The Ledoux Criterion for Convection in a Star [620515]

The Ledoux Criterion for Convection in a Star
Marina von Steinkirch, [anonimizat]
State University of New York at Stony Brook
August 2, 2012
Contents
1 Mass Distribution and Gravitational Fields 2
2 Conservation of Momentum 2
3 Conservation of Energy 3
4 Transport of Energy 4
5 Stability and the Ledoux Criterion 5
6 How a Supernova Explodes 8
1

1 Mass Distribution and Gravitational Fields
The Eulerian Description
For gaseous, non-rotating, single stars, without strong magnetic elds, the only
forces acting on a mass element are from pressure and gravity, resulting on
a spherically symmetric conguration. If we use tandras the independent
variables, we have the Eulerian description . For instance, density would be
=(r;t).
We want to represent the mass distribution inside the stars and its eect on
the gravitational eld. We dene the function m(r;t):
dm(r;t) = 4r2dr4r2vdt:
Partially dierentiating this equation, keeping either torr, gives relations
such as@r
@m= (4r2)1: (1.1)
The Lagrangian Description
If we take a Lagrangian coordinate instead ofr,i.e., one that is connected to
the mass elements, we have the new independent variables as mandt. The
partial derivatives with respect to the new derivatives are
@
@m=@
@r
@r
@m;
and@
@t
m=@
@r
@r
@t
m+@
@t
r:
The new recipe for the transformation between the two operators is then
@
@m=1
4r2@
@r:
Inside a spherically symmetric body, the absolute value gof the gravitational
acceleration at a given distance rfrom the center does not depend on the mass
elements outside of r, but it does depend on rand the mass inside:
g=Gm
r2: (1.2)
2 Conservation of Momentum
The mechanical equilibrium in a star is called hydrostatic equilibrium when
without rotation or magnetic elds: they are in such long-lasting phases of their
evolutions that no changes can be observed at all.
2

Considering a thin spherical mass shell with an innitesimal thickness drat
radiusrin the star. The mass per unit of area is dr, and the weight (gravita-
tional force towards the center) of the shell is gdr. This is counterbalanced
by the pressure of the same absolute value, outwards. Moreover, the shell must
feel a larger pressure Piat its interior than in the outer boundary Po. The total
net force per unit area acting on the shell due to this pressure is:
PiPe=@P
@rdr:
The sum of forces arising from pressure and gravity has to be zero,
@P
@r+g= 0;
giving the condition of hydrostatic equilibrium,
@P
@r=g: (2.1)
Together with Eq. 1.2, this equation becomes
@P
@r=Gm
r2; (2.2)
which is the second basic equation describing the stellar structure in the Eulerian
form. If we take mas the independent variable, as in Eq. 1.1,
@P
@m=Gm
4r4; (2.3)
the second of the basic equations in the Lagrangian form.
3 Conservation of Energy
The rst law of thermodynamics relating heat per unit mass is
dq=du+Pdv: (3.1)
Assuming a general equation of state, =(P;T) andu=u(;T) we can
dene the derivatives as
=@ln
@lnP
T;=P
v@v
@P
T;; (3.2)
=@ln
@lnT
P;=T
v@v
@T
P;; (3.3)
and
'=@ln
@ln
P;T: (3.4)
3

For an ideal gas with =P=T , one has=='= 1. The internal
energy in the Eq. 3.1 is rather
du=@u
@v
Tdv+@u
@T
vdT:
Plugging it back in that equation, we have
dq=@u
@T
vdT+h@u
@v
T+Pi
dv=@u
@T
vdT+T@P
@T
vdv:
4 Transport of Energy
The energy the star radiates away from its surface is generally replenished from
reservoirs situated in the hot central region. This requires an eective transfer of
energy through the stellar material, this is possible due the non-vanishing tem-
perature gradient in the star. This transfer can occur due radiation, conduction ,
andconvection .
Radioactive Transport of Energy
We rst estimate the free path l
of a photon at some point in the star:
l
=1
;
whereis a mean absorption coecient (radioactive cross-section over fre-
quency). For the sun, l
2 cm, i.e.,the matter is very opaque.
The typical temperature gradient in the star can be roughly estimated by
averaging between center and surface,

T

rTcTs
R:
The radiation eld at a given point is emitted from a small isothermal sur-
rounding, where the dierence of temperature is of the order of
T=l
(dT=dr ).
The energy of radiation is uT4, the relative anisotropy of the radiation at
some point is 4
T=T. Stellar interiors are very close to thermal equilibrium,
and the radiation very close to of a blackboard. However, the small anisotropy
can be the carrier of the star's luminosity. Radioactive transport of energy
occurs via the non-vanishing net
ux.
Since the radioactive transport in stars is very small compared to the charac-
teristic length, i.e.,the stellar radius, the transport can be treated as a diusion
process. The diusive
ux jof particles of dierent particle density nis
j=Drn;
whereD=1
3vlpis the coecient of diusion, determined by the average values
of mean velocity vand mean free path of the particles. We can replace nby the
energy density of radiation U=aT4, whereais the radiation-density constant.
4

Since we have spherical symmetry, Fhas only the radial component and we
have@U
@r= 4aT3@T
@r;
so that
F=4ac
3T3
@T
@r;
which is the equation for heat conduction
F=radrT:
If we replace Fby the local luminosity l= 4r2,
@T
@r=3
16acl
r2T3:
If we usemas an independent variable instead of r, as in Eq. 1.1, the
equation for radioactive transport of energy is
@T
@m=3
642acl
r4T3: (4.1)
Dividing the last equation by Eq. 2.3, we have
@T=@m
@P=@m=3
16acGl
mT3: (4.2)
The ration of the derivatives on the left, ( dT=dP )rad, is a gradient describing
the temperature variation with depth,
rrad=dlnT
dlnP
rad=3
16acGlP
mT4: (4.3)
5 Stability and the Ledoux Criterion
In stars, sometimes small perturbations may grow and give rise to macroscopic
local (non-spherical) motions that are also statistically distributed over the
sphere. These motions can have a strong in
uence in the stellar structure,
such as mixing stellar material and transporting energy. Stability criteria will
dene whether small perturbation will grow or keep small.
Dynamical instability happens when the moving mass elements have no time
to exchange large amounts of heat with the surroundings and move adiabatically.
In the surface of a concentric sphere, physical quantities such as temperature,
density, etc, may not be exactly constant, but show certain
uctuations [3].
For any physical quantity A, the dierence between the element and its
surroundings is dened as DA=AeAs. In the case of pressure, we can as-
sume that the element always remains in pressure balance with its surroundings,
DP= 0.
5

If we assume DT > 0, for a ideal gas with P=TnR=V , we have
D < 0. This means that the element is lighter than the surroundings and
buoyancy forces will lift it upwards.
To test the instability of a layer we can take a radial shift
r >0 of the
element. Considering this element to be lifted from rtor+
r, its density will
dier from the surroundings by
D=hd
dr
ed
dr
si

r:
A niteDgives the radial component of a buoyancy force (per volume),
K=gD;
wheregis the absolute value of the acceleration of the gravity. Both cases can
happen:
D< 0, the element is lighter and Kis upwards, generating an unstable
situation, the original perturbation being increased.
D> 0, the original element is heavier and Kis directed downwards, the
perturbation is removed, the layer is stable.
The condition for stability can be written as:
d
dr
ed
dr
s>0: (5.1)
However these criteria is impractical and we need to rewrite it in terms of
gradients of temperature. We can rewrite the equation of state =(P;T; )
in the dierential form, where ;are giving by Eqs. 3.2, 3.3, and 3.4.
d
=dP
PdT
T+'d
: (5.2)
Rewriting Eq. 5.1 with 5.2, we have

PdP
dr
e
TdT
dr
e
PdP
dr
s+
TdT
dr
s'
d
dr
s>0: (5.3)
The terms containing the pressure gradient cancel each other since DP= 0.
We then dene the scale height of pressure ,
Hp=dr
dlnP=Pdr
dP;
whereHPhas the dimension of length, being the length characteristic of the
radial variation of P. With Eq. 2.1,
HP=P
g>0;
6

sincePdecreases with increasing r.
Multiplying Hgback to Eq. 5.3 yields as a condition for stability:
dlnT
dlnP
s<dlnT
dlnP
e+'
dln
dlnP
s: (5.4)
We can dene three new derivatives,
r=dlnT
dlnP
s;re=dlnT
dlnP
e;r=dln
dlnP
s;
wherePis taken as a measure of depth. We can rewrite Eq. 5.3 as:
r<re+'
r: (5.5)
Recovering the denition of rradfrom Eq. 4.3, which describes the tem-
perature gradient for the case that the energy is transported by radiation (or
conduction) only. In a layer that transports all energy by radiation, r=rrad.
Recovering alsore=rad, the radiation layer is stable if
rrad<rad+'
r;
theLedoux criterion . In a region with homogeneous chemical composition,
r= 0, which is the Schwarzschild criterion ,
rrad<rad:
In the interior of evolving stars, the heavier elements are produced below the
lighter ones, and the molecular weight increases inwards so r>0, so this
element has an stabilizing eect. If these criteria admits stability, no convection
motions will occur and the whole
ux will be carried by radiation
7

6 How a Supernova Explodes
A supernova is an outcome in the sequence of nuclear fusion reactions which is
the life story of a star. Heat given by the fusion generates pressure to counteracts
the gravitational attraction that would make the star collapse [1].
Core collapse supernovae are explosions that mark the death of a massive
star, releasing energy of order 1053erg at rates of 104546Watts. These events
are the factory of most of the heavy nuclei found elsewhere the Universe. The
neutron-rich wind that emanates from the protoneutron star after the explosion
allows trans-iron elements to be synthesized by rapid neutron capture processes.
In addition, nucleosynthesis may occur by neutrinos can cause protons and
neutrons from heavier nuclei to produce rare isotopes. After the explosion,
supernovae can cool down becoming neutron stars and black holes.
Stars' Chain of Fusion
The rst series of fusion reactions have the net eect of four atoms of hydrogen
into a single atom of helium:
4H!1He4+ Energy:
When the core of the star runs out of H, it contracts due the gravitation. The
core and the surrounding material are then heated, causing Hydrogen fusion to
begin in the surrounding layers and other fusion reactions in the core:
He!Ca!Ne!O!Si!56Fe:
The iron nucleus is the most strongly bound of all nuclei and further fusion
would absorb energy instead than releasing it. At this point the star has an
onionlike structure, where the outer envelope is mostly hydrogen. How far in
these elements depends on the star sizes, e.g., the Sun would not burn further
than He [1].
Chandrasekhar Mass
When the fusion ends, a small star will shrinks to a white dwarf. The quantity
that denes whether the star will stop to burn further is called Chandrasekhar
mass,mC. This mass represents a limit to how much pressure can be resisted
by the electrons' mutual repulsion: when the star contracts, the gravitational
energy increases together with the energy of the electrons, and whether these
two forces are in balance or not depends on the mass of the star, if it is larger
thanmC, the star collapses [1]
The value of mCdepends in the relative number of electrons and nucleons,
where the higher the proportion of electrons, the larger the electron pressure
and themC. In small star with the chain of fusion reactions stopping at carbon,
the radio is around 1 =2 and the Chandrasekhar mass is 1.44 solar masses.
8

Type I Supernovae
White dwarfs in a binary star system are the origin of the Type I supernovae.
Matter from the binary companion is attracted by the gravitational eld of the
dwarf star and gradually falls onto its surface, increasing the mass of carbon
and oxygen core [1].
Type II Supernovae
The Type II supernovae arrives from very massive stars, where the lower limit
is around eight solar masses. When the nal fusion reaction begins, the core
made up of iron and other few elements begins to form in the center of the star,
within a shell of silicon [1]. The core now is inert under great pressure and it
can resist contraction only by electron pressure, subject to the Chandrasekhar
limit.
The Implosion of the Core
The compression raises the temperature of the core, however the pressure raised
by it does not help to slow down the collapse. The pressure is determined by the
number of particles in a system and their average energy. The pressure by the
electron is much bigger than of the nuclei. When the core is heated, some iron
nuclei are broken into smaller nuclei, increasing the number of nuclear particles
and raising the nuclear component of the pressure. This dissipation absorbs
energy, which is taken from electrons, decreasing their pressures. The net result
is that the collapse accelerates.
The entropy of the core, which has groups of 56 nucleons bonds, is lower
than when it was composed of hydrogen. The high density in the collapsing
core favors the reaction know as electron capture , which liberates a neutrino
which carries entropy and energy out of the star. The loss of the electron also
diminishes the electron pressure.
The rst stage of the collapse comes to an end when the density of the
stellar core reaches about = 41011grams per cubic centimeter. At this
point, matter becomes opaque to neutrinos.
The role of the Chandrasekhar mass here is for the analysis of how the
supernova changes: it is the largest mass that can collapse as a unit, the so
called homologous core and collapse . Ares withing the core communicate by
means of sound waves andpressure waves .
Within the homologously collapsing part of the core, the velocity of infalling
material is directly proportional to the distance from the center, where the
density and speed of sound decreases with distance from the center.
The radius at which the speed of the sound and the infall velocity coincide
is called sonic point and is the the boundary of the homologous core. For a
fraction of millisecond the sound waves at the sonic point build pressure there
and slow the material falling through. This creates a discontinuity in velocity
which forms the shock waves . The passages of the shock wave induces changes
in density, pressure and entropy and it moves faster than the speed of the sound.
9

The shock wave will advance through the onionlike structure until erupts on the
surface.
Core Collapse Supernova
The core collapse supernova begins with the collapse due to the force of gravity
of the iron core of a massive star at the end of its thermonuclear evolution, a
variant of the shock reheating mechanism
The rebound of the inner core is brought by a rapid increase of pressure
with density when this rises above the nuclear matter density, generating a
shock wave at a radius of 20, driving into the outer core. The shock stalls
and turns into an accretion shock at a radius of 100 to 200 km [2].
The energy transferred between neutrinos and matter before the shock is by
charged currents that producing cooling are
e+p!n+e
e++n!p+ e
which are proportional to the matter's temperature
cooling
nucleons/T6
matter
where heating is produced by the inverse:
n+e!e+p
p+ e!e+n
proportional to the luminosity Le;e, energye;e, and the inverse of a geo-
metric
ux factor, of the neutrino and anti-neutrino
heating
nucleons/Le;e
D
2
e;eE

D1
Fe;eE
: (6.1)
1. Nuclear dissociation behind the shock lowers the ratio of pressure to en-
ergy, lowering the strength to the inner core.
2. There is a reduction in both the thermal and lepton numbers contribution
in the shock pressure, resulting the burst of eradiation.
3. As the matter continues to
ow inward, neutrinos and heating increases
the core temperature until the cooling rate, which goes to sixth power
of the temperature, exceeding the heating rate. The radius at which the
heating and cooling rates are equal is gain radius .
4. The in
owing matter cools and add onto the core.
5. Ifheating is suciently rapid in the region between the shock and the
gain radius, the increased thermal pressure behind the shock allow it to
overcome the accretion and propagate out, producing a supernova.
10

Figure 1: Stellar core after bounce from [2].
Convection
The
ux-limited diusion neutrino transport was investigate by [2] to study the
role of the types of convection in core collapse supernovae. In special, two type
of convention, protoneutron star convection andneutrino-driven convention .
In the presence of neutrino transport, protoneutron star convention velocities
are too small to the bulk in
ow to result to any signicant convective transport
of entropy and leptons. Moreover, neutrino-drive convection in stars with more
than fteen solar masses (compact iron core) still does not have a model that
reproduces the explosions.
Convection Mechanism
Observations of supernovae indicates an extensive mixing thorough some of the
ejected material with points to
uid instabilities from the explosion itself. The
degree of mixings varies only from Rayleigh-Taylor instabilities in the expanding
envelope so that these instabilities are preceded by former instabilities occurring
11

during the explosion.
The postcollapse stellar core can be divided into two regions by the neutri-
noshpere, where convection in the region below it can enhance the reheating
mechanism by enhancing transporting of lepton from deep within the core, pro-
toneutron star convection .
The factors Landfrom Eq. 6.1 depend on the conditions at the neutri-
nosphere and can be aected by the protoneutron star convection.
Neutrinosphere at a Negative Entropy Gradient
The entropy driven protoneutron star convection will advect high entropy ma-
terial from deeper regions up to the vicinity of the neutrinosphere, raising the
temperature, hence increasing Land, until the limit of zero electron degener-
acy, at the e-sphere.
The region between the neutrinosphere and the shock is divided into two
regions: a region above the neutrinosphere of net neutrino cooling and a region
above that and below the shock of net neutrino heating. these regions are
divided by the gain radius.
The lepton fraction also will be increased by convection, from the fact that
rapide,e+capture and e, eescapes at the neutrinoshpere ling at a min-
imum lepton fraction Yl. The entropy and lepton drive convection will advect
lepton matter to the border of the neutrinoshpere, increasing edegeneracy,
hence increasing Landneutrinos decreasing antineutrino. This results heat-
ing rate of the material behind the shock.
Protoneutron Star Convention
Material in the vicinity and below the neutrinosphere are in nuclear statistical
equilibrium and its thermodynamic state and composition can be specied by
three variables: entropy per baryon s, lepton fraction Yland pressure P. Above
the neutrinosphere, we use Yeinstead.
With gradients in these variables, a strong gravitational eld and the neu-
trino transport of energy and leptons, the material become subject to many
uid
instabilities. The simplistic case, with no neutrino transport, is the Raylength
Taylor convective instability, where the criterion for convective instability, the
Ledoux condition, is

@
@lnYl!
s;P
@lnYl
@r!
s;P+
@
@lns!
Yl;P
@lns
@r!
Yl;P>0
where

@
@lns!
Yl;Pis negative for all thermodynamic states: entropy driven pro-
toneutron star convection.
12


@
@lnYl!
s;Pis negative (positive) for large (small) Yl: lepton driven pro-
toneutron star convection.
Adding neutrino transport of energy and leptons, the above convection have
reduced growth rates and two additional modes of instabilities are possible:
neutron ngers and semi-convection, occurring on a diusion scale (not dynam-
ical), requiring that one of the gradients become destabilizing and the other
stabilizing.
The
uid instabilities in the region below the neutrinoshpere play a roles in
the shock reheating mechanism:
instabilities tend to drive
uid motions that tend to circulate through the
unstable region.

uid motions with entropy driven convection tend to advect high entropy
material from deeper in the core to the neutrinosphere, increasing its tem-
perature and e, eemission rates.
lepton drive convection tend to advect lepton material from the core to the
neutrinosphere, increasing eand decreasing  eemission rates, producing
the deleptozinzation of the core.
Neutrino Driven Convection
As infalling material encounter the shock, it is shock dissociated into free neu-
trons and protons if the shock is within a radius of 200 km. As the material
continues to
ow inward, it will be heated by the charged-current reaction until
reaching the gain radius. Neutrino heating is strongest beyond the gain radius
and decreases farther out as the neutrino
ows becomes diluted. these factors
create a negative entropy gradient between the gain radius and shock, unsta-
ble to entropy-drive convection/ this will persist until the explosion develops
(neutrino heat)
If neutrino driven convection is able to develop, it plays a crucial role in the
generating an explosion. ecient way of conveying low entropy matter from the
shock to the gain radius and high entropy matter back to the shock.
Simulating Supernoave
Eects of gravity: the increased gravitational potential will pull the core into a
deeper potential well the resulting redshift of the neutrino radiation will reduce
the energy of the neutrinos in the heating region gravity will make successful
explosions more dicult.
13

References
[1] Bethe, H.A. & Brown, G., How a Supernova Explodes , Scientic American,
1985.
[2] Calder, A.C., Multidimensional Simulations of Core Collapse Supernovae
Using Multigroup Neutrino Transport , 1997.
[3] Kippenhahn, A. & Weigert, A., Stellar Structure and Evolution , 1994.
14