Introduction to Cosmology Ryden · Problem 2.6

Fundamental Observations

Physics — Astrophysics · blackbody-low-energy-fraction

Problem

Show that for an energy threshold

E_0 \ll kT

, the fraction of blackbody photons that have energy

hf < E_0

\frac{n(hf < E_0)}{n_{\gamma}} \approx 0.21 \left(\frac{E_0}{kT}\right)^2. \tag{2.44}

Microwave (and far infrared) photons with a wavelength

\lambda < 3\,\mathrm{cm}

are strongly absorbed by

\mathrm{H_2O}

and

\mathrm{O_2}

molecules. What fraction of the photons in today's cosmic microwave background have

\lambda > 3\,\mathrm{cm}

, and thus are capable of passing through the Earth's atmosphere and being detected on the ground?