Principles and perspectives in cosmochemistry lecture notes of the kodai school on synthesis of elements in stars held at kodaikanal observatory india april 29 may 13 2008

Part I of the book deals with stellar structure, osynthesis and evolution of low and intermediate-mass stars.. Amanda Karakas’s lectures dis-cuss nucleosynthesis of low and intermediate-

For further volumes:

http://www.springer.com/series/7395\

Lecture Notes of the Kodai School

on ‘Synthesis of Elements in Stars’ held

at Kodaikanal Observatory, India,

or parts thereof is permitted only under the provisions of the German Copyright Law of September 9,

1965, in its current version, and permission for use must always be obtained from Springer Violations are liable to prosecution under the German Copyright Law.

The use of general descriptive names, registered names, trademarks, etc in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use.

Cover design: eStudio Calamar S.L.

Printed on acid-free paper

Springer is part of Springer Science+Business Media (www.springer.com)

B Eswar Reddy

ereddy@iiap.res.in

ISBN 978-3-642-10351-3 e-ISBN 978-3-642-10352-0

DOI 10.1007/978-3-642-10352-0

 Springer-Verlag Berlin Heidelberg 2010

Library of Congress Control Number: 2010921800

The origin of elements is among the fundamental aspects of our universe;cosmochemistry tries to answer when, how and where the chemical elementsarose after hydrogen was created during primordial nucleosynthesis followingthe Big Bang However, quantitative answers to these fundamental questionsbegan to emerge only in the late fifties, with the pioneering works of Bur-bidge, Burbidge, Fowler and Hoyle, and Cameron Since then there had beensignificant progress in the understanding of synthesis of elements in stars.Cosmochemistry, however, remains a fertile area of research, as thereremain many outstanding problems A comprehensive approach to cosmo-chemistry requires a combination of a number of topics like primordialnucleosynthesis, stellar nucleosynthesis, explosive nucleosynthesis and solarabundance The Kodai school on ‘Synthesis of elements in stars’ was orga-nized to provide a glimpse of this exciting area of research to astrophysicists

of tomorrow, motivated young students from India and abroad The lecturesare thus aimed at researchers who would like to venture deeper into this ex-citing arena

The school drew strength from considerable in-house expertise at IIA in anumber of areas critical for the school A highlight of the school, however, wasthe faculty participation by a number of leading astrophysicists from differentparts of the world

Following a traditional and inspiring invocation from Upanishad and abrief inaugural function, the school was opened for technical sessions DavidLambert set the tone of the scientific sessions with the lead talk on ‘Synthesis

of elements in stars: an overview’ The basic properties of nuclei were plained by Arun Mangalam in a series of lectures The lectures by C Sivaramput the primary issue in cosmochemistry in perspective through a discussion

ex-on cosmological nucleosynthesis of light elements Aruna Goswami discussedsome current issues in the present understanding of the Galactic chemicalevolution Gajendra Pandey explained how stellar spectra can be analyzedusing ‘Curve of growth technique’ Kameswara Rao of IIA talked about thehigh resolution Echelle spectrograph at VBO, Kavalur and discussed someresults obtained from analysis of data acquired using this instrument These

lectures provided the background for the series of lectures by other speakersthat followed Apart from the regular class room lectures, students had ampletime for hands-on sessions coordinated by Goswami, Reddy and Pandey.The book has been organized into three parts to address the major issues

in cosmochemistry Part I of the book deals with stellar structure, osynthesis and evolution of low and intermediate-mass stars The lectures bySimon Jeffery outline stellar evolution with discussion on the basic equations,elementary solutions and numerical methods Amanda Karakas’s lectures dis-cuss nucleosynthesis of low and intermediate-mass stars covering nucleosyn-thesis prior to the Asymptotic Giant Branch (AGB) phase, evolution duringthe AGB, nucleosynthesis during the AGB phase, evolution after the AGB

nucle-and massive AGB stars The slow neutron-capture process nucle-and yields from

AGB stars are also discussed in detail by Karakas The lectures by S Giridharprovide some necessary background on stellar classification

Part II deals with explosive nucleosynthesis that plays a critical role in mochemistry The lectures by Kamales Kar provide essential background ma-terial on weak-interaction rates for stellar evolution, supernovae and r-processnucleosynthesis He also discusses in detail the solar neutrino problem Mas-sive stars, their evolution and nuclear reaction rates from the point of view

cos-of astronomers and nuclear physicists are discussed by Alak Ray His tures also describe the various stages of hydrostatic nuclear fuel burning withillustrative examples of how the reactions are computed He also discussedcore-collapse (thermonuclear vs core-collapse) and supernovae in brief Thelectures by Marcel Arnould address the phenomena of evolution of massivestars and the concomitant non-explosive and explosive nucleosynthesis Hehighlights a number of important problems that are yet unresolved but cru-cial for our understanding of Galactic chemical evolution The p-process nu-cleosynthesis attributed to the production of proton-rich elements, a topic ofgreat importance but yet less explored is also discussed in his lectures.The third and the final part of the book addresses use of solar system abun-dances to probe cosmochemistry quantitatively The lectures by Bruce Fegleyaddress cosmochemistry of the major elements; while the lectures by Katha-rina Lodders discuss elemental abundances in Solar, meteoritic and outsidethe solar system

lec-Cosmochemistry is still an evolving branch of astrophysics, with manychallenges The book is expected to serve as a contemporary reference materialfor research in cosmochemistry We would like to take this opportunity tothank all the contributors for making this book a reality

This school would not have been possible without the dedicated support of many We extend our sincere thanks to professor Siraj Hasan, Director, In- dian Institute of Astrophysics and professor Vinod Krishan for their all round support for the school.

We are particularly grateful to the school faculty from India and abroad for readily accepting to participate, prepare lecture notes and spend time with the students.

The organization of the school is a collective effort of the coordinators, the convener, the members of the local organizing committee and many others We are thankful to the administrative department of IIA and the staff members of Kodaikanal Solar observatory for their help and support in various activities

of the school.

Part I Stellar Structure, Nucleosynthesis and Evolution of Low and Intermediate-mass Stars

Stellar Structure and Evolution: An Introduction

Part II Massive Stars, Core Collapse, Explosive Nucleosynthesis

Weak Interaction Rates for Stellar Evolution, Supernovae

Kamales Kar 183

Massive stars as thermonuclear reactors and their explosions following core collapse

Alak Ray 209

The Evolution of Massive Stars and the Concomitant

Non-explosive and Explosive Nucleosynthesis

Marcel Arnould 277

Part III Cosmochemistry and Solar System Abundances

Cosmochemistry

Bruce Fegley, Jr., Laura Schaefer 347

Solar System Abundances of the Elements

Katharina Lodders 379

Cosmochemistry: A Perspective

Aruna Goswami 419

C Simon Jeffery

Armagh Observatory, College

Hill, Armagh ET61 9DG,

Saha Institute of Nuclear Physics,

Bidhannagar, Kolkata 700064, India

kamales.kar@saha.ac.in

Alak Ray

Tata Institute of Fundamental

Research, Mumbai 400005, India

1169, One Brookings Drive, SaintLouis, MO63130, USA

lodders@wustl.edu

Marcel Arnould

Institut d‘Astronomie et d‘Astrophysique,

Universite‘ Libre de Bruxelles, CP-226, B-1050 Brussels, Belgium

marnould@astro.ulb.ac.be

Bruce Fegley

Planetary chemistry Laboratory

Department of earth and planetary sciences, Washington University

St Louis, MO63130-4899, USA

Planetary chemistry laboratory

Department of earth and planetary sciences and McDonnell centre for thespace sciences

Washington University, Campus box, 1169, One Brookings Drive, Saint Louis,MO63130, USA

lodders@wustl.edu

Tiago Mendes de Almeida

Cidade Universitaria, Sao Paulo-SP-Brazil

Department of Physics, Panjab University, Chandigarh, India

Singh Abhishek Indrajit

University of Mumbai, Kalina Campus , Santa Cruz, Mumbai - 400098, India

Indian Institute of Astrophysics, Bangalore - 560034, India

Vinicius Moris Placco

Cidade Universitaria, Sao Paulo-SP-Brazil

Ananta C Pradhan

Indian Institute of Astrophysics, Bangalore - 560034, India

Yogesh Prasad

Dept of Physics, H.N.B Garhwal University

Srinagar Garhwal- 246174, Uttarakhand, India

Ashish Raj

Physical Research Laboratory, Navrangpura, Ahmedabad - 380009, India

N G Rudraswami

Physical Research Laboratory, Navrangpura, Ahmedabad - 380009, India

Krishna Prasad Sayamanthula

Indian Institute of Astrophysics, Bangalore - 560034, India

Indian Institute of Astrophysics, Bangalore - 560034, India

Bharat Kumar Yerra

Indian Institute of Astrophysics, Bangalore - 560034, India

Stellar Structure, Nucleosynthesis and Evolution of Low and Intermediate-mass Stars

Keywords: Stars: interiors – Stars: evolution – Stars: horizontal-branch –

Stars: AGB and post-AGB – HR and C-M diagrams – Equation of state –Convection – Atomic Processes – Nucleosynthesis

1 Introduction

What are the stars? How do they shine? What are they made of? Thesequestions have challenged mankind ever since he began to explore the worldaround him and appreciate the awesome splendour of the night sky Just aschallenging are questions about what we ourselves are made of, and where wecome from Only in the last hundred years have we started to find answersthat approach a coherent understanding of the universe we inhabit

Fundamental to understanding the stars are measurements of distance andbrightness, colour and constancy Any theory of what stars are and how theybehave must be able to explain these observations Deeper insight is gainedfrom measurements of chemical composition and the relationships betweenstars and the interstellar medium The big story will show how elements aremanufactured by nuclear reactions deep inside the stars – nucleosynthesis –and then transported to the stellar surface and into the interstellar medium

A Goswami and B.E Reddy (eds.),Principles and Perspectives in Cosmochemistry,

Astrophysics and Space Science Proceedings,

DOI 10.1007/978-3-642-10352-0 1, c Springer-Verlag Berlin Heidelberg 2010

Table 1.The Sun

kg

mSurface gravity: g = 2.7397(5) × 102m s−2

Effective temperature: Teff, = 5770(6) K

Surface hydrogen mass fraction: X  = 0.71

Surface helium mass fraction: Y  = 0.265

Surface metal mass fraction: Z  = 0.025

y

The object of these lectures is to explain the physics of stellar interiors,

to use this physics to make stellar models and hence to understand how starswork and evolve The lectures will demonstrate how models of stars are con-structed, and explain how these models predict stars should evolve

This article starts by introducing some of the fundamental observationalmaterial (Sect 2), and by providing an early preview of the stellar evolutiontheory (Sect 3) Fundamental timescales and the equations of stellar structureand evolution are derived in Sects 4–6 The micro-physics (equation of state,opacity and nuclear physics) are discussed in Sects 7–9 Some methods forcalculating approximate solutions and full numerical solutions are presented(Sects 10–11) Subsequent sections deal with the evolution of main-sequencestars (Sect 12), white dwarfs and supernovae (Sect 13), horizontal-branchstars (Sect 14) and hydrogen-deficient stars (Sect 15)

The text is based on a series of six lectures given at the 2008 Kodai School

on Synthesis of the Elements in Stars1 and on a more extended course given

in the University of St Andrews and Trinity College, Dublin over a period

of some twenty years At Kodaikanal, the core material comprised four tures Two more advanced lectures covered horizontal-branch stars (Sect 14)and hydrogen-deficient stars (Sect 15) Development of the core material wasoriginally drawn from several seminal texts [1, 2, 3, 4, 5, 6, 7, 8, 9] Section 15and parts of Sect 14 are based on [10, 11]

lec-Variables

In considering stellar structure, we will meet a number of quantities The mostimportant are:

• stellar mass M : often given in solar units M ,

• stellar radius R: often given in solar units R ,

• stellar luminosity L: often given in solar units L ,

1

Kodaikanal Observatory, Indian Institute of Astrophysics, 2008, April 29 - May13

Table 2.Physical Constants: CODATA 2006speed of light in vacuum c = 2.997 92458 × 108 m s −1

gravitational constant G = 6.674 28(67) × 10 −11 m3

kg−1s−2Planck constant h = 6.626 068 96(33) × 10 −34 J s

• surface gravity g: a force measured in m s −2,

• effective temperature Teff: usually given in degrees Kelvin

• mass-fractions of hydrogen X, helium Y and other elements Z,

where G is the gravitational constant The conservation equation for chemical

composition can be simply written

In situations where the abundances of individual elements or nuclides are

important (e.g nuclear reaction rates), relative abundances by mass will be given as x i , where i is either the atomic number, or denotes the nuclide in some

other distinct way Values for the Sun are given in Table 1; errors on the lastdigits are shown in parentheses Table 2 provides constants used throughoutthe text and enables many equations to be evaluated

2 The Hertzsprung–Russell Diagram

The most important correlations amongst stellar properties are contained in

a type of diagram developed independently by Ejnar Hertzsprung and Henry

Fig 1. Hertzsprung-Russell diagram: a plot of luminosity (absolute magnitude)against the colours of the brightest stars ranging from the high-temperature blue-white stars on the left side of the diagram to the low temperature red stars onthe right side Original image by Richard Powell licensed for derivative works andredistribution under the Creative Commons Attribution ShareAlike 2.5 License

Russell [12, 13, 14] The original Hertzsprung–Russell (HR) diagram showed

the distribution of spectral type and absolute magnitude (or brightness) for

stars with known distances (Fig 1) The latter is required to convert an

ap-parent brightness (e.g mV) to an absolute magnitude (MV) The diagram

demonstrates that stars do not appear with any combination of spectral type and brightness, but fall on well-defined sequences, e.g the main sequence, the

giant branch, the white dwarfs, and so on

A more convenient form is the colour – magnitude diagram, in which either

an apparent or absolute magnitude is plotted against a photometric colourindex, being the ratio of brightness at one wavelength to that at another.Such a diagram is particularly useful for comparing the properties of stars in

a cluster, which may be assumed to lie at approximately the same distance.The additional supposition that all stars in a cluster are of the same age has

Fig 2. Colour-magnitude diagram for the young galactic open clusters NGC 869

and NGC 884 (=h and χ Persei; based on [15])

important consequences for understanding stellar evolution – although thesupposition may not always be correct!

Wien’s displacement law states that there is a relation between the perature of a black body and the wavelength at which the maximum energy

tem-is emitted: λmax = b/T From this it was recognised that there should be a

connection between the colour of a star and its effective (or surface) ture With a theory of stellar atmospheres, the relationship between spectraltype, colour index and effective temperature became concrete In addition, acorrection to account for light emitted at unobserved wavelengths allows theapparent visual magnitude to be converted to a bolometric magnitude, and

tempera-hence to luminosity The use of an effective temperature – luminosity diagram

is common in theoretical work (cf Fig 8) It is important to recognise and

understand the connections and differences between all three types of diagram

2.1 Cluster Diagrams

Figure 1 contains stars of widely varying mass, age and composition, and canonly be constructed for stars whose distances can be measured directly In thecase of Galactic and globular clusters, we assume that all stars formed at ap-proximately the same time, from a gas-cloud of roughly uniform composition.Because the cluster members are at the same distance, their relative bright-nesses provide an HR diagram in which only the zero-point of the luminosityaxis is unknown

Galactic or open clusters are normally associated with the hydrogen-rich

disk of the Galaxy, hence are young with metallicities (Z ≈ 0.01−0.03) similar

to the solar value Their colour-magnitude diagrams (cf Figs 2 – 3) typically

show:

• Most stars on the main-sequence (MS)

• A turn-off (TO) point somewhere between F and O stars (depends oncluster age, Fig 3)

• A Hertzsprung gap between stars leaving the MS and Giants (Fig 2)

• A relatively flat Giant Branch (GB) (Fig 3)

Globular clusters are associated with the gas-poor halo of the Galaxy, and

are very old objects with generally low metallicities (Z ≈ 0.0001−0.01) Their

colour-magnitude diagrams (cf Figs 4 – 5) typically show:

• Only late-type stars on the MS

• A turn-off-point around spectral-type G (depends on age and metallicity)

• A continuous sequence from the TO through subgiants up to a steep GB

• A horizontal branch (HB), red in some clusters, blue in others (usuallydepends on metallicity)

• An asymptotic giant branch (AGB) sequence above the HB and parallel

to the GB

Fig 3.Colour-magnitude diagram for the old galactic open cluster M67 (based on[16])

Fig 4.Colour-magnitude diagram for the metal-rich galactic globular cluster 47 Tuc

(Z = 0.004, [Fe/H] = −0.76; based on [17])

Fig 5.Colour-magnitude diagram for the moderately metal-poor galactic globular

cluster ω Cen (Z = 0.0006, [Fe/H] = −1.62; based on [18])

Fig 6. The mass-radius relation for both components in each of fifty eclipsingbinary stars The primaries and secondaries are shown as squares and circles (based

on [19])

Fig 7. The mass-luminosity relation for the same stars as in Fig 6

2.2 The Temperature–Luminosity Relation

The most obvious empirical relation between stellar properties is provided bythe main-sequence in the HR diagram, where there exists a direct correlationbetween stellar luminosity and effective temperature of the form

L ∝ T α

where on average α ≈ 6, although the value is higher at both the upper and

lower ends of the main sequence

2.3 The Mass–Luminosity and Mass-Radius Relations

Another important empirical relation between stellar properties is provided

by eclipsing binary stars, for which direct measurements of mass, radius and

luminosity can often be obtained for both components Plotting log R and log L against log M for main-sequence stars (Figs 6 and 7), we find straight lines which mean that L follows a relation

where on average β ≈ 3.8 Our theory of stellar structure must reproduce

these results

3 Stellar Evolution – A Sneak Preview

We have seen that the luminosity of a star depends on its mass Since the

luminosity L determines the rate at which a star uses up its available fuel, and L goes as M β , β > 1, it is evident that stars of different mass effectively

age at different rates The cluster colour-magnitude diagrams are essentiallysnapshots of stellar evolution in which all the stars have the same age, butcover a wide range of mass

We cannot observe the passage of an individual star through the magnitude diagram, but we can attempt to model its evolution from birth todeath A simplified picture of the evolution of two stars is shown in Fig 8.This diagram provides a convenient way to outline the basic features of stel-lar evolution These are common to a majority of stars, with the greatestdifferences occurring at the extremes of mass

colour-Star Formation

At the current time in our Galaxy, it appears that star formation takesplace in massive interstellar clouds, with dimensions ≈ 10 pc, density ≈ 5 ×

109atoms m−3, and temperature ≈ 10 K The Galaxy is pervaded by a

netic field aligned approximately parallel to the galactic plane This netic field is strongly tied to the ionized plasma in the interstellar medium

mag-Fig 8.Schematic L −Teff diagram showing the evolution of 1 and 5 Mmass stars.(Based on a figure by J Lattanzio Evolution tracks computed by R G Izzard.)

Random turbulent process or other small perturbations in the field lead tolocal potential wells where the ISM can condense, pulling the magnetic fieldwith it, and leading to a Rayleigh-Taylor (or runaway) instability It is thoughtthat this provides the initial mechanism for the formation of dark clouds ofinterstellar matter

A number of processes including refrigeration (to keep the clouds cool) andaccretion disks (to dispose of angular momentum) enable these dark clouds

to collapse under their own gravity until they are sufficiently dense and hotfor nuclear reactions to begin At this point, we consider a star to have beenborn

The Main Sequence

Stars spend over 90% of their lives on the main sequence Hydrogen is verted to helium in the stellar core (Fig 9) Since four protons are slightlyheavier than one helium nucleus, the excess mass is converted into energywhich keeps the core hot, maintains internal pressure and allows the star toshine During main-sequence burning, the star increases its radius slightly asthe hydrogen content in the core drops More detail will be given in Sect 12

con-The Giant Branch

As hydrogen in the core is depleted, nuclear reactions switch off, and the corecontracts Hydrogen-rich material outside the core is compressed and heated

H > He

He

H > He H

H He C+O

Fig 9. A simplified view of the internal chemical structure of a star during themajor phases of stellar evolution The panels represent the main-sequence, throughthe giant-branch, helium-burning, asymptotic-giant branch and white dwarf phases.Filled circles and thick lines represent nuclear-burning regions Not to scale

by the contraction, so hydrogen-burning shifts to a shell In low-mass stars

(0.5 − 3M ), the outer layers of the star expand and the star becomes a red

giant During this phase of evolution, the hydrogen-burning shell adds helium

to the core beneath it, which consequently contracts and heats At the sametime, the shell becomes hotter, thinner and more luminous The hydrogen-richouter layers (the envelope) expand, and the entire star evolves upwards alongthe giant branch

Low-mass Helium Burning

Once the helium core reaches a critical mass, nuclear burning of helium begins,producing carbon and oxygen in a relatively long-lived phase of evolution.The energy derived from helium-burning heats the helium core and forces it

to expand Thus the hydrogen-burning shell may actually get weaker at this

point, the overall luminosity drops and the star contracts At this stage, the

star may be either blue or red, corresponding to the horizontal-branches inFigs 4 and 5 We shall learn more about these stars in Sect 14

Intermediate-mass Helium Burning

In stars heavier than about 2.3M  and with Z ≈ 0.02, helium burning

reac-tions start before the core becomes compact and before the hydrogen-burningshell gets very thin Core expansion following helium ignition is therefore mildand the total luminosity does not drop by much However the radius does be-come smaller, producing a blue-loop in the Hertzsprung-Russell diagram forthe duration of core-helium burning

Asymptotic Giant-Branch (AGB) Stars

With core-helium exhaustion, the focus of nuclear-burning in low- and mediate-mass stars shifts to a double-shell structure Because the He-burningshell is thermally unstable, a process known as thermal-pulsing is established.The helium-burning shell burns faster, and so keeps going out When sufficientfresh helium has accumulated below the hydrogen-burning, the helium-shellreignites, the intershell region is forced to expand, and hydrogen-burning isextinguished The interaction between these thermal pulses and convection indifferent layers of the star makes this one of the most important processes incosmology; more detail is presented by Karakas (these proceedings)

inter-Of consequence here is that when the hydrogen-burning shell is operating,

it is very thin and very powerful, so the star becomes very luminous and verylarge The hydrogen envelope is thus consumed from below (by the shell) andblown away from the surface by a strong stellar wind

Planetary-Nebula Formation and White-Dwarf Cooling

Ultimately, the expansion of the AGB star results in a dynamical instabilitywhich causes the expulsion of the outer layers as a planetary nebula Unable

to sustain nuclear reactions, whatever is left of the envelope starts to contract.When it has cooled sufficiently the star becomes a white dwarf: Sect 13 Wewill discuss whether the white dwarf stage is the end of the story in Sect 15

Massive Stars

Following main-sequence burning, the ultimate evolution of stars more massivethan ≈ 8M  is quite different Initial evolution is similar, the star expands

rapidly and starts to cross the HR diagram towards the red giant region.

For M ≥ 20 M , core helium ignition occurs at a progressively earlier stage

(i.e closer to the main-sequence) as the mass increases Core-helium burning

arrests the redward evolution until core helium is exhausted, and redward lution resumes As the inactive carbon-oxygen core contracts, it is sufficientlymassive that its temperature increases to the point where carbon-carbon re-actions can occur There follows a rapid sequence of nuclear-burning episodesproducing elements with increasing atomic weight Hydrostatic equilibrium

evo-is maintained until the core consevo-ists primarily of iron (56Fe), at which pointfurther nuclear reactions require more energy than they release The stellarcore collapses and a supernova explosion follows This explosion is responsiblefor the production of quantities of heavy nuclei quite distinct from those cre-ated in AGB stars These processes are discussed in greater detail by Arnould(these proceedings)

Binary Stars

This very simple description applies to those stars which evolve as single stars

or as members of a wide binary system which do not interact It is increasinglyclear that a large fraction of stars are born in binary or multiple systems inwhich two stars exchange material at some point during their evolution Thepossibilities of what can happen thereafter are too numerous to be able tocover here, but some of the more bizarre possibilities will be considered later

4 Stellar Time Scales

We want now to put some physical argument behind our cartoon picture ofstellar evolution It is evident that stars such as the Sun do not change theirproperties rapidly It is important to understand and appreciate the relativemagnitudes of the timescales on which a star can change

4.1 Dynamical (Free-Fall) Time

Consider a star as a ball of gas which is held up by the pressure forces within

it How long would the star survive if the pressure forces were removed? The

answer is known as the free-fall or dynamical time (tff or tdyn) A simple timate is given by the time required for a body to fall through a distance of

es-the order R (es-the stellar radius) under es-the influence of a (constant) tional acceleration equivalent to the surface gravity g = GM/R2 of a star of

gravita-mass M From Newton’s 2nd law,

R = 1

2gt

2

ff = 12

GM

R2 t2ff

⇒ t ≈ 2.2 × 103(R3/M ) 1/2s

where R and M are in solar units Writing the mean density as ρ = (3/4π)M/R3, we also have:

tff is also the characteristic time for a significant departure from hydrostaticequilibrium to alter the state of a star appreciably, the time taken for a bodyorbiting at the surface of the star to make one complete revolution, and thetime for a sound wave to propagate through the star

4.2 Thermal (Kelvin) Time

Next, consider a star as a ball of hot gas which acts as reservoir of heatenergy How long would it take for this energy to radiate away if it were

not replenished? This is the thermal or Kelvin-Helmholtz timescale (tK or

tth) If the total kinetic (thermal) energy of the star is Ekin, the timescale isapproximately

GM2

LR ≈ 3 × 107qM2

where M , L and R are in solar units.

The thermal time is the relaxation time for departure of a star from mal equilibrium It is also the time that would be required for a star to con-

ther-tract from infinite dispersion to its present radius if L were to remain constant

during its entire contraction

2

q = 3/5 for a sphere of uniform density, and becomes smaller with increasing

central condensation

energy Q, where Q ≈ 26 MeV The total available energy is thus

Enuc= qM/4mpQ,

where again q is a dimensionless factor of order unity representing the fraction

of the star available as nuclear fuel The nuclear time is simply the time taken

to radiate this energy

stellar material by a series of scattering collisions, mainly with electrons Sincescattering is an isotropic process, energy transport is most correctly described

by the diffusion equation Section 5 will examine this further, but essentially

the photon-path can be described by a random-walk consisting of N steps, each of length λ Whilst the total distance travelled by the photon is N λ, it may be shown that the nett distance travelled is r = √

N λ, because of the

scatterings Thus to escape from the star, the photon must travel a distance

R, which it will do in a time

with R in solar units.

Table 3.Comparative timescales

Star M R L Teff tesc tff tdiff tkin tnuc

5 Equations of Stellar Structure

The simplest picture of a star is that of an isolated body of gas sufficiently sive that the only significant forces are self-gravity and internal pressure In thesimplest case, we can assume spherical symmetry and neglect the influence ofrotation, magnetic fields and external gravitational influences (in fact, underthese assumptions, equilibrium solutions will enforce spherical symmetry)

0,0,Pc,Tc

X,Y,X

Fig 11.The variables of stellar structure

Consider a spherical system containing a fluid of mass M and radius R

(Fig 11) The internal structure of this system is described by the distribution

of various quantities, including

• r: radius

• m(r): mass of fluid contained within a spherical shell of radius r

• l(r): luminosity or total energy passing through a shell of radius r

• T (r): temperature of the fluid at radius r

• P (r): pressure in the fluid at radius r

• ρ(r): fluid density at radius r

These quantities are expressed as a function of radius r, running from r = 0

at the stellar centre to r = R at the stellar surface However, we could have equally chosen a different independent variable, for example mass, m In order

to keep the notation simple, stellar structure variables will usually be written

in short form, i.e m ≡ m(r), l, P, T, ρ The problem of stellar structure is

then to determine the run of these quantities throughout the star, for which

we require a system of equations

ρ δr

r

Fig 12. Derivation of the density equation

5.1 Mass Continuity

Consider a spherical shell of radius r, thickness δr (δr  r) and density ρ

(Fig 12) It has a mass (volume× density)

g

Fig 13. Gravitational and pressure forces in hydrostatic equilibrium

5.2 Hydrostatic Equilibrium

We must next consider the forces acting at any position within this sphere

(Fig 13) Consider an element of material at radius r The sphere of radius

r acts as a gravitational mass situated at the centre giving rise to an inwardgravitational force

If there is a pressure gradient through the sphere, there will be an additional

force If the element has a thickness δr and cross-section δA, then a net force

arises if the pressure on the inner and outer surfaces are unequal, so that the

inward force is, since δP = (dP/dr)δr,

But the mass of the volume element is δm = ρδrδA, then the sum of inward

forces due to gravity and pressure will be

F = δm



g +1ρ

dP dr



=−δm d2r

For the element considered in Fig 13 to be in equilibrium, the parenthesis in

Eq (16) must be zero, and by substitution from Eq (15),

5.3 Virial Theorem

A very important result can be obtained when an entire self-gravitating

sys-tem is in equilibrium, i.e if Eq (17) is satisfied at all r, since it becomes

possible to derive a simple relation between average internal pressure and thegravitational potential energy of the system This is done by multiplying both

sides of Eq (17) by 4πr3 and integrating from r = 0 to r = R to obtain

 M Gm

r dm.

Since P (R) = 0, the first term is zero Substituting 4πr2dr = dv

is the volume of the system, and the rhs is simply the gravitational potential

energy of the system Egrav Thus the average pressure needed to support a

system with gravitational energy Egrav and volume V is given by

13

Egrav

The statement that the average pressure needed to support a self-gravitatingsystem is one third of the stored gravitational energy is called the VirialTheorem The physical meaning of pressure depends on the system itself, but

it can be applied to clusters of galaxies as well as to individual stars Let usconsider two cases for gas in a star, where the equation of state relates the

pressure of the gas (P ) to the translational kinetic energy of the gas particles (Ekin)

Etot= Ekin+ Egrav=−Ekin,

and

These equations are of fundamental importance Note the following If a tem is in hydrostatic equilibrium and it is tightly bound, the gas particles

sys-have high kinetic energy They are hot If the system evolves slowly, close

to hydrostatic equilibrium, then changes in kinetic and gravitational energiesare simply related to changes in the total energy We shall see some of theconsequences later in considering the collapse of gas clouds

Thus hydrostatic equilibrium is possible only if the total energy is zero As

the ultra-relativistic limit is approached, e.g as the gas temperature increases,

the binding energy decreases and the system is easily disrupted This type ofinstability occurs in stars where the radiation pressure is very high (pressure

comes from photons which are, by definition, relativistic) e.g supermassive

stars, or where pressure comes from degenerate electrons and the electron

temperature become very energetic, e.g massive white dwarfs Again, we shall

examine these later

Jean’s criterion for gas-cloud collapse

An important corollary to the Virial theorem is connected with the conditionfor gravitational collapse Stars are believed to form when large clouds ofgas and dust contract However, this contraction can only take place if theinternal pressure of the cloud due to the thermal motion of the cloud particles

is weaker than the gravitational forces compressing the cloud Formally, a

cloud is bound if the total energy is less than zero Knowing the gravitational binding energy (Eq (8)) and adopting q = 1, and setting the thermal energy

per particle to 32kT , the total energy of the cloud is:

3

where the critical values are known as the Jeans mass, temperature anddensity

Equation (24) illustrates an important consequence of cloud contraction; if

the system is to remain bound, i.e if it is to continue to contract and become

a star, then the internal temperature of the collapsing cloud or protostar,

Fig 14. Derivation of the energy equation

5.4 Energy Conservation

Since energy moves through the star, the consequences for stellar structureimposed by the law of conservation of energy must be considered, whether thatenergy comes from the release of nuclear or gravitational energy Consider a

volume element dv = 4πr2dr at radius r (Fig 14).

The law of conservation of energy states that the total amount of energy

leaving a volume element dv must equal the total amount of energy entering

that element plus the energy lost or produced within that element

Assuming spherical symmetry, it is convenient to consider a shell of mass

δm Let l be the amount of energy entering the bottom of shell, and l + δl be

the amount of energy leaving the top of the shell Let us denote the energy

absorbed or produced within the shell, per unit mass, by ε Then we can

equate energy out to energy in plus energy produced or lost:

The fact that there is a temperature difference between the interior and surface

of the Sun implies that there must be a temperature gradient and hence aflux of energy Conservation of energy constrains the energy flux, but the wayenergy is transported establishes the temperature gradient There are threemajor transport mechanisms:

• radiation: generally by photons, but also by neutrinos

l

Equation (29) represents the temperature gradient within the star whenthe star is in radiative equilibrium Combining with Eq (17) (hydrostaticequilibrium) and taking logs, we define a gradient

An element of gas is at some radius r Consider its upward displacement by

a distance dr, allowing it to expand adiabatically until the pressure within

is equal to the pressure outside (Fig 15) Then release the element If itcontinues to move upwards, the layer in question is convectively unstable andconvective motions will persist

Let the pressure and density within and without the sphere be denoted by

P ∗ , ρ ∗ , and P , ρ respectively Then initially

2> ρ2so that the net force (bouyancy+gravitation)

is downwards and the element will return to its starting position Eliminating

asterisks and writing P1= P2+ dP , we obtain

P ρ

dρ

dP >

1

for radiative equilibrium This condition is related to the temperature gradient

assuming some equation of state (e.g P = ρkT /μm) so that Eq (31) becomes

1 In the centre of main-sequence stars, the radiation flux l/4πr2 can

be-come very large, whilst κρ remains small Thus the temperature ent d ln T /d ln P required for radiative equilibrium (Eq (30)) becomes

gradi-large, and the material becomes convectively unstable This gives rise tonuclear-driven convective cores in massive stars, and also convective zones

Equations (30) and (32) give the temperature gradient ∇ in radiative and

convective equilibrium respectively Where there are situations when it is clearwhich equation to use, the situation frequently arises when material may be

naturally convective, but the convective efficiency ξ is sufficiently low that radiation carries a substantial fraction of the flux In these cases, ξ must be

derived from a suitable theory of convection

With this in mind, it is useful to rewrite the equation for the temperaturegradient

• ξ = 1 : fully adiabatic convection

5.6 The Equations of Stellar Structure

We have now derived the four basic (time-independent) equations of lar structure These are mass continuity (Eq (14)), hydrostatic equilibrium(Eq (17)), conservation of energy (Eq (28)), and energy transport (Eq (33)).These form a set of coupled first order ordinary differential equations relating

stel-one independent variable, e.g r, to four dependent variables i.e., m, l, P, T ,

which uniquely describe the structure of the star, note that any variable could

be used as the independent variable In an Eulerian frame, the spatial

coordi-nate r is the independent variable For most problems in stellar structure and

evolution it is usually more convenient to work in a Lagrangian frame, withmass as the independent variable Transforming, we obtain:

dr

1

4πr2ρ , dl

dm = ε,