0 

e 

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0 

FINAL REPORT 

SOLAR DYNAMICS AND G-MODES 

NASA Contract NAS5-28196 

Jane B. Blizard, Principal Investigator 

Astrophysical, Planetary, and 

Atmospheric Sciences, Box 391 

University of Colorado 

Boulder, CO 80309 

Charles L. Wolff, Technical Director 

Greenbelt, MD 20771 

NASA Goddard Space Flight Center, Code 610.1 

February 10, 1987 

[NASA-CR-180737) SOLAR DYlaBXCS AID G-HODES 

Final Report (Colorado UniV.) 32 pCSCL 03B 

188-1 49 11 

Unclas 

63/92 0064792 

e 
Table of Contents 

Page 

Table of Contents . . . . . . . . . . . . . . . . . . . . . iii 

Acknowledgements ii 

Disclaimer ii 

Publications Issued Under Contract ii 

c, List of Figures i 

List of Tables i 

e - 

1 

m 

2 

3 

4 8 

5 

e 

R-Mode Oscillations in the Convective Zone of the Sun 

1.1 Introduction 

1.2 Eddy Viscosity 

2.1 Zurich Sunspot Number Record 

2.2 Greenwich Sunspot Area Record 

Sunspot Record and Fast Fourier Transforms 

Evaluation of FFT's with Wolff & Hickey (1986) Model 

External Force 

4.1 Magnitude in terms of Luminosity 

4.2 Time Span of Torque in External Sector 

4.3 Half-power Calculation 

4.4 North-South Asymmetry 

Bibliography 

1 

1 

1 

6 

6 

6 

16 

20 

20 

22 

25 

26 

28 
Acknowledgements 

The author expresses gratitude to Charles L. Wolff of NASA 

Goddard Space F 1 ight Center, Greenbe 1 t, Mary1 and, under whose 

sponsorship this work was performed as partial fulfillment of 

Contract NAS5-28196. Further acknowledgement is due to the APAS 

Dept. of the University of Colorado for assistance in preparation 

Colorado for copying and reproduction. 

reference material supplied by) Charles L. Wolff and John P. Cox. 

Many thanks to L. Jane Twigg for conversion of Zurich sunspot 

data from magnetic tape to floppy disks. 

Dean Pesne 1 1, E.R.Benton, C.Sawyer, J.W.Warwick, K.D.Wood, Che 1 a 

Kunasz, and Olivia Briggs of CU, and to H.H.Sargent 111, P. 

McIntosh, & M. Dryer of NOAA. 

Finally, many thanks to Mark Lund for extensive computer 

operation and programming during the course of the investigation, 

and for reading the entire manuscript. 

I of the line drawings and typing of the text, and to JILA, U. of 

The author acknowledges many useful discussions with (and much 

Many thanks for helpful discussions are due to C.J.Hansen, 

Disclaimer 

The contents of the report reflect the viewpoint of the 

author who is responsible for the accuracy of the results 

presented. Publication of this technical report does not 

necessarily reflect the views or policy of NASA Goddard Space 

Flight Center or the U.S.Government. 

Publications issued under NASA Contract NAS5-28196: 

1. Blizard, J. B., "Viscosity Losses in the Solar Convection 

Zone," B.A.A.S. 12#2 p. 610 (1985) 

2. Wolff, C. L. & Blizard, J. B., "Properties of r-Modes in the 

Sun," Sol. Phys. 105 1-15 (1986) 

3. Blizard, J. B., & Wolff, C. L., "Short Periods in the Power 

Spectrum of Sunspot Number--16 days to one year, and 

r-mode oscillations in the sun," P.A.S.P.98 1100 (1986) 

** 4. Blizard, J. B., & Wolff, C. L., "R-mode Oscillations in the ** Sun, 

Lecture Notes in Physics # :Stellar Pulsation, 

Springer-Verlag, NY 1987. 
e 

a 

m 

a 

* 

m 

-- List of Figures Page 

1.1 Schematic fluid motion of r-modes . . . . . . . . . . . . . 3 

2.1 Fast Fourier transforms, sunspot area: cycles 14, 15 8 

2.0 Twentieth century sunspot cycles 7 

2.2 cycles 16, 17 9 

2.3 cycles 18, 19 10 

2.4 cycles 20, 21 11 

2.5 Fast Fourier transforms, sunspot number: cycles 14, 15 12 

II 11 II I1 

It II It 11 

I1 11 11 II 

II II I1 II 

I1 I1 II 0 

II I1 I1 II 

2.6 'I cycles 16, 17 13 

2.7 It cycles 18, 19 14 

2.8 I' cycles 20, 21 15 

3.1 Mean Fourier spectrum, cycles 14-21, spot area, and 

model lines from Wolff & Hickey (1987) 18 

20 

4.2.1 Plan view of solar system, 1976 23 

4.3.1 Two point masses; half-value of gmax 25 

4.4.1 Relative spotted areas of N-S hemispheres; latitude 

of outer planets, 1830-1960 27 

4.4.2 Asymmetry of N-S hemispheres 1943-1977; latitude of 

outer planets relative to solar equator 27 

4.1 Duration of Fh over 5 day period 

4.2 Torque exerted on sun's equatorial bulge by a planet 20 

-- List of Tables 

1.1 Spherical harmonic terms of stress tensor . . . . . . . . . 3 

1.2 Eigenfunctions (l,m,n) numerically calculated for 140 

zones in solar convection zone 4 

1.3 Sidereal rotation of r-modes; repetition of identical 

flow fields viewed from Earth 5 

3.1 Expected frequencies & observed amplitudes for 8 solar 

cycles: spectra of spot area 17 

4.1 Days two planets are within 1/2 radian (Allen, 1973) 24 

4.2 Days two planets are within 1/2 radian (Astron. Almanac) 24 

4.3 Periods of the planets viewed from the subsolar point 24 

4.4 Relative amplitudes of torque of planets on sun 24 

4.4.1 Longitude of the outer planets, 1790-2000 27 
0 

0 

D 

1 R-UODE OSCILLATIONS IN THE CONVECTIVE ZONE OF THE SUN 

1.1 INTRODUCTION. P-modes and g-modes have been actively studied 

since.it first appeared that the Sun might be oscillating in 

modes of global scale. But there is another type of global mode, 

called r-modes from their similarity to the Rossby waves in 

meteorology. These toroidal modes have very long periods 

comparable to the sun's rotation period, while the p- and g-modes 

have periods of minutes or hours, and their motion is controlled 

by the Coriolis force, whereas for the p- and g-modes the 

restoring forces are compression and buoyancy. 

Also the motion of r-modes is almost purely toroidal (nonradial), while the p- and g-mode motions are mainly spheroidal 

(radial). Disturbance in the Sun would excite both toroidal and 

spheroidal components. We cannot continue to ignore the r-modes 

in solar physics. The r-modes are thought to be damped in the 

Sun. But damping does not remove them from observational 

interest. The solar envelope is in constant motion. Differential 

rotation drives eddies whose vertical component would be a source 

for r-modes. Lower angular harmonics (Fig. 1.1) would also be 

expected to be generated by the eddy motion of horizontal flows 

at the top of large convection cells and especially the cells of 

global scale. Any detection of r-modes would give an important 

new probe of the sun's interior rotation. 

1.2 EDDY VISCOSITY, We computed various properties of r-modes 

that would occupy the solar convection zone. Large scale motions 

in the solar envelope are non-adiabatic since they are immersed 

in a turbulent medium. 

computed from the coefficient of eddy viscosity 

where Vcon = L/tc (mixing length/conv. overturning time). We 

calculated values of this using the solar convective zone model 

of Baker c Temesvary (1966). Following Landau & Lifshitz, viscous 

damping acting alone will change the energy of an r-mode in one 

oscil lation period by 

If simple diffusion theory applies, an energy loss can be 

Tl= 1/3 9 VconL, 

where the deformation tensor for large scale velocity is 

In spherical coordinates for 1=1 and both 1=2 cases, the 

summation of Dij terms give compact expressions which can be 

integrated over angle. Assuming that - 72 depends only on radial 

distance, x=r/RO, 

- 

- (1.3) 

1 
where 6 is the term for the angular harmonic (Table 1.1). This 

approach requires the r-mode to have an oscillation period, P >> 

tc (conv. overturning time), a long wavelength X >> L (mixing 

length), and a viscous energy dissipation E, << E (total energy). 

Special attention must be given to layers close to the base 

of the convective zone because of a possible anisotropic 

viscosity coefficient there. In the lower part of the convection 

zone, mostoftheenergywould be lost to viscosity in a single 

oscillation period. Fortunately, this lower region is not very 

important tothemode as awhole; only about 10% of the undamped 

linear mode energy would lie below this region (at about 0.85 R,, 

where the base of the CZ = 0.75 Ro). An estimate of the total 

viscous damping per period is one-third for the (1,l) mode. All 

other modes are damped more strongly. We can conclude the 

following if current models of the sun's envelope are realistic. 

When any transient disturbance excited r-modes in the CZ, the 

modes cannot retain their initial amplitudes for more than a few 

solar rotations without a fresh injection of energy at the proper 

phase (see Table 1.2 for damping rate). 

their horizontal motion. For this approach, it is best to go from 

a coordinate frame rotating with the sun to an inertial frame. In 

an inertial frame, the middle columns of Table 1.3 give the 

rotation rate and period of various modes, based on a convective 

zone rotating with a period of 25.7 days. Viewed from the frame 

rotating with the earth (synodic) at 365.3 days two rates may be 

of interest: the synodic rate and the repetition rate, at which 

phenomena caused by one r-mode repeats itself on the apparent 

solar disk. Table 1.3 shows the range of periods by listing the 

lowest and highest angular harmonics. The values for frequency 

and period apply to any surface feature varying linearly with 

displace-rnent, such as velocity. Convection is likely to be 

modulated by r-modes. The overturning times of the large cells in 

the lower half of the convection zone are comparable to the 

oscillation periods of all r-modes with low 1 values. 

Also the length scales of these deep seated cells are 

similar to those of r-modes with moderate 1. Finally, convection 

One way to seek r-modes is to attempt direct detection of 

I has vorticity in the proper plane to couple with a toroidal I I velocity field. Thus the physical conditions for coupling are 

close to ideal. If convection is modulated on very large 

horizontal scales at time scales about one month, one should 

expect similar modulation in the various forms of solar activity 

like sunspots and flares. Wolff (1974) early suggested a relation 

between large scale convection and solar activity and since has 

provided additional observational material based on modes in the 

interior. Surface features would be affected more efficiently by 

the envelope r-modes, if they are indeed excited. This gives 

reason to search the solar observational record for evidence of 
3 

c Fig. 1.1 Schematic fluid motion of r-modes with various values 

of 1 and m, near surface of the sun (after Saio, 1982). 

Table 1.1 Spherical harmonic terms of the stress tensdr for low 

order r-modes (after Wolff). r=radial dist; f=angular func(din eq 1.3) 

Oi f,m = (1.1) (2.1) (2.2) j 

ve = -r-fcos@ -rfcosecosB rfsinecos(ia) 

0 0 0 

-2fsl necose +*fcosecosm 

-rf 'cos0 -rf'cosecose rf'ainecor20 I 

8 

'W-3 - 1 -+-- dve dvO dvpJt@)l 0 

DBe=r sine d0 de 
4 

Table 1.2 Eigenfunctions (l,m,n) numerically calculated for 

,140 radial zones in the solar convection zone. E DOT=viscous 

energy change=Ev in eq. 1.3; 

damping rate. 

K.E.=kinetic energy:SIGMA PRIME= 

E DOT SIGMA PRIME 

EIGEN l,l,O - 

------------ 

2.501470E+11 

K.E. 

7.532208E+17 

E DOT K.E. 

EIGEN 1,1,1 - 

------------ 

2.955946E+14 

SIGMA PRIME 

~~ ~ 

3.480709E-05 

E DOT 

EIGEN 1,1,2 - 

------------ 

8.322445E+17 

K.E. SIGMA PRIME 

E DOT 

EIGEN 2,1,l - 

------------ 

9.838837E+11 

K.E. SIGMA PRIME 

E DOT SIGMA PRIME 

EIGEN 2,1,2 - 

------------ 

2.622679E+15 

K.E. 

E DOT 

EIGEN 2,2,0 - 

------------ 

4.828180E+11 

K.E. SIGMA PRIME 

~~ ~ 

1.040677E-05 

K.E. SIGMA PRIME 

2.083029E-05 

E DOT SIGMA PRIME 

EIGEN 2,2,2 - 

------------ 

2.597350E+18 

K.E. 

------------ 

2.922550E-05 
5 

modulations at the rotation rates, repetition rates and beat 

frequencies of envelope r-modes (in the CZ). To first order, the 

r-modes- oscillate and rotate at rates independent of all solar 

interor conditions except for the rotation of the fluid in which 

they are embedded. Thus the r-mode rotation rates are as predictible 

as the g-modes. Each r-mode sequence is determined by a single 

free parameter: a mean rotation rate of the cavity in which the 

e modes are trapped. This implies regular but complex behavior in 

solar characteristics modulated by the r-modes. In summary, some 

periods of r-modes have been tabulated. Viscous damping is strong 

and removes one-third or more of the mode energy each oscillation 

period. 

Table 1.3. Sidereal rotation of r-modes in the convection zone 

and the repetition of identical flow fields as viewed from Earth 

(Wolff & Blizard, 1986). 

0 

- 

4 

1m ROTATION REPETITION, Vim 

Rate Period Rate Period 

(nHz) (days) (nHz) (days) 

11 0. 00 -31.7 -365.3 

* 2 1 300.0 38.6 268.3 43.1 

2 2 300.0 38.6 536,6 21.6 

3 1 375.0 30.9 343.3 33.7 

3 2 375.0 30.9 686.6 16.9 

3 3 375.0 30.9 1030. 11.2 8 

00 1 450.0 25.7 418.3 27.7 

00 2 450.0 25.7 836.6 13.8 

............................................. 

e 

e 
6 

2.1 FAST FOURIER TFUWSFORMS OF THE ZURICH DAILY SUNSPOT NUMBER 

AND GREENWICH DAILY WHOLE SPOT AREA 

A-fast Fourier transform (FFT) of daily Zurich sunspot 

number was made for eight sets of data, taken near peaks of solar 

cycles 14-21. The FFT accepts 2048 bits of data, so 2048 days 

were included, starting with 1 Jan except for the most recent 

cycle, when the data set began 16 Nov 1978, when data was 

available from the ACRIM irradiance sate1 lite experiment. 

with a starting date of: 

I) 

Thus the eight files contained 2048 days each, 

D 

File 1 16 Nov 1978 File 5 01 Jan 1935 

File 2 01 Jan 1966 File 6 01 Jan 1925 

File 3 01 Jan 1956 File 7 01 Jan 1915 

File 4 01 Jan 1945 File 8 01 Jan 1904 

For a sequence of 2048 days,the FFTwill display periods 

D between 2 days and 1024 days (2.8 yrs) due to the pairing of the 

data. Prominent in most of the spectra is the 27-day peak. 

Periods longer than 160 days would not be resolved. 

were obtained on magnetic tape from the National Geophysical Data 

Center of NOAA, In addition, an updated version of GPR 1982-1984 

was obtained from Douglas Hoyt of NCAR. The GPR tape was loaded 

onto a VAX and pertinent columns extracted (date, whole spot 

area). A program was written to average data over four days. 

Another program was written to fill in zeros of whole spot area 

(when no observations were made), by averaging the values of the 

day before and day after. Because of large data gaps in early 

years, the period 1904-1984 was selected for study with fast 

Fourier transform (FFT) (Fig. 2.0). 

as for the Zurich sunspot number. In this case, the plots were on 

sets, since there were 512 points rather than 2048 (4-day 

averages) (Figs. 2.1-2.4). The mean of the eight spectra showed 

very little detail. For the 4-day averages, the limits on the FFT 

are 8 days and 5.6 years, but in practise the lower limit is 16 

days because of the mirror image form of the Grafplot. Because 

the four-day smoothing resulted in improved spectra, it was 

applied to the Zurich sunspot number data, which were replotted 

(Figs. 2.5-2-8). 

The Greenwich Photoheliographic Results, 1874-1981, (GPR) , 

D 

D 

FFTs were run on eight data sets with the same starting dates 

D a single frame, rather than four frames for the previous data 

D 

. 

d 

0 

8 

P 

a 
9 

a 

16 17.1 18.3 19.7 21.3 23.3 25.6 28.4 32 36.6 42.7 51.2 64 85.3 128 256 

b 

Power Spectrum, Daily Whole 

Spot Area: 4-day averages 

Cycle 17: Jan 1935-Aug 1940 

Fig. 2.2 Fast Fourier transforms of Greenwich projected whole 

spot area (4-day averages) : solar cycles 16 and 17(a, b). 
D 

~ 

10 

B 

B 

D 

a 

-Power Spectrum, Daily Whole 

Spot Area: 4-day averages 

Cycle 18: Jan 1945-Aug 1950 

16 17.1 18.3 19.7 21.3 23.3, 25.6 28.4 32 36.6 42.7 51.2 64 85.3 128 256 

b 

mer Spectrum, Daily Whole 

Spot Area: 4-day averages 

Cycle 19: Jan 1956-Aug 1961 

-v-- T16 17.1 18.3 19.7 21.3 23.3 25.6 28.4 32 36.6 42.7 51.2 64 85.3 128 256 

Period, days 

Fig. 2.3 

spot area (4-day averages): solar cycles 18 and Wa,b). 

Fast Fourier transforms of Greenwich projected whole 
11 

a 

Fewer Spectrum, Daily Whole 

Spot Area: 4-day averages 

Cycle 20: Jan 1966-Aug 1971 

r-mode 

r-mode 

r-mode 

16 17.1 18.3 19.7 21.3 23.3 25.6 28.4 32 36.6 42.7 51.2 64 85.3 128 256 

b 

Power Spectrum, Daily Whole 

Spot Area, 4-day averages 

Cycle 21: Nov 1978-0ct 1984 

Period, days 

Fig. 2.4 

spot area (4-day averages): solar cycles 20 and 21(a,b). 

Fast Fourier transforms of Greenwich projected whole 
12 

a 

Power Spectrum. Zurich 

Daily Sunspot No.(rl-day ave) 

Cycle 14: Jan 1904-Aug 1909 

.. 

Frequency nHz 

Power Spectrum. Zurich 

Daily Sunspot No.(4-day ave] 

Cycle 15: Jan 1915-Aug 1920, 

.. 

Fig. 2.5. Fast Fourier transforms of Zurich daily sunspot number 

(4-day averages) for solar cycle 14 (a) and 15 (b). 
a 

Power Spectrum, Zurich . Daily Sunspot No. (4-day ava) ’ ’ 

Cycle 16: Jan 1925-Aug 1930 

.* 

t 

I ’. 

.. 

P b 

Power Spectni, Zurich 

Daily Sunspot No.(4-day ave) 

Cycle 17: Jan 1935-Aus 1940 

200 300 1 400 500 600 

IFrequencv nHz . 

Fig. 2.6. Fast Fourier transforms of Zurich daily sunspot number 

(4-day averages) for solar cycle 16 (a) and 17 (b). 
BD 

I 

D " 

#I1 

b Power Spectrum, Zurich 

Daily Sunspot No.(q-day ave) 

Cycle 19: Jan 19S6-Aug 1961 , 

I 

'I 

D 

Frequency nHz 

Fig. 2.7. Fast Fourier transforms of iuric); daily sunspot number, 

B (4-day averages) for solar cycle 18 (a) and 19 (b). 
15 

Power Spectrum, Zurich 

Daily Sunspot No.(.Q-day ave) 

Cycle 20: Jan 1966-Aug 1971 

b 

Power Spectrum, Zurich 

Daily Sunspot No.(4-day ave) 

Cycle 21: Nov 1,978-Jul 1984 

- Frequency nHz 

Fig. 2.8. Fast Fourier transforms of Zurich daily sunspot number 

(rl-dau averages) for solar cycle 20 (a) and 21 (b). 
16 

f 

B 

D 

B 

3 EVALUATION OF FFT'S WITH WOLFF & HICKEY (1987) MODEL 

Each theoretical frequency having measurable power which 

stands out prominently in any spectra is listed in Table 3.1. 

Frequencies prominent in the earlier cycles were primarily caused 

by r-modes. Only one of the 7 cases where a line was clearly more 

prominent in the earlier cycles did not involve r-modes; 5 of the 

7 were exclusively caused by r-modes. A different situation was 

found in the last cycle (2l)(later in the solar cycle), where 

only 2 of the 8 cases didnotinvolveg-modes.Thisis consistent 

with decaying r-modes which would be expected in the Sun. Wolff 

and Blizard (1986) found that turbulent viscosity would damp rmodes in a few months, and postulated that they might be excited 

irregularly by large convection events. Occasional injections of 

fresh energy would prolong r-mode oscillations. In any case, the 

above measurement showed r-modes to be a stronger influence in 

the first fewyears of solar cycles rather than later in the 

cycle, when their lessened influence a1 lowed the presumably more 

stable g-modes to have relative prominence (see Fig. 2.0 & 3.1). 

area (four-day averages). Below it lies the model lines from 

Wolff & Hickey (1987) and Table 3.1. The region between zero and 

50 nHz contains many theoretical lines due to high-1 modes, 

indicated here only as a group (labeled H). The data is 

unresolved but the high power measured there is expected. At 

frequencies above 50 nHz, there is a remarkable similarity 

between theory and the observed sunspot area spectra. Predicted 

lines occur where there is much observed power and minima in 

power tend to lie where there are no theoretical lines. All four 

of the prominent power maxima near 310, 380, 450 and 530 nHz are 

associated with the lowest angular harmonic. The first three 

correspond to beats involving the 1=1 r-mode. The fourth involves 

the 1=1 g-mode and the feature H. This supports the expectation 

that the lowest modes are the most influential on variations of 

sunspot area (and other indicators of solar activity). 

Only three peaks (100, 170 and 295 nHz) occur near no 

predicted line. Finally, considerable power is seen in the 7 

bands which all arise from feature H. All seven lie where there 

is considerable observed power and 5 of the 7 include observed 

maxima. If the model is correct, it implies that the combined 

effect of all high harmonic r-modes can be comparable to one low 

harmonic in modulating sunspot area. Overall, the agreement seen 

in Table 3.1 and Figure 3.1 is much better than one could expect 

by chance, since there were effectively fewer than two free 

parameters to adjust the 28 model frequencies (Wolff & Hickey, 

1987). 

Figure 3.1 shows a Fourier spectrum of projected sunspot 
I, 

a 

I' 

ORIGINAL PAGE IS 

OF POOR QUWm 

1 11 

U ma 

't 

m 

m+u" at 0' 

m" 

rB 

t 

m 

a 

2 

s 

2 
, 

18 

1 

-8 

.n aD 

U 

Lb b 

0 

--0 

b 

Q 

0 

0 

M 

8 

N 

- 8 

0 

i 0 
D 

B 

D 

B 

B 

D 

D 

D 

20 

t 

Fig. 4.1. Duration of F Over 5 d Fig. 4.2. Torque exerted on esuaPeriod during 15 d. half-cycle. torial bulge by a planet. 

4.1 MAGNITUDE OF EXTERNAL TORQUE 

force exerted by a planetary torque, does work on the photospheric layers at a rate comparable to 

energy seen released by solar activity). Assume that Fh is 

constant and acts for about 5 days (1/6 rotation of the sun) on a 

volume element of the photosphere, accelerating it away from the 

equator with no resisting force (for simplicity) (Fig. 4.1). Then 

the horizontal velocity is Vh = (Fh)t, where Fh is in units of 

force/mass, then (kinetic energy/volume) = 1/2 9vh . In choosing 

9, we do not go into the convecting interior because these layers 

may have a stiffness that resists the torque-induced flow, since 

they experience strong horizontal diffusion of momentum. This 

would give an upper limit toanenergy source fed into the photosphere. The long-term mean rate of energy deposition would be: 

(vol.equatoria1 bulge in photosphere)xp/2[vh(for t=5 days)] 

We wish to calculate if the horizontal component Fh of the 

L, (the mean rate of 

2 

2 

The force of an external body on the sun may be calculated from 

McCul lagh's formula, which expresses the gravitaional potential 

of any body at a distance R and with a mass M(see section 3.1): 

(4.1.1) 

15 days 

V = GM/R + GM(A+B+C-31)/2R3 + higher order terms 

where A, B and C are the principal moments of inertia of the 

Sun. It is usual to write J2=3/2(A-C)/MRO2, where J2 is the 

quadrupole moment. Since we assume that the sun is axisymmetric 

to a first approximation, A=B, and (A-C)/C is approximately 10 

(Hill, 1975). The torque of an external body is obtained by 

finding the force from the above equation (4.1.1) by differentiation; since the first term vanishes, the torque is then FxR, 

or, (see Fig. 4.2 for sin e), 

-3 

(4.1.2) 

M, sin9/R3 (4.1.3 

2 

f = 3GM A-C)sin 6/R3 Using 1,=3~10~~kg-m 1 , and G=6.67~10-"~2/kg , 

L' = 3 6. 67x10-"Nm2 /kg2) ( 3x104 6kg-m2 ) ( 

Substituting for Jupiter, Mj=19.1x10 27 kg, R,=7.78x10 11 m, and 

@AGE 17 ~R*~T!ow~\~,IY w.wK D 
21 

sin 6O=0.104, then the torque due to Ju iter is =24.0x10 N-m 

(Joules). For other planets, 

of the Sun: 

24 

=20.0x10 P6 n-m2/kg2(Mpsin8/R 3). 

P .. Next, calculate the volume vb of the oblate equatoriaf belt 

. Vb = (2nR,)dRAH (4.1.4) 

where AR/Ro = low5 and, arbitrarily, AH = 2x108m, then 

(4.1.5) 21 3 vb = 6.2~10 m 

Now, find the mass of the oblate belt, using a density for the 

photosphere p = 3~10-~kg/m~ (Abell, 3rd Ed). Then, 

Mb pvb = 18.6~10 16 kg (4.1.6) 

Consider the torque from one planet, say, Jupiter (equation 

4.1.3 and so on). Find the average horizontal force from the 

quadrupole term, 

Fh=rj/R,= (24.0~10~~N-m) /7.0x108m=3 .42x1016N (4.1.7) 

Calculate the horizontal velocity vh=(Fh)t, where t=5 days, and 

Fh is the force per unit mass (divide by mass of belt, Mb): 

Vh = Fh/Mb(5 days) = 79.4 km/s = 7.94x1O4m/s 

which is the same order of magnitude as the equatorial velocity 

of the sun, equal to 2 km/sec. 

(4.1.8) 

Now calculate the kinetic energy per unit time, 

Compare this figure to the energy of solar activity, 10-6L,r 

which is 4~10~~J/sec (Allen, 1973). Thus, 

(KE/sec) = 45.2~10~~J/s > (4x1020J/s (4.1.10) 

(KE/sec) > 10-6L, 

So the energy input is much greater than the estimated energy of 

solar activity. 
22 

D 

4.2 TIHE SPAN OF TORQUE IN EXTERNAL SECTOR OF SUN 

F-or-the case of two planets, the problem is to find how long 

both planets are located in a sector of given angular width as 

viewed from the sun, near heliographic conjunction. Analytically, 

one uses angular velocity, w, or the equivalent, 2n/P, where P is 

the planet period in convenient units, such as days. Then start 

from the two innermost planets and proceed outwards. Let t be the 

time both Mercury and Venus are located in a given sector of 

heliocentric longitude, 8, then, 

or using periods, 

when angles are expressed in radians. If the angles are in 

degrees, multiply by 360°/2H. If one uses an angular sector of 

1/2 radian (28.65O), then the time in eqiation (4.3.2) becomes 

t = 1 (87.969 d)(224.70 d) i 11.50 days 

4iE (224.701 - 87.969)d 

(4.2.3) 

D 

D 

B 

Similarly for all other combinations of two planets. Using 

sidereal periods in days (Allen, 1973) one obtains the values in 

Table 4.1 A graph can be used to check results. Find the 

degrees/day each planet moves in longitude (Astron. Almanac) and 

plot on a solar system chart. Diagramthetimetakenfora given 

planet to move 28.65O in heliocentric longitude, and compare with 

another planet when they are in conjunction (see Fig. 4.2.1, 

Tables 4.1 & 4.2). 

The axial rotation of the sun does not enter here. Using the 

axial angular velocity of the sun, wo, and the two planetary 

revolutionary velocities, w1 &w2, it can be shown that wo cancels 

out (where wl'and w2' are synodic velocities): 

(4.2.4) 

Table 4.3 shows the "synodic" periods of the planets as viewed 

from the subsolar point on the sun's surface. 

precessional torque of the various planets. 

Finally, Table 4.4 lists the relative amplitude of B 
23 

\\ 270" '/ 

-. I - 

=:- A 3 4 Dlan view of solar System, showing planet poslrlons ' 

n 2 planets are within a sector of half 

raw.q.A.i. v .. ___- lime span whei . 

bv 1976 dat.?. 

radian near conJunctlon. 

.. ,__- 
0 

., 

Mer 

Ven 

0 

P(days)* Venus . Earth Mars .Jupiter Saturn Uranus Neptune 

87.969 11.504 9.221 8.028 7.105 7.06 7.02 7-01 

220.701 46.47 26.572 18.86 18.26 18.01 17.9 

Ear 

Har 

Jup 

Sat 

.. a' L... 

I 

365.256 

686.780 

4332.589 

10759.22 

0 

Ura 

Nep 

0 

30685.4 4831.1 

60189. 

I - 

1976 dates Planets Calc. Graph P'/07 Daily P 

20 Jul Me V 11.5 11.0 11.5 7.78 

05 Jul Me J 7.15 9.0 7.15 4.83 

22 sep ne E 9.22 11.5 9.21 8.32 

15 JUn* VE 06.5 40.0 46.5 46.2 

62.065 31.74 30.09 29.42 29.2 

60.95 58.00 55.92 55.3 

577.21 401.06 371.5 

1318.50 1012.6 

19 May 

18 Nov 

VJ '18.9 20.0 22.4 18.9 

EJ 31.7 30.5 31.8 31.5 

longitude, viewed from the sun; four SOlUtiOnS (Astrc 

1 coni I i days 

Earth+Moon 

Mars 

Jupiter 

Saturn 

Neptune 

Uranus 

403503.22 1.2 x lo-* 3.0 x " 

42828.44 

126686.9 X lo3 

37938.5 x lo3 

686 x lo" 

0.072 " 

5.6 X 

0.24 

3.7 x 10-l' 

9 

2.7 x 10'" 

1.3 x 

585 x io4 2.5 x 10-'* 0.0055 

0.00016 " 7.5 x lo-= 

Table 4.3. Periods 

point. The sidereal 

25.80 days (Wolff. 

Mercury 

Venus 

Earth 

Mars 

Jupiter 

Saturn 

Uranus 

Neptune 

deg/day 

5.28-1.6O 

0.44-1 .oo 

6.06-0.09" 

1.62- 1. Oo 7 1.60-0.09' 

1.00-0.09" 

Df the planets, viewed from the subsolar 

rotation period of the sun Is taken as 

1976). (Allen, 1973) 

Sidereal Period I Svnodic! Period 1 

87.969 d 

224.701 

365.256 

686.790 

4332.589 

10759.22 

30695.4 

36.51 d 

29.15 

27.76 

26.81 

25.95 

25.86 

25.82 

60189. I 25.81 ! 

~ahie --I-- A -.- relative ..-----__ -__. nmnlltude -_____ of 

__ rirecessional torque of planets on ---- 

the sun. (Allen, 1973). J - -4 x lo-' (Hill & Stebbins,- 1975). I PLANET [gin (10 rn'//sec~> -A ~~ 
25 

FiQ.4.3.lca) Two point masses are situated on the y-axis at ra. 

Find'the gravitational field along the x-axis. when x>>a. 

(b) The half-value ga of om,,. (See text). 

4.3 THE COMBINED GRAVITY FIELD OF TWO PLANETS ACTING ON THE SUN 

We wish to calculate the angular separation in longitude of 

two gravitational bodies, such that the gravitational field at a 

large distance perpendicular to the line joining them is one-half 

of the value of the two bodies superimposed (in line). This would 

approximate two planets in conjunction, then separated by an 

angle 0 (both have the same mass and are at the same distance as 

a first approximation) (See Fig. 4.3.1). 

To find the half-value of gx, set gx = (1/2)gmax. Take a 

ratio 

zX = 1 = -Gmx/(x 2 +a 2 -1 3/2 

2 3/2 - (4.3.1) 

gmax 2 -Gmx/(x ) 

Solve for x, calculate tan 0 = a/x = 0.7665: 0 = 37.47O. 

Thus two planets ofthe samemass and in the same orbit separated 

by 37.47O longitude have one-half of the gravitational force that 

they would have if at the same longitude (in conjunction). 

conjunction on sunspot number extends to + 5 days, in the case of 

Venus and the Earth (Link et al, 1954), or about loo, not 37O, so 

the Gaussian or cosine formula to be used should be a much 

steeper function. 

In actuality, the duration of effect of two planets in 
B 

D 

D 

D 

D 

c 

B 

26 4.4 NORTH-SOUTH ASYWTRY OF SOLAR ACTIVITY 

Solar activity is not equally divided between the northern 

and southern hemispheres of the sun. Before 1850, sunspots were 

more numerous in the northern hemisphere. From 1865 to 1910 they 

predominated in the southern hemisphere. From 1910 to 1969 there 

was a northern excess, with a strong asymmetry in cycle 20 

(Dodson & Hedeman, 1970). Apparently in the 1980's the northern 

excess may have come to an end. 

If sunspots (and associated solar activity) are influenced 

by planetary positions, then their slow swing in N/S latitude may 

be related to the similar slow latitude swing in the combined 

influence of allplanetswhichexertaneffectonthe sun. Any 

such influence should refer to planetary latitude with respect to 

the solar equator. The slowest important planets, Uranus and 

Neptune, reached their greatest latitude north only 10 years 

apart in 1672 and 1682, respectively, in the solar equatorial 

system. The most extreme case of N/S sunspot asymmetry occurred 

between 1672 and 1704, when no northern spots were observed. 

During this entire time period two or more outer planets were 

north of the solar equator. Two years before Uranus reached its 

descending node, Saturn passed its ascending node. 

period of Saturn and Neptune is 36 years, or about half the 

observed long N/S period of sunspots. The Neptune orbital period 

is 164 years, or about twice the observed N/S sunspot period (See 

Figs. 4.4.1 & 4.4.2 and Table 4.4.1). 

Various observers find a long period N/S oscillation of 

sunspot positions of 55 years (Berdischevskaya, 1967) of 70 and 

84 years (Brunner-Hagger & Liepert, 1944), and 70 years (Wolbach, 

1960) (White & Trotter, 1971), depending upon how it is measured. 

decades for the entire historical record of sunspot measurement. 

In Fig. 4.4.2 is shown a comparison of the N/S asymmetry of 

sunspots for the period 1943-1977 (Fracastoro & Marocchi, 1978), 

and the latitude of four outer planets relative to the solar 

equator . Also relevant is the latitudinal position of the sun 

center (CS) relative tothecenter of Massofthe Solar System 

(Petrova et al, 1978). The outer planets contribute the following 

percentages to the displacement of the sun: Jupiter: 49%, Saturn: 

27%, Uranus: 8%, Neptune: 15%, and Pluto: 1%. The vector quantity 

(CS-CMSS) is related to the sun surface and sun center Coriolis 

acceleration (Pimm & Bjorn, 1969). 

The K-corona and the zodiacal light may rotate at an 

inclination to the solar equator. Evidence is seen from 

observations of the K-corona and zodiacal light, especially when 

observations are made near to the axis of the nodes of the 

planetary system on the solar equator. The inner K-corona is 

inclined about 3' to the solar equatorial plane. Farther out the 

zodiacal cloud is inclined between 4' and 5.7' (Misconi, 1977). 

There would be a resulting non-uniform shear around the solar 

equator, which might engender oscillations. Such oscillations may 

have been observed, with periods of half a year (180.5 days) and 

half of Jupiter's synodic period (199.3 days)(Czada, 1983). 

The orbital period of Uranus is 84 years. The conjunction 

Shown in Table 4.4.1 is the latitude of three outer planets by 
27 

1790 

1600 ' 

1810 

1820 

1830 

1640 

1850 

1860 

1870 

1680 

re90 

1900 

1910 

1920 

1930 

1940 

19150 

1960 

1970 

8 980 

1990 

2000 

* 

*e 

Fio.4.42 Monthly values of Parameter A, the asymmetry between 

the activity in the north and south hemispheres 1943-1977 (left) 

(Fracastoro 6 Marocchi. 19781, compared to latitude of outer 

Planets relative to the solar equator (right). 

N 

S 

S 

N 

N 

S 

N 

S 

S 

N 

S 

N 

N 

S 

N 

N 

S 

N 

N 

S 

N 

N 

Tablc4.41Lonoltude of the outer planets, Jan 1 of year given, relative POSittOn north or 8outh of the solar equator, and 

PDroxirete Latitude (B) 

Jan 1 I Saturn 

Year f N/S')r 6 

353' 4' 

128' -3' 

218. -3' 

3' +6' 

135' -4' 

ZOO' -0.5' 

350' *6* 

130' -0' 

250' -0.5. 

S* *6. 

140. -4.5. 

251. *0.5* 

10. +lo 

1SO. -5. 

270' el' 

160. -6. 

-0. *l' 

00. +lo 

179, -5' 

47' e0.2' 

25. *I' 

286' *1.s' 

mom the solar equator. 

Uranus I Neptune 

S 

S 

S 

N 

N 

N 

N 

N 

s 

s 

s 

S 

N 

N 

N 

N 

s 

s 

s 

8 

N 

N 

N/S X B I N/S )r €3 

128' -3' I s 201' -3' 
0 

0 

0 

B 

28 

7 BIBLIOGRAPHY 

A1 len, C. W., "Astrophysical Quantities," Athlone Press, London, 

Alfven; H. & Arrhenius, G., "Evolution of the Solar System," NASA 

"Astronomical Almanac," US GPO, Washington, DC (yearly) 

Baker, N. & Temesvary, S., "Tables of Convective Stellar Envelope 

Models," NASA Inst for Space Studies, NY, 2nd Ed., 1966 

Berdichevskaya, V.S., "A Periodicity in the North-South 

Asymmetry of the Sunspot Distribution at the Minimum of New 

Cycles," Sov. Astron. 11 283 (1967) 

"Climate: History, Periodicity and Predictibil ity," 

Hutchinson-Ross, NY 1987 

Blizard, J.B., "Long Range Solar Flare Prediction," Final 

Report, NASA Contract NAS8-21436, 169 pp., 1969 

Blizard, J.B., "Precessional Effect on an Oblate Sun," B.A.A.S. 

13 981 (1983) 

Brunnz-Hagger, V. W. & Liepert, A., "Comparison of the Sunspot 

Curve for the Years 1853-1939 for the North- and South 

Hemispheres," Astr. Mitt. - 141 556 (1941) 

Bursa, M., "Precessional-Nutational Torque of the Planets," 

B.A.C. 35 87 (1984) 

Campbell, L.& Mof fat, J.W., "Quadrupole Moment of the Sun and 

the Planetary Orbits," Ap. 3. - 275 L77 (1983) 

Cox, J.P., "Theory of Stellar Pulsation," Princeton U Press, 

Princeton, NJ 1980 

Cox, J.P., & Guili, R.T., "Principles of Stellar Structure," 

Gordon & Breach, Inc. NY 1967 

Czada, I. K., "Evidence for the Phi-dependent Rotationoscillation of the Sun," Sol. Phys. 82 439 (1983) 

Dodson, H.W. & Hedeman, E.R., "Time Variations of Solar 

Activity," p 151 in "Solar Terrestrial Physics/l970," 

Dyer, ed., D. Reidel, Dordrecht-Hol land, 1972 

Hickey, J.R., et al, "Solar Variability Indications from Nimbus 7 

Satellite Data," in NASA CP-2191, Washington, DC 1981 

Hill, H.A. & Stebbins, R., "The Intrinsic Visual Oblateness of 

the Sun," Ap. J. - 200 471-483 (1975) 

Hoyt, D. & Eddy, J., "Solar Irradiance Modulation by Active 

Regions, 1969-1981," Geophys.Res.Lett.10 509-512 (1983) 

Jackson, S., "Classical Electrodynamics," Wzey NY 1976 

Jose, P.D., "Sun's Motion and Sunspots," A.J. 70 193 (1965) 

Kaula, W.M., "Introduction to Planetary PhysicT" Wiley NY (1968) 

Khlystov, A.I., "On the Possibility of Resonance Enhancement in 

Link, F., "Planetary Influences on the Sun, XII," B.A.C. - 6 133 

Melchoir, P., "The Earth Tides," Pergamon Press, NY 1966 

3rd ed., 1973 

SP-345, Washington, DC 1976 

Blizard, J. B., "Solar Activity: Long-Range Prediction," in 

the Convective Zone," Soln. Dannye #10 78 (1978) 

(1954) 
29 

Misconi, N.Y., "The Symmetry Plane of the Zodiacal Light near 1 

AU," in "Solid Particles in the Solar System," J. Halliday & 

B.A. McIntosh, Ed., Reidel, Boston, 1980 

of the CMSS Relative to the Solar Center and Solar 

Activity," Soln. Dannye #12 89 (1978) 

using Dynamic Relations between the Sun and Planets," NAS8- 

21445 Final Report, April, 1969 

Oscil lations of a Slowly Rotating Star: Quasi-toroidal 

Modes," Astr. Astrophys. 94 126 (1981) 

Saio, H., "R-mode Oscillation in Uniformly Rotating Stars," Ap. 

J. - 256 717 (1982) 

Sanders, J.E., "Short-term Periodicity of Solar Orbital Accelerations," in C1 imate: History, Periodicity and Predictibility," Hutchinson-Ross, NY 1987 

Takahashi, K., "Solar-terrestrial Disturbances of August, 1972," 

J.Radio Res. Lab. Japan - 21 459 (1974) 

Tuominen, J. et al, "Meridional Motions Derived from Positions of 

Sunspot Groups," MNRAS - 205 691 (1984) 

Unno, W., et al, "Nonradial Oscillations of Stars," Univ. Tokyo 

Press, Tokyo, Japan, 1979 

Ward, F., "Circulation of the Solar Atmosphere and Maintenance 

of the Equatorial Acceleration," Ap. J. - 141 534 1965) 

Warwick, J., "Magnetic Fields in the Solar System," in "Magnetospheric Particles and Fields," McCormac, ed. Reidel NY 1976 

White, O.R. 6, Trotter, D.E., "Distribution of Sunspots Between 

North and South Solar Hemispheres," Ap.J.Supp. 33 391 (1971) 

wil liams, G.E. & Sonett, c.P., e8Solar Signature in Sedimentary 

Cycles from the Late Precambrian Elatina Formation, 

Australia," Nature - 318 523-322 (1985) 

Winkler, C., et al, "Plane of Symmetry of the Zodiacal Light and 

Structure of the Gegenshein," Astr Astrophys. 143 194 (1985) 

Wolbach, J., "On the Unequal Spottedness of the Two Solar Hemispheres," Smiths. Contrib. 5 195 (1962) 

Wolff, C.L., "Rigid and Differential Rotation Driven by 

Oscillations Within the Sun," Ap.J. 194 489 (1974) 

Wolff, C.L., "Timing of Solar Cycles by Rigid Internal Rotations," Ap.J. - 205 612 (1976) 

Wolff, C.L., "Rotational Spectrum of G-modes in the Sun," Ap.J. 

- 264 667 (1983) 

Wolff, C.L., "Solar Irradiance Changes Caused by G-modes and 

Large Scale Convection," Sol. Phys. 93 1 (1984) 

Wolff, C.L. & Blizard, J.B., "Properties of R-modes in the Sun," 

Sol. Phys. - 105 1 (1986) 

Wolff, C.L. & Hickey, J.R., "Multiperiodic Irradiance Changes 

Caused by r-modes and g-modes," Sol.Phys. - 108 (1987) 

Petrova, N.C., et al, "Comparison between Time-spatial Position 

Pimm, R.S. & Bjorn, T., "Prediction of Smoothed Sunspot Number 

Provost, J., Berthomieu, G., & Rocca, A., "Low Frequency