A Hot Rarefied Gas Will Produce a Continuous Spectrum

An Introduction to Atmospheric Radiation

In International Geophysics, 2002

1.3.2.2 Doppler Broadening

Assuming that there is no collision broadening in a highly rarefied gas, a molecule in a given quantum state radiates at wavenumber v ₀. If this molecule has a velocity component in the line of sight (the line joining the molecule and the observer), and if υ ≪ c, the velocity of light, the wavenumber

Note that because of the conventional use of notation the wavenumber v and the velocity υ appear indistinguishable. Let the probability that the velocity component lies between υ and υ + dυ be p(υ) dυ. From the kinetic theory, if the translational states are in thermodynamic equilibrium, p(υ) is given by the Maxwell-Boltzmann distribution so that

(1.3.16) $p\left(\upsilon \right)d\upsilon ={\left(m/2\pi KT\right)}^{1/2}\text{exp}\left(-m{\upsilon }^2/2KT\right)d\upsilon ,$

where m is the mass of the molecule, K is the Boltzmann constant, and T is the absolute temperature.

To obtain the Doppler distribution, we insert the expression of v in Eq. (1.3.15)] into Eq. (1.3.16)], and perform normalization to an integrated line intensity S defined in Eq. (1.3.11)]. After these operations, we find the absorption coefficient in the form

(1.3.17) $k_v=\frac{S}{{\alpha }_D\sqrt{\pi }}\text{exp}\left[-{\left(\frac{v-v_0}{{\alpha }_D}\right)}^2\right],$

where

(1.3.18) ${\alpha }_D=v_0{\left(2KT/m{c}^2\right)}^{1/2},$

is a measure of the Doppler width of the line. The half-width at the half-maximum is ${\alpha }_D\sqrt{\ln 2}$ . The Doppler half-width is proportional to the square root of the temperature.

A graphical representation of the Doppler line shape is also shown in Fig. 1.11. Since the absorption coefficient of a Doppler line is dependent on $\text{exp}\left[-{\left(v-v_0\right)}^2\right]$ , it is more intense at the line center and much weaker in the wings than the Lorentz shape. This implies that when a line is fully absorbed at the center, any addition of absorption will occur in the wings and will be caused by collision effects rather than Doppler effects.

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Open Problems and New Trends

Cédric Villani , in Handbook of Mathematical Fluid Dynamics, 2002

Bibliographical notes

General references. Standard references about the kinetic theory of rarefied gases and the Boltzmann equation are the books by Boltzmann [93], Carleman [119], Chapman and Cowling [154], Uhlenbeck and Ford [433], Truesdell and Muncaster [430], Cercig-nani [141,148], Cercignani, Illner and Pulvirenti [149], as well as the survey paper by Grad [250], The book by Cercignani et al., with a very much mathematically oriented spirit, may be the best mathematical reference for nonspecialists. The book by Uhlenbeck and Ford is a bit outdated, but a pleasure to read. There is no up-to-date treatise which would cover the huge progress accomplished in the theory of the Boltzmann equation over the last ten years.

For people interested in more applied topics, and practical aspects of modelling by the Boltzmann equation, Cercignani [148] is highly recommended. We may also suggest the very recent book by Sone [407], which is closer to numerical simulations.

The book by Glassey [233] is a good reference for the general subject of the Cauchy problem in kinetic theory (in particular for the Vlasov–Poisson and Vlasov–Maxwell equations, and for the Boltzmann equation near equilibrium). Also the notes by Bouchut [96] provide a compact introduction to the basic tools of modem kinetic theory, like characteristics and velocity-averaging lemmas, with applications.

To the best of our knowledge, there is no mathematically-oriented exposition of the kinetic theory of plasma physics. Among physicists' textbooks, Balescu [46] certainly has the most rigorous presentation. The very clear survey by Decoster [160] gives an accurate view of theoretical problems arising nowadays in applied plasma physics.

There are many, many general references about equilibrium and non-equilibrium statistical physics; for instance, [49,227]. People who would like to know more about information theory are advised to read the marvelous book by Cover and Thomas [156]. A well-written and rather complete survey about logarithmic Sobolev inequalities and their links with information theory is [16] (in french).

Historical references. The founding papers of modem kinetic theory were those of James Clerk Maxwell [335,336] and Ludwig Boltzmann [92], It is very impressive to read Maxwell's paper [335] and see how he made up all computations from scratch! The book [93] by Boltzmann has been a milestone in kinetic theory.

References about the controversy between Boltzmann and his peers can be found in [149, p. 61], or Lebowitz [293]. Some very nice historical anecdotes can also be found in Balian [49].

Certainly the two mathematicians who have most contributed to transform the study of the Boltzmann equation into a mathematical field are Torsten Carleman in the thirties, and Harold Grad after the Second World War.

Derivation of the Boltzmann equation. For this subject the best reference is certainly Cercignani, Illner and Pulvirenti [149, Chapters 2 and 4]. A pedagogical discussion of slightly simplified problems is performed in Pulvirenti [394]. Another excellent source is the classical treatise by Spohn [410] about large particle systems. These authors explain in detail why reversible microdynamics and irreversible macrodynamics are not contradictory – a topic which was first developed in the famous work by Ehrenfest and Ehrenfest [202], and later in the delightful book by Kac [284], Further information on the derivation of macroscopic dynamics from microscopic equations can be found in Kipnis and Landim [287].

Hydrodynamic limits. A very nice review of rigorous results about the transition from kinetic to hydrodynamic models is Golse [239]. No prerequisite in either kinetic theory, or fluid mechanics is assumed from the reader. Note the discussion about ghost effects, which is also performed in Sone's book [407]. The important advances which were accomplished very recently by several teams, in particular, Golse and co-workers, were reviewed by the author in [441].

There is a huge probabilistic literature devoted to the subject of hydrodynamic limits for particle systems, starting from a vast program suggested by Morrey [352]. Entropy methods were introduced into this field at the end of the eighties, see in particular the founding works by Guo, Papanicolaou and Varadhan (the GPV method, [263]), and Yau [466]. For a review on the results and methods, see the notes by Varadhan [439], the recent survey by Yau [467] or again, the book by Kipnis and Landim [287].

Mathematical landmarks. Here are some of the most influential works in the mathematical theory of the Boltzmann equation.

The very first mathematical steps are due to Carleman [118,119] in the thirties. Not only was Carleman the very first one to state and solve mathematical problems about the Boltzmann equation (Cauchy problem, H theorem, trend to equilibrium), but he was also very daring in his use of tools from pure mathematics of the time.

In the seventies, the remarkable work by Lanford [292] showed that the Boltzmann equation could be rigorously derived from the laws of reversible mechanics, along the lines first suggested by Grad [249]. This ended up a very old controversy and opened new areas in the study of large particle systems. Yet much remains to be understood in the Boltzmann–Grad limit.

At the end of the eighties, the classical paper by DiPerna and Lions [192] set up new standards of mathematical level and dared to attack the problem of solutions in the large for the full Boltzmann equation, which to this date has still received no satisfactory answer. A synthetical review of this work can be found in Gérard [231].

Finally, we also mention the papers by McKean [341] in the mid-sixties, and Carlen and Carvalho [121] in the early nineties, for their introduction of information theory in the field, and the enormous influence that they had on research about the trend to equilibrium for the Boltzmann equation.

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Radiative Transfer1

With Qiang Fu , in Atmospheric Science (Second Edition), 2006

a. Absorption continua

Extreme ultraviolet radiation with wavelengths ≲0.1 μm, emitted by hot, rarefied gases in the sun's outer atmosphere, is sufficiently energetic to strip electrons from atoms, a process referred to as photoionization. Solar radiation in this wavelength range, which accounts for only around 3 millionths of the sun's total output, is absorbed in the ionosphere, at altitudes of 90 km and above, giving rise to sufficient numbers of free electrons to affect the propagation of radio waves.

Radiation at wavelengths up to 0.24 μm is sufficiently energetic to break O₂ molecules apart into oxygen atoms, a process referred to as photodissociation. The oxygen atoms liberated in this reaction are instrumental in the production of ozone (O₃), as explained in Section 5.7.1. Ozone, in turn, is dissociated by solar radiation with wavelengths extending up to 0.31 μm, almost to the threshold of visible wavelengths. This reaction absorbs virtually all of the ∼2% of the sun's potentially lethal ultraviolet radiation. The ranges of heights and wavelengths of the primary photoionization and photodissociation reactions in the Earth's atmosphere are shown in Fig. 4.20.

Fig. 4.20. Depth of penetration of solar ultraviolet radiation in the Earth's atmosphere for overhead sun and an average ozone profile.

[Adapted from K. N. Liou, An Introduction to Atmospheric Radiation, Academic Press, p. 78, Copyright (2002), with permission from Elsevier.]

Photons that carry sufficient energy to produce these reactions are absorbed and any excess energy is imparted to the kinetic energy of the molecules, raising the temperature of the gas. Since the energy required to liberate electrons and/or break molecular bonds is very large, the so-called absorption continua associated with these reactions are confined to the x-ray and ultraviolet regions of the spectrum. Most of the solar radiation with wavelengths longer than 0.31 μm penetrates to the Earth's surface.

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Fundamentals of Volcanic Seismology

Vyacheslav M. Zobin , in Introduction to Volcanic Seismology (Third Edition), 2017

3.1.1.3 A Gas-Particle Dispersion Magma Flow Regime

A gas-particle dispersion magma flow occurs above a narrow region of fragmentation that separates a zone of high-density, high-viscous magma from a zone of low-density gas-particle dispersion, the resistance of which is determined by turbulent viscosity of the gas phase and is negligibly small. When Δp exceeds a critical value, fragmentation of the bubbly media occurs. A competing process is the coalescence of the bubbles with development of a permeable porous structure and outflow of gas from the magma through a system of interconnected pores. This process reduces gas pressure and can also lead to a collapse of the porosity to form dense magma. The total resistance of the conduit and average weight of the mixture are determined by the position of the fragmentation region.

The actual process of fragmentation is probably different in magmas of different rheologies. The likely fragmentation mechanism in low-viscosity magmas, such as Hawaiian fire fountains and Strombolian bubble bursts, is the fluid instability. The fragmentation of an already vesiculated magma occurs by sudden decompression and bubble rupture. This mechanism may be applied to the failure of volcanic domes, and may also describe explosive disruption of viscous melts during Vulcanian eruptions. Relative rate of bubble growth, magma transport, and gas loss will control the style of eruptive behavior (explosive vs. effusive; Cashman et al., 2000).

The interaction of magma (or lava) with groundwater or surface water can lead to phreatomagmatic fragmentation. The depth of fragmentation is likely to be determined by the depth of the aquifer. Phreatomagmatic fragmentation results in a spectrum of eruptive activity ranging from passive quenching of the magma/lava to explosive ejection of ash (Morrissey et al., 2000).

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Interstellar Matter

Donald G. York , in Encyclopedia of Physical Science and Technology (Third Edition), 2003

VI.D Evolution

The evolution of the intercloud medium depends on the injection of ionization energy through supernova blast waves, UV photons, and stellar winds. A single supernova may keep a region of 100-pc diameter ionized for 10⁶ years because of the small cooling rate of such hot, low-density gas. Ionizing photons from O stars in a region free of dense clouds may ionize a region as large as 30–100   pc in diameter for 10⁶ years before all the stellar nuclear fuel is exhausted. Thus in star-forming regions of galaxies with low ambient densities and with supernova rates of 1 per 10⁶ years per (100   pc³) and/or comparable rates of massive star formation, a nearly continuous string of overlapping regions of ∼10⁴–10⁶  K can be maintained. When lower rates of energy input prevail, intercloud regions will cool and coalesce, forming new clouds. In denser regions, comparable energy input may not be enough to ionize the clouds, except perhaps near the edges of the dense region, for periods as long as 10⁸ years.

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Collision-Induced Spectroscopy

Lothar Frommhold , in Encyclopedia of Physical Science and Technology (Third Edition), 2003

IV.A Equation of State

A gas in thermal equilibrium obeys the equation of state,

$\text{p}=k\text{T}\left\{\frac{{\text{N}}_{\text{A}}}{\text{V}}+\frac{\text{B}(\text{T})}{{\text{V}}^2}+\frac{\text{C}(\text{T})}{{\text{V}}^3}+\cdots \right\},$

where p, T, V, k designate pressure, temperature, molar volume, and Boltzmann's constant, respectively; N _A is Avogadro's number, and B(T), C(T), … are the second, third, … virial coefficients of the equation of state. The infinite series to the right is called a virial expansion.

In the limit of a highly diluted gas (V→∞), the right-hand side of this expression is equal to ρkT, where ρ   = N _A /V is the density of the gas. This is the ideal gas approximation of the equation of state which approximates the gas as a collection of non-interacting point particles—which is quite a reasonable model for rarefied gases. The second and higher terms in the virial expansion represent the effects of the intermolecular interactions. In particular, the second virial coefficient B(T) expresses the effect of the strictly binary interactions upon the pressure of the gas. The third coefficient C(T) describes the effect of the ternary interactions, and so forth. With increasing gas densities (e.g., in compressed gases) these virial coefficients become more and more important. At densities as high as those of liquids, the virial expansion becomes meaningless. Under such conditions every particle of the fluid is in simultaneous interaction with quite a number of other particles nearby.

The second virial coefficient is expressed in terms of the pair-interaction potential V(R),

$\text{B}\left(\text{T}\right)=-2\pi {\text{N}}_{\text{A}}^2{\displaystyle \underset{0}{\overset{\mathrm{\infty }}{\int }}}\left({\text{e}}^{-\text{V}\left(\text{R}\right)/\text{kT}}-1\right){\text{R}}^2\text{dR}.$

Similarly, C(T) can be expressed in terms of ternary interactions, of both the pairwise and the irreducible kind. Valuable information on intermolecular interactions has been obtained from measurements of the virial coefficients. The second, third,   … virial coefficients are functions of temperature only and can be calculated in terms of the interactions of two, three,   … molecules in the volume V. In other words, the N _A -body problem of the imperfect gas has been reduced to a series of one-, two-, three-body, … problems which are much more tractable than the very difficult many-body problem of an amorphous fluid.

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Magnetic Fields in Astrophysics

Steven N. Shore , in Encyclopedia of Physical Science and Technology (Third Edition), 2003

III.B Interstellar Magnetic Fields

The source for stellar magnetism, at least in the protostellar phase, must lie in the environment. Measurements of dust-induced polarization in nearby molecular clouds, especially Taurus-Auriga and ρ Ophiuchus, show that the magnetic field is complex and pervasive throughout the parent cloud in which star formation is occurring. Additional evidence for large-scale magnetism in the interstellar medium comes from the radio observation of supernova remnants, strong synchrotron sources that show the effects of the expanding blast wave in the redirection and shock-amplification of the ambient galactic field.

There are two methods for measuring the magnetic field of the interstellar medium. The most direct method is the detection of the Zeeman splitting of the 21-cm line of neutral hydrogen, or the splitting of the ground state transitions of the OH molecule. Both of these are present in low-density clouds that pervade the interstellar medium and that have low enough internal temperatures that the lines are not too broad for the measurement.

Individual HI clouds having temperatures of order 100   K display internal fields of order 10   μG, larger than that inferred for the low-density gas and consistent with flux-freezing in the clouds. OH transitions, which are collisionally excited and arise from masers that are pumped in the presence of strong infrared sources, provide similar field measurements.

The magnetic field in the diffuse interstellar medium is significantly lower than that observed in the clouds. One indication of the presence of magnetic fields in molecular clouds comes from the observation that line widths in these clouds are far larger than would be expected from the cloud temperatures. The sound speed in a typical molecular cloud that has a temperature of 30 or 40   K (measured using infrared emission from the embedded dust) is about 0.5   km s⁻¹. Yet line widths are often observed in excess of 2   km s⁻¹, and it is possible that this indicated the presence of turbulent Alfvén waves within the cloud. Such turbulence is also apparently needed to support the clouds against gravitational collapse. Fields of order milligauss appear to be required to supply this turbulence. Theoretical models of collapse of magnetic clouds show that flux conservation during cloud formation is capable of amplifying the microgauss fields observed in the diffuse medium to these large values.

Observations of the galactic center show filamentary structures extending from the core nonthermal source, Sgr A, that extend radially to about 50 pc and have field strengths of order 1 mG. These structures, which show both clustered arclike emission regions and isolated filaments extending up to 50   pc from the plane, may be similar to the structures observed in the active nuclei of external galaxies, and their connection with the nuclear molecular clouds argues for the ubiquity of magnetic activity in all portions of the interstellar medium and its central role in the structuring of galactic scale activity.

The Orion Molecular Cloud (OMC-1) is perhaps the best studied galactic region of recent star formation. The magnetic field has been measured in the cloud by a variety of methods. All of these methods show that the weakest fields are of order of a few microgauss, while the strongest fields exceed about 100   μG. The condition of flux-freezing provides for such large amplifications, assuming that they grow like ρ^1/2.

The strongest interstellar fields are observed in OH masers. These sources are especially difficult to measure with Zeeman effect because beam smearing usually limits the accuracy of the polarization determination. Since the sources are very compact, it is necessary to use very long baseline interferometry to separate out the various (circularly) polarized components. The fields observed in a number of maser knots range from 2 to 10   mG. The conversion between magnetic field strength and velocity width for the normal Zeeman pattern for the OH lines, which have a wavelength of about 18   cm (1.6   GHz) is approximately 0.3   km s⁻¹ mG⁻¹ so that for a 10   mG field, the lines are completely resolved even in the total intensity. This means that it is possible to measure the total field—much as in the case of Babcock's star HD 215441—not just the projected field along the line of sight. In fact, the field is responsible for the desaturation of the maser. OH observations are the most certain test to date of the scaling of the magnetic field of any interstellar region with ambient density, and show that the field follows the ρ^1/2 scaling law.

The role played by magnetic fields in the support and structuring of molecular clouds is highlighted by molecular observations. These indicate that the line widths observed in CO, CS, and other strong molecular tracers of the density and dynamics of the cloud cores are far in excess of the thermal widths. The primary mechanism for support of the clouds against gravitational collapse is turbulence supported by Alfvén waves trapped within the clouds, which provides the equivalent of an added pressure (δB ²  >/   4π). It is also possible that the fields help regulate star formation by serving as the primary mechanism for transferring angular momentum away from the collapsing cores of the clouds and assisting accretion of matter. As previously mentioned, the observation of bipolar molecular outflows associated with newly formed stars supports the contention that magnetic fields, amplified in the process of collapse, are responsible for the growth of the protostellar object.

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Multiphoton Spectroscopy

Y. Fujimura , S.H. Lin , in Encyclopedia of Physical Science and Technology (Third Edition), 2003

I Introduction

In Fig. 1, several typical multiphoton processes are shown. The result of the material–multiphoton interaction is usually detected through direct absorption, fluorescence, ionization current, or a photoelectron detection system. The excited-state structures of these materials in gases, liquids, or solids, such as electronic, vibrational, or rotational states or fine structure, which are not found in ordinary single-photon spectroscopy because of their difference in selection rules and low transition intensity, can be seen in a wide frequency range from lower electronic excited states to ionized continua.

FIGURE 1. Several multiphoton processes seen in atoms and molecules: (a) a nonresonant two photon absorption process; (b) a resonant two-photon absorption process; (c) a two-photon resonant three photon ionization; and (d) a four-wave mixing process. Solid lines and broken lines represent real and virtual states, respectively; ω _i denotes photon frequencies.

Multiphoton spectroscopy requires an intense light source. The first experimental observation of the simplest multiphoton transition, two-photon absorption of an Eu²⁺-doped CaF₂ crystal in the optical region by Kaiser and Garrett (1961), was made possible only after a high-power monochromatic ruby laser was developed as the intense incident light source, although the possibility of simultaneous two-photon absorption or stimulated emission was pointed out in 1931 by Goeppert-Mayer.

The main reasons for wide interest in the multiphoton spectroscopy are due to the advent of dye lasers for tunability and of multiphoton ionization technique for detecting information from the excited state created by the multiphoton excitation.

The tunability of dye lasers is particularly important for multiphoton excitation because one can obtain an excitation source by using only a single-frequency laser beam rather than the two or more lasers of different frequencies. The multiphoton ionization technique consists of collecting free electrons produced by the multiphoton ionization process after irradiation by a tunable laser pulse, amplifying ion currents, and recording the signal as a function of the laser frequency. In general, even ion currents of only a few charges per second can be detectable. Therefore, by using this method, one can detect and characterize extremely small amounts of atoms or molecules, even in a rarefied gas. The sensitivity exceeds that of fluorescence and other detections. The multiphoton ionization technique is also important in practical applications such as isotope separation, laser-induced fusion, and the dry etching process.

A Ti:Sapphire laser makes it possible to generate pulses whose intensity is stronger than 10¹³  W/cm² in an ultrashort time. Application of such intense laser pulses to atoms and molecules is expected to open up new fields of study on multiphoton processes, such as high-order harmonic generation, above-threshold ionization, and above-threshold dissociation. These cannot be explained by using a simple perturbative treatment. Nonperturbative treatments should be used to explain the mechanisms of such multiphoton processes. A direct method for solving the time-dependent chrödinger equation as well as other theoretical methods is being developed.

Multiphoton transitions related to the multiphoton spectroscopy have several characteristic features: laser intensity dependence, resonance enhancement, polarization dependence, and so on. For example, the transition probability of the nonresonant two-photon absorption process shown in Fig. 1a with ω₁  =   ω₂, W ⁽²⁾ _i→f , can be written as:

(1) ${\text{W}}_{\text{i}\to \text{f}}^{\left(2\right)}={\text{I}}^2{\sigma }^{\left(2\right)}/{\left(\mathrm{\hslash }{\omega }_{\text{R}}\right)}^2$

where σ⁽²⁾, I, and ω_R denote the cross section for the two-photon absorption, the laser intensity, and the laser frequency, respectively. Equation (1) indicates that the two-photon transition probability is proportional to the square of the laser intensity applied. This is called the formal intensity law. If no saturation of photon absorption takes place during the multiphoton processes, the order or the multiphoton transition can be experimentally determined from measuring the slope of log–log plots of the transition probability as a function of the laser intensity as shown in Fig. 2:

FIGURE 2. The formal intensity law for a nonresonant two-photon process. Here I and W ⁽²⁾ _{i  → f} denote the laser intensity and two-photon transition probability from the i to the f state, respectively.

(2) $\text{In}{\text{W}}_{\text{i}\to \text{f}}^{(2)}=2\text{In}\text{I}+\text{C}$

for the nonresonant two-photon process.

Multiphoton spectroscopy usually utilizes resonance enhancement; that is, a dramatic increase in the multiphoton transition ability can be seen when the exciting laser is tuned and its frequency approaches a real intermediate electronic state called a resonant state. In this case, the level width of the resonant state plays a significant role in determining the transition ability. It is well known that photons can be regarded as particles of mass 0 and spin 1. Polarization dependence of multiphoton processes is associated with the spin angular momentum. Polarization dependence and symmetry selection rules of multiphoton transitions are of great importance in characterizing the multiphoton transition process and in determining the symmetry of the states relevant to the transitions. For example, for a two-photon transition of a molecule with a center of symmetry, the initial and final states have the same parity, which is in contrast to the parity selection rule of one-photon spectroscopy governed by the opposite parity. Therefore, one- and two-photon spectra are complementary for measuring vibronic states of the molecule. This relationship between one- and two-photon spectroscopic techniques is similar to that between the infrared (IR) absorption governed by the opposite parity and Raman spectroscopy by the same parity. These characteristic features mentioned briefly are described in detail in Section IV after the theoretical treatment and experimental techniques for the multiphoton spectroscopy are introduced. Not only structures but also dynamic behaviors in electronically excited states of atoms and molecules are widely studied using the multiphoton spectroscopic methods. Typical examples of such applications of multiphoton spectroscopy are presented in Section V.

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Discoveries and Concepts

Jack B. Zirker , Oddbjørn Engvold , in The Sun as a Guide to Stellar Physics, 2019

2 The Sun's Chemical Composition

The chemical abundance of the Sun is a fundamental yardstick in astronomy. Knowing the Sun's chemical composition became essential for discovering energy generation in the Sun and stars. The final breakthrough came in 1936 with the discovery by Hans Bethe, Charles Crichfield, and Carl Friedrich von Weizäcker of nuclear reactions taking place under the extreme pressure and temperatures in the core of the Sun (Foukal, 2004).

2.1 Spectroscopic Methods

Spectral observations of the solar photosphere are currently possible and available with very high spectral resolution and signal-to-noise ratio because of the great brightness of the source, allowing the profiles of a multitude of weak or blended absorption lines to be measured accurately. Element abundances of essentially all astronomical objects are referenced to the solar composition and basically every process involving the Sun and stars depend on their compositions. The abundance of elements in the Sun has become more extensively and reliably known than in any other star.

The German optician Joseph von Fraunhofer was the first to observe and describe the multitude of dark lines in the emission spectrum of the Sun. He designated the principal absorption features with the letters A through K, and weaker lines with lowercase letters. Physicist Gustav Kirchoff, also from Germany, realized that the dark lines corresponded to the emission lines that he and his colleague Robert Bunsen observed in emission from heated gases. Kirchhoff concluded that the lines on the spectrum of the Sun were dark because they resulted from absorption by cooler layers of gas in the Sun's atmosphere above hotter layers where the continuous emission spectrum originated. Kirchhoff's formulated the following three laws that enabled solar scientists to exploit the potential of spectrometry in chemical analysis of the Sun and subsequently in stars: (1) A solid, liquid, or dense gas excited to emit light will radiate at all wavelengths and thus produce a continuous spectrum; (2) a low-density gas excited to emit light will do so at specific wavelengths, and this produces an emission spectrum; and (3) if light composing a continuous spectrum passes through a cool, low-density gas, the result will be an absorption spectrum.

The emission spectra of elements, which could be vaporized by the Bunsen burner, were examined and compared with solar absorption line spectra. This became a truly fundamental astrophysical tool and a breakthrough in the science of astronomy. Kirchoff and Bunsen discovered lines from cesium and rubidium in the Sun. Swiss mathematician and physicist Johann Jakob Balmer observed the visible line spectrum of hydrogen and determined its wavelengths. The dominant red Fraunhofer line C, at wavelength 6563   Å, is referred to by astronomers as Hα of the Balmer series.

At a solar eclipse in India in 1868, French astronomer Pierre-Jules Janssen recorded the emission spectrum of a solar prominence, which contained a yellow line (Fraunhofer's D3) at 5875   Å, which had not yet been seen in laboratory spectra. This led Janssen and his contemporaries to conclude that it must represent a purely solar element, which soon was named helium after helios, the Greek word for "Sun." In 1895, Swedish chemists Cleve and Langlet could confirm the presence of terrestrial helium gas coming out of a uranium ore called cleveite.

The availability of spectral data and an understanding of the origin of the absorption lines stimulated development of analytical techniques to determine the constitution and structure of the solar atmosphere, including its chemical composition.

2.2 Modeling of the Sun's Atmosphere

Before around 1940, calculations of solar spectral lines were based on the "Schuster–Schwarzschild" model of the atmosphere, in which the photosphere radiated a continuous spectrum and was overlaid by a cooler layer that resulted in pure absorption. This crude approximation, which was most appropriate to use for strong resonance lines, was often applied in combination with the so-called curve of growth technique developed by Dutch astronomer Marcel Minnaert and collaborators C. Slob and G.F.W. Mulders in the early 1930s (Goldberg et al., 1960). The curve of growth is a graph showing how the equivalent width of an absorption line, or the radiance of an emission line, increases with the number of atoms producing the line and depends on the oscillator strength of the transition.

The Milne–Eddington model is considerably more sophisticated. Here, the condition for spectral line formation, i.e., the ratio of the emission coefficient to the absorption coefficient, which is denoted the line source function, may vary with optical depth in the atmosphere. However, solar and stellar abundance determinations are only as accurate as the modeling ingredients. The most recent determinations of the solar chemical composition are based on the use of state-of-the art three-dimensional atmospheric modeling and the calculation of spectral line formation, which also accounts for departures from local thermodynamic equilibrium (Asplund et al., 2009).

A comprehensive listing of element abundances in the solar photosphere and in meteorites is provided by Nicolas Grevesse and Jacques Sauval (1998).

2.3 Settling of Light Elements

When the solar abundances of lithium, beryllium, and boron are compared with their abundances in carbonaceous chondrite meteorites, in younger stars, and in the interstellar medium, it is found that the current solar lithium abundance is about a factor of 160 lower than in the primordial material, whereas the abundances of beryllium and boron are about normal.

The variations in abundance of light elements with stellar age is associated with the existence of a subsurface convective layer in solar-type stars. The core region where nuclear fusion takes place is followed by the radiative zone out to 70% of the radius where the energy is transported outward by radiation whereas the remaining outer layer is the convection zone. A layer of thickness 0.02   R_ʘ between the base of the convection zone and the top of the radiative zone is termed the solar tachocline (Elliot and Gough, 1999). This layer occurs because the inner radiative region rotates as a solid body while the convection zone rotates faster at the solar equator than near the poles.

Because lithium burns at about 2.4   ×   10⁶K whereas beryllium requires 3.5   ×   10⁶K, its surface abundance is considerably affected, because the surface convection zone reaches down to the dynamic tachocline layer, at a temperature around 2   ×   10⁶K, where some exchange of material with the radiative zone takes place. This process also explains the observed increased lithium depletion in cooler, low-mass stars, which are expected to have deeper convection zones than the Sun (Vauclaire, 1998).

An additional settling of elements will also result from the migration of elements through the interface between the convection zone and the radiative zone. A 10% reduction in helium abundance relative to hydrogen from the solar surface downward to the tachocline has been demonstrated from helioseismic studies and is explained as element migration (Grevesse and Sauval, 1998). This effect is active in both the Sun and stars.

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In situ Volcano Monitoring

Gilberto Saccorotti , ... Alessandro Aiuppa , in Volcanic Hazards, Risks and Disasters, 2015

7.3.3 Case Studies

7.3.3.1 Etna Volcano, Italy

Of particular interest is the continuous gravity sequence collected at Etna Volcano (Italy) during a 2.5-month-long time interval, bracketing the onset of the 2002–2003 eruption (Carbone et al., 2007). The start of the eruption on the northeastern flank was marked by a rapid gravity decrease of 400   μGal in less than 1   h, followed immediately by a gravity increase at a rate of 100   μGal/h to near-background levels. This anomaly was interpreted as the consequence of the opening of a fracture system along the northeastern rift and subsequent magma intrusion from the central conduit toward the lower elevation eruptive vents. During the eruption, three smaller-amplitude, negative anomalies (10–30   μGal) were observed to occur simultaneously with increases in the amplitude of volcanic TR. Detailed analysis of these data showed a clear inverse correlation between gravity decrease and TR increase, suggesting a joint source for the two geophysical observables. These anomalies occurred during periods of transition from intense lava fountaining to mild Strombolian activity. Carbone et al. (2006) thus hypothesized that the joint tremor-gravity source could be associated with accumulation of a low-density gas cloud within the shallow volcanic plumbing system (see Figure 7.5 in Seismology Section).

FIGURE 7.5. (Top) Time evolution of tremor effective amplitude at frequency 1   Hz, compared with (sign-reversed) data from a continuous recording spring gravimeter. Data are from the 2002–2003 Etna eruption, and are associated with the transition from lava fountain to pulsating Strombolian activity. After normalization for the respective maximum amplitudes, the two time series overlap almost perfectly. (Bottom) Tremor amplitude at frequency 10   Hz compared with light intensity from an infrared camera pointed to the eruptive vent for the subperiod marked by the gray rectangle in the top panel. Light changes are associated with pulsating Strombolian explosions, whose intensity is well tracked by the high-frequency tremor signal. Gravity and thermal data were provided by Daniele Carbone and Mike Burton, respectively.

7.3.3.2 Nisyros Volcano, Greece

At Nisyros Volcano (Greece), short-term gravity variations of 20–30   μGal occurred over time scales of 40–60   min, in conjunction with seismic bursts and rapid ground deformation (Gottsmann et al., 2007). Independent data indicate that the local dynamics is governed by shallow aqueous fluid movement within the hydrothermal system. Gottsmann et al. (2007) suggested that the data could be best explained by assuming thermo-hydro-mechanical variations caused by the release and upward migration of hydrothermal fluids during anomalous degassing of a deeper magmatic reservoir. The study by Gottsmann et al. (2007) also highlights a main issue concerning the interpretation of gravity changes during periods of unrest, i.e., the possibility of discriminating between shallow hydrothermal and deep magmatic contributions. To this purpose, additional constraints can be gained from the joint inversion of ground deformation and gravity signals, as demonstrated for the Phlegrean Fields Caldera, Italy, case study (e.g., Battaglia et al., 2006).

7.3.3.3 Kilauea Volcano, Hawaii (USA)

Carbone and Poland (2012) reported gravity oscillations with a period of ∼150   s, observed at two continuous stations, on the summit of Kilauea Volcano, Hawai'i. Comparison with simultaneous seismic signals allowed Carbone and Poland (2012) to exclude inertial acceleration as the cause of the gravity fluctuations. Rather, simplified source modeling suggests that the oscillations are caused by density inversions in a magma reservoir located ∼1   km beneath the east margin of Halema'uma'u Crater in Kilauea Caldera—a location of known magma storage. This result may thus represent rare evidence that convection in a shallow magma chamber is able to produce measurable geophysical signals, as theoretically postulated by Vassalli et al. (2009). In a subsequent work, Carbone et al. (2013) observed a strong gravity decrease (about 200   μGal) accompanying a marked change in lava level within Kilauea's summit eruptive vent. Using an appropriate model of the eruptive vent geometry, and from comparison with optical recordings of the lava free-surface, Carbone et al. (2013) were thus able to invert the observed gravity changes for lava density, obtaining values as low as 950   ±   300   kg/m³. Representing the very first in situ geophysical determination of a fundamental property of magmatic fluids, these data open important perspectives toward the use of continuous gravity observations for constraining dynamical processes in active magmatic systems.

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