The Sun's Eleven Year Magnetic Reversal Theory

8/3/2019 The Sun's Eleven Year Magnetic Reversal Theory

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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 .

The Sun's Eleven Year Magnetic Reversal Theory

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