Question: You are observing a spherical cloud with radius a of glowing interstellar gas, which is imaged and resolved by your telescope. The cloud has a

You are observing a spherical cloud with radius a of glowing interstellar gas, which is imaged and resolved by your telescope. The cloud has a uniform, isotropic monochromatic emission coefficient, j0 (defined as the energy per unit volume, time, frequency and solid angle). Assuming that the cloud is optically thin for all frequencies of interest, calculate how the monochromatic specific intensity, I (defined as the energy per unit area, time, frequency and solid angle), of the image depends on the projected radial distance from the center, r. The figure below shows the 2D geometry in the plane of the image as seen by your telescope (left) and a cross section of the 3D cloud (right). Because the cloud is at a very large distance from you, assume that all rays reaching your telescope are effectively parallel. [Hint: Find the solution for the equation of radiative transfer as a function of r and properties defined in the text only.]

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