1. Rotation of 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 )

2. Radiative Transfer dIn/dtn=-In + Sn = -In +jn/kn
Radiation is the dominant mechanism for energy transport in photospheres
Convection can be important below the surface, rarely carries much flux in the photosphere
Hence – assume radiative transfer
Formal solution of the eqn of radiative transfer is not easy because kn and jn depend on temperature (ionization and excitation)
dIn/dtn=-In + Sn
= -In +jn/kn

3. The Integral Form: How to we get there?
A plausible form for the solution of the differential transfer equation is
Differentiate this wrt tn:
The terms on either side are equal if we set b=-1 and define Sn as:

4. If
then f must equal:
Substituting this expression for f into our guess at the solution to the transfer equation(and setting the constant = In(0)) gives the integral
c0 must be In(0) at tn=0 (at the top of the atmosphere), and we can pull e-tn inside the integral to get:

5. This is the integral form of the transfer equation
One must know the source function to solve the transfer equation. For LTE, Sn(tn) is just the Planck function Bn(T)
In LTE, the solution of the transfer equation becomes just T(tn) (or T(x))

6. The Transfer Equation tn (looking into the star)
Intensity originates at each point on the line of sight
The intensity from each point suffers exponential extinction that depends on the optical depth to that point
The intensity at the “surface” is the sum of the extincted intensities at each point.
In(0)
tn=0 tn
tn - tn
(looking into the star)

7. Toss in Geometry
In real life, we are interested in In from an arbitrary direction, not just looking radially into the star. We must convert line of sight coordinates to polar coordinates.
Here is a slab of a plane parallel photosphere of thickness dr. The increment of the path along the line of sight in the direction “z” is dz = dr sec q (dr = cosq dz) r
r+dr dz q
to observer
local
“zenith” dr
Plane parallel atmosphere

8. More formally…
For most stars, the plane parallel approximation is reasonable (the depth of the solar photosphere is 0.1% of the solar radius), and this assumption leads to azimuthal symmetry
The projection along the line of sight of a distance along some angle q to the line of sight is dz = dr sec q
dI/krdr becomes cos q dI/krdz
The transfer equation becomes

9. Special Case
What do we call this assumption?
At the stellar surface, we can divide the specific intensity into two parts, Inin(0) and Inout
Inin (0)= 0
Inout(0) =
For the Sun, we can measure Inout(0) at points on the stellar disk
For stars, we must integrate Inout(0) over the stellar disk to compute the flux

10. Recall the three integrals defined in Chapter 5…
These three integrals also need to be converted to polar coordinates.

11. The Flux Integral (lots of math…) Assume no azimuthal dependence
Divide into Iin and Iout
If LTE, Sn is isotropic
Use exponential integrals
exponential integrals are monotonically decreasing functions of x

12. The Flux Integral
Defined per unit area – total radiation at frequency n is 4pR2Fn(0)
The surface flux is the sum of the source function at each depth times an extinction factor
Assumes kn and tn are isotropic – often true, but not always

13. Radiative Equilibrium
To satisfy conservation of energy, the total flux must be constant at every depth of the photosphere
Two other radiative equilibrium equations are obtained by integrating the transfer equation over solid angle and over frequency

14. Integrating over Solid Angle
Again, assume knr and Sn are independent of direction, and substitute the definitions of flux and mean intensity:
becomes:
Then integrate over frequency:

15. (integrating over frequency…)
LHS is zero in radiative equilibrium, so
In deep layers where the optical depth at all frequencies is large
Jn = Sn

16. The third form of the radiative equilibrium condition is also obtained by integrating over solid angle and frequency, but first multiply through by cos q. Then

17. 3 Conditions of Radiative Equilibrium:
Teff becomes a fundamental constant of a stellar atmosphere.
These three eqns are not independent. An Sn that satisfies one will satisfy all.
For radiative equilibrium Jn=Sn in deep layers.
In real stars, energy is created or lost from the radiation field through convection, magnetic fields, and/or acoustic waves, so the energy constraints are more complicated.

18. Solving the Transfer Equation in Practice
Generally, one starts with a first guess at T(tn) and then iterates to obtain a T(tn) relation that satisfies the transfer equation
The first guess is often given by the “gray atmosphere” approximation: opacity is independent of wavelength (it’s grey…, not really realistic, but useful)

19. Solving the Gray Atmosphere
Integrating the transfer equation over frequency:
gives or
The radiative equilibrium equations give us:
F=F0, J=S, and dK/dt = F0/4p

20. Eddington Approximation (again)
Assume that the intensity of the radiation (Il) has one value in all directions toward the outward facing hemisphere and another value in all directions toward the inward facing hemisphere.
These assumptions combined lead to a simple physical description of a gray atmosphere

21. Steps to a Grey Atmosphere
in practice, this means that
S(t) = J(t) = ½ [Iin(t) + Iout(t)]
Since dK/dt = F0/4p (radiative equil.)
and

22. Boundary Conditions From the K integral: To evaluate the constant:
at the outer boundary (t=0), Iin=0, so I=F0/2p
then

23. In Terms of Temperature
for the case of pure absorption
And we already know that

24. Eddington’s Solution (1926)
Using the Eddington Approximation, one gets
Chandrasekhar didn’t provide a rigorous solution until 1957
Note: One doesn’t need to know k since this is a T(t) relation

25. Limb Darkening
This white-light image of the Sun is from the NOAO Image Gallery. Note the darkening of the specific intensity near the limb.

26. Limb Darkening in a Gray Atmosphere
Recall that
so that as q increases the optical depth along the line of sight increases (i.e. to smaller tn and smaller depth and cooler temperature)
In the case of the gray atmosphere, recall that we got:

27. Limb darkening in a gray atmosphere
so that In(0) is of the form
In(0) = a + bcos q
One can derive that
and

28. Comparing the Gray Atmosphere to the Real Sun

29. Convective Energy Transport
Decline in density and opacity through the photosphere makes convection inefficient – only a small fraction of the energy is transported by convection
About convection
Stability criterion for convection
Adiabatic temperature gradient
When is convection important
Convection in the Sun
The Mixing Length Formalism

30. Criterion for Stability against Convection
If we displace an element of gas, will it continue to move in the same direction?
P2’ = P2
If r2’ < r2, the element will continue to rise.
P2’=P2
r2’
T2’
P2, r2, T2
Displaced gas
Initial gas:
P1, r1, T1

31. Stability against Convection
Since P2=P2’ (the gas will adjust to equalize the pressure), then
r2T2=r2’T2’
To be stable against convection, r2’ must be greater than r2
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

32. 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

33. 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, PVg = constant and
where g = 5/3 for ionized, monatomic gas and is less for neutral or incompletely ionized regions near the surface. g=4/3 for radiation trapped in a convective cell

34. What’s the g???
PVg = constant
g is the ratio of the specific heat of the gas under constant pressure to the specific heat of the gas under constant volume
g is related to the polytropic index as g = n/(n+1)
g is a fudge factor to describe the behavior of the gas with changing pressure and density

35. The 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 dlogm/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.

36. Class 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 g = 5/3.

37. 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

38. Convection 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
DT~200K
velocity~200 m s-1
The distance from t = 1 to t=25 is less than about 100 km, just a fraction of the size of a convective cell

39. Convection – the Movie!
SVST (La Palma)

40. The 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

41. 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” (a = 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, a=0
When convection is present, a is typically assumed to be about 1.5, although values from zero up to 2-3 are used.

42. Calibration of the Mixing Length Parameter
Using helioseismology
a = /
From cluster CMDs
Straniero et al. 1997
ApJ, 490, 425