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The Suns Eleven Year Magnetic Reversal Theory
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1. Introduction
The presented theory in this draft document uses the speed of the rotating magnetic fields of the
Sun in order to calculate the magnetic field activity of the Sun and the number of sunspots which
appear on the Suns surface. A sunspot is a place on the Suns surface which is characterized by a very
strong magnetic field. Therefore, the number of the sunspots on the Sun is a good indicator of the
intensity of the overall Suns magnetic activity. It is well-known that the magnetic field of the Sun
peaks every eleven years, a cycle known as the sunspot cycle. At the peak of magnetic activity, the sun
records maxima of sunspot numbers on its surface. It should be noted that the length of the sunspot
cycle is not always exactly eleven years, to the contrary, it varies as discussed by Mursula and Ulich
(1).
The presented theory tries to achieve the following (non-exhaustive) goals:
- To successfully calculate the length of the sunspot cycle based on the variability of the
speeds of the Suns magnetic fields as found by Callebaut (2)
- To be able to calculate the speeds of the polar magnetic fields based on the sunspot cycle
length and the equatorial speed
- To successfully calculate the varying hemispherical number of sunspots during each
sunspot cycle
- To calculate the positive and negative polarity of the sunspots
- To depict the polar magnetic reversal event
A correlation between the intensity of the Suns magnetic activity and the variability of the
speeds of its magnetic fields is presented in Long Term Variations of the Torsional Oscillations of the Sun
(2). The authors state that the differential rotation of the Sun is least differential during the magnetic
maxima and most differential during the magnetic minima. In other words, the speeds of the equatorial
and the two polar speeds (North and South polar speeds of rotation) have the most similar values
during the period when a maximal number of sunspots are recorded. The presented theory tries to go
along with this finding and tries to calculate speeds of the equatorial and polar fields which fit the
behavior described in reference (2).
This draft document is structured as follows: In Section 2, the magnetic field theory is
described, in Section 3 the calculated values are presented and finally this document is concluded with
the essential findings of the new theory.
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2. The Magnetic Field Theory of the Sun
As it has been previously stated, the theory bases its calculation on the variability of the speeds of
the magnetic fields of the Sun. The theory uses the model depicted in Fig. 1.
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Figure 1. The observers model
S marks the polar view of the Sun. Let Ae represent the angular rotation of the equatorial field, An the
angular rotation of the north polar magnetic field and A s the angular rotation of the south polar
magnetic field. O represents an individual observer which travels around the Sun with constant
angular speed of Aob.The observer travels the length of the mean of the north polar and equatorial
field in one day. Although aware that the speed and the length of the south magnetic field is slightly
different from the speed and the length of the north field, we use the length of the north field only for
simplicity. Practical evidence has shown that the obtained results do not differ to a great scale. As a
consequence of the polar fields change at the end of each sunspot cycle, we observe different travel
lengths of the observer if two consecutive sunspot cycles are simulated. For instance, when the north
polar magnetic field switches with the south magnetic field and the length and speed of the north
magnetic field are lower than the length and speed of the south field, we observe shortened travel
length and consequently increased travel time of the observer.
The object m represents the direction of the magnetic fields of the Sun. The angular rotation speeds aremeasured in degrees per day.
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The number of sunspots is calculated at each equilibrium point of rotational position (degrees)
between the equatorial field and the North or South polar field. Therefore, {fi} represents the position
(degrees) of the ith equilibrium point recorded (for North or South polar magnetic field). The number
of the sunspots is obtained from Eq. (1):
obeqsouthnorth PosPosMSSN =| (1),
where Poseq is the position of the equatorial field (degrees) and Pos ob is the position of the individual
observer. MSSNnorth|south is the number of sunspots on the North and South hemisphere of the Sun
respectively. It should be noted that Eq. (1) is performed at each equilibrium point of position between
equatorial and polar north field and between equatorial and polar south field, in order to calculate
MSSNnorth and MSSNsouth respectively.
The length of each sunspot cycle is calculated as sum of the periods (days) between the
equilibrium points of position of the equatorial and the polar fields (the equilibrium points will be
referred to as taking-over events further on).
Since the theory is based on an external individual observer, this theory is referred to as The
Observers Magnetic Field Theory of the Sun.
3. The Results
Two types of evaluation of the observers magnetic field theory were performed:
a) A static approach; the input parameter of the static approach is the average speed of theequatorial magnetic field. This approach is called static since the A eq, An and As are simulated with
constant values during the known length of the cycle. Considering the differential rotation of the Sun,
An and As are calculated as in Eq. (2) and Eq. (3):
An =Aeqx FactPN (2)
As= Aeqx FactPS (3)
where FactPN and FactPS represent the factors of difference in speed of rotation between the
equatorial magnetic field and the north and south magnetic fields respectively (see reference 3, The
Internal Rotation of the Sun).
.
b) A dynamic approach; in this approach a more realistic scenario is simulated. The input
parameters in this scenario are the length of the sunspot cycle and the monthly equatorial speed.
Therefore, the value of the speed of the equatorial field is not constant anymore, it is changed every
month. The values of the changing equatorial speed of the Sun in the period of the year 1920 till the
year 1990 are depicted in Fig. 2.
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Figure 2. The dynamic equatorial speed.
From Long Term Variations of the Torsional Oscillations of the Sun (2).
At each taking-over event, a speed of the North and South polar field is searched in the
interval of 36 to 38 days for the North polar field, and in the interval of 36.5 to 38.5 days for the South
polar field. The chosen polar speed for that taking-over event is the speed which gives a minimal
difference between the real hemispheric sunspot number and the calculated hemispheric sunspot
number according to Eq. (1).
The simulation efforts were made with specially crafted software application created for this
purpose.
3.1.The Static Approach
The static approach is an approach which concentrates on the ability of the observers magnetic
field theory to successfully calculate the length of the cycle, not the magnetic intensity. Hence, by thealgorithm of the static approach presented in Section 3, it can be easily concluded that the simulation
of constant averaged values of the equatorial and polar fields cannot successfully calculate the real
hemispherical number of sunspots since the movement of the equatorial and therefore polar fields is
certainly not constant (Fig. 2).
The intensity of a sunspot cycle (s) used for calculation of the length of the cycle only, is depicted
in Fig. 3 and Fig. 4, for North and South polar fields respectively. The calculated sunspot numbers are
calculated from averaged equatorial speed of 25.75 days. On the Y-axis of both Fig. 3 and Fig. 4 are
presented the sunspot numbers calculated at each taking-over event. On the X-axis a time scale is
represented, where the time is measured in bits. One bit represents the time interval measured in days
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between each taking-over event. Since all fields of the sun have constant values, accompanied by the
constant value of the observer, one bit for the North field has value of 83.78 days and 82.63 for the
South field for equatorial speed of 25.75 days and values of FactPN and FactPS 1.443744 and
1.4527033 respectively.
As it has been stated earlier, the static approach of this theory does not successfully present the
numbers of sunspots on the Suns surface, it is merely used to successfully calculate and visually
present the length of the sunspot cycle. Additionally, the sudden peaks evident in Fig. 3 and Fig. 4 are
also undesirable byproduct of the static unrealistic nature of the simulation and should be neglected for
observation.
Figure 3. The static approach Sunspot cycle of the North hemisphere
Figure 4. The static approach Sunspot cycle of the South hemisphere
For several known sunspot cycles the static approach was deployed. For each year in the cycle
the appropriate yearly equatorial speed was used in order to calculate averaged equatorial speed
for the whole sunspot cycle. The averaged equatorial speed was used as input parameter to the
static approach in order to obtain the length of the sunspot cycle.
Based on the calculations for many sunspot cycles, certain correlations between the FactPN
and FactPS and the cycle length was found. We map the values of FactPN and FactPS for different
average equatorial speeds in order to calculate series of lengths of sunspot cycles of 11 years in
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Table 1. Furthermore, in Table 2 we present the values of FactPN and FactPS when the average
equatorial speed of rotation of the Sun is 25.75 days for lengths of sunspot cycles of different
magnitude.
Table 1. Values of FactPN/FactPS for the same sunspot cycle length and different equatorial rotation
speeds (separated in two sub-tables for clarity). Note that the same length is not always possible
because the length of the bits is between 81 and 87 days (the mean of the length is calculated after 10
cycles)
Equatorial rotation
speed (day)
Length of sunspot
cycle (year)
FactPN North polar
rotation speed
(day)
25.75 11.01 1.443744 37.17641
25.35 11.00 1.44458 36.620126.3 11.01 1.44259 37.9401
Equatorial rotation
speed (day)
Length of sunspot
cycle (year)
FactPS South polar
rotation speed
(day)
25.75 11.00 1.4527033 37.40711
25.35 10.99 1.45342 36.84419
26.3 10.99 1.45173 38.1805
Table 2. Values of FactPN/FactPS for different sunspot cycle lengths and equatorial rotation speed of
25.75 days (separated in two sub-tables for clarity)
Length of sunspot cycle
(year)
FactPN North polar rotation speed
(day)
11.01 1.443744 37.17641
13.56 1.44458 37.19793
8.73 1.44259 37.14669
Strange observation
As you can see there is something very strange with the south polar field.
Length of sunspot cycle
(year)
FactPS South polar rotation speed
(day)
11.00 1.4527033 37.40711
14.02 1.45173 37.38205
9.49 1.45342 37.42556
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When the factor is decreased this should lead to a shorter sunspot length. We calculate theopposite.... And vice versa
Based on the observations presented in Table 1 and Table 2, it can be easily concluded that even a
small change in the equatorial rotation speed of the Sun (and consequently in the polar rotation speed)
can result in considerable change in the length of the sunspot cycle. Additionally, we calculated that a
difference of only 0.0807 percent in the speed of rotation of the north polar field causes change in the
cycle length from 9.64 to 12.49. Generally, in Table 2 we present the interconnectivity between the
hemispherical polar speeds and appropriate factors in order to calculate the same cycle length (see
11.0 years, Table 2) we depict greater factor (1.4527033 vs. 1.443744) for slower polar speed
(37.40711 vs. 37.17641).
In Table 3, the calculated values of averaged sunspot length are presented for six sunspot
cycles. The average equatorial speed is calculated from the data presented in Long Term Variations of
the Torsional Oscillations of the Sun (2). We use fixed values of 1.443744 and 1.4527033 for FactPN and
FactPS respectively.
Before looking at this table, please remember that a difference of only 0.0807 percent in the
speed of rotation of the north polar field causes change in the cycle length from 9.64 to 12.49.
Table 3. Calculated sunspot lengths by using the static approach
Sunspot cycle duration Average
equatorial speed
(day)
Real sunspot
length (year)
Calculated sunspot length/Pole
(years)
August 1923
September 1933
25.70 10.1 N = 10.76 S = 11.22
September 1933
February 1944
25.77 10.4 S = 10.91 N = 11.11
February 1944 - April
1954
25.78 10.2 N= 11.16 S = 10.87
April 1954 October
1964
25.75 10.5 S = 11.00 N = 11.00
October 1964 June
1976
25.88 11.7 N = 11.73 S = 10.50
June 1976 September
1986
25.91 10.3 S = 10.35 N = 11.91
Conclusion:
If we flip poles in every new cycle, then we find the closest values for the length, except for cycle
number 3. The 1964 1976 and 1976 1986 periods match with the calculated data.
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Remarks:
1. In the timeframe from 1915 till 1990 the Sun's equatorial rate declined from 14.05 to 13.95. This
can lead to different fixed factors for the earlier calculated lengths.
2. We also note that the differences in speed from the equator field in the 1964 1976 and
1976 1986 periods are larger then in the previous four.
3. If we make the fourth cycle = 10.5 years for North and South, then we find the following fixedfactors: 1.44350 and 1.45293. If we use these new constants for the first 3 cycles we find the
following:
Table 4. Calculated sunspot lengths by using 1.44350 and 1.45293 as new fixed factors
Sunspot cycle duration Average
equatorialspeed (day)
Real sunspot
length (year)
Calculated
sunspotlength/Pole
(year)
August 1923 September 1933 25.70 10.1 N: 10.21
September 1933 February 1944 25.77 10.4 S : 10.41
February 1944 - April 1954 25.78 10.2 N: 10.57
April 1954 October 1964 25.75 10.5 S = 10.5
Conclusion: The difference between the cycle lengths can be solved by using other factors for the
fixed factors.
Calculation off New Polar Factors for Changing Length Sunspot Cycles
Use the constant values of 1.443744 and 1.4527033 of the 2 sunspot cycles for a speed of 25.75days for the equator. Then increase the speed of the equator to 25.35 days, the fastest value found by
Callebaut. This way we observe the maximal length for the southern field and the minimal length for
the northern field.Results:
1. The Sunspot cycle from 25.35 and 1.443744 decreases in length to 9.24 years
2. The Sunspot cycle from 25.35 and 1.4527033 increases in length to 13.10 years
Then decrease the speed of the equator to 26.30 days, the slowest value found by Callebaut. This
way we observe the maximal length for the northern field and the minimal length for the southern
field.
Results:
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1. The Sunspot cycle from 26.30 and 1.443744 increases in length to 14.76 years
2. The Sunspot cycle from 26.30 and 1.4527033 decreases in length to 9.06 years
CONCLUSION
Both hemispheres need to have the same length. So one of them has to change the value of hisfactor!Example:
North is the Dominant Field
Table 5. Calculated sunspot lengths by using new fixed factors for the Southern field
Average equatorial
speed (day)
FactPN Calculated sunspot
length/ North
FactPS Calculated sunspot
length/ South
25.35 1.443744 9.24 1.454250 9.24
25.75 1.443744 11.01 1.4527033 11.00
26.30 1.443744 14.76 1.450587 14.76
South is the Dominant Field
Table 6. Calculated sunspot lengths by using new fixed factors for the Northern field
Average equatorial
speed (day)
FactPS Calculated sunspot
length/South
FactPN Calculated sunspot
length/ North
25.35 1.4527033 13.10 1.445287 13.11
25.75 1.4527033 11.00 1.443744 11.01
26.30 1.4527033 9.06 1.441620 9.07
Study of the Known Solar Cycles
By studying the solar cycles we have following remarks:
1. Solar cycle 4 is the longest. Therefore we choose it as the northern
dominant field. This matches with our previous finding.
2. Solar cycle 2 falls slightly out of our calculated lengths with the data from
Callebaut. There are 2 possibilities:
A. The equator field was a bit faster
B. The length of the cycle is not completely right
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To have a length of 9 years for the dominant northern field we calculate a meanequator speed of 25.28 days (calculated length = 8.99 years). The southern fieldhas then a length of 13.55 years.
Table 7 . Some properties of solar cycle 123
Solar
cycle
number
Starting of
solar cycle
(mm/yyyy)
Solar
maximum
(mm/yyyy)
Ending of
solar cycle
(mm/yyyy)
Dominan
t polar
field
Length of
solar
cycle
(years)
1 3/1755 6/1761 5/1766 S 11.25
2 2 6/1766 9/1769 5/1775 N 9.00
3 6/1775 5/1778 8/1784 S 9.25
4 9/1784 2/1788 4/1798 N 13.67
5 5/1798 2/1805 7/1810 S 12.25
6 8/1810 4/1816 4/1823 N 12.75
7 5/1823 11/1829 10/1833 S 10.5
8 11/1833 3/1837 3/1837 N 9.67
9 7/1843 2/1848 11/1855 S 12.42
10 12/1855 2/1860 2/1867 N 11.25
11 3/1867 8/1870 11/1878 S 11.75
12 12/1878 12/1883 2/1890 N 11.25
13 3/1890 1/1893 12/1901 S 11.83
14 1/1902 2/1906 7/1913 N 11.58
15 8/1913 8/1917 7/1923 S 10.0
16 8/1923 4/1928 8/1933 N 10.08
17 9/1933 4/1937 1/1944 S 10.42
18 2/1944 5/1947 3/1954 N 10.17
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19 4/1954 3/1957 9/1964 S 10.5
20 10/1964 11/1968 5/1976 N 11.67
21 6/1976 12/1979 8/1986 S 10.25
22 9/1986 7/1989 5/1996 N 10.0
23 6/1996 7/2000 9/2007 S 11.33
Table fromStudy of sunspots and sunspot cycles 124 (4)
3.1.1. Change of Polarity of Sunspots
It is well known that sunspots change their polarity at each new cycle of the Sun. With the
observers theory one can successfully observe the change of polarity of the sunspots as one cycle
ends and another one begins. The different polarity of the sunspots is depicted in Fig. 5.
Figure 5. Different sunspot polarities in North and South hemisphere
We observe the change in the speeds of the northern and southern polar fields. In Fig. 6 we
observe the calculated intensity of the sunspot cycle using the static approach. However, since it is
obvious that the static approach cannot produce realistic sunspot numbers, we only pay attention to the
direction of the depicted magnetic intensity of the Sun, calculated using a modified version of Eq. (1).
To differ between the positive and negative polarity of the sunspots, we eliminate the absolute
function of Eq. (1).
In Fig. 6 we depict a situation where the polarity of the sunspots is N-S. It can be detected by
observing the positive values of the first calculated intensities, and by the trend of constant increasing
of values (whether positive or negative). Moreover, in Fig. 7 we present the opposite situation. The
polarity of the sunspots is S-N. It can be noted that the direction of change of the calculated sunspot
numbers is now downwards.
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Figure 6. Upward direction of the calculated magnetic intensity of the Sun. It marks N-S polarity of
sunspots
Figure 7. Downward direction of the calculated magnetic intensity of the Sun. It marks S-N polarity of
sunspots
The change of the sunspots polarity implies changes in the speeds of the polar magnetic fields
of the Sun. We observe such change relative to a fixed value of equatorial speed of 25.75 days. We
obtain N-S and S-N polarized sunspots on different hemispheres of the Sun, by calculated polar field
speeds of 37.17641 and 37.40711 days respectively. To conserve the natural law of changing polarity
of sunspots at each new cycle, we conclude that the polar speeds must also undergo change. If we
assume that the average equatorial speed of the next is also 25.75 days, then the polar speed of
37.17641 days of the previous cycle will have to decrease to 37.40711 and vice versa.
3.1.2. Observing Very Low Sunspot Activity
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The period of very low hemispheric sunspot activity during the change of polarity of sunspots
can also be observed with the magnetic field theory of the sun. If we add on to the previous example
and assume equatorial speed of 25.75, we conclude that change of polarity of the sunspots occurs at
polar speeds somewhere around 37.3 days. The graph produced by the static approach shows indeed
very low magnetic intensity (Fig. 7). We conclude that change in polar speeds cause the almost-zero
activity of the Sun. Therefore, we successfully model the difference in magnetic activity (and therefore
the length of the sunspot cycle) of the hemispheres; we model the situation when there is almost-zero
sunspot activity on one hemisphere while there is evident sunspot activity on the other hemisphere.
Figure 7. Low sunspot activity during sunspot polarity reversal
3.1.3. Polar Flip of the Sun
We also model the delay in the polar reversal of the poles of the Sun. Due to the well-known
fact that the poles of the sun flip at the peak of the sunspot cycle, we observe the peaks of the sunspot
intensity of the North and South hemisphere presented in Fig. 8 and Fig. 9 respectively. We used an
equatorial speed of 25.75 days and North and South polar speeds of 37.17641 and 37.40711 days
respectively.
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Calculation Observers Loop see The Internal Rotation of the Sun (ref 3)
Circumference Sun: 4,373,000 km
Circumference Polar field
After studying the possibilities we came to the conclusion that the polar fields can have a maximum andminimum circumference from 300,000 to 1,300,000 km. The mean loop for the polar fields has to be found by
further calculations. At the moment we took 2,586,5000 km for the mean value for the northern field. This gives
us an observers loop of 29.936 km/sec or plus minus 360 days.
Theoretical calculations place the mean for the northern field between 350 and 370 days and 365 to 385 days for
the southern. However, as the calculations show, the difference in the final result is small between an observers
loop of 355 or 375 days.
4,373,000 + 300,000 Mean value: 4,673,000 : 2 = 2,336,500 km
2,336,500 : 86,400 = 27.043 km/sec
4,373,000 + 1,300,000 Mean value: 5,673,000 : 2 = 2,836,500 km
2,836,500 : 86,400 = 32.83 km/sec
Formula
a = radius
a = GM/ v2
v = speed
G = 6.67428 x 10 ((power -11) m kg
Mass Sun = 1.9891 x 10 power (30) kg
GM = 13,275,810
Long loop
a = 13,275,810/ 27.043 x 27.043 = 181,533,356 km
Circumference = 181,533,356 x 2 x 3.141592 = 1,140,608,807 km
Days needed for loop=
1,140,608,807 : 2,336,500 = 488.17 days
Short loop
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a = 13,275,810/107.78 = 1,231,751 km
Circumference = 1,231,751 x 2 x 3.141592 = 773,931,842 km
Days needed for loop=
773,931,842 : 2,836,500 = 272.85 days
References:
1. Mursula, K and Ulich, T., A new method to determine the solar cycle length. Geophys. Res. Lett, 1998,
25, 1837-1840
2. Valentine I. Makarov, Andrey G. Tlatov, Dirk K. Callebaut, Long term variationsof the torsional
oscillations of the sun.Solar Physics, 170:373-388, 1997
3. Michael J. Thompson, Jrgen Christensen-Dalsgaard, Mark S. Miesch, Juri Toomre The InternalRotation of the Sun Astrophys. 2003. 41:599643
4. A. K. Tripathi, Aka Tripathi and S. C. Dubey, S. K. Pandey1, Rahul Shrivastava, L. K.
Borkar, Study of sunspots and sunspot cycles 124SCIENCE, VOL. 98, NO. 11, 10 JUNE 2010 .