Stellar Structure
Section 4: Structure of Stars
Lecture 7 – Stellar stability
Convective instability
Derivation of instability criterion …
… in terms of density or temperature
Generalisation to include radiation pressure
Conditions where convection is likely
Energy carried by convection
Dynamical stability of stars
• Static, equilibrium stellar models need to be checked for stability
• If an unstable star changes on a dynamical timescale, it is
dynamically unstable
• Typical timescale is hours to days – easily observable; most stars
completely stable against large changes on such timescales
• Some stars definitely dynamically unstable – the regular variable
stars, such as Cepheids; the instability grows until limited by nonlinear effects to a large-amplitude oscillation, usually radial but
sometimes non-radial
• Other stars eject mass sporadically, either gently (cool giants) or
violently (novae, supernovae)
• All such stars are important, but beyond the scope of this course
Unstable stars
Cepheid variables in IC1613:
SN1987A light curve:
Convective instability – a vital
component in most stars
• Convection – localised instability, leading to
large-scale motions and transport of energy
• Rising element of gas:
 pressure balance with surroundings,
provided rise is very subsonic
• Density changes (not in thermal balance):
 if density is more than in surroundings,
element falls back: stable
 if density is less than in surroundings,
element goes on rising: unstable
Mathematical treatment – key
assumptions
• Plane geometry – elements small
•  = constant
•  = constant
• Neglect radiation pressure (include later)
• Element rises adiabatically: no heat exchange with
surroundings; this means no change in the heat content of the
element (see blackboard for mathematical formulation) and
over-estimates the instability
Mathematical treatment – criterion for
convection (see blackboard for detail)
• Apply
pressure balance
to find the pressure
change in the element
• Use adiabatic condition
to relate density change to
pressure change
• Write down condition for
density in element to be
less than in surroundings
• Re-write in terms of
density and pressure
gradients in surroundings
Element:
Surroundings:
+
P+P

d
z
dz
P
dP
z
dz
z


P
P
Other forms of criterion for instability
(see blackboard for detail)
• Pressure gradient is negative => instability if density gradient is
positive – not very likely.
• If density gradient is negative, can re-write criterion in terms of
the gradient of density with respect to pressure, d/dP, or in
terms of the variable polytropic index n.
• For ideal gas, with constant , can re-write criterion in terms of
temperature gradient with respect to pressure, dT/dP, and
relate it to the adiabatic value of the gradient, involving .
• Radiation pressure can also be included, and gives a similar
criterion, with  replaced by (), where  is ratio of gas
pressure to total pressure.
Where does convection occur?
• Convection starts if
• i.e. for
P dT   1

T dP

 PdT/TdP large (for normal  ~ 5/3, ( -1)/ ~ 0.4)
 or  -1 <<  (for normal gradient, PdT/TdP ~ 0.25)
• Large T gradient needed where a large release of energy
occurs – e.g. nuclear energy release near centre of a star
•  -1 small occurs during ionization, where latent heat of
ionization is important and cp → cp + latent heat ≈ cv + latent
heat, =>  ≈ 1. Occurs near surfaces of cool stars (gas in hot
stars is ionized right up to the surface)
Energy carried by convection
• Convection usually involves turbulence, and sometimes
magnetic fields, and is very hard to simulate numerically, even
under laboratory conditions
• Detail of convective energy transport remains a major
uncertainty in stellar structure – what can be said?
• Must replace L by Lrad in energy transport equation (see
blackboard)
• Must add
where
L = Lrad + Lconv
(4.40)
Lconv = ?
(4.41)
• Energy carried by convection depends on conditions over a
convective cell, not purely on local conditions
• Can we estimate Lconv? See next lecture!