Review of Lecture 4 Forms of the radiative transfer equation Conditions of radiative equilibrium Gray atmospheres –Eddington Approximation Limb darkening

Convective Energy Transport/Chapter 7 Stability criterion for convection Adiabatic temperature gradient When is convection important Convection in the Sun The Mixing Length Formalism

Criterion for Stability against Convection If we displace an element of gas, will it continue to move in the same direction? P 2 ’ = P 2 If  2 ’ <  2, the element will continue to rise. P 2 ’=P 2  2 ’ T 2 ’ Initial gas: P 1,  1, T 1 P 2,  2, T 2 Displaced gas

Stability against Convection Since P 2 =P 2 ’ (the gas will adjust to equalize the pressure), then  2 T 2 =  2 ’T 2 ’ To be stable against convection,  2 ’ must be greater than  2 Thus, T 2 must be greater than T 2 ’ That is, the temperature in the moving element must decrease more rapidly than in the surrounding medium: dT/dr element > dT/dr surroundings

Stability Criterion in Terms of Pressure Since pressure falls upward in the atmosphere, the stability criterion can be rewritten as: Take the derivative and multiply by P/T to get: or

Adiabatic Equilibrium If the surroundings are in radiative equilibrium, and no heat is transferred between the element and the surrounding gas, the rising gas is said to be in adiabatic equilibrium (i.e. no energy transfer). For gas in adiabatic equilibrium, PV  = constant and where  = 5/3 for ionized gas and is less for neutral or incompletely ionized regions near the surface. (Recall that  is related to the polytropic index as  = n/(n+1) and is the ratio of the specific heat of the gas under constant pressure to the specific heat of the gas under constant volume.)

The Temperature Gradient If the temperature gradient then the gas is stable against convection. For levels of the atmosphere at which ionization fractions are changing, there is also a dlog  /dlogP term in the equation which lowers the temperature gradient at which the atmosphere becomes unstable to convection. Complex molecules in the atmosphere have the same effect of making the atmosphere more likely to be convective.

When Is Convection Important? When opacities are high, temperature gradients become steep (i.e. the opacity is so large that the transfer of energy by radiation is inefficient) Stars of F and cooler spectral type have surface convection zones Surface convection zones become deeper with later spectral types until the cool M dwarfs, which are fully convective Surface convection drives the formation of chromospheres, and acoustic or magnetic transport may play a role in carrying energy above the temperature minimum at the top of the photosphere Convection is also important in stellar interiors, and will probably be covered in that course

Class Investigation Using the Kurucz models provided, map out the effective temperatures and surface gravities at which significant flux is carried by convection at optical depth 1 for main sequence stars and for supergiants. Again, assume  = 5/3.

Conclusions Convection is (generally) not present for Teff>9000K For A stars, a thin convective zone is present ( km) below the visible layers Giant stars are less likely to be convective at a given temperature than main sequence stars because the low density of the atmosphere makes convection inefficient for transferring energy

Convection in the Sun We observe convection in the Sun in the form of granulation. Each granule is the top of a rising column of hot gas, and the granules are surrounded by cooler falling gas The granules are typically 1000 km in size Note that the distance from t = 1 to t=25 in the Sun is less than about 100 km, just a fraction of the size of a convective cell

Solar Granulation

The Mixing Length Formalism We have developed a test for the stability of an atmosphere against convection, but we also need a mathematical tool to compute the flux carried by convection in a stellar atmosphere. Convection is a hard problem The mixing length formalism was developed in the 1950’s (e.g. Erika Bohm-Vitense) and is still the most widely used treatment of convection A proper theory of convection is beginning to come from 2D and 3D hydrodynamical calculations

Definition of the Mixing Length The mixing length L is the distance traveled by a convective cell before merging into the surrounding medium The “mixing length to pressure scale height ratio” (  = L/H) just expresses the assumed mixing length in terms of a characteristic atmospheric length H (the distance over which the pressure is reduced by the factor e) In the case of no convection,  =0 When convection is present,  is typically assumed to be about 1.5, although values from zero up to 2-3 are used.