Energy Transport

Energy Transport • Formal solution of the transfer equation • Radiative equilibrium • The gray atmosphere • Limb darkening

The Transfer Equation Recall: for radiation passing through a gas, the change in In is equal to: dIn = intensity emitted – intensity absorbed dIn = jnrdx – knrIndx or dIn/dtn = -In + Sn

The Integral Form • A solution usually takes the form • Where • One must know the source function to solve the transfer equation • For LTE, Sn(tn) is just the Planck function Bn(T) • The solution is then just T(tn) or T(x)

Toss in Geometry • In real life, we are interested in In from an arbitrary direction, not just looking radially into the star • In plane parallel geometry we have azimuthal symmetry, so that

Radiative Equilibrium • To satisfy conservation of energy, the total flux must be constant at all depths of the photosphere • Two other radiative equibrium equations are obtained by integrating the transfer equation over solid angle and over frequency

Integrating Over Solid Angle • Assume knr and Sn are independent of direction, and substitute the definitions of flux and mean intensity: • becomes: • Then integrate over frequency:

(integrating over frequency…) • LHS is zero in radiative equilibrium, so • The third radiative equilibrium condition is also obtained by integrating over solid angle and frequency, but first multiply through by cos q. Then

3 Conditions of Radiative Equilibrium: • 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

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

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

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

Class Problem • The opacity, effective temperature, and gravity of a pure hydrogen gray atmosphere are k = 0.4 cm2 gm-1, 104K, and g=2GMSun/RSun2. Use the Eddington approximation to determine T and r at optical depths t = 0, ½, 2/3, 1, and 2. Note that density equals 0 at t = 0.

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.

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:

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

Comparing the Gray Atmosphere to the Real Sun

Energy Transport