Stellar structure
Stars = self-gravitating, speherical bodies powered by nuclear reactions.
If star is system in virial equilibrium its gravitational potential energy
Ug ~ GM2/R must be balanced by its thermal energy UT ~ NkBT -
-
T ~ GMmp/kBR.
For a sufficiently large value of M/R (e.g. take one solar mass and one solar
radius) T is large enough (> 107 K) that nuclear reactions will take place
(high density also satisfied because also M/R3 very large)
-- nuclear reactions establish a pressure/temperature gradient that
supports the star against collapse
We study stars assuming (1) steady state holds (so fundamental equations
of stellar structure time independent), (2) chemical homogeneity (i.e. same
composition everywhere) and (3) rotation and magnetic fields negligible
The star will eventually radiate the thermal energy produced by nuclear
reactions. If the star remains in virial equilibrium (does nor contract or
expand) it means that the energy produced by nuclear reactions balances
that lost at the surface via radiation (will be one of structure equations…)
Equations of stellar structure - I
The first three equations of stellar structure are trivial:
(1)Hydrostatic equilibrium 1/r x grad(P)= -grad(f)  dP/dr = -GM(r) r(r)/r2
(2)Conservation of mass
dM(r)/dr = 4pr2r
(3) Equation of state (ideal gas + photons) P=P(r, T ) in general (will
reduce to barotropic equation of state P=P(r) for white dwarfs/neutron stars)
P = rkBT/mmH + 1/3aT4
a = 4s/c (radiative constant)
Composition determines equation of state through value of m. Assuming
(1)Fully ionized gas (ok because collisional ionization efficient at high r)
(2) X+Y+Z = 1, X, Y, Z = mass fractions of hydrogen, helium and metals and
(3) Z << X, Y ~ 1 – X, X ~ 0.71 -- m ~ 0.5 X -0.57 ~ 0.6
Note: discarding metals would be perfectly right for the first generation of stars
that formed at high redshift in the first galaxies. These stars, called Population III
stars, formed from the collapse of metal-free molecular clouds with primordial
composition (X ~ 0.75, Y ~ 0.25, from the Big Bang because metals were first
synthesized by massive stars via nuclear reactions in their interior and then
released as they explode as supernovae.
Without metals cooling is inefficient (recall carbon and oxygen crucial to cool
ISM) so cloud collapse when they are still massive because Jeans mass high-
first stars were massive (M > 10 Mo) and therefore exploded as supernovae.
Today’s Pop I stars (e.g, the Sun) have X ~ 0.71, Y ~ 0.27, Z ~ 0.02 (measured)
and Pop II stars (in stellar halo and globular clusters) slightly more metal poor.
We now have three equations for four variables M(r), r(r), P(r) and T(r).
We need one more equation giving T(r).
This will depend on what is the process that transports thermal energy from
the center to the surface of the star. There are two possibilities and they
will be more or less important in (1) different regions of the same star and
(2) stars of different masses;
(1) Radiation
(2)Convection
Equations of stellar structure – II
Radiation and convection
Radiation
Stars have very hig-h densities in their interior so t > > 1. The energy flux
due to radiation can be calculated using the diffusion approximation:
F(r) = L(r)/4pr2 = - 4acT3/3rk(r) x dT/dr  dT/dr = - 3k(r)r L(r)/(16acT3pr2)
One needs the auxiliary opacity function k(r). For a fully ionized gas this
is easy to calculate since the opacity is the combination of just a few
processes. Their relative importance will vary as function of distance from
the center because some of the opacity sources are function of r, T + all
depend on the composition of the star;
Kes = 0.2 (1 + X) cm2g1
electron/photon scattering (Thomson/Compton)
Kbf ~ 4 x 1025 cm2g-1 Z(1 + X)rT-3.5 cm2/g-1 absorption of electron by atom/ion
Kff ~ 4 X 1022 (X+Y)(1+X)rT-3.5 cm2g-1 absorption of photon by free electron
kH- ~ 2.5 x 10-31(Z/0.02)r1/2T9
(inverse brehmsstrahlung)
absorption of photon by H3000 K < T < 6000 k – envelope)
(important at
Convection
The star will be convectively unstable in regions where dlnT/dlnP > (g – 1)/g,
where (g-1)/g is the temp. gradient expected in an adiabatic gas (4/3 < g < 5/3)
depending on degrees of freedom f of atoms/molecules/ions, g = ((f+2)/f)
The conditions of the gas in the star will not be adiabatic in general
because the star is losing energy via radiation.
We can write dlnT/dlnP = P/T(dT/dr/dP/dr) - if radiative losses create a
rapid variation of temperature with radius (i.e steep dT/dr) convection
becomes important. Hence it is radiation that establishes the gradient.
Since there is no general theory of convective energy transport that works
in all regimes (i.e. for arbitrary convective fluxes) we will assume the
maximum possible convective flux is establshed as soon as convection
starts --- this implies that the gradient will tend to reach the adiabatic
value because convection is efficient, hence:
dlnT/dlnP = P/T(dT/dr/dP/dr) = (using hydrotsatic equilibrium equation
for dP/dr) = -r2/GM(r) (P/rT)dT/dr = (1 – 1/g)
- dT/dr = -(1 – 1/g)(rT/P)(GM(r)/r2)
For simplicity one assumes that convection is the dominant energy transport
mechanism in regions where dlnT/dlnP > (g – 1)/g.
Note that; (1) the condition is local, so the star can switch between radiative and
convective transport in different regions; (2) the transition from one regime to
the other across radius is not only controlled by radiation (which changes the
temperature gradient) but also by the local degree of ionization/composition
which can change the value of g (we have seen it changes m as well).
-------------We now have three differential equations plus the equation of state P=P(r,T)
but the variables to solve for are 5 - T(r), P(r), r(r), M(r) and L(r)
We need one more equation to close the set. This has to express conservation
of energy, namely the relation between the energy lost at the stellar surface
and the energy produced by nuclear reactions;
dL(r)/dr = 4pr2re
where e=e0rlTn.
The specific form of e(r, T) depends on the dominant nuclear reactions. These
change depending in the central density and temperature of the star (e.g.
p-p chain (l=1, n=4), CNO cycle (l=1, n=15) or helium burning)
Basic nuclear fusion reactions
Low mass stars: pp-chain
High mass stars : CNO cycle
Equations of stellar structure – III
The Final set
We now have 4 differential equations + 3 auxiliary functions (for P, e and k),
and in total seven variables (L(r), T(r), r(r), M(r), k(r,T), e(r,T), P(r, T))
We have thus completed the set of equations and we can proceed to solving
them once we adopt a set of boundary conditions. These we can express as:
M(r)=0
M(r) = M
at r=L=0
at r=R where r=P=T=0
And we can rewrite the 4 differential equations using Mr=M(r) as the
independent variable in place of r by dividing them by the mass conservation
equation:
(1) dP/dMr = -GMr/4pr4
(2) dr/dMr = 1/(4pr3r)
(3) dT/dr = -3k/64pac2(1/T3)(Lr/r4) (radiative regime)
(4) dLr/dMr = e
-(1 - 1/g) (T/4pP)(GMr/r4) (convective regime)
Note: boundary conditions at the origin trivial but not those at the surface. In principle
no radiation is emitted at the surface if T=0 (think stellar surface as blackbody)!
In reality one has to solve separately the problem of stellar atmospheres and the
problem of stellar interiors that we are solving here and then define a new set
of boundary conditions between atmosphere and interior at which the two solutions
have to match. Stellar atmospheres difficult to study because not optically thick (no
simple approximation to radiative transfer equation!). Similar problem exists also
for protoplanetary disks.
Given the mass of the star M we can use the equations of stellar structure
+ boundary conditions to determine all the properties of a star (such as
its temperature, density and luminosity as a function of radius). Since now
r(Mr) is one of the structure equations the radius R of the star will also be
automatically determined once M is given.
The steady-state, homogeneous composition model that we have constructed
is meaningful only for timescales shorter than the nuclear evolution timescale
of the star. For example, when the star switches from p-p chain to helium
burning both its composition and the rate of energy production change and
a new equilibrium has to be recomputed.
Our description is valid for a zero-age main sequence star (ZAMS) because
a star spends most of its life along the main sequence (p-p chain relevant)
Solutions of stellar structure equations
Existence:: there is no rigorous way (e.g. a theorem of prompt use) to
determine a priori that a solution exists and is unique because the equations
of stellar structure are quite complex.
Explicit numerical integration (with non trivial methods, e.g. problem is to
avoid divergence of derivatives at the “surface” where boundary conditions
impose several variables to vanish) shows that solutions exist for M > 0.1 Mo
and X~0.7-1. Note however brown dwarfs (stars with masses 0.01-0.08 Mo)
exist but they are a special case (powered by deuterium burning early on
but later supported by the pressure of degenerate electrons)
Uniqueness: there is also no way to prove uniqueness of solutions but usually
one can prove (case by case, not in general!) that situations that admit more
than one solution are unphysical!
We will now study the solutions obtained from the numerical integration of
stellar structure equations for stellar masses in the range 0.08 – 60 Mo and
with homegenous composition given by X=0.74 and Y=0.24
Properties of solution
In general structural properties vary significantly between stars with M < Mo
and stars with masses M > Mo
(1)The radius of a star increases with mass
(growth weaker than linear)
(2) The central temperature increases slowly as
the mass grows but increase is stronger as mass
goes beyond 1 Mo. When T > 1.4 x 107 K the
dominant nuclear reaction process changes from
p-p chain to CNO cycle. e(r, T) is more sensitive
to T for CNO cycle  steeper
temperature gradients are possible near the
center -- convective core for massive
stars (only).
(3) Luminosity is a strong function of mass (logarithmic axes in
the plot) for M >~ Mo the approximate scaling is given by
L/Lo ~ (M/Mo)3.5.
This steep dependence is the result of the steep growth of the nuclear
energy production rate for stars with M >~ Mo
(CNO cycle).
We can estimate the lifetime of the star as a
steady-state object sustained by hydrogen
burning as the time it takes to convert 10% of
the stellar mass from hydrogen into helium:
tnuc ~ 0.1 e XM/L ~
(e = 6 x 1018 ergs, X ~ 0.7) 1010 (M/Mo)(L/Lo) yr ~
~ (using the L(M) relation) 1010(M/Mo)-2.5 yr
This is the time the star spends on the main
sequence, ~ 1010 yr for a star with M ~ Mo.
more massive stars will stay on the main
sequence much less.
(4) assuming that the star radiates as a blackbody one can derive a relation
between L and the effective temperature of the blackbody Teff
-- L/Lo = 8.973 x 10-16 (R/Ro)2 T4eff
Using the M-R relation and the approximate L-M relation for M >~ Mo
one gets:
Teff ~ Toeff (L/Lo)h
h=0.12-0.17
Teff can be measured from stellar spectra (fitting a blackbody to the
spectrum) and if the distance of the star is know one can measure L
--- one can construct the L-Teff plane for many stars, also known as
the Hertzsprung-Russell (HR) diagram (color-magnitude diagram)
The slope of the H-R diagram is given by h.
More massive stars have a higher effective temperature because
they are brighter (from L-M relation). They are also less dense
because the radius grows almost linearly with mass
The outer regions of all stars with M > Mo are also of very low density
compared to the central regions as shown by cumulative mass profiles
-- gravitational force in those regions is essentially from the core.
The HR diagram
(5) Most of the energy, and thus most of the luminosity, is generated
in the central region of stars, for quite a range of masses (very little
energy generated at r > 0.4 R)
(6) Mechanism of energy transport depends on stellar mass
The central regions of massive stars (M > 1.3 Mo) are convective
while those of low mass stars (up to 1 Mo) are radiative.
However low mass stars have
convective envelopes, and the size
of the convective envelope increases
as the mass of the star decreases
(for M < 0.3 Mo) entire star is convective)
Note: the atmosphere of the star,
close to the surface, is always radiative
but to see this one needs to model
the stellar atmosphere
Main reason for convective core in
massive stars; CNO cycle produces
high energy flux in central region,
hence high dT/dr from diffusion
equation -- convective instability
sets in!
At larger radii flux lower, hence dT/dr
lower - can be radiative.
Reason for convective envelope in low-mass stars:
Again from the diffusion equation, at fixed temperature gradient the
radiative flux decreases with increasing opacity. Hence if the star has
to keep its luminosity, i.e. energy flux constant in outer region, it
implies that dT/dr must increase -- towards convective instability!
Why does opacity increase in envelope?
In envelope T decreases and approaches T ~ 105 K, at which point bound
free and free-free opacity begin to dominate over electron scattering
because temperature close to first ionization potential of hydrogen
(recall thermal balance condition between recombination and ionization
of hydrogen gives T=105 K).
Note:adiabatic gradient for ideal, fully ionized gas ~ 0.4, but drops to
0.27 in cores of massive stars where radiation pressure is important