2. Radiation Field Basics I

1. 2. Radiation Field Basics I • Rutten: 2.1 • Basic definitions of intensity, flux • Energy density, radiation pressure

2. In(r, n) s dW q dA Specific Intensity Pencil beam of radiation at position r, direction n, carrying energy dEn, passing through area dA, between the times t and t + dt, in the frequency band between n and n + dn. s is normal to dA Units of In: J/m2/s/Hz/sr (ergs/cm2/s/Hz/sr)

3. p’ p s s’ q q’ r dA’ dA No sources or sinks of radiation. Pencil beam of radiation passing through dA at p and dA’ at p’. Radiant energy passing through both areas is the same: Solid angle dW subtended by dA’ at p, dW’ subtended by dA at p’: Therefore: Specific intensity independent of distance when no sources or sinks.

4. Mean Intensity Jn is a very important quantity. It determines level populations and ionization state throughout atmosphere. Like In, it is a function of position. For a plane parallel atmosphere (no f dependence) with m = cos q: We’ll later use moments of the radiation field. Jn is the 0th moment Moment operator M operating on f :

5. R* q* r I* What is Jn at distance r from a star with uniform specific intensity I* across its surface? I = I* for 0 < q < q* (m* < m < 1) I = 0 for q > q* (m < m*) w is the dilution factor At large r, w = R2/4r2

6. Flux The flux enables us to calculate the total energy,E, passing through a surface in a given time, i.e., integrated over all directions. The energy transport can be positive or negative. Monochromatic Flux, Fn : The net flow of radiant energy per second through an area dA in time dt in frequency range dn. This is used for specifying the energetics of radiation through stellar interiors, atmospheres, ISM, etc. In principle, flux is a vector.

7. In stellar atmospheres, the outward radial direction is always implied positive, so that With both the outward flux, Fn+, and the inward flux, Fn-, positive. Isotropic radiation has Fn+ = Fn- = pIn and Fn = 0. Axisymmetry:

8. The flux emitted by a star per unit area of its surface is Fn = Fn+ = pIn* where In* is the intensity, averaged over the apparent stellar disk, received by an observer. This equality is why that flux is often written as pF = F ,so that F = I*, with F called the Astrophysical Flux. This explains the often confusing factors of p that are floating about in definitions of flux: F = Monochromatic Flux or just the Flux; F = Astrophysical Flux They are related by pF = F . In terms of moments of the radiation field, the first moment is defined as the Eddington Flux, Hn . For plane parallel geometry:

9. n q dA dV ds = c dt Energy Density The energy flow in a beam of radiation is The flow has velocity c (photons) and travels a distance ds in time dt = ds/c through volume dV = dA ds cos q. Thus, each beam carries dEn = (1/c) In dW dV. If multiple beams pass through a small volume DV, integration over DV and over all beam directions gives the radiant energy En dn contained in DV across bandwidth dn as:

10. For sufficiently small DV, the intensity is homogeneous, so the two integrations (V, W) are independent. The energy density is

11. p = hn/c q n dA hn/c cos q ds = c dt Radiation Pressure Each photon has momentum p = hn/c. Component of momentum normal to a solid wall per time per area is Re-write in terms of In and integrating over solid angle gives: A Isotropic radiation has pn = un/3. Radiation pressure is analogous to gas pressure, being the pressure of the photon gas.