What is the observed intensity toward the centre of cloud

Question

Assignment:

1. Radiation Quantities

X-ray photons are produced in a cloud of radius R at a uniform rate photons s-¹ cm^-3. The cloud is a distance d away from the observer. The intervening medium is optically thin (neglect absorption). A detector at Earth records x-ray photons arriving from an angle within ±Δθ (the so-called "acceptance half-angle" or "beam" of the instrument) of normal to the detector, which has an e↵ective area ΔA.

a. Assume that the source is completely resolved. What is the observed intensity (photons s^-1 cm^-2 sr^-1) toward the centre of the cloud?

b. Assume that the source is completely unresolved. What is the observed average intensity when the source lies within the beam of the detector?

2. Photoionization

Photoionization is the process in which a photon is absorbed by an atom or molecule and an electron is ejected. An energy at least equal to the ionization potential is required. Let this energy be hv₀ and let  σ_v be the cross-section for photoionization. Show that the number of photoionizations per unit volume per unit time is

4πn _vo∫^∞ σ_vJ_vdv/hv = cn _vo∫^∞ σ_vu_vdv/ hv, 

where n is the number density of absorbing atoms, u is the energy density of the radiation field at frequency v, and J_v is the corresponding mean intensity.