Feb. 2, 2011 Rosseland Mean Absorption Poynting Vector Plane EM Waves The Radiation Spectrum: Fourier Transforms

1. Feb. 2, 2011Rosseland Mean AbsorptionPoynting VectorPlane EM WavesThe Radiation Spectrum: Fourier Transforms

2. Rosseland Mean Opacity Recall that for large optical depth In a star, is large, but there is a temperature gradient

3. Plane parallel atmosphere How does Fn(r) relate to T(r)? z dz ds so Equation of radiative transfer: emission scattering absorption

4. = Planck function For thermal emission: so Where the source function

5. Rewrite: (1) “Zeroth” order approximation: Independent of m 1st order approximation: Equation (1)  Depends on m

6. = 0

7. Integrate over all frequencies: Now recall: s = Stefan-Boltzman Constant

8. Define Rosseland mean absorption coefficient: Combining, Equation of radiative diffusion In the Rosseland Approximation • Flux flows in the direction opposite the temperature gradient • Energy flux depends on the Rosseland mean absorption coefficient, • which is the weighted mean of • Transparent regions dominate the mean

9. Conservation of Charge Follows from Maxwell’s Equations Take charge density But so current density

10. Poynting Vector One of the most important properties of EM waves is that they transport energy— e.g. light from the Sun has traveled 93 million miles and still has enough energy to do work on the electrons in your eye! Poynting vector, S: energy/sec/area crossing a surface whose normal is parallel to S Poynting’s theorem: relates mechanical energy performed by the E, B fields to S and the field energy density, U

11. Mechanical energy: Lorentz force: work done by force rate of work =0 since

12. but also so...

13. More generally, where U(mech) = mechanical energy / volume

14. Back to Poynting’s Theorem  Maxwell’s Equation use the vector identity But and

15. (1) = field energy / volume Now So (1) says rate of change of field energy per volume rate of change of mechanical energy per volume +

16. Plane Electromagnetic Waves

17. Maxwell’s Equations in a vacuum: (3) (1) (4) (2) These equations predict the existence of WAVES for E and B which carry energy Curl of (3)  use (4) 

18. Use vector identity: =0 from (1) Vector Wave Equation SO Similarly, for B:

19. Note: operates on each component of so these 2 vector wave equations are actually 6scalar equations So, for example, one of the equations is Similarly for Ey, Ez, Bx, By and Bz

20. What are the solutions to the wave equations? First, consider the simple 1-D case -- Wave equation A solution is A = constant (amplitude) [kx] = radians [kvt] = radians So the wave equation is satisfied Ψ is periodic in space (x) and time (t)

22. The wavelength λ corresponds to a change in the argument of the sine by 2π frequency angular frequency