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Solar Astrophysics

AA2055

Available as a module for a University Advanced Certificate, CertHE, DipHE and BSc in Astronomy.

Sample Notes

These sample pages from the Course Notes for the module Solar Astrophysics have been selected to give an indication of the level and approach of the course. They are not designed to be read as a whole, but are intended to give you a flavour of the syllabus, style, diagrams, images, equations, mathematical content and presentation. They are a subset of the colour, navigable on-line version of the learning materials. All enrolled students will be sent a CDRom with all the Course Notes, videos and learning materials, in addition to having access to them via the course website. · All sections of notes will be available in modest colour and basic navigation in pdf format suitable for downloading and printing at home. It is anticipated that most students will prefer to use the notes in the colour pdf files on the CDRom.

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1.4 How we observe the Sun today

Observations of the Sun today take place over a wide range of wavelengths, from the gamma rays through to the radio region of the spectrum. Critically, we can zoom in on narrow parts of the Sun's spectrum to sample particular elements and temperature regions within the Sun's atmosphere and upper-most layers. 1.4.1 The Solar Spectrum Gas emits or, where conditions are favourable, absorbs preferentially at certain wavelengths, known as spectral lines. In practice each element has its own characteristic sets of spectral lines, with the preferred wavelengths depending upon the electronic energy levels and the degree of ionisation. Spectroscopy of the Sun shows continuum emission, which is a smooth brightness variation across all wavelengths, with super-imposed absorption lines that can be used to identify the elements present. The strongest and most obvious lines (Figure 1.5) are called the Fraunhofer lines, and are attributed to hydrogen, iron, sodium, calcium, magnesium and molecular oxygen. Note however there are a multitude of other lines.

Figure 1.5 The solar spectrum, showing the Fraunhofer lines labelled A to K and then a to h. The wavelengths are in Ångstroms. 1 Ångstrom = 0.1 nm (See www.coseti.org/image s/ospect_1.jpg for a more detailed version.) Image credit: Wabash Instrument Corporation.

If we observe the Sun at ever greater sensitivity and spectral resolution (Figure 1.6) we find more and more absorption lines of varying width and intensity, many of which remain unidentified. This may appear unlikely ­ after all there are only around 92 elements from which they could originate. However at the temperatures encountered close to the Sun, each of these elements can suffer ionisation. Furthermore each element has a number of electrons equal to its atomic number, Z and can therefore exist in different degrees of ionisation ­ atoms can be singly, doubly, triply, etc. ionised. For example atomic oxygen (chemical symbol O) has 8 electrons. The corresponding neutral oxygen spectrum is referred to by astronomical spectroscopists as O I, pronounced "Oh-one" (Spectroscopists use roman numerals to denote the ionisation state.) When the oxygen atom loses one electron, it is said to be singly ionised. Chemists call this ion O+, pronounced "ohplus" and spectroscopists describe its spectrum as O II, (because it is one step up from neutral oxygen labelled O I.) If it has lost five electrons it is five-times ionised (O5+, O VI). Once it has lost all its electrons, it is fully ionised (O8+, O IX). Ions with more electrons tend to have complex characteristic spectra (e.g. Iron has 26 electrons) and each element will have a characteristic spectrum for each of the ionisation levels available to it. Many of these species are present in the solar atmosphere. So we can see that lines in the solar spectrum can originate from an enormous range of ions. The primary determinant of the level of ionisation is the temperature of the plasma. From this we may see that observations of specific lines sample plasma at different temperatures.

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Figure 1.6 A high resolution solar spectrum, showing numerous lines in the visible region. Wavelength runs left to right, but spectra are stacked vertically to ensure a compact plot. See www.noao.edu/image _gallery/html/im0600. html for larger versions of this figure. Image credit: NOAO/AURA/NSF.

Calculations of the spectra of strongly ionised ions show that many lines are in the ultraviolet (UV). Thus by observing in very narrow ultraviolet wavelength bands we can isolate emission from specific ions. In the solar atmosphere each ion is generated by a narrow range of combinations of temperature, pressure and magnetic field. Thus such narrow band images allow us to home in on regions of the solar atmosphere where different conditions pertain. As UV light is absorbed by the Earth's atmosphere, we need observatories in space to take advantage of these spectral lines.

Spectroscopy

It is appropriate at this point to look briefly at how these spectral images are created. Emission from the Sun is separated by wavelength using a spectrograph. This takes advantage of the refractive or diffractive properties of light. In refractive spectrographs the light is bent through a prism or similar optical device. Because the refractive index changes with wavelength the light emerges separated into different colours. Higher resolution can be achieved using a diffraction grating. Very fine lines are etched onto a filter in the instrument so that transmitted light is diffracted (bent when encountering an edge) in a way that splits it into far finer fractions of the wavelength range. Diffraction gratings generally deal with much narrower wavelength ranges than refraction spectrographs ­ there is a trade-off between spectral resolution (how finely we can split the light up) and wavelength bandwidth (what wavelength range the spectrograph can accept). Observations of the narrow transition region (typically less than 30 km across) can be made over a very large number of spectral lines because there is such a dramatic variation in temperature. Many of the ultraviolet images of the Sun are dominated by features in the transition region. A second technique of interest here is simply to interpose an opaque disc between the detector and the Sun. Effectively we cause an artificial eclipse, with the added advantage of being able to increase the size of the obscuring disc to screen out the brightest parts of the corona as well. We find that much of the activity in the outer corona and solar wind are impossible to observe with the Sun in the field of view. This is because the detectors are overwhelmed by the Sun's intensity. Using a

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we can remove the Sun from the observations and concentrate on the coronal and solar wind features.

coronagraph

1.4.2 Solar observatories Because much of the interesting emission from the Sun is in the UV, most solar observatories are space-based. Nonetheless some important observations are made from the ground.

Ground-based observations

Ground-based observatories continue to play a very important roll in solar astrophysics. Two key advantages of these observatories are the much greater resolution they can achieve and the number of locations available. Ground-based facilities can also test new or improved instruments, one of the major advantages ground-based astronomy has over space-based observatories.

White light

The most straight-forward forms of ground-based observation are made in white light. These are simply observations across the full visible band. They are white not in the sense of equal quantities of all colours, but rather in the sense that they detect light across all colours without distinguishing between the different wavelengths. The primary solar features observed in white light are the sunspots and the granules. Hydrogen emits radiation in a large number of narrow lines. In most circumstances the strongest line in the visible band is Hydrogen Alpha (H for short). As hydrogen is the most abundant element in the Universe, observations of the H line are widely used to examine astronomical objects. By using a spectrometer or optical filter that excludes light outside a narrow wavelength range centred on 656.3 nm, we can isolate the light from H. This allows us to make images showing where the H emission is strongest and take advantage of the Doppler shift effect. Modern solar telescopes use tuneable filters, allowing observers to rapidly scan between different velocities and obtain images that can track changes in velocity across the solar surface. From this we can see how the gas is moving on a given part of different solar structures. There are considerable disadvantages to ground-based observatories. The most important ones are: 1. The restriction to optical and a few infrared wavelengths. 2. The inability of any single observatory to observe the Sun continuously. The first of these means that emission from many elements and ions cannot be viewed from Earth, because many of the most interesting emission lines occur in the ultraviolet region of the spectrum. The second point makes it extremely difficult to follow processes that take place over a few hours, such as the development and motion of granules.

Hydrogen Alpha

Solar and Heliospheric Observatory

Space-based observatories

SOHO is a satellite observatory optimised for continuous observations of the Sun. It was launched in 1995 in time for the solar minimum, but has observed continuously since. SOHO will now hopefully observe for the whole solar cycle.

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In order to allow continuous observation of the Sun, SOHO is located at the L1 or Inner Lagrangian point of the Sun-Earth gravity system, 1.5 million km towards the Sun. This is the point at which the forces due to Earth's and the Sun's gravity balance, so that in principle an object placed there will move in neither direction. In practice, however, the L1 point is unstable so that the slightest additional force in any direction will cause an object to move away from the point. In order to maintain its position, SOHO is in a halo orbit around the L1 point (Figure 1.7). Essentially this results in the observatory orbiting about the line between Earth and the Sun. Careful control of the satellite's orientation allows it to keep the Sun in view at all times.

Figure 1.7 SOHO's insertion and current halo orbit. The satellite's orbit now lies within the hatched region. Image credits: SOHO

As a true observatory, SOHO has a large number of instruments, 12 in all. A brief description of each and its applications follows: · CDS (Coronal Diagnostic Spectrometer) ­ imaging the Sun simultaneously between 15 and 80 nm (extreme ultraviolet) to understand the dynamics and structure of the solar atmosphere CELIAS (Charge, Element, and Isotope Analysis System) ­ sampling and analysing the solar wind in order to better understand its origin and the heating mechanisms of the corona. COSTEP (Comprehensive Suprathermal and Energetic Particle Analyzer) ­ measuring the properties of high energy particles, including electrons, in CMEs and the solar wind. EIT (Extreme ultraviolet Imaging Telescope) ­ imaging the entire solar disc at a number of ultraviolet wavelengths. ERNE (Energetic and Relativistic Nuclei and Electron experiment) ­ studying the most energetic particles during explosive solar events. In between, ERNE examines cosmic rays from the Milky Way.

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Section

2

Probing the Solar Interior

In this Section, we will be looking at how the Sun's energy is generated and how that energy is transported through the solar interior to the surface. Considering the equations that describe the structure of the solar interior, we will see how the temperature, density and pressure vary throughout the interior of the Sun. We will also discuss the key results from the new research field of helioseismology ­ using oscillations within the Sun to examine its interior.

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2.1 Standard Solar Model

The standard solar model is a mathematical description of how the temperature and density of the solar interior varies from the centre to the solar surface. The solar model needs to able to explain the age, temperature, luminosity, mass and radius of the Sun. This model goes beyond simply describing the internal temperature and density distribution discussed in Section 1. The standard model makes the following assumptions: 1. The Sun is a sphere (all physical quantities depend on radius) and is in a steady state (it is not expanding or contracting). Hence this ignores rotation which causes the equator to bulge. 2. The Sun was made out of a primordial cloud of gas that consisted primarily of hydrogen and helium. 3. The energy produced by the core is transferred by convection is more efficient.

radiation,

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Encycl. of the Sun Important Equation Important Reaction

unless

4. The internal rotation is sufficiently slow and the internal magnetic field is weak enough that the resultant forces are negligible.

The standard solar model also needs to describe the abundance of the elements present in the Sun. This is simply the relative fractions by mass of the different elements (hydrogen, helium, etc.) The equation that describes the abundance is

X +Y + Z =1

Equation 2.1

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where X is the fractional abundance of hydrogen, Y is the fractional abundance of helium and Z is the fractional abundance of the heavier elements (generally referred to as the "metals"). The elemental abundances vary with radius and time. The abundance values normally used in the solar model are: X = 0.71 , Y = 0.27 and Z = 0.02 . This means that 71% (by mass) is hydrogen, 21% is helium and all the other elements together contribute the remaining 2%. The value of Z is taken to be the abundance of heavy elements found in the solar atmosphere.

2.2 Modelling Solar Structure

The structure of the Sun can be examined using set of five equations that describe: the balance of forces on the gases in the Sun; how the mass of the Sun is distributed throughout the interior; the generation of energy in the Sun; how it is transported through the interior, together with an equation for the state of the gas within the Sun. This set of equations is valid for all stars. See also Chapter 3 of the Encyclopaedia of the Sun, particularly Focus 3.2.

Mathematical notation

Throughout this section we refer to rates of change. These are denoted mathematically using differential notation. A rate of change can be considered as a gradient. The slope of a hill that increases in height by an amount h over a horizontal length l would have a gradient of h l This means that for every change in horizontal distance of length l , the height changes by an amount h . This is fine if the slope is constant. However a more common situation is that the gradient is different at different points on the slope as illustrated in Figure 2.1.

gradient =

h l

gradient =

dh dl

Figure 2.1 Use of differential notation to represent the gradient of a point on a variable slope.

h

constant slope

l

variable slope

To determine the gradient at any point mathematically requires differentiation, which is beyond the scope of this course. However, this leads to the gradient being represented by the differential notation. Thus for our hypothetical slope of variable gradient, the gradient is represented by dh dl where dh and dl represent an infinitesimal (very small) change in each value. This is simply a mathematical concept that allows us to consider changes as if they take place over a range of values as that tends to zero. So for example we can talk about the gradient of a curved surface at just one point.

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2.2.1 Balance of forces By considering the balance between outward gas pressure and inward gravitational pressure we can determine the hydrostatic equation, which describes the rate of change of internal pressure ( P ) with radius ( r ). To understand the physical basis of this equation, we start by noting that pressure is force spread over a surface. Let us calculate the gravitational force on the shell of thickness dr at a radius r due to the mass it encompasses, M (r ) as illustrated in Figure 2.2. Note that the mass outside this shell has no (gravitational) effect.

Figure 2.2 Thin shell of radius r and thickness dr . This diagram applies to any shell for 0< r R .

dr M(r)

r

centre

The mass of the shell can be determined to be the density times the volume. The volume for a very thin shell is approximately the surface area of the shell times its thickness, so that the mass is 4r 2 dr (r ) . From Newton's law of gravitation, this gives the force dF on the shell due to the mass it encompasses as GM (r )4r 2 dr (r ) r2

dF =

Equation 2.2

where G = 6.6725985 × 10 -11 m3 kg-1 s-2 is the gravitational constant. The pressure can then be determined by dividing dF by the area that it acts across, that of the entire shell, i.e. 4r 2 . This is the pressure acting on the shell due to the gravity pulling inwards. For hydrostatic equilibrium (i.e. for outward gas pressure to match inwards gravity pressure) this must be equal and opposite to the gas pressure. If we call the gas pressure dP then we find

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dP = -

GM (r )4r 2 dr (r ) GM (r )dr (r ) =- , Equation 2.3 2 2 4r r r2

where the negative sign appears because the gas pressure opposes the gravitational pressure. From this we get the hydrostatic equation

Mean molecular weight is the ratio of the average mass per molecule of the naturally occurring form of an element to the sum of the relative atomic masses of all the atoms that comprise a molecule.

dP GM (r ) =- . dr r2

Equation 2.4

This can be used to determine the pressure at any radius within the Sun. 2.2.2 Equation of state Recall that the equation of state for a perfect (or ideal) gas gives pressure as P(r ) = nkT (r ) Equation 2.5

where T (r ) is the temperature at radius r . From this we can see that pressure also depends on the radius. In order to relate this to the conditions inside the Sun, it is necessary to determine the particle density n in terms of physical conditions inside the Sun. In Equation 2.2 we used the density of the gas within the Sun. If we could divide this by the mass of each particle, we would have the particle density. A common way to approach this problem is to introduce a mean molecular weight µ for the gas. Then the mass of each particle can be approximated by µmH where mH is the mass of a hydrogen atom. From this the particle density can be determined from

n=

µmH

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Equation 2.6

The pressure within the Sun is then given by

P(r ) =

(r ) kT (r ) µm H

Equation 2.7

Within the Sun we can use µ in a manner that was not intended when the concept was originally introduced. The Sun consists almost entirely of hydrogen, and all that hydrogen is ionised. This means that, for each hydrogen nucleus, there is also an electron. The pressure defined by Equation 2.5 does not distinguish between particles of such significantly different masses, so the value of n must

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Section

3

The Sun's Magnetic Personality

Here we explore the Sun's magnetic personality, an aspect of our nearest star that we have only been able to study since 1907. We will look at the photosphere, where the visible light from the solar spectrum we can see comes from, and the phenomena that are observed there. Also, we will see how surface magnetic fields are observed, how they are created in the solar interior, and how the variation of these magnetic fields produces the solar cycle.

3.1 The Photosphere

In Section 1 we defined the photosphere as the layer at which light of wavelength 500 nm can escape. In practice visible light has wavelengths between 350 to 950 nm. Hence in the following we will describe the photosphere as a layer of approximately 400-500 km thick. The surface of the sun as defined within Section 1 lies within this layer. The photosphere is named from the Greek `photos', meaning `light'. The light that we see comes from above this surface and escapes into space. Light from the photosphere takes 8.3 minutes to travel from the Sun to the Earth. Section 5.1 of the Encyclopaedia of the Sun discusses the photosphere and the associated magnetic field.

I C O N K E Y

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The photosphere has a temperature of 5780 K and is the densest part of the solar atmosphere (1023 particles m-3). As we shall see in the following sections, the photosphere is physically connected to both the convective zone beneath and the chromosphere above.

Limb darkening

Encycl. of the Sun Important Equation Important Reaction

The white light image in Figure 3.5a shows how the visible surface of the Sun is not uniformly bright. The intensity of the photosphere decreases from disc centre to the limb, with the limb being less intense. This effect is called limb darkening, and is illustrated schematically in Figure 3.1.

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Figure 3.1. Limb darkening in the lower chromosphere. Near the limb photons have to be further from the photosphere to escape because of the increased path-length through the atmosphere. Here the temperature is cooler, and the observed continuum spectrum is fainter and redder.

Limb darkening is the decrease in intensity at visible wavelengths from the centre to limb of the Sun's disc. Light from the central portion of the disc is radiated radially towards us (directly along our line of sight). However, light from closer to the limb has to pass obliquely through a greater thickness of the solar atmosphere, and therefore appears fainter and redder.

Granulation

Granulation is a pattern of small, highly dynamic irregular polygons that match the pattern of the convection occurring beneath the Sun's surface. The bright centre of each granule is the highest point of a column of hot gas. The dark lanes around the granules are the cool gas sinking. This is apparent in Figure 3.6, where the photosphere shows the granulation pattern outside of the sunspots and pores. The speed at which the convection flows occur in granules is up to 2 km s-1. The mean diameter of a granule is 1100 km, and the mean distance between granules is 1330 km. Inter-granular lanes are up to 230 km wide. Granules have an approximate lifetime of 5 minutes. Granulation is uniform over most of the solar disc, but is inhibited by strong magnetic fields, such as in sunspots. A larger (up to 30,000 km) cellular pattern, called supergranulation, also exists. There are typically 2500 supergranules visible on the solar disc, and they have lifetimes of 1-2 days. The speed of the convection flows that give rise to the observed supergranules is less than those associated with granules. The flows associated with supergranulation sweep the magnetic field to the edges of the supergranules, where the magnetic field collects and strengthens, and forms magnetic networks. Figure 3.2 shows a schematic illustrating how granules and supergranules reflect the size and position of convection cells below the surface. The larger,

Supergranulation

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deeper convection cells interact with the smaller, shallower cells so that granules move within supergranules.

Figure 3.2. Schematic of solar granulation and supergranulation. The convection cells governing the movement of the two types of granulation are labelled with the type governed.

magnetic field

Solar surface

granules on surface

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supergranule

Exercise

Given the size of a supergranule and a lifetime of 1.5 days, show that the convective motions will sweep any magnetic field from the centre of the supergranule to the edge with a velocity of around 0.1 km s-1 (to one decimal place). Discuss any assumptions that you have to make to do this calculation.

3.2 Observing the Sun's Magnetic Field

The Sun's magnetic field through the photosphere is observed with a magnetograph, and the resulting images of the magnetic field are called magnetograms. Magnetographs observe specific wavelengths where the spectral lines are sensitive to magnetic fields. If the spectral line is split into multiple components, then it is evidence of the Zeeman effect (see below) occurring, and hence the presence of magnetic fields. The displacement of the line component is proportional to the strength of the magnetic field. A number of ground based observatories and the SOHO/MDI instrument produce magnetograms. In the images, black and white regions illustrate regions of negative and positive magnetic field, respectively. MDI uses the neutral nickel (abbreviated Ni I) line at 676.8 nm (6768 Å) to produce its magnetograms, which only shows the line of sight component of the magnetic field to the solar surface.

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The upcoming Solar-B mission (see Section 8) will have a vector magnetograph on board, which will allow us to measure the directionality of the magnetic field as well as its strength. However in this module, we will only be considering line-of ­sight magnetograms. The Sun's magnetic field is discussed in Section 5.1 of the Encyclopaedia of the Sun.

Normal component

A magnetic field at a point has a strength and a direction as indicated by field lines. It can emerge in any direction from the Sun's surface. The diagram here shows that only the component of the field along the line of sight is observed.

Figure 3.3. A line of sight component of the magnetic field as it emerges from the solar surface.

Tangent to surface Line of sight Normal component Normal to surface

Magnetic field line 90° angle

3.2.1 Zeeman effect Electrons in an atom adjust their energy when they are in the presence of a magnetic field. This change in energy produces a shift in the wavelength of the energy emitted by the electrons due to orbital transitions. We will consider the case where the line is split into three components: the component is at the usual line wavelength 0 seen when no magnetic field is present, and the other two components are observed with a separation, ± , from the original position 0 (Figure 3.4). If the magnetic field is pointing directly towards or away from the observer, only the two components are seen. However, at any other angle to the observer's line of sight, all three components are seen, with the component being the strongest when the field lines are perpendicular. Page 97 of the Encyclopaedia of the Sun discusses the Zeeman effect. This process is called Zeeman splitting. It was predicted by Hendrick Lorentz and measured experimentally by Pieter Zeeman. Both physicists won the Nobel Prize for Physics in 1902 for their work.

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Figure 3.4. Schematic of Zeeman splitting. Note absence of component parallel to field lines.

No magnetic field

Observed perpendicular to field

0

Observed parallel to field satellite

0

0

Magnetic field lines

The change in wavelength due the magnetic field, , depends on the line wavelength in the absence of the magnetic field 0 and the strength of the magnetic field B , such that

=

e 0 2 B = 46.7 0 2 B m 4cme

Equation 3.1

where me is the mass of an electron and e the charge on an electron. In this equation, the units of the wavelength and magnetic field are metres and tesla, respectively. From the observed size of we can determine the magnetic field strength. By examining the relative strengths of the and components, we can deduce the direction of the field lines. Thus Zeeman splitting is a powerful tool for understanding magnetic fields. 3.2.2 Sunspots Sunspots are produced where strong magnetic fields from the interior pierce the surface, and are typically observed in white light or H. Figure 3.5 shows (a) a full disc magnetogram, (b) a full disc white light image of the Sun and (c) a full disc H image all taken on the 28th October 2003. The alignment between the sunspots and the large concentrations of magnetic field is clearly visible.

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Section

4

Facing the Universe

In this Section we will be examining the atmosphere of the Sun from the chromosphere through to the corona. In order to enable solar astronomers to explore and consequently, diagnose the physical state of the solar atmospheric environment, techniques within solar spectroscopy are employed. The Coronal Diagnostic Spectrometer (CDS) on-board SOHO is described as a working example of a solar spectrometer and we will outline how we can remotely sense the temperature and density of the atmospheric plasma.

4.1 Solar Atmosphere

I C O N K E Y

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Test yourself Worked example Encycl.. of the Sun Important Equation

In previous Sections, we have concentrated upon the solar interior, the generation of the Sun's magnetic field and the solar activity cycle (in particular on the effect of the creation and evolution of sunspots in the photosphere). In this Section, we will come to see that the Sun's magnetic field plays a vital role in the structure of the Sun's outer layers and in the features we observe there. Specifically, we will concentrate upon how the basic properties of the plasma vary through the atmosphere and how these properties are deduced by using the important technique of solar spectroscopy. In the last decade, these diagnostic methods have allowed major advances to be made in our understanding of the plasma state in this part of the Sun; this is due mainly to the instrumentation we have on-board the SOHO satellite. 4.1.1 Lower atmosphere: Chromosphere and Transition Region Immediately above the photosphere lies the chromosphere. The temperature drops to a minimum value of 4300 K and then rises slowly to a temperature of 10000 K at the top of this region. It is about 10 000 km thick and highly nonuniform due to the presence of the Sun's magnetic field. Section 5.2 of the Encyclopaedia of the Sun discusses this portion of the atmosphere. H emission (see Figure 3.5(c)) is radiated at chromospheric temperatures and reveals the same intriguing structures throughout the chromosphere. For example, the network of supergranule cells caused by convective motions below the photosphere can also be seen in the chromosphere. Figure 4.1(b)

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and (d) shows clearly the bright centre and dark boundaries of the supergranule cells which give the speckled appearance. Also, sunspots can be seen as dark spots and the bright regions surrounding the sunspots are called plage. Filaments (marked as F in Figure 4.1(a)) are long dark absorption features observed on the disk; they are called prominences (P in Figure 4.1(d)) when observed at the limb as bright structures in chromospheric lines. We will cover these features and their magnetic structure in more detail in Section 5. One very distinct feature of the chromospheric limb of the Sun is the presence of many small finger-like structures called spicules.

Figure 4.1 Prominences seen in H and He II (chromospheric) emission at the limb (P) and on the solar disk as a filaments (F). Image Credits: NJIT/BBSO and SOHO/EIT.

(a) F

(b)

F

F

(c

(d) P

P

Figure 4.2 shows a series of close up H images of the limb of the Sun, where the spicules can be seen clearly. On the right of each image strip is indicated the different shifts in the wavelength detected away from the centre of the H line. Therefore, using the Doppler velocity formula, it is possible to compare these images to investigate the dynamics of the spicules (each frame illustrates material that is red (+x Å) or blue-shifted (-x Å) as indicated).

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Figure 4.2 Spicules seen at the limb of the Sun in H emission. Wavelength shifts away from the rest H wavelength are indicated on the right hand side. Image Credits: NJIT/BBSO

Spicules propel material upwards up to 20 km s-1 with an estimated 100,000 of them occurring at any one time on the Sun. Recent research has shown that these jets appear to reoccur at the same location with a five minute period. It is surmised that the spicules are caused by sound waves "leaking" from the solar interior along emerging magnetic field that is inclined to the surface at the chromosphere (from Figure 4.2, one can see the spicules pointing outwards in all directions). These waves encounter the lower densities of the atmosphere above and develop something called a shock wave. The shock wave then rapidly pushes the cooler chromospheric material upwards to form the spicule jet.

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Shock Waves

When a disturbance moves faster than the local sound speed (ie, travels supersonically), material near the disturbance cannot get out of the way before the disturbance arrives. A very large change in the local pressure over a very short timescale can occur and these immense pressure changes can selfsteepen into a shock wave. A sound wave is a small amplitude compression wave that travels at the local sound speed and leaves the medium through which it is propagating unchanged. However, for a travelling shock, the properties of the medium (pressure, temperature, density) can be dramatically different on either side of the shock wave front (e.g. Figure 4.3). The shock wave changes the state of the medium through which it passes with generally, the medium being at a higher temperature once the shock wave has passed through.

Pressure t1 P P0 c-c t2 c c+c Direction of travel Shock front

Figure 4.3 Development of gas pressure in a shock.

t3

t4

As an example, Figure 4.4 illustrates the prominent shock wave pattern (the thick black curve line) around a free-flight model of a space capsule as it travels through the Earth's atmosphere.

Figure 4.4 Space capsule shock wave. Image credit: NASA History Division

A sharp rise in temperature occurs as one travels from the chromosphere up into the corona. Traditionally, the narrow layer where this large temperature gradient is located is called the transition region. However, recent images of complete magnetic loop features as well as the presence of many small-scale

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5.3.1 Maxwell's Equations The theory of electromagnetism, as set out in Maxwell's Equations, represents a clear way of defining and exploring the fundamental nature of electricity and magnetism. Maxwell's Equations describe the behaviour of electric and magnetic fields and how they each evolve over time ­ that is, if the electric field changes in size and/or direction, the equations predict the effect that this has on the magnetic field. Here we present a simplified approach to applying these in MHD. For the astrophysical plasma in the solar corona, a number of assumptions are made to reduce further the standard Maxwell's Equations. These include assuming that:

·

the plasma is treated as a continuous fluid (which is valid as long as the theory only describes things happening at much larger scales than any "internal" to the plasma; for example, the ion gyroradius); most of the plasma properties (for example, the magnetic permeability described in Equation 5.3) are isotropic ­ this means that they have the same value in all directions. One exception is thermal conduction that is much greater along the magnetic field direction than perpendicular to it; in spite of the fact that the Sun is rotating, this motion is ignored. These rotational effects could become important if one is trying to model a very large coronal structure that stretches out over several solar radii above the solar surface (for example an extended coronal streamer); any velocities in the plasma are much smaller than the speed of light and thus relativistic effects can be ignored.

·

·

·

In particular, a consequence of the last assumption above is that it allows us to take Maxwell's Equations and write down a direct relationship between the magnetic field and the current in the plasma. This is,

× B = µ0 I

Equation 5.14

where I is the current vector. Essentially the current is related directly to the amount of twist in the magnetic field. This highlights an important difference between electromagnetism and MHD. In electromagnetism, the magnetic field is the consequence of the presence of the electric field (the magnetic field appears when we switch on the current in the wire say). However in MHD, the magnetic field and the plasma velocity are specified and the current and electric field are then calculated. The current and electric field are not independent of the magnetic field and velocity but rather are a consequence of them. 5.3.2 The Induction Equation The induction equation is of central important to MHD as it links changes in the magnetic field to the motion in the plasma itself. In words, the induction equation can be expressed as:

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Induction Equation

Rate of change of the magnetic field with time is a consequence of the balance between: Equation 5.15

Plasma/magnetic field Advection term Plasma/magnetic field Diffusion term

v 0 B0 l0

0 B0 2 l0

The term advection refers to the transport of something from one region to another. This term describes how the magnetic field and the plasma would "move together" if some external force is applied. On the other hand, diffusion is a term you might have come across before in regard to atoms or molecules suspended in a liquid ­ in that case, it is the spontaneous tendency of a substance to move from a more concentrated to a less concentrated area. The same process can occur for magnetic fields, diffusing through the plasma from regions of strong field to regions of weak field. Here we have introduced B0 as a typical magnetic field strength, v0 as a typical plasma speed, l 0 is a typical lengthscale within the system and 0 as the magnetic diffusivity of the plasma (in units of m2s-1). In advection, we have the situation where the magnetic field is connected intimately with the plasma motions (this is explained in more detail below). In diffusion, the opposite occurs in that the magnetic field lines are allowed to slip through the plasma from regions of strong to weak magnetic field. Taking the ratio of the advection term to the diffusion term yields the unitless quantity called the Magnetic Reynolds Number Rm :

Worked Example 1

Rm =

v 0 B0 l 0 l 0 v0 = 0 B0 l 02 0

.

Equation 5.16

The value of Rm helps us decide which of the advection term or the diffusion term is more important.

·

If Rm is very small, then the diffusion term in the plasma must be more dominant (larger) than the advection term. The magnetic field can easily diffuse through the plasma. On the other hand, if Rm is very diffusion in that system.

large,

·

then advection dominates over

For a hydrogen plasma at a temperature of 10 4 K in the laboratory, it is found that the magnetic diffusivity 0 = 10 3 m2s-1 . Find the magnetic Reynolds number Rm .

Solution

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Taking a laboratory lengthscale of l0 = 1 m and a velocity v0 = 0.1 ms-1, we calculate Rm 10 -4 which is much smaller than one. Thus, in this case, the diffusion term is large and dominates over the advection term in the induction equation.

Worked Example 2

For the solar corona at a temperature of 10 6 K, it is calculated that 0 = 1 m2s-1 . Find Rm .

Solution

Taking a lengthscale of l 0 = 10 7 m (about the size of a typical sunspot umbra) and a velocity v0 = 10 4 ms-1, we calculate Rm 1011 which is much greater than one. Thus, in this case, the advection dominates over diffusion in the induction equation. This is typical in most astrophysical plasmas. The magnetic field lines are said to be frozen to the high Rm plasma and move with it. One important aspect to note is that the above calculations are a crude approximation to the whole coronal plasma environment. As we shall see in Section 6, the magnetic diffusion term can become important in small spatial regions, particularly when the magnetic field is changing very rapidly over a very short distance (that is, when the plasma lengthscale l 0 is greatly reduced). 5.3.3 Conservation of mass, momentum and energy The set of MHD equations also has three conservation equations as outlined below:

Conservation of mass This is an equation of mass continuity Conservation of momentum This is an equation of motion

which states simply that mass cannot be created or destroyed but is conserved within the system. that considers all the forces at work within the magnetised plasma. For the case when there are no flows (the plasma velocity is zero), this equation is reduced to a competition between the pressure force in the plasma itself, a global Lorentz force and the force of gravity.

Concentrating on the global Lorentz force, this is the same force outlined in Section 5.2.3 but now placed here in the context of MHD. It can be written as

FLorenz = I × B

Equation 5.17

and after applying some vector algebra (see Appendix A if you are interested in the details), the Lorentz force breaks down into:

·

a magnetic tension force which tries to straighten curved field lines. It acts along the magnetic field in the direction of B, parallel to the field with magnitude FT given by:

FT = B2

µ0

Equation 5.18

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where µ 0 was defined in section 5.2.2. In Figure 5.9a), the sketched magnetic field lines are curved (semi-circular) and the magnetic tension force acts in a direction so as to straighten them.

·

Figure 5.9 a) a semi-circular magnetic field with the magnetic tension force shown; b) parallel magnetic field lines but with the magnetic field strength decreasing across the field- the magnetic pressure force is shown.

a magnetic pressure force acting in the magnetic field from regions of strong to regions of weak magnetic field. Its magnitude pm is equal to

pm = B2 2µ 0

Equation 5.19

In Figure 5.9b) the space between the field lines increases from left to right across the field. The magnetic field strength is decreasing across the field and thus the magnetic pressure force acts from left to right trying to equalize this magnetic pressure difference.

a)

b)

Tension force

Pressure force

We can define the parameter, plasma beta ( ) at any point in the plasma:

is the ratio of the plasma pressure to the magnetic pressure.

It indicates which of the two pressures is dominant. We have already seen from the induction equation in Section 5.3.2 that for the corona, the magnetic field is said to be frozen to the plasma. The plasma parameter tells us whether it is magnetic field or the plasma that governs the motion:

· ·

if the plasma pressure is greater than the magnetic pressure ( greater than 1) then the magnetic field follows the motion of the plasma; if the magnetic pressure is greater than the plasma pressure ( less than 1) then the plasma follows the motion of the magnetic field.

The solar corona is a low plasma and the magnetic field dominates the structures we observe. This can be seen from the typical active region loop images where the plasma is clearly following and flowing along the three dimensional topology of the coronal magnetic field (Figure 5.10).

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Section

6

The Coronal Heating Problem

In this Section we will investigate one of the most perplexing of astrophysical problems; namely, the counter-intuitive presence of a million degree plasma environment (the corona) enveloping the cooler (~5780K) photospheric surface of the Sun. It is now accepted widely that the magnetic field in the corona coupled to the turbulent motions in the photosphere play vital roles in understanding the reasons for this unusual temperature inversion. Both sophisticated theoretical modelling and high resolution observations must be employed to solve fully this solar conundrum.

6.1 What is the coronal heating problem?

I C O N K E Y

Valuable information Test yourself Worked example

Encycl. of the Sun Important Equation Important Reaction

6.1.1 Historical background As we outlined in Section 4, the solar corona was first observed during solar eclipses as a faint halo surrounding the Moon. Initially it was surmised that this glowing region was the result of the refraction of light from the photosphere by the lunar atmosphere. However, the subsequent discovery that the Moon is atmosphere-less lent strong evidence to the fact that the observed emission was solar in origin. In the mid-to-late 1800s, the science of spectroscopy was flourishing and from an eclipse observation in 1869, bright emission was observed in the corona from the green part of the electromagnetic spectrum (at 530.3 nm; see an example in Figure 6.1). However at that time, this emission line could not be associated with any known element on Earth ­ hence, it was thought that the corona must contain an as yet unknown ingredient which was the given the name coronium. This might not be as strange a thing to do as it seems ­ at around that time, similar studies of the eclipse spectrum from the chromosphere by Sir Norman Lockyer revealed a strong, yellow line emission (at ~587.5 nm) which was termed helium (from the Greek "helios" for Sun). It was around thirty years later that helium itself was isolated on our planet by the British chemist, Sir William Ramsay.

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Figure 6.1 The inner corona as seen by the LASCO C1 coronagraph in the green forbidden coronal line of Fe XIV. Coronal structures can be seen as far as 1 million km above the solar surface. Image credit: SOHO/LASCO.

However, unlike helium, there were major problems with this new, hypothetical element ­ in particular it was difficult to see where it fitted into the periodic table of elements. During an eclipse, the corona is observed stretching out far above the solar surface (Figure 6.1) over a solar radius say ( 6.96 × 10 8 m). If the pressure in the material making up the corona is balanced only by gravity then the resulting rate at which the pressure decreases through the atmosphere can be modelled by the equation:

dp = - g dr

Equation 6.1

where r is the radial distance from the centre of the Sun, p is the gas dp pressure (and thus is the rate of change of the pressure with distance from dr the solar centre), is the density of the unknown coronal material and g = 274 ms­2 is the gravitational acceleration at the solar surface (for simplicity we are assuming here that the value of g will not change very much over the distances we are considering). For this unknown element, let us take = mu nu where mu is the mass and nu is the number density of the coronal particle. Thus, by also using the gas law: p = nu k b T we can eliminate from Equation 6.1 to give us:

m g dp =- u p dr kbT

Equation 6.2

Equation 6.3

If we assume that the corona has a constant temperature T, then this equation can be integrated from above the photosphere to give:

- (r - R ) p = p photo exp H

Equation 6.4

where R is the solar radius, p photo is the gas pressure at the photosphere and

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Volume of corona =

4 8 R + 10 3

(

)

3

-R

3

7 × 10

26

m3

and with an average coronal density of 1014 m­3, the mass of the corona will be 1.2 × 1014 kg. Thus, if about ten prominences are present on the Sun at any one time, then their formation would deplete the corona completely! Therefore, some other mechanism must be continually supplying the prominence material.

Ballistic injection

Essentially this mechanism postulates that chromospheric material is somehow "injected" up into the coronal magnetic field where it gathers in pre-existing dips (see Figure 7.7b). Spicules (Figure 4.2) have been suggested as examples of this process.

Heating at the loop base

This mechanism is in part a combination of the two mechanisms outlined above. Computer simulations of coronal loop plasma have shown that if the loop is heated more at one foot-point than the other, the resulting pressure difference sets up a siphon flow of material up from the chromosphere and along the loop (Figure 7.7c). After a time, the thermal structure along part of the loop becomes quite uniform and a local condensation process occurs. This causes a localised, prominence-like region to form in the loop. In the next section, we will go onto to describe the role that filaments play in an important type of solar eruption.

7.3 Coronal Mass Ejections

One type of solar eruption is termed a coronal mass ejection or CME. These are huge expanding clouds of magnetised plasma that often dwarf the Sun itself. Our understanding of CMEs has been revolutionised by continuous observations taken by the Large Angle Spectrometric Coronograph Experiment (LASCO) instrument on board SOHO. LASCO employs an occulting disc to block out the light from the photosphere, allowing the observation of extremely faint emission from the outer corona over many solar radii. Currently, LASCO has two operating coronagraphs ­ C2 (which takes observations stretching out from 1.5 to 6 solar radii at a resolution of 7980 km) and C3 (which observes from 3 to 32 solar radii with a spatial resolution of 39,200 km). Figure 7.9 displays a "typical" CME eruption captured initially in LASCO C2 and then observed some 6 hours later by LASCO C3.

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Figure 7.9 SOHOLASCO C2 (left, red) and subsequent C3 (right, blue) observations of a CME (Image credit: SOHO/LASCO).

Table 7.2 outlines some average properties of CMEs, presenting us with an idea of the size and scale of these events. For the more energetic events, up to 5 × 1013 kg of plasma can be expelled, travelling at a maximum velocity of over 10 6 ms­1. Even after 2 to 3 solar radii, their approximate width is comparable to the diameter of the Sun and it has been shown clearly that their rate of occurrence varies substantially over the solar cycle.

Table 7.2 Typical physical characteristics of CMEs.

CME property

Average mass carried Rate of occurrence Average speed of bright front Average energy content

Value

1013 kg Solar minimum: 0.8 per day Solar maximum: 3.5 per day

4 × 10 5 ms­1 10 23 to 10 24 J

7.3.1 Relationship to flares and filaments Another important class of solar eruption is that of a solar flare. These are observed as very large, sudden increases in intensity across virtually the entire electromagnetic spectrum. A schematic diagram of the typical main physical processes operating and the associated observational signatures occurring in these features is shown in Figure 7.10.

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Figure 7.10 Sketch of a typical flare: a) Physical processes b) Observations.

a) Physical processes Magnetic reconnection site Accelerated electrons and particles Gamma ray and hard X-ray Gamma ray and hard X-ray

Chromospheric evaporation

b) Observations

Hard X-rays Soft Xray cusp Gamma ray and hard X-ray signature

Hard X-rays

Cooling EUV flare loops Gamma ray and hard X-ray signature

H brightening

The main energy release mechanism for a flare is believed to be magnetic reconnection. The reconnection site lies at the top of a cusp-like magnetic 20 22 structure, releasing energy for an average flare at a rate of 10 - 10 J per second! As shown in Figure 7.10a, electrons, protons and other ions are accelerated from the reconnection region down the magnetic field and hit the chromosphere. These impacts heat the chromospheric plasma to millions of degrees. This plasma evaporates up into the overlying magnetic structure.

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