The energy density u of starlight in the solar neighbourhood is about 0.45 eV cm−3. It is sometimes convenient to approximate the density of the background starlight as a “diluted blackbody” of “dilution factor” W and “color temperature” Tc. The dilution factor is defined to be the ratio of the actual energy density u to the energy density of (undiluted) blackbody radiation of temperature Tc. Take Tc = 5000 K as characteristic of the starlight background.

a) Estimate W for the starlight background. (And what is W for the CMB?)

I used the Stefan-Boltzmann law here: u_c = 4 \sigma T^4 /c, to calculate the energy density of the (undiluted) blackbody radiation of temperature Tc. I took the ratio of the energy densities (definition of W) en I got 0.17 $ 10^-12. I was wondering if any one have an idea if this result is acceptable? because I have no idea of a typical value of this quantity

b) The cosmic background has a temperature of 2.7 K. Estimate the ratio of the number density of microwave background photons to the number density of starlight photons in the solar neighborhood.

Here using the average photon energy is 3kT = 1.13 × 10−22 J, I calculated the number density of CMB: n_b = u_cmb / 3kT_cmb. Using the Stefan-Boltzmann equation, I derived the corresponding temperature to be able to calculate the number density with n = u / 3kT . My question is: Is it acceptable to calculate the temperature for the starlight with S.B. law? I got a result of T = 2.44K, is that even possible?

Thank you guys in advance