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The Sun as a Star
Internal Structure and
Evolution
Simon Jeffery, Armagh Observatory
Monday, 17 September 12
Outline
• Properties of Stars
• Equations of Stellar Structure
• Plasma Physics
• Solar Models
• Equations of Stellar Evolution
• Future of the Sun
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1. Properties of Stars
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Monday, 17 September 12
ω Cen - Kitt Peak
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ω Cen - HST
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The Hertzsprung-Russell diagram
Spectral type
<> colour
<> effective temperature
Magnitude
<> luminosity
White Dwarfs
<> Dwarfs
<> Subgiants
<> Giants
<> Supergiants
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Spectral Type
O
50 kK
B
20 kK
Hγ
Hβ
Hα
A
10 kK
F
7500K
G
6000K
K
4000K
M
3500K
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Image Credit: N.A.Sharp, NOAO/NSO/Kitt Peak FTS/AURA/NSF
Monday, 17 September 12
Stellar Properties
The Sun
M
stellar mass (M / M)
M = 1 M = 1.99 1030 kg
R
stellar radius (R / R)
L
stellar luminosity (L / L)
R = 1 R = 6.96 108 m
L = 1 L = 3.86 1026 W
Teff
effective temperature (K) = ∜L/4πσR2
g
surface gravity = GM/R2
X,Y,Z mass fractions of H, He and other
elements
t
age
Monday, 17 September 12
Teff = 5780 K
g = 274 m s-2
X = 0.71
Y = 0.265
Z = 0.025
t ~ 4.6 109 y
Big Questions
What is the Sun made of?
What happens inside the Sun?
Why does the Sun shine?
What holds the Sun up?
How did the Sun form?
How long will the Sun last?
How will the Sun end?
Can we “see” inside the Sun?
Where did the Sun’s constituents come from?
What will happen when the Sun runs out of fuel?
What governs the size and brightness of the Sun?
...
Ask the same questions for any star ....
but the answers are NOT the same for every star!
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2. Equations of Stellar Structure
Objective
Assumptions
Conservation Laws
Mass Continuity
Energy Conservation
Total Energy of a Star
Hydrostatic Equilibrium
Virial Theorem
Energy Transport
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a stellar interior
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definitions
Surface: r=R
Structure:
r
radius
m(r)
mass within r
l(r)
flux through r
T(r)
temperature at r
P(r)
pressure at r
[ ρ(r)
r
M,L,0,Teff
density at r ]
Centre: r=0
0,0,Pc,Tc
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X,Y,Z
m,l,P,T
assumptions / approximations
Isolated body:
Thermodynamic equilibrium
Conservation Laws:
Mass
Energy
Angular Momentum
Forces:
self-gravity
internal pressure
Approximations:
spherical symmetry (1D) **
no rotation **
no magnetic fields **
no chemical diffusion **
** Many models now take some or all
of these into account, but in
setting up the basic equaitons,
and for comparing with other
stars, it is more practical to make
these approximations.
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conservation of mass
(or the continuity equation)
Consider a spherical shell of radius r
thickness δr (δr <<r)
density ρ
Its mass (volume x density):
δm = 4 π r2 ρ δr
As δr→0:
dm/dr = 4 π r2 ρ
Also:
m= 4 π ∫ r2 ρ dr
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δr
r
ρ
conservation of energy
Consider a spherical volume element
dv=4 π r2 dr
Conservation of energy demands:
energy out = energy in + energy
produced or lost
If ε is the energy produced/lost per
unit mass, then
l+δl = l + ε δm
⇒ dl/dm = ε
Since dm = 4πr2 ρ dr
dl/dr = 4πr2 ρ ε
We will consider the nature of energy
sources, ε, later.
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l+δl
δr
l
r
εδm
hydrostatic equilibrium
Consider forces at a radius r
acting on an element of thickness δr, area
δA and mass δm = ρ δr δA
A sphere of mass m acts as a gravitational
mass situated at the centre, giving rise to
an inward force:
δm g = δm Gm/r2
P+δP
g
ρ
zP
δr
If a pressure gradient (dP/dr) exists, there
will be a nett inward force
δr δA dP/dr ≡ δm / ρ dP/dr
The sum of inward forces is then
δm ( g +1/ ρ dP/dr ) = - δm d2r/dt2
r
In order to oppose gravity, pressure must
increase towards the centre.
For hydrostatic equilibrium, forces must
balance:
dP/dr = - Gm ρ / r2
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energy transport
Stars shine. What does this tell you about them ?
A temperature difference between the centre and
surface of a star implies a temperature gradient, and
hence a “flux” of energy.
How can energy be transported ?
What methods are important in stars ?
How does the choice of energy transport affect the
temperature gradient – or vice versa ?
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radiative equilibrium
If transported by radiation (photons), then the flux
density F obeys Fick’s law of diffusion:
F = -D ∇Urad
Urad =aT4 is the radiation energy density.
For radiative transfer, identify the diffusion coefficient
D = c/3κρ.
In spherical symmetry ∇ = d/dr. Hence
F = - c/3κρ d(aT4)/dr
Multiply by 4πr2 to obtain a total flux, whence
l= 4πr2 (-c / 3κρ) d(aT4)/dr
⇒ dT/dr = - 3 κ ρ/4acT3 l/4πr2
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convective equilibrium
An element of gas has pressure P and
density ρ.
Displace upward by distance δr.
Allow to expand adiabatically
until pressure within is equal to pressure
outside.
Release the element.
If it continues to move upwards, the layer
is convectively unstable.
If it remains stationary or sinks, the layer
is stable against convection.
It may be shown that the condition for convective equilibrium is related to
the temperature gradient assuming some equation of state (eg P =
ρkT/µmH) so that:
d ln T
⌘ r < rad ⌘
d ln P
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1
3. Plasma Physics
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Stellar Structure Equations
We have derived four time-independent first-order
differential equations in four variables, with one
independent variable r. They can also be rewritten in
terms of any of the other variables; m is useful.
dm
dr
dl
dr
dP
dr
dT
dr
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2
= 4⇡r ⇢
2
= 4⇡r ⇢"
=
=
Gm⇢
r2
Gm⇢ T
r
2
r P
dr
dm
dl
dm
dP
dm
dT
dm
1
=
4⇡r2 ⇢
="
=
=
Gm
4⇡r4
Gm T
r
4
4⇡r P
Boundary Conditions and Constitutive Relations
Four first order differential equations require four
boundary conditions:
Centre: m(r=0) = 0, l(r=0) = 0
Surface: P(r=R) = 0, T(r=R) = Teff
To close the system, we need expressions for:
ρ
ε
κ
∇
density
nuclear energy generation
opacity
energy transport
Plasma
Physics
In practice, these are all functions of P, T, and Xi.
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Equation of State
ρ = n/N0µ
P=1/3E
PR=aT4/3
8
log T/K
P=nkT
6
P=KNRn5/3
4
P=KURn4/3
P=2/3E
-4
0
4
log ρ/kg m-3
8
Stellar Structure: TCD 2009: 7.
Monday, 17 September 12
12
photon-electron interactions
bound-bound
bound-free
free-free
hν = ½m(v22-v12)
½mv2
χion
n=3
hν = χ3-χ2
n=2
Excitation
energy
hν = χ2
hν = χion+½mv2
n=1
Stellar Structure: TCD PYA301: 8.
Monday, 17 September 12
Opacity
In regions where
specific atoms are
ionized, opacity - due to
bound-free transitions has a local maximum.
H+He
H+He
He+
He+
Fe,…
C,N,O
κ= κ(ρ,T,Xi)
e-s
Stellar Structure: TCD PYA301: 8.
Monday, 17 September 12
Proton
Rest mass: mp = 1.672 x 10-27 kg = 1.00728 u
E = mpc2 = 1.5027 x 10-10 J
+
+
The
electrostatic
force
like charges
repel!!
Stellar Structure: TCD PY3A03: 9.
Monday, 17 September 12
Atomic Mass Unit
1u = m(12C)/12
Nuclear Reaction Rates
The cross-section <σv> for a fusion reaction is represented by the
product of the particle energy distribution and the tunnelling
probability
∝exp(-E/kT)
∝exp(-(EG/E)1/2)
E0
∝exp(-E/kT - (EG/E)1/2)
Δ
Stellar Structure: TCD PY3A03: 9.
Monday, 17 September 12
Proton
rest mass: mp = 1.00728 u
E=mpc2 = 1.5027 x 10-10 J
+
+-
νe
d
+
Deuterium nucleus
rest mass: md = 2.01355 u
Eexcess = δmc2 = (2mp - md)c2 = 1.502 x 10-13 J
Stellar Structure: TCD PY3A03: 9.
Monday, 17 September 12
Proton
rest mass: mp = 1.00728 u
E=mpc2 = 1.5027 x 10-10 J
+
+
νe
d
+
3He
+
Helium-3 nucleus
rest mass: m3He = 3.01603 u
Eexcess = δmc2 = (3mp - m3He )c2 = 8.697 x 10-13 J
Stellar Structure: TCD PY3A03: 9.
Monday, 17 September 12
Proton
Helium-4 nucleus
rest mass: mp = 1.00728 u
rest mass: m4He = 4.00260 u
E=mmpc2 = 1.5027 x 10-10 J
Eexcess = δmc2 = (4mp - m4He )c2
+
+
1.4 x 1010 y
d
6s
+
+
3He
106 y
3He
+
Stellar Structure: TCD PY3A03: 9.
Monday, 17 September 12
J
4He
νe
d
+
-12
+
+
+
= 3.956 x 10
=24.7MeV
νe
+
Nuclear energy production
Stellar Structure: TCD PY3A03: 9.
Monday, 17 September 12
Convection
MHD model for opacity-driven convection zone in the Sun showing superganulation -- click above !!!
http://www.pa.msu.edu/~steinr/research.html#convection
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4. Solar Models
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Solutions - 3 families
Approximate solutions
Polytropes
Cowling point source model
** Clayton model
1D Numerical Solutions
Henyey (difference) schemes (Lagrangian)
3D Numerical Solutions
Stellar Structure: TCD PY3A03: 10.
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The Clayton model
Guess a form for the Pressure profile P(r)
Use this as starting point for an approximate solution.
Outline:
Tackle equations sequentially, starting with mass
continuity and hydrostatic equilibrium to obtain the
pressure structure.
Use an equation of state to obtain the temperature
structure.
Introduce an opacity κ(ρ,T) to obtain the power flow from
the transport equation .
Compare this with the power flow obtained from the
nuclear power generation ε(ρ,T) and energy
conservation.
Stellar Structure: TCD PY3A03: 10.
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Clayton model for 1 solar mass star
Stellar Structure: TCD PY3A03: 10.
Monday, 17 September 12
Standard solar model
Spherically symmetric
quasi-static model
Constrained by mass,
luminosity, radius, age
and composition...
Necessary to tune
physics (eg mixing
length for convection,
diffusion of 3He, ...) in
order to match
constraints
Model can be tested...
Sun is only star for
which internal
properties can be
measured directly...
Stellar Structure: TCD PY3A03:
Monday, 17 September 12
The Solar Interior
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pulsation across the HR diagram
•
~ hours
• main-sequence
stars:
β Cep, PG1716,
δ Sct
• horizontal-branch
stars:
RR Lyr, PG1716
•
~ minutes
• main-sequence:
ro-Ap, Sun-like
• hot subdwarfs:
EC14026, PG1159
• white dwarfs:
DAV, DBV, CVs
•
< 0.1s
• neutron stars ?
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the sun
tens of millions of modes
periods ~ 2 - 20 min
colour shows energy at each
frequency for every mode
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add rotation, magnetic fields, 3d convection, ...
• differential rotation, mapped by helioseismology
• tachocline - shear layer at radiative / convective interface
Gough & McIntyre 1998, Garaud & Garaud 2008
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Measuring
neutrinos
Helioseismology
says solar models
are right.
Stellar Structure: TCD PY3A03:
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Solar Neutrinos
Solution (particle physics):
all neutrinos generated in the Sun are
electron neutrinos
if neutrinos have small mass, they may
change type on path from Sun to Earth
from electron to µ or τ neutrinos
vacuum oscillations: ✘
matter oscillations: ✔
Above: The Sun in neutrinos from Super K2
Stellar Structure: TCD PY3A03:
Right: Sudbury Neutrino Observatory
Monday, 17 September 12
Other Main-Sequence Stars
How do overall properties vary with mass?
How does internal structure vary with mass?
What happens as hydrogen is converted to helium?
What do observations tell us?
Stellar Structure: TCD PY3A03:
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F - K dwarfs: 1.8 - 0.7 M
About 13% of stars in Galaxy
Convective envelopes and radiative
cores
H-burning mostly by PP-chains
Convective envelopes associated with
starspots, magnetic fields, and other activity
About one third of F - K stars are predicted to have at least one planet
Subdwarf F - K stars (sdF, ...) are metal-poor
or Population II main-sequence stars.
They lie below the main-sequence
because the lower opacity makes the
temperature gradient steeper, and hence
reduces their radii.
Well-known F - K stars:
The Sun
Alpha Centauri
Stellar Structure: TCD PY3A03:
Monday, 17 September 12
M dwarfs: 0.7 - 0.1 M
M dwarfs make up some 80% of stars
in the Galaxy
They are completely convective
As a consequence, M dwarfs have strong magnetic fields, which drives a lot of
activity, including giant spots and giant flares and bright X-ray coronae.
Hydrogen-burning is by the PP chains;
Brown Dwarfs (L, T, ...) are found below
H-burning limit, so are not real stars;
limited deuterium burning operates.
Stellar Structure: TCD PY3A03:
Monday, 17 September 12
B - A stars: 20 - 1.8 M
A and B stars make up less than 1%
of the stars in the Galaxy
Exercise: which emits more light in total:
B stars or G stars ?
A and B stars have convective cores and radiative envelopes.
Hydrogen-burning is by the CNO-cycle.
They do not have active atmospheres (no spots, flares, etc.)
Some have strong surface magnetic fields
Some important sub-types:
Ap stars: chemically peculiar atmospheres
produced by radiatively driven diffusion
Be stars: B stars show hydrogen-emission
lines, from a shell or disk
HgMn stars: B stars with strong lines of
mercury and manganese (see Ap stars)
Well-known B and A stars:
Sirius A
Pleiades
Stellar Structure: TCD PY3A03:
Monday, 17 September 12
O stars: >20 M
The brightest main-sequence stars.
O stars have convective cores and
radiative envelopes.
Hydrogen-burning is by the CNO-cycle.
O stars are always observed close to sites of current or recent star formation.
Because of their high luminosities, O stars have very strong stellar winds,
observed as asymmetric emission/absorption lines. Material is driven off the
star at velocities of several hundred km s-1, with mass-loss rates of 10-7 10-8 M yr-1.
With lifetimes of ~107 yr, an O star can lose most
of its mass during its main-sequence lifetime.
Well-known O stars:
P Cygni
Trapezium cluster in Orion
Stellar Structure: TCD PY3A03:
Monday, 17 September 12
5. Equations of Stellar Evolution
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time
So far, none of our equations have used time: t.
An equilibrium solution of the time-independent
equations is simply a model of stellar structure,
not evolution.
Why does a star change during its evolution ?
Change in size (expansion / contraction)
Change in entropy (heating / cooling)
Change in chemical composition
nucleosynthesis
mixing
Monday, 17 September 12
size
If a region of a star changes in volume rapidly, then the material must
obey laws of motion
Normally we consider changes in stellar dimensions to be very slow,
but in the case of explosions, this equation becomes very important.
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entropy
If a region of a star changes in volume it will release or
absorb energy according to the 1st law of TD
dE = -T dS = - (dU + PdV)
Hence the energy equation needs to consider
dL/dm = ε – T dS/dt
Where ε represents other sources/sinks of energy within
a unit volume
Monday, 17 September 12
nucleosynthesis
Nuclear reactions produce energy and transmute
elements from one species to another
A reaction i(j,k)l destroys species i and j, and creates
species k,l at a rate rij.
The rate of change of abundance of species i will then
be given (in mass fraction) by
dxi/dt = - (1+δij) rij / NA
Where the bracket allows for more than one particle of
species I being destroyed in a reaction.
Thus for the pp chain:
dX/dt = -4 rpp / NA
dY/dt = - dX/dt
Monday, 17 September 12
mixing
Convection provides an efficient energy transport
process as well as a chemical transport mechanism.
In stellar evolution calculations it is normally assumed
that a fully convective region is fully mixed.
During a timestep, nucleosynthesis may transform the
mixture in part of a convection zone, or else a
convective region may move to incorporate material
of different composition. The new material must then
be mixed throughout the convection zone.
In 1D, the composition in a convective region
<Xi> = ∫ [Xi(m) + δt dXi(m)/dt] dm / ∫ dm
BUT....
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Monday, 17 September 12
6. Future of the Sun and other Stars
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Hertzsprung Russell diagram
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Evolution of a 1 M star
Stellar Structure: TCD PY3A03:
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main-sequence stars
H
H > He
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giant-branch stars
H
H > He
He
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3D convection in a rotating red giant
Velocity structures in a rotating model star with convection in equilibrium in its unstable outer half,
PPM simulation by Porter, Anderson, Habermann, and Woodward on two 64-processor SGI Origins at NCSA, 1997.
Monday, 17 September 12
white dwarfs
H He
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horizontal-branch stars
H
H > He
He
He > C/O
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asymptotic giant-branch stars
H
H > He
He
He > C/O
C/O
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planetary-nebula formation
Stellar Structure: TCD PY3A03: 13.
Monday, 17 September 12
Evolution of a 5 M star
Stellar Structure: TCD PY3A03:
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white dwarfs
H He
C/O
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The Sun’s Final Show
Stellar Structure: TCD PY3A03: 13.
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Evolution of 1 M star
Stellar Structure: TCD PY3A03: 13.
Monday, 17 September 12