The Sun And Its Spots

THE SUN AND ITS SPOTS ABSTRACT Human beings have observed dark spots on the surface of the sun for centuries, yet we have only come to understand what these spots actually are since the invention of the telescope. These so-called sunspots are magnetic indentations in the solar surface. Sunspots are visibly distinct from the rest of the surface because their intense magnetic fields inhibit the upward flow of hot material from the solar interior, making them relatively cool and dark. This paper provides a detailed description of what a sunspot is, examines the physics behind their formation, and discusses their role in the relationship between the earth and the sun.

Lukas Fried Integrative Exercise Final Version April 8, 2009

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INTRODUCTION Our own sun is a dynamic celestial body. Among the most prominent indicators of its evolving, non-static nature are sunspots. Sunspots appear as dark patches on the solar surface and have been observed to peak in number roughly every 11 years. The Athenian astronomer Theophrastus first noted the existence of these dark patches during the 4th century B.C.1 However, it was not until the invention of the telescope in the early 17th century A.D. that astronomers such as Galileo and Scheiner, both credited with the discovery of sunspots, truly began to study their physical behavior and characteristics in depth.2 Indeed, many consider the first telescopic observation of sunspots in 1611 to be the birth of astrophysics as a whole. Some important discoveries concerning sunspots that have been made since then include their indented geometry (Wilson, 1769), regular 11-year spike in number (Schwabe, 1843), tendency to drift towards the equator and poles following this spike (Spörer, 1873), magnetic bipolarity (Hale, 1908), and the unified model for solar magnetism that explains their formation (Babcock, 1960).3 Since human beings have known of their existence for so long, and have spent nearly four centuries probing their physical characteristics, we have come to understand quite a lot about sunspots. The goal of this paper is not to provide an exhaustive summary of all sunspot research conducted since the beginning of time. Nor is it a guide to sunspot observation methods, though observation is the primary source of our knowledge about sunspots. Rather, this paper seeks to provide physically rigorous answers to the most fundamental questions about sunspots: what are they? How and why do they form? Why are they relevant? These questions will be answered through an exploration of the sun’s internal structure, the physical characteristics of sunspots, the time evolution of the sun’s magnetic state, and theories that link the presence of sunspots to terrestrial phenomena.

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SOLAR STRUCTURE AND ENERGY TRANSPORT To comprehend the formation, structure, and behavior of sunspots, we must first have a rudimentary understanding of the environment in which they exist. Therefore, we begin our journey into the nature of sunspots with the structure of the sun, itself, and the energy transport processes that characterize its structural elements. Much like our own planet, the sun can be thought of as a series of concentric spherical layers. The internal layers of the sun, in outward order from the solar center, are termed the core, radiative zone, convection zone, and photosphere, collectively depicted in Fig. 1.

Figure 1. The solar structure.4

The core extends radially outward from the center of the sun to about 0.2 solar radii, or 140 000 km. It is characterized by high temperatures – 15 million K, compared to temperatures of around 5800 K at the solar surface – and high mass density – 150 g/cm3, three orders of magnitude higher than the average density of the convection zone.5 These intense conditions are due to the exothermic conversion of hydrogen into helium in a fusion process known as the proton-proton (p-p) chain, which fuels the sun. The p-p chain is described by the following set of chemical equations:6

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1

H + 1H → 2D + e+ + νe + 1.44 MeV 2 D + 1H → 3He + γ + 5.49 MeV 3 He + 3He → 4He + 1H + 1H + 12.85 MeV

(1) (2) (3)

In Eqs. (1)-(3), 1H is a proton, 2D is a deuteron, e+ is a positron, νe is an electron neutrino, 3He and 4He are electron-bare helium nuclei, and γ is a γ ray. Note that none of the nuclei have bound electrons, since the core’s high temperatures provide electrons with enough energy to be stable in unbound states. In Eq. (1), two protons overcome their strong electrostatic repulsion by coming within femtometers (10-15 m) of each other, where the strong force, rather than coulombic forces, dominate interactions. At that point, one of the protons spontaneously decays into a neutron and a positron. There is only a one in 14 billion chance of this occurring, on average, but the presence of many free protons in the core ensures that the decay products form in appreciable amounts.7 The positron produced by the decay will annihilate with the first electron it encounters, releasing more energy,8 while the neutron fuses with the remaining proton to form a deuteron. In Eq. (2), that deuteron fuses with another proton to form the nucleus of an isotope of helium with two protons and one neutron, releasing a γ ray. Finally, in Eq. (3), two helium nuclei formed by way of Eqs. (1) and (2) fuse into an α particle, a “normal” helium nucleus consisting of two protons and two neutrons, ejecting the two remaining protons in the process. The net result is that four protons fuse to make a helium nucleus: 41H → 4He + 2e+ + 2νe + 2γ + 26.71 MeV

(4)

An important consequence of the p-p chain is that it releases enormous amounts of energy into the core of the sun. Energy is liberated in the form of γ rays and in additional kinetic energy imparted to the various massive particles involved in the chain. The quantities on the right side of Eqs. (1)-(4) account for both. The liberated energy escapes the core through two

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heat transport processes: radiation and convection. These transport processes define the remaining interior solar layers. That brings us to the next layer of the sun, the radiative zone, which reaches from the edge of the core out to approximately 0.7 solar radii (490 000 km). As its name suggests, the radiative zone is where energy produced from the fusion reactions in the core is shuttled around by electromagnetic radiation, in the form of photons. The photons are continually scattered as they encounter free electrons, protons, and electron-bare atomic nuclei. As a result, it takes roughly 50 million years for energy produced in the core to escape through the sun and out into space, due to the random-walk motion of the photons by continual scattering.9 Moving radially outward, temperatures in the radiative zone gradually fall to around 1 million K, which enables some atomic nuclei to acquire electrons, and thus become proper atoms.10 The bound electrons can easily absorb photons they encounter, entering excited states. When an excited electron drops to a less-excited state, it releases a photon, which can be quickly absorbed by electrons in other atoms. The ease of photon absorption and emission by bound electrons makes it difficult for energy to radiate past the outer edge of the radiative zone. The gases in this location become opaque from the high photon concentrations, and energetically unstable due to their inability to expel energy outward via radiation.11 Fortunately, radiation is not the only way for the gases to transfer their energy. At the point where the gases become opaque (around 0.7 solar radii), in order to prevent an unstable buildup of energy, convective energy transfer processes take over, and we enter the convection zone of the sun. Convection is a consequence of heating a fluid from one side (usually “from below”). Rather than heat being uniformly transferred from the heated side to an unheated side, it is more energetically favorable for hot fluids to form bubbles – called convection cells – that are

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hotter than their surroundings. Convection cells tend to rise through their cooler surroundings because of a vertical pressure imbalance that results directly from the ideal gas law and Archimedes’ principle. The ideal gas law,

ρ=

µm u P , kB T

(5)

tells us that higher temperature gases are less dense than cooler gases of the same chemical € are less dense than the gases around them. In Eq. (5), ρ is composition,12 so convection cells

mass density, kB is Boltzmann’s constant, T is temperature, µ is the average mass of particles in the fluid, mu is 1 atomic mass unit, and P is pressure. Archimedes’ principle states, “the buoyant force on an object submerged in a fluid is equal to the weight of the fluid displaced by that object.”13 Take a convection cell to be the submerged object and the gases surrounding it to be the fluid. The fact that the convection cell displaces gases denser (i.e. – heavier) than itself means that the buoyant force acting on the cell exceeds the cell’s own weight. Assuming that gravity and buoyancy are the only considerable forces, the convection cell will rise. This rising action is what we mean by convection. Whenever there is sustained convection, regular convective currents are established in which convection cells rise, then release their energy, cooling them to the point that they become denser than their surroundings and sink.14 This is exactly what occurs in the outer 20% - 30% of the sun, as indicated by the circular arrows in Fig. 1.

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Figure 2. Photospheric granulation. The area enclosed by the circle is 5 arcseconds in diameter.15

A thin outermost layer known as the photosphere encloses the convection zone. Here, radiation once again becomes the dominant form of energy transport, as the convection cells that have risen through the convection zone reach the solar surface and emit photons out into space.16 This constitutes the final link in the chain of energy transfer processes that ensures the ultimate escape of energy produced at the core, maintaining the sun’s state of dynamic equilibrium. The largest photospheric convection cells – termed supergranules – are an average of 32 000 km in diameter and can be found in the lower regions of the photosphere.17 By the time they emerge on the surface and radiate their energy, the cells have shrunk to about 1100 km in diameter and are called granules.18 Figure 2 shows an image of the photospheric granulation. It is important to understand the solar structure and the energy transport processes of radiation and convection in any discussion of sunspots. Radiation is why we see sunspots in the first place; spots are visually distinct from their surroundings because they radiate different wavelengths of light than the rest of the photosphere. This wavelength difference results from how sunspots affect photospheric convection, which will be explained shortly. Furthermore, if we understand the p-p chain and the processes responsible for expelling the energy it liberates,

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then we understand why the sun is so hot. The sun’s high temperatures ensure the physical conditions necessary for sunspot formation: the partial ionization of the sun’s gases and the radiation pressure needed to prevent the sun from collapsing under its own gravity.19

WHAT IS A SUNSPOT? We can now begin our physical discussion of sunspots, keeping in mind the solar structure and energy transport processes we have established in the previous section. Sunspots appear as dark spots on an otherwise yellowish photosphere, hence their apt name. They consist of a dark central region called the umbra, surrounded by a lighter-colored, annular penumbra. It should be noted that the shape of a particular spot can vary greatly from the ideal round prototype – indeed, the prototype is the exception rather than the rule – and that even some large spots lack penumbrae.20 Typical spots have umbral diameters of perhaps 10 000 km, though occasionally there are some 20 000 km wide, five times the distance between New York and Los Angeles. There are also other small dark areas around 2500 km in diameter that have no penumbrae; these are called pores and are considered to be undeveloped sunspots, though they can sometimes enlarge and develop into full-fledged spots.21 Within some penumbrae, we can sometimes see thin filaments of brighter material extending into the umbra, called light bridges. An image of this general structure appears in Fig. 3.

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Figure 3. A dark sunspot umbra (center), surrounded by a lighter penumbra and light bridges extending into the umbra. Photospheric granulation can clearly be seen in the areas lacking spots.22

When we look at the sun from the earth, we have no visual depth perception of its surface. What we see is essentially a two-dimensional projection of the side of the sun facing us, called the disk, which is bounded by the apparent “edge” of the sun, the limb. Consequently, sunspots look like flat features as they traverse the photosphere. In actuality, they are not flat, but rather they are indentations in the photosphere. Wilson determined this in 1769, when he observed the penumbra of a large spot contract and disappear as it approached the limb.23 He concluded that this behavior was evidence for a depressed, three-dimensional spot structure. This constituted the first physical (rather than observational) inquiry into the nature of sunspots and is termed the Wilson effect, after its discoverer. As we will see later, magnetism plays a primary role in the behavior and formation of sunspots. Scientists know that sunspots are magnetic because of the Zeeman effect, which is the splitting of atomic spectral lines by an external magnetic field.24 The Zeeman effect occurs because most atoms possess a magnetic moment, due to the configuration of their electron orbitals. For a given atomic energy level l, there are 2l + 1 orbitals, each oriented differently. Every orbital has a corresponding magnetic moment associated with it, though that moment is 9

zero when the magnetic quantum number m is zero (m can take the value of all integers between –l and l, inclusive). The potential energy U of an object with magnetic moment µ in a magnetic field B is U = –µ⋅B = –µB cos θ,

(6)

where θ is the angle between µ and B.25 Since the orbitals of a particular level l have different magnetic moments, they must have different potential, and therefore, total, energies, when a magnetic field is applied. An applied field shifts the energy of an orbital by an amount mµBB from its normal, degenerate level (µB is the Bohr magneton, 5.79 ×10−5 eV/T).26 As the energy of a photon emitted in an atomic transition is equal to the energy gap between the two states

€ involved in the transition, the energy shift creates frequency shifts in the light emitted by transitioning electrons. Hence, there is spectral splitting, depicted in Fig. 4. The degree to which a spectral line is split directly corresponds to the magnitude of the applied field. The spectra of inactive (i.e. – spotless) regions of the photosphere exhibit only marginal Zeeman splitting, indicating the presence of weak magnetic fields, on the order of 1 G (100 µT).27 However, the high degree of splitting in sunspots indicates that strong magnetic fields, generally of magnitude 4000 G (400 mT), exist in sunspot umbrae, normal to the photosphere.28 Weaker magnetic fields exist in sunspot penumbrae, as well, though these fields are more horizontal than vertical because of the 6 km/s radial outflow of material from umbrae – the Evershed outflow.29 Zeeman-Doppler imaging takes advantage of the Zeeman effect and Doppler shifting – the alteration of emitted photon frequencies due to the motion of the emitting object, relative to the observer – to explain both the Evershed outflow and sunspots’ magnetic geometry.30

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Figure 4. The Zeeman effect in the helium atom.31 (a) An external magnetic field B splits the normally threefold-degenerate l = 1 atomic state into three distinct states, separated by energy µBB. (b) There is corresponding splitting of the spectral line associated with a transition from the l = 1 excited state to the l = 0 (ground) state. In the absence of the field, only photons of frequency f0 are emitted in this transition; with the field present, two additional frequencies f0 ± µBB/h are possible.

Maxwell’s law ∇ ⋅ B = 0 states compactly that magnetic fields are not emitted from a source, as electric fields are, but rather loop around and re-enter any point they exit from. In € other words, there is no such thing as a magnetic monopole; magnetism is always manifested in

dipole behavior. It therefore seems suspicious that sunspots, which from the preceding paragraph sound like magnetic monopoles, would form in isolation. It turns outs that sunspots ensure that ∇ ⋅ B = 0 by forming in bipolar pairs, with a leader spot of one magnetic polarity being followed

by a follower spot of opposite polarity, relative to the direction of the sun’s rotation. This is €

known as Hale’s polarity law and is explained by the sunspot formation mechanism, which will be discussed later.32 Why are sunspots dark? The short answer is because they are cooler than the surrounding photosphere. Like sunspots’ magnetic properties, this has been determined by comparing the light emitted by sunspots with the light emitted by the inactive photosphere. The connection between light and temperature is best explained by the Planck distribution function for blackbody radiation,

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2πhc 2 1 I( λ,T) = , 5 hc / λkB T λ e −1

(7)

where I is the intensity of light, λ is the wavelength of light, h is Planck’s constant, and c is the

€33 Equation (7) shows that the spectrum of light emitted by an ideal speed of light in a vacuum. blackbody – an object that absorbs all incident light – is temperature dependent. Hotter objects will emit most of their light at shorter wavelengths (correspondingly, at higher frequencies and energies) and in greater amounts than cooler objects, making them brighter. We can therefore draw conclusions about the temperature of an object from the spectrum of light it emits. It turns out the radiative peaks of sunspot umbrae occur at longer wavelengths and lower intensities than the peaks of the inactive photosphere; thus, sunspots are lower in temperature (see Fig. 5). Spectroscopic analysis has determined that the average temperature of the photosphere is 5800 K, while sunspot umbrae are around 4200 K, meaning that sunspots are 1600 K cooler than the

Intensity !J s "1 m"2 sr"1 m"1 "

photosphere!34 2.5 !10"32

2. !10"32

5800 K !Photosphere"

1.5 !10"32

1. !10"32

4200 K !Umbra"

5. !10"33

0 0

500

Wavelength !nm" 1000

1500

2000

Figure 5. A sunspot umbra and the inactive photosphere, idealized as perfect blackbodies. Note that the umbra’s radiative peak lies at a higher wavelength and lower intensity than that of the photosphere, as the umbra is lower in temperature.

This, however, is an incomplete answer to the darkness question because it fails to account for the mechanism by which sunspot umbrae become cooler than the photosphere in the first place. A more complete answer as to why sunspots are dark is that their intense magnetic 12

fields inhibit photospheric convection – they hinder the upward flow of hot material from within the sun – which translates to lower temperatures within the umbra and therefore a darker appearance. Convection normally mixes hot subsurface gases with the cooler surface matter in the photosphere, making the surface hotter than it would be otherwise.35 It still occurs in sunspots, evident in the presence of umbral granulation (albeit small granules, 30% smaller than normal36), but to a lesser degree. The physical explanation for convection inhibition requires a better comprehension of the behavior of solar magnetic fields, particularly the “freezing-in” of magnetic fields in plasmas. “Frozen-in” fields are explained in the following section of the paper. Once we understand this concept, we can build a picture of solar magnetism that elucidates the sunspot formation mechanism, which will allow us to better account for the relative darkness of sunspot umbrae.

FROZEN-IN FIELDS Thus far, in describing the structure of the sun and sunspots, we’ve treated the sun’s matter as a hot ideal gas. This treatment is certainly useful for conceptualizing the thermal processes of convection and radiation, but it neglects something critical – ionization. Due to the sun’s hot temperatures, much of the sun’s matter is ionized, meaning that the solar gases are gases of electrically charged particles. A gas consisting of ionized particles, in which, on average, there is no net charge, is called a plasma.37 Plasmas are considered to be the fourth state of matter in the universe, the first three being solids, liquids, and gases. The presence of charge in plasmas necessitates the inclusion of both electric fields and magnetic fields (created, of course, by the movement of charge) in any “complete” physical model of their dynamics. It turns out that magnetic fields in plasmas tend to get dragged around by the plasmas themselves. This is

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known as the “freezing-in” of magnetic fields and is vital to our understanding of how sunspots form. This section draws on a derivation carried out by Foukal,38 establishing the equations of motion for a plasma subjected to gravitational and magnetic fields, and then shows how “freezing-in” results from them. We deal with large quantities of plasma when we talk about solar dynamics, and so it is logical to treat solar plasma as a continuous medium, rather than a collection of discrete ions. For a plasma of density

, moving with velocity v, the requirement of continuity can be stated as

∂ρ + ∇ ⋅ ρv = 0 ∂t .

(8)

Essentially, Eq. (8) is a statement of conservation of mass flow. It explains that the density of a

€ plasma at a point in space decreases as the plasma spreads outward to other locations. This establishes the plasma as a fluid, since the motion of its constituent matter at a certain point mechanically influences behavior at all adjacent points. The equation of motion for a plasma is given by

ρ

dv = −∇P + ρg + j × B , dt

(9)

where P is pressure, g is the uniform gravitational acceleration at the location of the plasma, j is

€ B is an external magnetic field.39 Equation (9) can be thought of as electric current density, and an expression of Newton’s Second Law, ΣF = ma ,

(10)

generalized for a particular density of plasma, rather than an explicit mass.40 The plasma’s € acceleration is accounted for by the left side of Eq. (9), and the forces (per unit volume) on the plasma are summed on the right. The term −∇P is a pressure gradient. It is the net force per unit volume on the plasma from pressure imbalances surrounding it. These pressure imbalances can

€ 14

arise from different parts of the surrounding plasma being at different temperatures or densities, as the ideal gas law [Eq. (5)] suggests.

is the gravitational force per unit volume. Finally,

is the magnetic volume force. The movement of the plasma in a magnetic field B creates a current density j, and the magnetic volume force arises from the interaction of this current density with the same magnetic field. Figure 6 depicts the geometry of this configuration, and shows that the magnetic volume force will oppose the plasma’s motion.

Figure 6. Geometry of j, B, and j × B , relative to the plasma’s velocity v.41

Current density j can be calculated from Ohm’s law, €

j = σ (v × B) ,

(11)

where σ is the electrical conductivity of the plasma.

€ quantities of plasma, we cannot neglect the modification of As we are dealing with large €

B by the currents induced in the plasma. We therefore must rewrite Eq. (11) as

j = σ (E + v × B) ,

(12)

where E is the induced electric current given by Faraday’s law of induction,

€

∇×E =−

€ 15

∂B ∂t .

(13)

Equations (12) and (13) are built on the assumption that the small electrostatic fields between charges in the plasma are negligible, that is, ∇ ⋅ E = 0 over the dimensions we are interested in. Taking the curl of Eq. (12) and applying a special vector identity, we determine that € ∂B 1 = ∇ × (v × B) + ∇ 2B . ∂t 4 πσ

(14)

This is an induction equation. It explains how the magnetic intensity at a point in space evolves

€ plasma of uniform conductivity σ , moving in a velocity field v. In in time within a magnetized theory, we can combine the induction equation [Eq. (13)] with the equation of motion [Eq. (8)], € the continuity statement [Eq. (8)], and the ideal gas law [Eq. (5)] to completely determine the

dynamics of a magnetized plasma. Our work, however, is not over. The sun’s high degree of ionization makes it an excellent electrical conductor; halfway through the convection zone, its conductivity is on the order of

10 6 A /V ⋅ m,42,43 one tenth the conductivity of the best conducting metals known to man.44 This means that σ is quite large, and as σ appears in the denominator of the rightmost term in Eq.

€

(14), that term becomes negligible. We are left with a slightly simpler expression of the € induction equation,

€

∂B ≅ ∇ × (v × B) . ∂t

(15)

Equation (15) indicates a remarkable result of the sun’s high electrical conductivity " . It

€ of a magnetic field in a solar plasma is almost entirely dependent explains that the time evolution ! on the motion of the plasma itself. The plasma is literally dragging the magnetic field around

with it as it moves! Magnetic fields become part of the plasma; this is what is meant by a “frozen-in” field. The “freezing-in” of magnetic fields in plasmas forms the basis for the physical

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discipline of magnetohydrodynamics, or MHD, and is of special importance in the formation of sunspots, which we discuss in the next section. Prior to moving on, though, we must make one clarification: Only the components of plasma motions perpendicular to magnetic fields will move the field lines, which follows from the cross product v × B in Eq. (15). Put differently, if you shear a plasma volume element parallel to the field lines in it, as in Fig. 7(b), the field lines will remain unchanged. If, instead, € you shear the plasma perpendicular to the field lines, as in Fig. 7(c), the direction of the field

lines will shift with the shearing, decreasing the spacing between them and thus increasing the magnetic flux density.

Figure 7. (a) A rectangular plasma volume element with imbedded field lines. (b) Shearing the rectangle parallel to the field lines leave the field lines unchanged. (c) Shearing perpendicular to the field lines decreases the spacing between them by a factor cos γ.45

THE BABCOCK MODEL AND SUNSPOT FORMATION In 1960, H.W. Babcock published a paper entitled, “The Topology of the Sun’s Magnetic Field and the 22-Year Cycle,” which established the Babcock model for solar magnetism, based on the work of Hale, Maunder, Alfvén, Cowling, and other forefathers of MHD and sunspot theory.46 The Babcock model takes advantage of “frozen-in” magnetic fields to provide what is still considered to be the most fundamental, unified explanation for sunspot formation.

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Here, we summarize the Babcock model, which describes the evolution of the solar magnetic field as a 22-year cycle. Central to the model is the sun’s differential rotation. On the earth, we experience uniform solid body rotation, since the solid landmasses on which we live are more or less anchored to one spot. No matter where you stand on the crust, you can be sure that, in a period of about 24 hours, the earth will rotate you back to your original position (relative to the earth’s axis). The sun, however, is made up of gas and not solid matter, so different parts of it rotate at different angular velocities. From observations of the motion of sunspots themselves, it has been determined that angular velocity ω at a point on the photosphere is a function of latitude φ :47 € ω = 14.38° − 2.77°sin 2 φ

(16)

It follows that the equator € rotates with a period of 26 days and regions near the poles rotate with a period of 37 days.48

€

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Figure 8. The Babcock model of solar magnetism. (a) Field lines are initially poloidal. (b) Field lines begin to distort and wrap around the equator. (c) Fields have become largely toroidal. (d) Dipolar magnetic regions move to neutralize and invert weak extant poloidal field.49

The Babcock model is built around the idea that the sun’s differential rotation causes a time evolution of solar magnetic fields through the shearing of “frozen-in” field lines in convection zone plasma. In the first stage of the model, the sun is taken to be an axisymmetric magnetic dipole,50 a sphere of rotating plasma with magnetic field lines running parallel to lines of longitude from one pole to the other – an entirely poloidal field [see Fig. 8(a)]. This initial field is thought to be the product of rotations of convection zone plasmas about the solar axis, known as the dynamo mechanism, though the details of its operation are still not well

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understood.51 The number of sunspots present is at a mimimum in the first stage, so this stage is conventionally called the solar minimum. In the second stage, differential rotation causes shearing of plasma in the convection zone, since plasma closer to the equator moves faster than at higher latitudes. The shearing thus occurs perpendicular to the field lines. Recall from the previous section that shearing perpendicular to field lines alters the direction and spacing of the field lines. In Fig. 8(b), we see this alteration, as the fast-moving equatorial plasma begins to distort the field lines, dragging them so that they begin to wrap around the equator. The wrapping continues into the third stage, shown in Fig. 8(c). At this point – called the solar maximum, which occurs after 3 years – the sun’s magnetic field has been dragged so much at the equator that its field lines have been wound into spirals of 5.6 turns each in the northern and southern hemispheres, roughly at latitudes φ = ±55° .52 The field lines are no longer poloidal, but rather they are almost completely toroidal – they are oriented parallel to lines of latitude, with lines in the northern hemisphere€pointing in the opposite direction of the lines in the southern hemisphere. Again, recalling the previous section, shearing a plasma perpendicular to its imbedded field lines shrinks the distance between lines, increasing its magnetic flux density. At the solar maximum, the field lines have been squeezed together to the extent that they behave as long cylinders of strong magnetic flux, called flux tubes. Flux tubes, Babcock states, will inevitably encounter some kind of non-uniformity in their non-magnetized surroundings, such as an area of differing viscosity or a rising convective cell, and so it is only natural that they will develop kinks and twists.53 The most twisted-up regions contain the strongest magnetic fields, as their field lines have been wound tightly together, decreasing line spacing and therefore increasing

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flux density. These extremely twisted regions experience not only internal and external gas pressures of the normal variety, but significant magnetic pressure from the tightly-packed field lines, as well. A pressure balance between the twisted flux tubes and their surroundings implies that

Pext = Pmag + Pint

,

(17)

where the P’s are the external, magnetic, and internal pressures, respectively.54 Therefore, we know that

€

Pext > Pint

(18)

and therefore, given the ideal gas law [Eq. (5)], if the temperature inside the flux tubes is similar to that outside,

€

ρ ext > ρ int .

(19)

The density of the plasma inside the twisted tubes is less than the outside plasma density, and so

€ the tubes will rise through the convective zone. This is termed magnetic buoyancy.55 Given the right conditions, a twisted bundle of tubes will rise fast enough that it will erupt out of the photosphere, creating an Ω-shaped column of plasma called a prominence above the photosphere. Figure 9 diagrams the formation of a prominence, while Fig. 10 is an image of an actual prominence.

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Figure 9. A highly twisted bundle of flux tubes rises through the convection zone and erupts out of the photosphere, forming a solar prominence.56

Figure 10. An image of a prominence, taken by the International X-ray Observatory (IXO). 57 Note the striking clarity of the field lines.

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Figure 11. Sunspots are created by flux tube emergence.58

At the points where the prominence meets the photosphere, the eruption creates a pair of dark, bipolar magnetic regions – sunspots! This satisfactorily accounts for both the sunspot formation mechanism and sunspots’ bipolarity. Bipolarity is explained by the fact that a pair of sunspots are connected by the flux tubes that created them – the field lines exiting the photosphere through one spot will re-enter the photosphere through the other spot, and thus the spots will be of opposite magnetic polarity. Now that we have seen why many sunspots begin to form in the third stage of the Babcock model, the solar maximum, we can return to our summary of the model. In the fourth and final stage of the Babcock model, the follower spots (see the definitions of leader and follower spots on p. 11) migrate towards the poles, which aligns the bipolar sunspot pairs with the poles, as in Fig. 8(d). To quote Babcock, “the cause of [this] migration in latitude . . . remains obscure.”59 In any case, the poleward migration re-establishes a dominant poloidal magnetic field, oriented in the opposite direction of the field present in the first stage of the model.60 The lack of strong toroidal fields makes it difficult for new sunspots to form, and so there are few present. We have returned to solar minimum, roughly 11 years since the beginning of the cycle.

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At this point, the cycle restarts, but the initial poloidal field is now oriented opposite its direction at the start of the previous cycle. Because the initial poloidal magnetic field is oriented the same direction every 22 years, and not every 11 years, some, including Babcock, conceptualize the evolution of the sun’s magnetic field as a 22-year cycle (see Fig. 12).

Figure 12. The photospheric magnetic field as a function of time (horizontal axis) and latitude (vertical axis).61 The field strength and orientation at the poles (90N and 90S) reverses every 11 years, and thus have a period of 22 years, justifying the notion a 22-year solar magnetic cycle. The curves extending from the equator to the poles between 1988 and 1993 show the poleward migration of the location of the follower spots, corresponding to the re-establishment of a dominant poloidal field.

RETURN TO THE DARKNESS QUESTION With the framework of “frozen-in” field lines, flux tubes, and magnetic buoyancy in mind, we can finally tackle the question of why sunspots are dark. Earlier in the paper, we left off by saying that sunspots are dark because their magnetic fields inhibit convection, making them cooler and therefore less radiant than the surrounding photosphere. The explanation for this is derived from the magnetic pressure, and is very similar to the explanation of flux tube magnetic buoyancy. Within a sunspot, there must be a lateral balance between the external pressure on the spot and the combination of internal gas and magnetic pressures. Otherwise, the 24

spot would either expand outwards (if the external pressure were weaker than the other two pressures) or contract (if the external pressure were stronger than the other two pressures). Given this balance, the internal gas pressure must be less than the external gas pressure. The ideal gas law [Eq. (5)] tells us, then, that the plasma inside a sunspot is either less dense or cooler than the plasma outside. In reality, it is a combination of both. A less dense plasma would tend rise more quickly up through the sunspot because of vertical gravitational pressure imbalances (see the explanation of convection on pp. 5-6). This would actually aid convection. Therefore, it makes sense that the cooling of the internal plasma is the dominant effect. Cooler plasma is less likely to rise up and thus convection is inhibited.

LARGER IMPLICATIONS It is sometimes difficult to find relevance in a pure field like astrophysics. Pure science pursues pure truth without needing a purpose other than to deepen mankind’s understanding of nature. This fact is disquieting for those unsatisfied with the pursuit of knowledge exclusively for knowledge’s sake, because it makes science seem irrelevant to their own lives. While we have taken a pure approach to exploring sunspots, they are far from being irrelevant. Rather, sunspots play a role in the complex relationship between the earth and the sun, having possible effects on the earth’s climate, economics, politics, terrestrial radiowave propagation, and the appearance of the night sky. The Maunder Minimum is textbook evidence for a potential connection between sunspots and climate. As we have established, the number of sunspots follows an 11-year cycle with marked regularity. Between the years of 1645 and 1715, however, astronomers observed only about 50 sunspots over a 30-year period, compared to a normal range of 40 to 50 000.62 This

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period of relatively few sunspots is known as the Maunder Minimum, and is shown in Fig. 13. Interestingly enough, the Maunder Minimum coincided with the Little Ice Age, in which winter temperatures in earth’s northern hemisphere were reduced by 1 to 1.5 K, enough to freeze over normally ice-free rivers in Europe. The temperature reduction has been estimated from analysis of tree rings and ice cores, and is quite evident in period paintings of outdoor winter scenes.63 Why would sunspot scarcity promote a colder climate? The current theory is that sunspots increase the energy output of the sun, allowing more sunlight to reach and warm the earth. Fewer sunspots translates to less sunlight, and therefore lower temperatures. The idea that sunspots increase solar energy output seems counterintuitive, given that sunspots hinder the upward flow of hot convection zone plasma. Nevertheless, the sun radiates 0.2% more energy at solar maximum (when the highest number of spots are present) than it does at solar minimum.64 This phenomenon may be attributable to an increase in hot matter ejected from the sun during magnetically active periods, which will be described momentarily.

Figure 13. The Maunder Minima and other sunspot extrema, in the context of normal fluctuations in sunspot number.65

While there is clearly a convincing argument for a connection between sunspots and climate, it is harder to rationalize the effects sunspots could have on economics and politics. It is conceivable that the effects on climate could influence agricultural output, triggering changes in

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crop supply, consumer prices, and other economic indicators, all of which could be tied to levels of civil unrest and voting habits. By and large, though, studies on economics and politics shy away from suggesting logical connections, and instead consist primarily of statistical correlations between the sunspot number and various economic or political trends. One study shows that all of the major American economic recessions or depressions from 1926 to 2003 occurred near solar maximum, with the notable exception of the Great Depression, which occurred at solar minimum (Fig. 14). Another notes that the Dow Jones Industrial Average has a tendency to spike shortly before solar maximum (Fig. 15). In a third study, A.J. Tchijevsky, a 20th-century Russian professor of Astronomy and Biological Physics, determines that 80% of the most significant events in human history, mainly violent, occurred during 5-year periods of the most intense solar activity.66 This is based on the compiled histories of 72 countries from 500 BC to 1922 AD. Finally, one tenuous study finds a connection between the number of Republicans in the United States Senate and the sunspot number (Fig. 16).

Figure 14. Sunspot cycles and depressions/recessions, 1926-2003.67

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Figure 15. The Dow Jones Industrial Average (DJIA) peaks before solar maximum.68 Sunspot numbers are listed as a percentage of an 11-year moving average. The arrows indicate “significant” DJIA peaks, with the most recent arrow being a prediction made when the graph was originally printed.

Figure 16. The number of Republicans in the U.S. Senate spiked just after solar maximum from 1965 to 1986.69

As rational or irrational as the hypothesized climatic, economic, and political effects of sunspots may be, none can be definitively confirmed or denied, due to their complexity. The only real certainty is that sunspots increase the amount of ions ejected towards the earth by the sun. 28

Ionic ejection, as it pertains to sunspots, is a result of magnetic reconnection – when a solar prominence becomes twisted, its magnetic field lines of opposite polarity may reconnect, forming a closed loop of plasma that ejects forcefully out into space at speeds of up to 1000 km/s.70,71 These so-called coronal mass ejections (CMEs) create shock waves in the solar wind – the steadier, “normal” outflow of charged matter from the sun – causing bursts of plasma to stream towards earth. Upon reaching the earth, these bursts collide with and distort the ionosphere, the atmospheric layer of charged particles. This can disrupt radio communications and generate terrestrial electromagnetic storms that damage power stations and electronics.72 In 1989, one such storm was powerful enough to knock out power to most of the Canadian province of Quebec.73 Since sunspots are created in the formation of prominences, and prominences provide the plasma needed for these violent events, the presence of many sunspots on the photosphere is a good predictor of when these events are most likely to occur. Rarely are electromagnetic storms as powerful as the 1989 behemoth, but the influx of ions from the sun can still be experienced by listening to a radio at solar maximum – the ions may create crackles and fuzziness in the signal. A pleasanter consequence of the increased incidence of charged matter at periods of high solar activity is the enhancement of polar aurorae. These beautiful swirls of color are collections of ions that become confined in the atmosphere near the earth’s poles because of interactions with the earth’s magnetic field there. When sunspot numbers are high, more ions stream out from the sun and become trapped in these polar regions, creating more brilliant aurorae. Figure 17 compares the sunspot number to auroral activity, while Fig. 18 is a picture of the aurora borealis, the northern lights. It is hard to question the relevance of sunspots when the dazzling light show they create hangs over one’s head.

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Figure 17. Auroral activity, measured by the AA index, is in phase with the 11-year-periodic number of sunspots.74

Figure 18. The aurora borealis above Bear Lake, Eielson Air Force Base, Alaska.75

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CONCLUSION Sunspots are striking evidence that the sun is a dynamic, evolving body. Through careful thought and observation, mankind has gone from merely noting the existence of dark spots on the sun to rigorously probing their inner nature. By developing specialized ways of observing the sun, taking data, and applying reasoning based on the known laws of physics, those who have studied the sun have assembled a wonderful picture of what sunspots are and how they most likely form. In this paper, we have captured the essentials of their work. We know that sunspots are magnetic indentations in the surface of the sun that appear to be dark because they inhibit the convective energy flow upwards from within the sun. They are created by the warping of the magnetic field lines frozen into the solar plasma, by the sun’s differential rotation. As the field lines get wound around the sun, flux tubes form. These magnetic tubes are less dense than the unmagnetized plasma surrounding them, and so they can erupt out of the photosphere, forming a bipolar pair of sunspots. The interplay of magnetic fields, convection, gas dynamics, and fluid dynamics here can only be called beautiful.

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WORKS CITED 1

R.J. Bray and R.E. Loughhead, Sunspots (Wiley, New York, 1965), pp. 1-2. Ibid., pp. 1-2. 3 Ibid., pp. 4-10. 4 P. Foukal, Solar Astrophysics (Wiley, New York, 1990), p. 189. 5 Ibid. 6 H. Zirin, “Sun,” McGraw-Hill Encyclopedia of Science & Technology, 10th ed. 17, 672 (2007). 7 K.J.H. Phillips, Guide to the Sun (Cambridge, Cambridge, 1992), p. 49. 8 N. Christensen (private communication). 9 Sunspots: Current Research 2 of 7 from Exploratorium, http://www.exploratorium.edu/sunspots/research2.html. 10 K.J.H. Phillips, Guide to the Sun (Cambridge, Cambridge, 1992), p. 51. 11 Ibid. 12 Ideal gas law from Wikipedia, http://en.wikipedia.org/wiki/Ideal_gas_law. 13 D.C. Giancoli, Physics for Scientists and Engineers, 4th ed. (Prentice Hall, Upper Saddle River, NJ), p. 349. 14 K.J.H. Phillips, Guide to the Sun (Cambridge, Cambridge, 1992), p. 51. 15 P.R. Wilson, Solar and Stellar Activity Cycles (Cambridge, Cambridge 1994), p. 33. 16 K.J.H. Phillips, Guide to the Sun (Cambridge, Cambridge, 1992), p. 73. 17 G.W. Simon and R.B. Leighton, “Velocity Fields in the Solar Atmosphere. III. Large-Scale Motions, the Chromospheric Network, and Magnetic Fields,” Astrophys. J. 140, 1120 (1964). 18 K.J.H. Phillips, Guide to the Sun (Cambridge, Cambridge, 1992), p. 74. 19 C. Blaha (private communication). 20 P. Foukal, Solar Astrophysics (Wiley, New York, 1990), p. 189. 21 Ibid. 22 sunspot.jpg from Light and Matter, http://www.lightandmatter.com/html_books/7cp/ch06/figs/sunspot.jpg. 23 R.J. Bray and R.E. Loughhead, Sunspots (Wiley, New York, 1965), p. 4. 24 M.A. Dubson, J.R. Taylor, and C.D. Zafiratos, Modern Physics for Scientists and Engineers, 2nd ed. (Prentice Hall, Upper Saddle River, NJ), p. 291. 25 Ibid. 26 Ibid., p. 293. 27 E.R. Priest, “The Sun and its Magnetohydrodynamics,” Introduction to Space Physics (Cambridge, New York, 1995), p. 60. 28 K.J.H. Phillips, Guide to the Sun (Cambridge, Cambridge, 1992), p. 103. 29 E.R. Priest, “The Sun and its Magnetohydrodynamics,” Introduction to Space Physics (Cambridge, New York, 1995), p. 71. 30 M. Semel, “Zeeman-Doppler imaging of active stars,” Astron. Astrophys. 225, 456 (1989). 31 M.A. Dubson, J.R. Taylor, and C.D. Zafiratos, Modern Physics for Scientists and Engineers, 2nd ed. (Prentice Hall, Upper Saddle River, NJ), p. 294. 32 K.J.H. Phillips, Guide to the Sun (Cambridge, Cambridge, 1992), p. 105. 33 M.A. Dubson, J.R. Taylor, and C.D. Zafiratos, Modern Physics for Scientists and Engineers, 2nd ed. (Prentice Hall, Upper Saddle River, NJ), p. 141. 2

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C.W. Allen in K.J.H. Phillips, Guide to the Sun (Cambridge, Cambridge, 1992), p. 107. E.R. Priest, “The Sun and its Magnetohydrodynamics,” Introduction to Space Physics (Cambridge, New York, 1995), p. 73. 36 P. Foukal, Solar Astrophysics (Wiley, New York, 1990), p. 189. 37 M.G. Kivelson, “Physics of Space Plasmas,” Introduction to Space Physics (Cambridge, New York, 1995), p. 27. 38 P. Foukal, Solar Astrophysics (Wiley, New York, 1990), pp. 106-21. 39 Ibid., p. 119. 40 D.C. Giancoli, Physics for Scientists and Engineers, 4th ed. (Prentice Hall, Upper Saddle River, NJ), p. 87. 41 P. Foukal, Solar Astrophysics (Wiley, New York, 1990), p. 119. 42 Ibid., pp. 189-191 43 M. Stix, The Sun: An Introduction, 2nd ed. (Springer, New York, 2004), p. 308. 44 Electrical conductivity from Wikipedia, http://en.wikipedia.org/wiki/Electrical_conductivity. 45 T.E. Faber, Fluid Dynamics for Physicists (Cambridge, Cambridge, 1995), p. 415. 46 H.W. Babcock. “The Topology of the Sun’s Magnetic Field and the 22-Year Cycle,” Astrophys. J. 133, 572-86 (1960). 47 Newton and Nunn in H.W. Babcock. “The Topology of the Sun’s Magnetic Field and the 22Year Cycle,” Astrophys. J. 133, 573(1960). 48 E.R. Priest, “The Sun and its Magnetohydrodynamics,” Introduction to Space Physics (Cambridge, New York, 1995), p. 63. 49 K.J.H. Phillips, Guide to the Sun (Cambridge, Cambridge, 1992), p. 69. 50 H.W. Babcock. “The Topology of the Sun’s Magnetic Field and the 22-Year Cycle,” Astrophys. J. 133, 575 (1960). 51 K.J.H. Phillips, Guide to the Sun (Cambridge, Cambridge, 1992), p. 68. 52 H.W. Babcock. “The Topology of the Sun’s Magnetic Field and the 22-Year Cycle,” Astrophys. J. 133, 576 (1960). 53 Ibid. 54 E.R. Priest, “The Sun and its Magnetohydrodynamics,” Introduction to Space Physics (Cambridge, New York, 1995), p. 73-4. 55 H.W. Babcock. “The Topology of the Sun’s Magnetic Field and the 22-Year Cycle,” Astrophys. J. 133, 581 (1960). 56 Ibid. 57 solar_prominence.gif from the NASA Goddard Space Flight Center, http://ixo.gsfc.nasa.gov/images/resources/imageGallery/science/solar_prominence.gif. 58 Emerging of a flux tube, http://www.lund.irf.se/Helioshome/fluxtube.html. 59 H.W. Babcock. “The Topology of the Sun’s Magnetic Field and the 22-Year Cycle,” Astrophys. J. 133, 583 (1960). 60 K.J.H. Phillips, Guide to the Sun (Cambridge, Cambridge, 1992), p. 68. 61 Synoptic-solarmag.jpg from Wikimedia, http://upload.wikimedia.org/wikipedia/commons/3/33/Synoptic-solmag.jpg. 62 The Sun’s Chilly Impact on Earth from NASA, http://earthobservatory.nasa.gov/Newsroom/view.php?old=200112065794. 63 Ibid. 35

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64

Sunspots and climate from the U. of Wyoming, http://wwwdas.uwyo.edu/~geerts/cwx/notes/chap02/sunspots.html. 65 Sunspot_Numbers.png from Wikimedia, http://upload.wikimedia.org/wikipedia/commons/2/28/Sunspot_Numbers.png. 66 D. Wilcock, Sunspot Cycles and Human History from Kondratieffwinter.com, http://www.kondratieffwinter.com/kw_esoteric_sunspot_cycles.html. 67 Ibid. 68 T. Modis. “Sunspots, GDP, and the stock market,” Tech. Forecasting & Soc. Change 74, 1510 (2007). 69 lindzen_sunspots.jpg from RealClimate, http://www.realclimate.org/images/lindzen_sunspots.jpg. 70 P.R. Wilson, Solar and Stellar Activity Cycles (Cambridge, Cambridge 1994), p. 51. 71 K.J.H. Phillips, Guide to the Sun (Cambridge, Cambridge, 1992), pp. 211-9. 72 P.R. Wilson, Solar and Stellar Activity Cycles (Cambridge, Cambridge 1994), p. 51. 73 Scientists probe northern lights from all angles from CBC News, http://www.cbc.ca/health/story/2005/10/22/northern_lights_051022.html. 74 Sunspots and climate from the U. of Wyoming, http://wwwdas.uwyo.edu/~geerts/cwx/notes/chap02/sunspots.html. 75 Polarlicht_2.jpg from Wikimedia, http://upload.wikimedia.org/wikipedia/commons/a/aa/Polarlicht_2.jpg.

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ANNOTATED BIBLIOGRAPHY R.J. Bray and R.E. Loughhead, Sunspots (Wiley, New York, 1965). Sunspots is supposed to be the Bible of the field. I found it largely useless. If you are interested in sunspot astronomy – observation methods, etc. – this is the place to turn. They go into endless detail about astronomical techniques and sunspot morphology. If, like me, you are curious about sunspot astrophysics, there are better sources than this one. Try Foukal. H.W. Babcock. “The Topology of the Sun’s Magnetic Field and the 22-Year Cycle,” Astrophys. J. 133, 572-84 (1960). A primary reference for my section on the Babcock model. These are the words of Babcock himself. I found this article very accessible for a scholarly paper, and after rereading it several times, it worked the kinks out of my understanding of sunspot formation. M.A. Dubson, J.R. Taylor, and C.D. Zafiratos, Modern Physics for Scientists and Engineers, 2nd ed. (Prentice Hall, Upper Saddle River, NJ). Great for a review of the Zeeman effect and the Planck distribution. This was the textbook I used in Physics 228: Atomic and Nuclear Physics. P. Foukal, Solar Astrophysics (Wiley, New York, 1990) This was my favorite source and quite possibly the most useful. It provided the derivation for the “frozen-in” field approximation that forms the equation-heavy section of my paper. I had some problems trying to read other books on magnetodynamics, and this one turned out to be right on my level. Plus, it always brought the MHD back to how it works on the sun. K.J.H. Phillips, Guide to the Sun (Cambridge, Cambridge, 1992) This book was a lifesaver. Excellent presentation of the solar structure and the Babcock model. It clarified many of the conceptual issues I had because it did not go into the (unwanted) detail that the Bray and Loughhead text did. E.R. Priest, “The Sun and its Magnetohydrodynamics,” Introduction to Space Physics (Cambridge, New York, 1995) I didn’t always like the brevity of this textbook’s explanations, but it was helpful for understanding the pressure balance requirements for flux tube buoyancy.

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