Ch. 5 - Basic Definitions

1. Ch. 5 - Basic Definitions • Specific intensity/mean intensity • Flux • The K integral and radiation pressure • Absorption coefficient & optical depth • Emission coefficient & the source function • Scattering and absorption • Einstein coefficients

2. We want to characterize the radiation from Specific Intensity • Area DA • at an angle of view q from the normal to the surface • through an increment of solid angle Dw Normal q to observer Dw DA Assume no azimuthal dependence

3. Specific Intensity • Average Energy (Eldl) is the amount of energy carried into a cone in a time interval dt • Specific Intensity in cgs (ergs s-1 cm-2 sr-1 Å-1) • Intensity is a measure of brightness – the amount of energy coming from a point on the surface towards a particular direction at a given time, at a frequency n • For a black body radiator, the Planck function gives the specific intensity (and it’s isotropic) • Normally, specific intensity varies with direction

4. In vs Il • The shapes of In and Il are different because dn and dl are different sizes at the same energy of light: dn = -(c/l2) dl • For example, in the Sun, Il peaks at ~4500Å while In peaks at ~8000 Å

5. Mean Intensity • Average of specific intensity over all directions • If the radiation field is isotropic (same intensity in all directions), then <In>=In • Black body radiation is isotropic and <In>=Bn

6. Flux • The flux Fn is the net energy flow across an area DA over time Dt, in the spectral range Dn, integrating over all directions • energy per second at a given wavelength flowing through a unit surface area (ergs cm-2 s-1 Hz-1) • for isotropic radiation, there is no net transport of energy, so Fn=0

7. On the physical boundary of a radiating sphere… • if we define Fn =Fnout + Fnin • then, at the surface, Fnin is zero • we also assumed no azimuthal dependence, so • which gives the theoretical spectrum of a star

8. One more assumption: • If In is independent of q, then • This is known as the Eddington Approximation (we’ll see it again)

9. Specific Intensity vs. Flux • Use specific intensity when the surface is resolved (e.g. a point on the surface of the Sun). The specific intensity is independent of distance (so long as we can resolve the object). For example, the surface brightness of a planetary nebula or a galaxy is independent of distance. • Use radiative flux when the source isn’t resolved, and we're seeing light from the whole surface (integrating the specific intensity over all directions). The radiative flux declines with distance (1/r2).

10. The K Integral • The K integral is useful because the radiation exerts pressure on the gas. The radiation pressure can be described as

11. Radiation Pressure • Again, if In is independent of direction, then • Using the definition of the black body temperature, the radiation pressure becomes

12. Luminosity • Luminosity is the total energy radiated from a star, at all wavelengths, integrated over a full sphere.

13. Class Problem • From the luminosity and radius of the Sun, compute the bolometric flux, the specific intensity, and the mean intensity at the Sun’s surface. • L = 3.91 x 1033 ergs sec -1 • R = 6.96 x 1010 cm

14. Solution • F= sT4 • L = 4pR2sT4 or L = 4pR2 F, F = L/4pR2 • Eddington Approximation – Assume In is independent of direction within the outgoing hemisphere. Then… • Fn = pI n • Jn = ½ In (radiation flows out, but not in)

15. The Numbers • F = L/4pR2 = 6.3 x 1010 ergs s-1 cm-2 • I = F/p = 2 x 1010 ergs s-1 cm-2 steradian-1 • J = ½I= 1 x 1010 ergs s-1 cm-2 steradian-1 (note – these are BOLOMETRIC – integrated over wavelength!)

16. The K Integral and Radiation Pressure • Thought Problem: Compare the contribution of radiation pressure to total pressure in the Sun and in other stars. For which kinds of stars is radiation pressure important in a stellar atmosphere?

17. Absorption Coefficient and Optical Depth • Gas absorbs photons passing through it • Photons are converted to thermal energy or • Re-radiated isotropically • Radiation lost is proportional to • absorption coefficient (per gram) • density • intensity • pathlength • Optical depth is the integral of the absorption coefficient times the density along the path (if no emission…)

18. Class Problem • Consider radiation with intensity In(0) passing through a layer with optical depth tn = 2. What is the intensity of the radiation that emerges?

19. Class Problem • A star has magnitude +12 measured above the Earth’s atmosphere and magnitude +13 measured from the surface of the Earth. What is the optical depth of the Earth’s atmosphere at the wavelength corresponding to the measured magnitudes?

20. Emission Coefficient • There are two sources of radiation within a volume of gas – real emission, as in the creation of new photons from collisionally excited gas, and scattering of photons into the direction being considered. • We can define an emission coefficient for which the change in the intensity of the radiation is just the product of the emission coefficient times the density times the distance considered. Note that dI does NOT depend on I!

21. The Source Function • The “source function” is just the ratio of the absorption coefficient to the emission coefficient: Sounds simple, but just wait….

22. Pure Isotropic Scattering • The gas itself is not radiating – photons only arise from absorption and isotropic re-radiation • Contribution of photons proportional to solid angle and energy absorbed: • Jn is the mean intensity: dI/dtn = -In + Jv • The source function depends only on the radiation field

23. For pure isotropic scattering • Remember the definition of Jn • So Jn = jn/kn • Hey! Then Jn = Sn for pure isotropic scattering

24. Pure Absorption • No scattering – all incoming photons are destroyed and all emitted photons are newly created with a distribution set by the physical state of the gas. • Source function given by Planck radiation law • Generally, use Bn rather than Sn if the source function is the Planck function

25. Einstein Coefficients • For spectral lines or bound-bound transitions, assumed isotropic • Spontaneous emission is proportional to Nu x Einstein probability coefficient, Aul jnr = NuAulhn • (Nu is the number of excited atoms per unit volume) • Induced emission proportional to intensity knr = NlBluhn – NuBulhn

26. Induced (Stimulated) Emission • Induced emission in the same direction as the inducing photon • Induced emission proportional to intensity knrIn = NlBluInhn – NuBulInhn True absorption Induced emission

27. Radiative Energy in a Gas • As light passes through a gas, it is both emitted and absorbed. The total change of intensity with distance is just • dividing both sides by -knrdx gives

28. The Source Function • The source function Sn is defined as the ratio of the emission coefficient to the absorption coefficient • The source function is useful in computing the changes to radiation passing through a gas

29. The Transfer Equation • We can then write the basic equation of transfer for radiation passing through gas, the change in intensity In is equal to: dIn = intensity emitted – intensity absorbed dIn = jnrdx – knrIndx dIn /dtn = -In + jn/kn = -In + Sn • This is the basic equation which must be solved to compute the spectrum emerging from or passing through a gas.

30. Special Cases • If the intensity of light DOES NOT VARY, then Il=Sl (the intensity is equal to the source function) • When we assume LTE, we are assuming that Sl=Bl

31. Thermodynamic Equilibrium • Every process of absorption is balanced by a process of emission; no energy is added or subtracted from the radiation • Then the total flux is constant with depth • If the total flux is constant, then the mean intensity must be equal to the source function: <I>=S

32. Simplifying Assumptions • Plane parallel atmospheres (the depth of a star’s atmosphere is thin compared to its radius, and the MFP of a photon is short compared to the depth of the atmosphere • Opacity is independent of wavelength (a gray atmosphere)

33. Eddington Approximation • 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