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 non- linear 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) Element:Surroundings: zz  +  P+  P  P P P 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

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  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 c p → c p + latent heat ≈ c v + 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 L rad in energy transport equation (see blackboard) Must add L = L rad + L conv (4.40) where L conv = ?(4.41) Energy carried by convection depends on conditions over a convective cell, not purely on local conditions Can we estimate L conv ? See next lecture!