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The Relationship of Sunspot Cycles to Gravitational Stresses on the Sun: Results of a Proof-Of-Concept Simulation

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The Relationship of Sunspot Cycles to Gravitational Stresses on the Sun: Results of a Proof-Of-Concept Simulation Chapter 14 The Relationship of Sunspot Cycles to Gravitational Stresses on the Sun: Results of a Proof-of-Concept Simulation Edward L. Fix 1395 Quail Ln., Beavercreek, OH 45434, USA Chapter Outline 1. Introduction 335 4.1. Details 345 2. Approach 337 5. Discussion 347 3. Method 340 5.1. Weaknesses 347 3.1. Barycenter Ephemeris 5.2. Predictions 348 Preparation 341 6. Conclusion 349 4. Result 342 1. INTRODUCTION The 11-year sunspot cycle is well known. With recent minima occurring in early 1965, 1976, 1986, and 1996, the sunspot cycle was expected to bottom out in 2006 or 2007, and begin an upswing into Solar Cycle 24. The sun, however, had other ideas. The numbers continued to decline through 2006, 2007, 2008, and 2009, only beginning a slow climb in late 2009 to early 2010. Since none of the many models had predicted this, sunspot prediction suddenly became a hot topic again. Understanding the sun’s dynamics has become important in the space age, because sunspots bring solar storms, which affect satellites giving us communications, weather prediction, and navigation. And, of course, the sun’s activity has a direct effect on the earth’s climate, although the climate’s exact sensitivity to solar activity is a subject of considerable controversy. Evidence-Based Climate Science. DOI: 10.1016/B978-0-12-385956-3.10014-2 Copyright Ó 2011 Elsevier Inc. All rights reserved. 335 336 PART j IV Solar Activity There are other characteristics of the sunspot cycles that are less well known. Sunspots have magnetic fields (Hale, 1908), and eastewest oriented pairs of sunspots appear magnetically bound. The eastern sunspot of the pair might have a north-leading polarity (relative to the sun’s rotation) while its western mate has a north-trailing polarity, and in the other hemisphere, that relationship will be reversed. During the next cycle, Hale discovered that the polarities of the sunspot pairs were reversed from the previous cycle (Hale et al., 1919). At first, he mistakenly attributed this to the difference in the sunspot’s latitude, but it is now known that the vast majority of sunspots of each cycle have polarity reversed from those of the previous (and succeeding) cycle. This approximately 22-year cycle of pairs of 11-year sunspot cycle has become known as the Hale cycle. The sun’s total magnetic field is much more fluid than that of the earth. In the mid-twentieth century it was observed that the sun’s global magnetic field reverses near the peak of each sunspot cycle (although the polarity of the sunspots themselves remain the same throughout the cycle) (Babcock, 1961). Finally, sunspots have long been observed occurring in an 11-year cycle appear at the mid-latitudes (roughly midway between the equator and the pole in both hemispheres), and as the cycle progresses, they appear closer and closer to the equator. This is another way of distinguishing between sunspots of the previous cycle and those of the next, as sunspots often overlap during a sunspot minimum. Motion of the planets has long been suspected as the driver of the 11-year sunspot cycle. Attempts to relate the two date back at least 150 years (Wolf, 1859). Results of various analyses range from disappointing to tantalizing. Three models using essentially pattern matching techniques (Jose, 1965; Fairbridge and Shirley, 1987; Cole, 1973) did not predict the last half of the 20th century. Bracewell (1986) used precambrian varves to develop a sum-of-sines mathe- matical model which did a good job of predicting Solar Cycles 22 and 23. It matched the actual peak sunspot counts of the cycles, as well as the dates of the minima. Due to its nature, it was inherently incapable of predicting the long minimum between Cycles 23 and 24, and it failed to reproduce the Maunder Minimum. Bracewell also did no more than speculate on the genesis of the pattern. Many other models do not attempt to reproduce individual sunspot cycle amplitude or timing, but relate the motion of the sun about the barycenter of the solar system to large-scale patterns in the sunspot cycle such as the Maunder Minimum (e.g., Landscheidt, 1999). It is important to distinguish between tidal forces on the sun, and the motion of the sun about the barycenter of the solar system. Tidal forces are due primarily to the inner planets, and involve differential gravitational force between the planet’s subpoint on the sun, and the limb of the sun. Correlation between solar activity and tidal forces has been shown (Hung, 2007; Bigg, 1967), but not with the overall sunspot cycle. On the other hand, the sun’s Chapter j 14 The Relationship of Sunspot Cycles to Gravitational Stresses 337 FIGURE 1 Position of the barycenter of the solar system relative to the sun, 1900e2000. motion about the barycenter, shown in Fig. 1, is determined mainly by the four giant planets: Jupiter, Saturn, Uranus, and Neptune. This distinction is also emphasized by Fairbridge and Shirley (1987). 2. APPROACH This study departs in a number of ways from earlier efforts. First was the selection of format for representing the historical sunspot data. Bracewell (1953) advocated displaying the sunspot cycle as a zero-mean sinusoid by reversing the sign of every second cycle, reflecting the 22.3-year Hale cycle. This conclusion was independently reached early in this current study. It was noted that the minima in the sunspot cycle tend to be sharper and better defined than the maxima, a characteristic of a graph of the absolute value of a sinusoid. This is not an entirely accurate analogy, owing to the fact that during the sunspot minima, spots from both the previous and succeeding cycle appear simulta- neously. However, it is at least a good first approximation, and there seems no harm in plotting the sunspot data in this way, to see where it leads. Starting with 338 PART j IV Solar Activity the average monthly sunspot number data downloaded from the ftp server of the National Geophysical Data Center of the National Oceanic and Atmospheric Administration (NOAA/NGDC), Solar-Terrestrial Physics Division (ftp://ftp. ngdc.noaa.gov/STP/SOLAR_DATA/SUNSPOT_NUMBERS/MONTHLY), the sign of every second cycle is reversed, arbitrarily picking Cycle 23 as positive. I picked the crossover points by simple inspection. The result is called the “polarized sunspot number” in Fig. 2. Another unique feature of this effort is the way the motion of the sun about the barycenter of the solar system is related to the sunspot cycle. Earlier efforts relate the solar cycle to parameters like spin-orbit coupling (Wilson et al., 2008; Javaraiah, 2005; Juckett, 2000) and changes in angular momentum, torque, or curvature of the sun’s orbit (Palus et al., 2007; Landscheidt, 1981, 1999; Wood and Wood, 1965; Javaraiah, 2005). These all primarily involve the sun’s angular motion around its orbit, and the attendant changes in orbital speed. In contrast, this study focuses primarily on the radial component of the sun’s motion about the barycenter, and was initiated after making the following observation. An ephemeris of the barycenter of the solar system was obtained from the server at NASA JPL (http://ssd.jpl.nasa.gov/horizons.cgi) (Giorgini et al., 1996). For purposes of this study, the Z component of the barycenter’s position was ignoreddthat is it was always assumed to lie on the ecliptic planedand a month is taken as 1/12 of 365.25 days. The barycenter’s coordinates were plotted in polar format (r, q), and various comparisons made to the polarized sunspot plot. The dark curve in Fig. 2 is the velocity (r_) of the radial distance of the barycenter relative to the sun; that is, the first derivative of the radial distance with respect to time, with dimension of km/month. The sunspot count and r_ data in Fig. 2 are from completely inde- pendent data sets. Intriguing similarities of the curves are evident between 1750 FIGURE 2 Polarized sunspot number vs. barycenter radial velocity. Chapter j 14 The Relationship of Sunspot Cycles to Gravitational Stresses 339 and about 1780, and again between 1891 and about 1951. It seemed unlikely that the correlations could be coincidental. The second derivative of the radial distance of the barycenter with respect to time (r€) is acceleration, and acceleration is simply a force scaled by mass (F ¼ ma). So, why not treat it as a force and see what happens? A third departure of this effort, relative to previous work, was the use of a dynamic simulation to investigate the possible effect of changes in the sun’s motion. A variety of analytical techniques have been used to predict sunspot cycle activity. Some investigators have used more or less purely mathematical analysesdphysically-based curve fitting efforts (Wolf, 1859; Bracewell, 1986). Others have performed frequency and/or phase analyses of the sunspot cycle (De Meyer, 1998; Palus and Novotna´, 1999; Mininni et al., 2000; Cole, 1973; Juckett, 2000). Many studies have sought to correlate features of the sun’s orbit with features of the sunspot cycle (Landscheidt, 1981; Landscheidt, 1999; Wilson et al., 2008; Zaqarashvili, 1997; Charva´tova´, 1990a,b; Fairbridge and Shirley, 1987; Jose, 1965). These are mainly static analyses, and demonstrate correlation between the sun’s orbit and the sunspot cycle. However, correlation does not necessarily imply causation. A dynamic oscillatory model driven by the barycenter’s motion, however, does show causation by its nature. A positive result would lend support to the notion that the sunspot cycle could be driven by the sun’s motion. A simple resonant oscillator was chosen as a test platform for this simu- lation.
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