 338, 516,
695, ...,
BC
) and the ®rst millennium (159, 338, 516, 695,
...,
AD
). The centre of symmetry is the year 25
AD
(for
details see Charva
Âtova
Â, 1995b, c). In the centre of the
2402-year cycle, smaller irregularities occur in 178.7-
year cycle.
The exceptional segments of 370 years that regularly
recur in steps of 2402 years have served as a further tool in
searching for a response to solar inertial motion in solar
activity (Charva
Âtova
Â, 1998). The properties of the 370-
year segment (exceptionality and approximate stability)
are found in the corresponding intervals of the proxy
(radiocarbon) record and provide evidence for a solar
motion response and indicate that the solar motion could
be a cause of solar variability also on a millennial scale.
An analogy with a response of the 50-year trefoil
orbital pattern in solar activity (high solar activity cycles
with constant 10-year length) suggests that the behav-
Fig. 2. The orbit of the centre of the Sun around the centre of mass of
the solar system (in units of 10
)3
AU) separated into two basic types,
the ordered (in a JS-trefoil) (top) and the disordered (bottom). The
area in which the Sun moves has a diameter of 0.02 AU or 4.4 r
s
,this
being the solar radius, or 3 á10
6
km. The most disordered sections of
the intervals lying between the trefoils are plotted. The Sun enters into
the trefoils with a periodicity of 178.7 years, on the average (see the
times, years at the top of the respective ®gures). The value represents
the ®rst basic cycle of solar motion. While the trefoils are nearly
identical (after a rotation), the disordered orbits dier one from the
other. The Wolf, Spo
Èrer, Maunder and Dalton prolonged minima of
solar activity coincide with the intervals of disordered solar motion.
The Sun moves along a trefoil (along one of the loops), over 50 (10)
years, respectively. The two latest and the following trefoils are
denoted by triangles
I. Charva
Âtova
Â: Can origin of the 2400-year cycle of solar activity be caused by solar inertial motion? 401
iour of solar activity during the exceptional 370-year
orbital pattern can be characterized as trefoil to quasi-
trefoil, or could be close to that described for a trefoil.
Figure 5 shows radiocarbon
14
C records for the three
latest exceptional segments (the respective solar orbits
are plotted in Fig. 4) together with the adjacent millen-
nia, i.e. from 5800
BC
to 3800
BC
, from 3400
BC
to 1400
BC
and from 1000
BC
to 1000
AD
. In all three cases,
exceptionality and approximate stability (without pro-
longed minima) is seen. In this ®gure, local or regional
data (Irish or German oaks and pine) are taken from
Pearson and Quay (1993), Stuiver and Becker (1993)
and Stuiver and Pearson (1993). The most stationary
part of the record is seen precisely during the 370-year
exceptional segments of solar motion.
Figure 6 represents the whole series of data (marine
model
14
C ages calculated from atmospheric tree-ring
data with a smoothing spline through coral data) taken
from Stuiver and Braziunas (1993) since 6000
BC
. One
can see that the most stationary parts of the record
occurred precisely during the 370-year exceptional
segments of solar motion denoted there by the dashed
vertical lines and by three triangles.
4 Predictive assessments for solar activity
up to the year 2610 AD based on analogies
with the previous solar motion patterns
Predictions of solar cycle heights and lengths have
mostly been made on the basis of broad set of recent
solar and geomagnetic data and on the basis of relations
found only in the latest high activity solar cycles. Not
knowing the underlying source of solar variability, such
predictions can be false, as manifested at present in the
case of the current cycle number 23 when a high to
extremely high cycle has been predicted (R
max
within
140±225) (e.g. Wilson, 1992; Schatten et al., 1996; Kane,
1997). Ahluwalia (1999) predicted that cycle 23 will be a
moderate cycle (more like cycle 17). However, it is now
clear that the cycle 23 will be one of the lowest activity
cycles of this century.
Solar motion, computable in advance, oers instead
predictive possibilities, so far, of course, only as anal-
ogies with the results found for the previous six
millennia. The solar orbits plotted in the intervals of
50 to 65 years from 1680 to 2001 (or 2135) are seen in
Fig. 3b, 2. The Sun's orbit in the years 1985±2035 is of
disordered (nontrefoil) type and similar to that of the
second half of the nineteenth century. By analogy,
mostly weaker and longer solar cycles should occur.
Their lengths could vary between 9.6 and 12.3 years,
their R
max
could lie between 65 and 140. The predictions
of R
max
for the cycle 23 made during the last decade vary
between 140 and 225 (an unacceptable range). Our own
prediction made on the basis of solar motion was the
only opinion to express the opposite: i.e. a low solar
cycle was predicted (Charva
Âtova
Â, 1988, 1990a, b; Char-
va
Âtova
Âand Str
Ïes
Ïtõ
Âk, 1991). Charva
Âtova
Â(1990b) wrote:
``The current cycle 22 is probably the last of the high ones.
It should be followed by an epoch of about 40 years, in
which the solar motion will be chaotic (disordered) and
solar activity, therefore, should be low. The cycles will
probably be longer and irregular.'' Charva
Âtova
Â(1995a, b)
pointed out that the lengths of future cycles should be
variable. As indicated by the initial, 3-year part of the
cycle 23 in comparison with the respective part of the
cycle 22 (in smoothed form, about R
max
80±90 in
comparison with R
max
155 in the same time since the
Fig. 3a±d. This ®gure summarizes the results obtained for solar
motion ± solar activity relation around the latest basic 178.7-year
cycle. aThe lengths of sunspot cycles since
AD
1700: notice a constant
cycle length equal to 10 (JS/2) years during the trefoil interval 1913±
1964 (taken from minimum to minimum). During the previous trefoil
interval (1734±1785) the cycle lengths cannot be calculated more
precisely (see text). However, the mean value of cycle lengths is also 10
years there. In the surrounding and intermediate intervals, the cycle
lengths are variable and mostly greater. bThe solar orbit from
AD
1680 to 2001 in sections of 50±65 years. Two trefoil intervals are
denoted by the triangles.Theblack circles denote the positions of the
sunspot cycle maxima. cThe Wolf sunspot numbers since
AD
1700.
The letter A(A¢) denotes the trefoil interval of the eighteenth
(twentieth) century. dCoecients of correlation between the series of
®ve sunspot cycles that occurred during the trefoil intervals [1734±
1785 (A) and 1913±1964 (A¢)] and the successive groups of ®ve
sunspot cycles in steps of one cycle since 1700. The only signi®cant
coecient (r
k
= 0.81) is that between the groups of ®ve cycles
belonging to the trefoil intervals. The letter Ddenotes the Dalton
minimum of solar activity
402 I. Charva
Âtova
Â: Can origin of the 2400-year cycle of solar activity be caused by solar inertial motion?
minimum), our prediction was true. After 2085
AD
, the
Sun will move into the orbital trefoil. The activity series
of the cycles 15±19 should be repeated.
The next 370-year exceptional segment will occur
after
AD
2240. Until 2610
AD
, approximately stationary
activity should occur as a repetition of solar behaviour
during the previous exceptional segments (Figs. 5 and
6). High solar activity cycles with lengths of 10 years
could prevail, corresponding to the 10-year lengths
occurring for the trefoil type of solar motion.
5 Conclusion
These results show that the solar motion around the
centre of mass of the solar system could be considered as
the cause of solar variability. Not a static, but a dynamic
Sun should be taken into account. It appears that solar
motion is also the cause of the 2400-year cycle found in
proxy of solar activity. The results indicate that the so
far uncon®rmed source of the approximately 2400-year
cycle in solar activity, somewhat surprisingly, could be
in solar motion.
The discovery of the exceptional 370-year segments
of solar motion that recur in steps of 2402 years and
their imprint in the radiocarbon record is the main result
presented here. The basic cycle of 178.7 years, being the
interval between consecutive 50-year orbital trefoils, is
twice shortened by about 20 (JS) years, and during the
intermediate intervals the Sun moves along the trefoil to
a quasitrefoil orbit (Fig. 4). The 50-year orbital trefoils
in steps of 178.7 years and the 370-year exceptional
segments in steps of 2402 years represent the exceptional
patterns of solar motion (Figs. 2, 4). A response of the
50-year trefoil motion in solar activity is a series of ®ve
high, 10-year solar cycles (Fig. 3a). A response of the
370-year segments of trefoil to quasitrefoil motion is
approximately stationary (linear) with higher solar
activity (Figs. 5 and 6). The 10-year cycle length prevails
there.
Since solar motion is computable in advance, this
permits predictive assessments for future solar behav-
iour. Moving along the disordered orbit to 2035
AD
.
(Fig. 2, bottom), which is similar to that of the second
half of the nineteenth century (Fig. 3b), the Sun should
develop lower solar cycles (R
max
from 65 to 140) of very
variable length (from 9.6 to 12.3 years). The initial
development of the cycle 23, now in its third year,
con®rms this for the present cycle. Between 2086 and
2135
AD
, a set of cycles comparable to 15 to 19 should be
Fig. 4. The 370-year segments of the exceptional, stable pattern of
solar motion recurring in steps of 2402 years: notice the twice
shortened distance of 159 years between the three trefoils in each
segment (from 158
BC
to 208
AD
, from 2561
BC
to 2193
BC
and
from 4964
BC
to 4596
BC
). The next such segment will occur between
2240 and 2610
AD
I. Charva
Âtova
Â: Can origin of the 2400-year cycle of solar activity be caused by solar inertial motion? 403
repeated because the Sun will again move along a trefoil
orbit.
Predictive assessments for the following centuries can
be based on the 370-year exceptional segments and the
related results. The next such segment will occur from
AD
2240 to 2610. In correspondence with solar behav-
iour during the three previous exceptional 370-year
segments, approximately stationary, high solar activity
will be observed. The 10-year cycle lengths should
prevail. A very long term maximum of solar activity
comparable to that which was last observed during
classical antiquity should occur in the mentioned inter-
val (also see Stothers, 1979).
Statistical processing of the solar activity records
in context with two types of solar motion formations
is needed to estimate whether solar motion responds
in solar activity and to understand mutual relations
between the two phenomena. According to the results
obtained, the appropriate mechanism could eventually
be established. A key to this is likely hidden in the 50-
year trefoil interval. The mutual relations found between
solar motion and solar activity and the latest knowledge
about the internal structure of the Sun (obtained by
SOHO, e.g. Turck-Chieze et al., 1997; Kosowichev
et al., 1997) should be considered for this purpose.
The individual solar spheres must surely respond to
solar motion in distinct ways, provoking on the bound-
ary layers a solar dynamo. The thin layer at the
boundary between radiative and convection zones,
where a shear ¯ow was found by SOHO ± MDI,
`is likely to be the place where the solar dynamo operates'
(Kosowichev et al., 1997). The results indicate that
`solar dynamo' that was long sought in the solar
interior, operates more likely from the outside, by
means of the varying planetary con®gurations. As has
been shown in Charva
Âtova
Â(1995a, b, c, 1997a), the solar
motion could aid predictions also for terrestrial phe-
nomena including climate.
Acknowledgements. This research was performed under the sup-
port of the Grant Agency of the Czech Republic, grant 97/205/
0921.
Topical Editor E. Antonucci thanks R.W. Fairbridge and
G. Bonino for their help in evaluating this paper.
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