Interesting News… RegulusInteresting News… Regulus

•Age: a few hundred million years •Mass: 3.5 solar masses•Rotation Period: <1 day•Equatorial Velocity: 200 km s-1

•Axial tilt: 86 degrees from celestial north•Gravity darkening:

•poles 15,000K•equator 10,000K

•5 times brighter at the poles

Center for High Angular Resolution Astronomy (CHARA at Georgia State )

Review of Lecture 4Review of Lecture 4

• Forms of the radiative transfer equation

• Conditions of radiative equilibrium

• Gray atmospheres– Eddington Approximation

• Limb darkening

Convective Energy Transport/Chapter Convective Energy Transport/Chapter 77

• Stability criterion for convection• Adiabatic temperature gradient• When is convection important• Convection in the Sun• The Mixing Length Formalism

Criterion for Stability against Criterion for Stability against ConvectionConvection

If we displace an element of gas, will it continue to move in the same direction?

P2’ = P2

If 2’ < 2, the element will continue to rise.

P2’=P2

2’T2’

Initial gas:P1, 1, T1

P2, 2, T2

Displaced gas

Stability against ConvectionStability against Convection

• Since P2=P2’ (the gas will adjust to equalize the pressure), then

2T2=2’T2’• To be stable against convection, 2’

must be greater than 2

• Thus, T2 must be greater than T2’• That is, the temperature in the moving

element must decrease more rapidly than in the surrounding medium:

dT/drelement > dT/drsurroundings

Stability Criterion in Terms of PressureStability 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

gssurroundinblob dP

dT

dP

dT

gssurroundinblob TdP

PdT

Tdp

PdT

gssurroundinblob Pd

Td

Pd

Td

log

log

log

log

Adiabatic EquilibriumAdiabatic 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.)

1log

log

Td

Pd

The Temperature GradientThe Temperature Gradient• If the 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.

1log

log

Td

Pd

Class InvestigationClass Investigation

• Using the Kurucz models provided, map out the effective temperatures and surface gravities at which significant flux is carried by convection at a level where T=1.5 Teff for main sequence stars and for supergiants. Again, assume = 5/3.

When Is Convection Important?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

Convection in the SunConvection in the Sun• Each granule is the

top of a rising column of hot gas, and the granules are surrounded by cooler falling gas

• characteristics– 1000 km in size T~200K– velocity~200 m s-1

• The distance from = 1 to =25 is less than about 100 km, just a fraction of the size of a convective cell

Convection – the Movie!Convection – the Movie!

SVST (La Palma)

The Mixing Length FormalismThe Mixing Length Formalism• stability criterion for

convection • need a mathematical

formalism to compute the flux carried by convection in a stellar atmosphere (unsolved problem)

• Mixing length formalism (developed in the 1950’s by Erika Bohm-Vitense) is still the most widely used treatment of convection • A proper theory of

convection is beginning to emerge from 2D and 3D hydrodynamical calculations

Definition of the Mixing Definition of the Mixing LengthLength

• 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.

Calibration of the Mixing Length Calibration of the Mixing Length ParameterParameter

• Using helioseismology = 1.8473 +/- 0.002

• From cluster CMDs

Straniero et al. 1997ApJ, 490, 425