Stellar Structure

1. 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

2. 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

3. Unstable stars Cepheid variables in IC1613: SN1987A light curve:

4. 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

5. 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

6. Element: Surroundings: z 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 + P+P  P  P

7. 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.

8. 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 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)

9. 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 L = Lrad + Lconv(4.40) where 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!