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- Sprache: Englisch
A guide to the fascinating interplay between particle physics and astrophysics that highlights the discovery of neutrino oscillations
Written by three international experts on the topic, Solar Neutrino Physics offers a review of the status of solar physics with its strong link to neutrino physics. The book explores constitutive physics and the governing equations of standard solar models. The authors also review the theory of neutrinos in the Standard Model and the related detector experiments. The book contains a summary of the results from various experiments and develops a coherent view of the current state-of-the-art of solar neutrino physics.
Solar Neutrino Physics shows how solar models can be calibrated with the observational constraints of the age, mass, radius, and luminosity of the sun. The authors present general evolutionary properties of the sun as a star, past and future. They also discuss the solar neutrino production via the pp-chains and CNO-cycle, including the important role of the chemical composition of the sun. A very important source of information about the solar interior is offered by helioseismology, the study of solar oscillations. This important book:
- Presents a high-level overview of the field of solar neutrino physics
- Brings together data and their interpretation of results obtained at various solar neutrino observatories
- Combines the theory of nuclear reactions with solar neutrino experiments
- Contains a review of SNO+, JUNO, LENA, Hyper-Kamiokande, and DUNE.
Written for astronomers, physicists, and high energy physicists, Solar Neutrino Physics contains a review of the field of neutrino physics, the relevant equations, and the impact of matter on the behavior of neutrino oscillations.
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Seitenzahl: 444
Veröffentlichungsjahr: 2020
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Table of Contents
Cover
1: Introduction
2: Solar Structure and Evolution
2.1 Equations of Stellar Structure and Evolution
2.2 Constitutive Physics
2.3 Calibrating Standard Solar Models
2.4 Standard Solar Models
2.5 Solar Neutrinos
2.6 Helioseismology
2.7 Solar Abundance Problem
2.8 Uncertainties in SSMs
2.9 Solar Models Beyond the SSM
Notes
3 Neutrino Physics
3.1 Neutrinos in the Standard Model
3.2 Neutrino Oscillations
3.3 Matter Effects
3.4 Neutrino Oscillation Experiments
3.5 Conclusions and Open Questions
Notes
4: Solar Neutrino Experiments
4.1 Introduction
4.2 The Cl Experiment
4.3 Kamiokande‐II/III
4.4 The Ga Experiments
4.5 Super‐Kamiokande
4.6 SNO
4.7 Borexino
4.8 Summary and Open Questions
Note
5: Future Solar Neutrino Experiments
5.1 SNO+
5.2 JUNO and LENA
5.3 Hyper‐Kamiokande
5.4 DUNE
Notes
References
Index
End User License Agreement
List of Tables
2
Table 2.1 Astrophysical factors
, energy release
, and average neutrino energy ...
Table 2.2 Recommended abundances of the most abundant metals in two widely used ...
Table 2.3 Solar neutrino fluxes on Earth. Predicted by two solar models based on...
Table 2.4 Summary of properties of B16 SSMs and solar quantities.
Table 2.5 Dominant theoretical error sources for neutrino fluxes and the main ch...
3
Table 3.1 Elementary particles in the standard model.
Table 3.2 Overview about past and actual main neutrino oscillation experiments.
Table 3.3 Overview about double
‐experiments, their techniques, and achieved 90%...
4
Table 4.1 Summary of the main characteristics of solar neutrino experiments.
Table 4.2 Homestake experiment: expected signal, background, and measurement.
Table 4.3 Super‐Kamiokande: Different detector configurations and running modes.
Table 4.4 Super‐Kamiokande measured solar neutrino flux for the different detect...
Table 4.5 The three phases of the SNO experiment.
Table 4.6 Energy‐unconstrained fluxes in units of
from SNO‐I/II/III.
Table 4.7 Day–night asymmetry measurement in SNO.
Table 4.8 Calibration sources used in Borexino in the inner vessel and in the ou...
Table 4.9 Unoscillated rates in Borexino based on SSM (GS98‐2016).
Table 4.10 Summary of solar neutrino measurements in Borexino.
Table 4.11 Summary of 2018 solar neutrino measurements in Borexino.
Table 4.12 Summary of solar neutrino measurements.
List of Illustrations
2
Figure 2.1 (a)
,
, and
in the solar interior as a function of temperature....
Figure 2.2 Ionization fraction of all ionization states for H, He, and the mos...
Figure 2.3 Rosseland mean opacity as a function of
and
. The
–
profile of...
Figure 2.4 Fractional contribution of individual chemical elements to
at thr...
Figure 2.5 Complete set of reactions in the three pp chains (ppI, ppII, ppIII)...
Figure 2.6 Complete set of reactions in the CNO‐bicycle. Branching occurs at t...
Figure 2.7 (a) Penetration factors (dashed lines), Maxwell distribution (dashe...
Figure 2.8 Histogram of temperature and density as a function of depth in a 3D...
Figure 2.9 Hertzsprung–Russell diagram showing the evolution of a
stellar mo...
Figure 2.10 (a) Time evolution of energy sources and stellar luminosity, as in...
Figure 2.11 Internal chemical composition profiles for relevant nuclear specie...
Figure 2.12 Internal profiles for physical quantities and composition for the ...
Figure 2.13 (a) Temperature gradients and (b) radiative opacity in the solar i...
Figure 2.14 Distribution functions of the production of neutrinos in the pp ch...
Figure 2.15 (a) Distribution functions of the production of neutrinos in the C...
Figure 2.16
and
normalized to solar values. Black circle and error bars: s...
Figure 2.17 Individual contributions to the solar neutrino spectra. Fluxes and...
Figure 2.18 Spectrum of solar oscillations. Large symbols: data from the Miche...
Figure 2.19 Propagation diagram of oscillations for a solar model.
Figure 2.20 (a) Sound speed inversion results for B16‐GS98 and B16‐AGSS09 SSMs...
Figure 2.21 (a) Sensitivity of sound speed with respect to the
, the location...
Figure 2.22 Comparison of temperature gradients and opacity in B16‐GS98 and B1...
Figure 2.23 Lithium depletion in solar models of increasing amount of turbulen...
3
Figure 3.1 Fundamental couplings of neutrinos. The index
stands for a specif...
Figure 3.2 Oscillation probabilities as function of the distance between sourc...
Figure 3.3 Neutrino mixing and rotation scheme in case of vacuum (a) and in ca...
Figure 3.4 Adiabatic conversion of
to
in solar matter. High energy electro...
Figure 3.5 Schematic view of the survival probability
for an electron neutri...
Figure 3.6 Scheme of neutrino production in the upper atmosphere due to strong...
Figure 3.7 Calculated flux ratio between muon and electron neutrinos produced ...
Figure 3.8 Charged current cross sections for (a) neutrino and (b) antineutrin...
Figure 3.9 Schematic of the 50 kt Super‐Kamiokande water Cherenkov neutrino de...
Figure 3.10 Electron‐ (a) and muon (b)‐like neutrino interaction visible as si...
Figure 3.11 First evidence for neutrino oscillations in the atmospheric data o...
Figure 3.12
‐Analysis of atmospheric neutrino data obtained by Super‐Kamiokan...
Figure 3.13 Zenith angle distribution of
candidates in Super‐Kamiokande (Pha...
Figure 3.14 Schematic view of the neutrino gun used in the K2K experiment. A b...
Figure 3.15 Energy spectrum (a) and oscillation contour plots (b) obtained in ...
Figure 3.16 Allowed oscillation parameter space by MINOS as published in [159]...
Figure 3.17 The third
candidate in OPERA in projected topography. In the ins...
Figure 3.18 Disappearance oscillation probability,
, for T2K, NO
A, and DUNE ...
Figure 3.19 T2K disappearance
measurement at high statistics. Upper plot: th...
Figure 3.20 Oscillation probability
for T2K and for two
values in the norm...
Figure 3.21 The 68% and 90% CL allowed regions for
as a function of
and th...
Figure 3.22 Preliminary
spectrum with 33 events as obtained in the long base...
Figure 3.23 Shape of a non‐oscillated reactor neutrino spectrum observed via t...
Figure 3.24 Schematic scheme of the 1 kt liquid scintillator detector used in ...
Figure 3.25 Prompt event energy spectrum of electron antineutrino candidate ev...
Figure 3.26 KamLAND reactor neutrino data as function over
in comparison wit...
Figure 3.27 Schematically view of the Double Chooz far detector. As neutrino t...
Figure 3.28 Background subtracted positron spectrum of the Daya Bay far detect...
Figure 3.29 The antineutrino survival probability as a function of the effecti...
Figure 3.30 Regions of the oscillation parameter space allowed in the Daya Bay...
Figure 3.31 Combined oscillation analysis of all neutrino data available. The ...
Figure 3.32 Schematic view of a
spectrum and the impact of neutrino masses a...
Figure 3.33 Principle of a
‐spectrometer with MAC‐E filter. At the top the ex...
Figure 3.34 Averaged counting rate from the Mainz experiment using a frozen tr...
Figure 3.35 Schematic view of the KATRIN experiment with a windowless gaseous ...
Figure 3.36 Supernova 1987a optical observation (a) in the Large Magellanic Cl...
Figure 3.37 Evolution of the CMB,
, dark matter, baryonic and
energy densit...
Figure 3.38 Scheme of the double
decay of
Ge to
Se. The single
decay to
Figure 3.39 Feynman diagram for the neutrino mass mechanism of neutrinoless do...
Figure 3.40 Allowed values for the effective Majorana neutrino mass
as a fun...
Figure 3.41 Spectrum of the total kinetic energy of
Ge originating from the
Figure 3.42 Energy spectrum shown by the GERDA experiment after an initial exp...
Figure 3.43 Normal and inverted mass ordering or hierarchies (abbr. NMO) of ne...
Figure 3.44 Scatter plot (energy versus azimuth angle) of the normal versus in...
Figure 3.45 Calculated reactor neutrino spectrum at a distance of 53 km with a...
Figure 3.46 Calculated oscillation probability for
to appear as
at a dista...
Figure 3.47 Calculated oscillation probability for
to appear as
at a dista...
Figure 3.48 Probabilities for accepting the correct NMO scenario while excludi...
Figure 3.49 Energy‐dependent oscillation probabilities for (a) neutrinos and (...
Figure 3.50 Exclusion plot for the mixing of a massive sterile neutrino with m...
Figure 3.51 Distribution of the ratio between observed and expected neutrino l...
4
Figure 4.1 Homestake detector and ancillary facilities [241]. This drawing is ...
Figure 4.2 Schematic view of the Kamiokande‐II detector.
Figure 4.3 Distribution in
, the cosine of the angle between the trajectory o...
Figure 4.4 Distribution in
corresponding to 1036 life days in Kamiokande‐III...
Figure 4.5 (a) Ratio of data to standard solar model in 200 days bins from Kam...
Figure 4.6 Count rate of selected L‐peak and K‐peak events averaged over the a...
Figure 4.7 Summary of 65 solar neutrino runs in GALLEX. The right‐hand scale c...
Figure 4.8 GNO energy spectrum of selected
Ge events for the whole data set. ...
Figure 4.9 (a) Count rate in SAGE [267] against energy and rise time for event...
Figure 4.10 Combined result for each year in SAGE from 1990 to 2007. The shade...
Figure 4.11 The LINAC calibration system in Super‐Kamiokande is shown. The dot...
Figure 4.12 Cosine of the angle between the electron direction and the directi...
Figure 4.13 Energy resolution is Super‐Kamiokande. Solid line: SK‐IV. Dashed l...
Figure 4.14 Vertex resolution for SK‐I (dotted line), SK‐II (dashed‐dotted lin...
Figure 4.15 Super‐Kamiokande IV energy spectrum measured as the ration of the ...
Figure 4.16 SK‐IV day–night asymmetry as a function of the solar zenith angle....
Figure 4.17 Sketch of the SNO detector [280]. The AV that contains the
is vi...
Figure 4.18 Distribution of the NCDs strings deployed inside the AV in SNO.
Figure 4.19 Solar zenith angle distribution of selected events with visible en...
Figure 4.20 Flux of
8
B solar neutrinos from SNO‐I and SK‐I that have been dete...
Figure 4.21 Data and Monte Carlo distributions for the main variables used in ...
Figure 4.22 Flux of
8
B solar neutrinos from SNO‐I, which have been detected as...
Figure 4.23 Solar neutrino survival probability determined using the polynomia...
Figure 4.24 Sketch of the CTF detector. The external water tank, the open stru...
Figure 4.25 Sketch of the Borexino detector.
Figure 4.26 Schematic view of pseudocumene unloading, purification, and fillin...
Figure 4.27 Electron recoil spectrum from solar neutrinos in Borexino. pp: sol...
Figure 4.28 Spectrum of recorded data in Borexino. The spectrum after the fidu...
Figure 4.29 Energy spectrum in Borexino for neutrino‐like events in the fiduci...
Figure 4.30 Gobal solar and KamLAND neutrino oscillations fit.
Figure 4.31 Electron neutrino survival probability. Red points: Borexino data....
Figure 4.32 Energy spectrum of CNO solar neutrinos and
Bi.
5
Figure 5.1 Current measurement of solar
B (
‐axis) and
Be (
‐axis) neutrino ...
Figure 5.2 Survival probability
for solar
B‐neutrinos in the standard MSW c...
Figure 5.3 Survival probability
for solar
B‐neutrinos in the standard MSW c...
Figure 5.4 Expected solar
B‐spectrum coming from elastic neutrino electron sc...
Figure 5.5 Expected recoil spectrum due to elastic neutrino scattering of elec...
Figure 5.6 Schematic view of the Hyper‐Kamiokande single cylindrical tank.
Figure 5.7 Schematic view of the DUNE far detector with the four 10 kt TPCs an...
Figure 5.8 Present (a) and future (b) precision measurements of solar neutrino...
Guide
Cover
Table of Contents
Begin Reading
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Solar Neutrino Physics
The Interplay between Particle Physics and Astronomy
Lothar Oberauer
Aldo Ianni
Aldo Serenelli
Copyright
Authors
Professor Lothar Oberauer
TU München
Physik Department
E15
James‐Franck‐Straße 1
85748 Garching
Germany
Dr. Aldo Ianni
I.N.F.N. Laboratori Nazionali del
Gran Sasso
Via Giovanni Acitelli 22
67100 Assergi (AQ)
Italy
Dr. Aldo Serenelli
Institute of Space Sciences
Carrer de Can Magrans, s/n
08193 Bellaterra
Spain
Lothar Oberauer
Aldo Ianni
Aldo Serenelli
All books published by Wiley‐VCH are carefully produced. Nevertheless, authors, editors, and publisher do not warrant the information contained in these books, including this book, to be free of errors. Readers are advised to keep in mind that statements, data, illustrations, procedural details or other items may inadvertently be inaccurate.
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The Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data are available on the Internet at <http://dnb.d‐nb.de=.
© 2020 Wiley‐VCH Verlag GmbH & Co. KGaA, Boschstr. 12, 69469 Weinheim, Germany
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Print ISBN: 978‐3‐527‐41274‐7
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Cover Design Grafik‐Design Schulz
The Sun is the source of energy for life on Earth. Solar neutrinos are the gift given to guide us in unlocking its most hidden secrets.
1Introduction
In 2015 Arthur B. McDonald and Takaaki Kajita were awarded with the Nobel Prize in Physics for their discovery of neutrino oscillations and the inevitable consequence that neutrinos have a mass. The perception of non‐vanishing neutrino masses implies that the standard model of particle physics has to be extended. This great discovery has been achieved by measuring solar and atmospheric neutrinos. The former are emitted as a product of thermonuclear fusion reactions inside the Sun; the latter are produced in weak decays of mesons, which are generated by the reactions of cosmic particles in the top layers of the Earth's atmosphere. Indeed, the origin of the idea of neutrino oscillations can be found in the pioneering Homestake solar neutrino experiment in the early 1970s (Nobel Prize 2002). Homestake was the first experiment able to detect solar neutrinos. In addition, it proved the basic assumption of thermonuclear fusion processes in the Sun and, at the same time, it recorded a deficit in the measured solar neutrino flux, when one compares it with theoretical predictions. Yet, it took almost 30 years and several experiments until the existence of neutrino oscillations could really be proven.
Solar neutrino experiments have a second aspect, in addition to particle physics. The production of neutrinos in the Sun is sensitive to the physical conditions in the solar interior. Therefore, solar neutrino measurements can be used to determine the physical properties of the solar core.
In this book we present and discuss the actual status of solar physics with its strong link to neutrino physics. Chapter 2 deals with the physics and basic equations that are relevant to understand the stellar structure and evolution. We show how solar models can be calibrated by confronting them to the observational constraints of the age, mass, radius, and luminosity of the Sun. We present general evolutionary properties of the Sun as a star, past and future. For the present‐day Sun, a detailed presentation of its internal structure is given. This is required for then discussing the solar neutrino production via the pp chains and CNO cycle, including the important role of the chemical composition of the Sun. A very important source of information about the solar interior is offered by helioseismology, the study of solar oscillations. The topic is introduced briefly, and then the most relevant results are given, again placing some emphasis on the differences arising from the assumptions made about the solar composition. The solar abundance (or solar modeling?) problem, a now more than 15 year old problem is discussed to some extent, both in the context of helioseismology and, very importantly, solar neutrinos. Chapter 2 closes with a description of model uncertainties and an overview about solar models beyond the standard case.
The neutrino physics is introduced in Chapter 3. First we describe neutrinos in the standard model of particle physics. Then, we introduce the concept of neutrino mass eigenstates and neutrino mixing leading to the phenomenon of neutrino oscillations. The basic equations in the case of two‐ and three‐neutrino oscillations will be derived. As solar neutrinos are influenced by matter effects inside the Sun and at least partially also by Earth matter effects, we describe the basic impact of matter on the behavior of neutrino oscillations. In Chapter 3, we give an overview about results of neutrino oscillation experiments, which are not using solar neutrinos as source. The chapter closes with a discussion about open questions in neutrino physics. We review the concept of Dirac and Majorana neutrinos, mass ordering, and CP violation in the framework of neutrino physics. We discuss the idea of sterile neutrinos.
Solar neutrino experiments are discussed in Chapter 4. Here, we follow a historical line beginning with radiochemical experiments, where the so‐called solar neutrino puzzle has been established. We continue with the description of real‐time experiments, namely, Kamiokande, Super‐Kamiokande, and Sudbury Neutrino Observatory (SNO). We underline the impact of these experiments on our current understanding of the solar neutrino puzzle in terms of neutrino oscillations. Finally, the performance and recent results of the Borexino experiment are reported and discussed. The chapter closes with a summary of the achievements after 50 years of solar neutrino physics and a discussion about open questions.
Chapter 5 reports a review of upcoming experiments and their capabilities to contribute to a better understanding of neutrino and solar physics. In particular, we briefly review SNO+, Jiangmen Underground Neutrino Observatory (JUNO), Low Energy Neutrino Astrophysics (LENA), Hyper‐Kamiokande, and Deep Underground Neutrino Experiment (DUNE).
2Solar Structure and Evolution
This chapter introduces the equations of stellar structure and evolution and the most important physical processes that determine the evolution of the Sun. We then present the important concept of standard solar model (SSM) and place it in the broader context of evolution of low mass stars, of which the Sun is a typical representative, by discussing the main characteristics of its evolution, from its formation to advanced evolutionary phases. The present‐day properties of SSM are then discussed in detail, including theoretical predictions for solar neutrinos and helioseismic probes of the solar interior and the current theoretical uncertainties in solar models. We close the chapter with a generic discussion of nonstandard solar models (non‐SSMs), i.e. in which physical processes considered nonstandard have been included in the models.
2.1 Equations of Stellar Structure and Evolution
The fundamental equations that determine the structure and evolution of stars reflect the conservation laws of mass, energy, and momentum and the physical processes that take place in stellar interiors. The resulting equations form a set of partial differential equations and the problem requires both initial and boundary conditions. Here, we derive the full set of equations and boundary conditions that underlie studies of solar models.
2.1.1 Mechanical Structure
The mechanical structure of stars, the Sun among them, is determined by the conservation laws of mass and momentum. In spherical symmetry, the equation of mass conservation can be established in a convenient form by considering the total variation of the mass inside a sphere of radius :
Here, the first term on the right is the mass contained in a shell of thickness d and the second term is the mass flow with radial velocity at the surface of the sphere in the time interval d. In this description, is the independent coordinate. However, in spherical symmetry it is more convenient to use the Lagrangian description in which is the independent coordinate, i.e. to express and other quantities as functions of and . The transformation between the two descriptions is obtained by considering the partial derivative:
Applying this transformation to , one readily obtains the equation of mass conservation in the Lagrangian description:
By comparing this expression with Eq. (2.1), it becomes clear that using the Lagrangian formulation equations can be written more simply.
The conservation of momentum in spherical symmetry is expressed as
where the net acceleration is the result of the gravitational acceleration and the force per unit mass exerted by the pressure gradient.
The secular evolution of stars driven by stable nuclear burning leads to very slow changes in stellar size, leading to a negligible net acceleration. This is especially true for the Sun, a low mass main sequence star that evolves appreciably only over timescales of years. Estimates of the acceleration based on the evolution of detailed solar models lead to maximum values of the order of a few times at the solar surface. This can be compared with the gravitational acceleration that, in the solar surface, is , i.e. about 28 orders of magnitude larger. Under these conditions, the conservation of momentum reduces to the equation of hydrostatic equilibrium:
The response timescale to departures from hydrostatic equilibrium is very short. This can be estimated by considering Eq. (2.3). If, for example, a very large increase in the pressure gradient were to occur such that the second term on the right‐hand side dominates, then the response time would be of the order . Here, is an average value of the sound speed in the solar interior, of the order of . On the other hand, if the pressure gradient were to become negligible, the response time would be that of free fall: , where is the solar mean density. Any large‐scale dynamical instability would relax in timescales extremely short compared to the evolutionary timescales of the Sun, justifying again the assumption of hydrostatic equilibrium in solar evolution models. Finally, transforming Eq. (2.4) to the Lagrangian formulation leads to
Equations (2.2) and (2.5) determine the mechanical structure of the Sun.
2.1.2 Energy Conservation and Transport
The next step in deriving the full set of stellar evolution equations is to consider the energetics of stars. The first law of thermodynamics states:
where and are respectively the heat and internal energy per unit mass for an element of specific volume . Here, the first term represents a change in the internal energy and the second term the work, by expansion or contraction, done on the element. Also, if represents a local rate of energy production per unit mass and is the radial component of the energy flux, the rate of change of heat in the volume element is
where the second term is the divergence of the energy flux per unit mass. This can be expressed as
Here is the energy flux across a sphere of radius , i.e. the luminosity. Using the Lagrangian transformation,
defining
and combining with the first law of thermodynamics:
In standard stellar evolution represents nuclear energy sources and neutrino energy losses, i.e. . The second term, , usually referred to as gravothermal energy, accounts for changes in internal energy and work. The final form of the equation of energy conservation is then:
2.1.2.1 Energy Transport by Radiation and Conduction
In standard stellar models, there are three mechanisms for transporting energy that occur because of the temperature gradient that exists between the inner and outer stellar regions: radiation, (electron) conduction, and convection. From a strict point of view, the first two are always present because they directly depend on the existence of a temperature gradient. Convection, on the other hand, occurs in stars when a region becomes dynamically unstable and macroscopic convective motions set in. The actual temperature gradient in the stellar interior will be determined by the combined action of these transport mechanisms.
Radiation transport in stellar interiors can be treated under the so‐called diffusion approximation, i.e. by consideration of Fick's law applied to radiation energy. This is justified because the mean free path of photons in stellar interiors is much smaller than the characteristic length scales over which physical conditions change significantly.1 The opaqueness of matter to radiation is conveniently represented in stellar interiors by the opacity, an absorption coefficient expressed in units of cross section per unit mass. The mean free path of photons is then given by
where is an appropriate average, to be discussed below, of the opacity taken over the whole spectrum of photons. Typical values of determined from detailed solar models range between and in the solar interior. The small values of also mean that the variation of temperature that a photon experiences over is very small and local thermodynamic equilibrium (LTE) is very well satisfied in solar interiors.
Fick's law applied to radiation energy density is
with as the speed of light, the radiation density constant, and the energy density of radiation. Expressing now the radiation flux in terms of the luminosity, replacing the energy density gradient by its corresponding temperature gradient, and with Eq. (2.7):
Applying the Lagrangian transformation, the equation of radiative transport is
Conduction can also be treated as a diffusive process, characterized by a conductive opacity . It is accounted for in the transport of energy by replacing the radiative opacity by
In the Sun, conduction is a very inefficient transport mechanism. In an ideal gas under non‐degenerate conditions, electrons have an extremely short mean free path in comparison to photons. In the solar center, where the density of free electrons is highest, the conductive opacity is , in comparison to a typical value for the radiative opacity , i.e. . The mean free path of electrons is then shorter by the same factor, rendering conduction a negligible energy transport mechanism in the solar interior.
The equation for radiative and conductive transport is then
It is useful to express the temperature gradient in a dimensionless form:
where the radiative gradient is defined as
so that Eq. (2.9) reads:
Before, it was mentioned that is an appropriate average over the whole spectrum of photon energies. In fact, the relation used above implicitly assumes LTE and that the radiation field is that of a black body, i.e. described by a Planck distribution. Under these conditions, it can be shown that is the Rosseland mean opacity, given by
where is the radiation frequency, the monochromatic opacity, and
is the Planck function for radiation intensity. The Rosseland mean opacity entering the equation of radiative transport is then the harmonic mean of the monochromatic opacity weighted by .
2.1.2.2 Criterion for Dynamical Instability
Radiative transport is inefficient when is sufficiently large and the temperature gradient required to transport energy is too steep. This typically occurs under two circumstances, when is large as it happens in regions where dominant chemical elements are partially ionized, and when the production of nuclear energy is concentrated in small volumes such that the ratio is large. The increase in inside a star can lead to the development of a dynamical instability in which macroscopic motions of the gas set in, a process known as convection.
At any given depth , there are always small fluctuations in the thermodynamic properties of the gas. The question of dynamical instability is whether or not these fluctuations grow and develop into large‐scale motions. Consider a region in the star where the actual temperature gradient is radiative, and consider a small parcel of material that experiences a small fluctuation, e.g. an increase, in its temperature:
where and are the temperatures of the small parcel of gas and the stellar surroundings at depth , respectively. There are pressure and density fluctuations associated with . Hydrostatic equilibrium between the surrounding medium and the parcel of matter quickly restores pressure balance, i.e. . Under this condition, and for an ideal equation of state (EoS), the density fluctuation will be
so the parcel is less dense than the medium and experiences a net buoyancy force. Which is the situation after the parcel of material has risen an infinitesimal distance ? In other words, is positive (stability) or negative (instability)? The change in is
Under almost every situation inside stars, and certainly in solar interiors, density gradients are negative, so that the stability condition is
The EoS , where is the mean molecular weight, relates changes among thermodynamic quantities such that
Replacing this expression in Eq. (2.14), and using and that there are no changes in the chemical composition of the parcel, the stability condition is now
By multiplying this relation by , which is a negative quantity, and extending the definition of the dimensionless temperature gradient, the stability condition is now expressed more simply as
For simplicity, the subindex can be dropped. Then, under the condition of stability, . Moreover, the limiting case is established when the there is no heat exchange between the moving parcel and the surrounding medium, i.e. , where
is the adiabatic dimensionless temperature gradient that is determined solely by the EoS. Taking all this into account, the stability criterion known as the Ledoux criterion reads
When the layers of the star have homogeneous composition, i.e. , this reduces to the Schwarzschild criterion:
The stability condition has been presented here in some detail because it is fundamental in the determination of the properties of the convective envelope of the Sun, particularly the location of its lower boundary that can be determined precisely using helioseismic techniques (Section 2.6).
2.1.2.3 Energy Transport: Convection
Consider a situation in which the composition is homogeneous () and in which the stability condition Eq. (2.17) is not fulfilled. Convection sets in such that hot bubbles rise and release their excess heat at outer regions. Simultaneously, cooler material is drafted downwards, also contributing to an outward transport of energy. There is always a temperature gradient present in the star, and some fraction of energy is always transported by radiation. But the presence of convection implies . Also, although the motion of the parcels, or bubbles, closely follows adiabaticity, there is always some heat exchange with the surroundings so that the gradient in the moving parcel is larger than the adiabatic one, i.e. . Combining these ideas with the stability condition in Eq. (2.15), the following inequality can be established in convective regions:
Here is the actual dimensionless temperature gradient in the star:
Combining this with the equation of hydrostatic equilibrium in its Lagrangian form (Eq. (2.5)):
This equation has the same structure as the equation of radiative energy transport (Eq. (2.12)) with the difference that has been replaced by . In practice, has to be determined from a theory of convection across all convective regions in a star.
In stellar interiors, convection is turbulent and there is no first principles solution of the hydrodynamics equations from which to determine the properties of convective flows such as convective velocities, size of convective eddies, and temperature gradient. In stellar models, the most widely used theory of convection is the mixing length theory (MLT) [1]. Despite its simplicity, MLT has been used with great success in stellar interior and atmosphere models that show very good agreement with observations. We present here a basic outline of MLT, following closely that in Ref. [2].
In a convective region the total energy flux, , can be expressed as the sum of the radiative and the convective fluxes, and respectively:
and is determined by the, yet unknown, actual gradient :
MLT provides an expression for based only on local quantities. If a rising bubble has an excess temperature with respect to the medium, then is
where , the specific heat per mass unit at constant pressure, is used because the pressure in the bubble and the medium are assumed equal: i.e. heat exchange takes place under the condition . is the velocity of the convective motions. In MLT, it is assumed all moving elements at a given radius travel at the same velocity and the same distance , the mixing length, before dissolving and releasing their excess heat into the surroundings. On average, then, at a given location in the star, it is assumed that the crossing elements have traveled . With these assumptions, the convective flux can be expressed as
where the pressure scale height, , has been used to relate to the dimensionless temperature gradients. can be related to introducing the so‐called mixing length parameter such that
Note that is a free parameter in the theory that needs to be calibrated using observational constraints. This is discussed further in Section 2.3.2.
In the expressions above, , , and are unknown but related by Eq. (2.20). Additional considerations are required to establish a closed system of equations. The first one is the balance between the kinetic energy acquired by the work done by the rising element and the energy deposited in the surroundings. The second one is related to the temperature changes taking place inside the bubble that are related both to adiabatic cooling due to expansion and to radiative heat transfer with the surroundings. The final set of expressions that allow the calculation of , given here without demonstration, are [2]
Then, is determined by solving for ξ in the cubic equation:
which has only one real solution. Note that all quantities entering the calculation of and are local. The gradient determined from MLT is therefore local as well.
The difference is sometimes referred to as overadiabaticity. There are two limiting solutions for a given value of W which correspond to cases where or . Convection is nearly adiabatic and a small overadiabaticity of the order – suffices to transport all the energy. This is typical of conditions in stellar interiors, and it is also the case at the base of the solar convective envelope. Physically, it occurs when the thermal radiation timescale is long compared to the convective timescale so heat exchange between the moving bubbles and the surroundings is negligible. In this limit, where convection is almost completely adiabatic, there is no need for a full solution of Eq. (2.22) as is determined from the EoS. The second case leads to , i.e. . Here, convection is very inefficient and heat exchange through radiation dominates. This is typical of very low density environments, for which the mean free path of photons is long. The intermediate case requires determining the actual solution of Eq. (2.22). In the Sun, this is needed in the outermost layers, at . This is discussed further in some detail in Section 2.4.2.
A more general and thorough discussion of the solutions of Eq. (2.22) can be found, for example, in [2,3].
2.1.3 Changes in Chemical Composition
2.1.3.1 Convective Mixing
An order of magnitude estimate of the typical velocity of convective flows can be done without recurring to theories of convection. Assume convective transport is efficient, such that the convective flux is . The flow of heat and kinetic energy density are comparable. The latter is proportional to , so at a given radius :
Typical values in the convective envelope of the Sun are , , and . From the relation above, the estimation of the convective velocity is . The extension of the solar convective envelope is . After neglecting factors of order unity, this yields an estimate of the global timescale for convection . Convection is a fast process in comparison with the secular timescales linked to solar evolution. Convective mixing will be comparably fast in maintaining chemical homogeneity in the convective regions. As a result, the solar convective envelope can be considered at all times to be chemically homogeneous.
Under certain circumstances, it is useful to use a more general approach in which convective mixing is treated as a time‐dependent process. This simplifies the inclusion of other types of mixing processes that might not be as fast as convection, or special situations in which nuclear burning and convective mixing occur on comparable timescales. A widely used approach is to consider convective mixing as a diffusive process. A convection theory, e.g. MLT, is used to determine the average velocity of the flow at a given location in the star and the characteristic length over which convective elements travel. From Section 2.1.2.3, these quantities are and . The diffusive coefficient for convective mixing is therefore . Convection is a macroscopic process and is the same for all chemical species present in the convective regions. In radiatively stable zones, formally because and no convective mixing occurs.
If is the mass fraction of nuclear species then, in spherical symmetry, the diffusion equation is [4,5]
and, in Lagrangian form,
An improved estimate of the convective mixing timescale in the solar envelope can now be obtained, where typical values are cm and was estimated above. The convective diffusion coefficient is then so the diffusive timescale across the convective envelope is , with factors of order unity neglected again.
2.1.3.2 Changes in Chemical Composition: Atomic Diffusion
In the Sun, there is strong evidence from helioseismology that the helium fractional mass abundance in the solar convective envelope is lower by 10–12% than in the stably stratified radiative interior below it and lower than the helium abundance with which the Sun formed [6–8]. Accordingly, the hydrogen fractional mass abundance in the solar envelope is higher that the initial solar abundance. This cannot be accounted for by any sort of macroscopic mixing and it leads to consideration of several other mixing processes, generically grouped under the name of microscopic mixing, that induce a separation of elements by acting differentially on individual chemical species.
Following [9
