Question #42163

A spherical blackbody source of proper temperature $T_0$ moves directly towards a stationary observer with a constant velocity $v = \frac{4}{5}c$. In the rest frame of the source, the spectral radiance peaks at a wavelength $\lambda_0$. The observer measures the incoming radiation. Let $\lambda_{obs}$ be the observed peak wavelength. Let $I_{obs}$ be the total observed radiance (energy per unit area per unit time per unit solid angle) integrated over all frequencies, and $I_0$ be the corresponding total proper radiance. Given that the quantity $\frac{I_\nu}{\nu^3}$ is Lorentz invariant (where $I_\nu$ is the spectral radiance and $\nu$ is the frequency), determine the values of $\frac{\lambda_{obs}}{\lambda_0}$ and $\frac{I_{obs}}{I_0}$.