Problems

#02

$\lambda = c/\nu$, with $c=3\times 10^8$ m/s and $\nu=100$ MHz (or $1\times 10^8$ Hz), the wavelength of FM 100 is 3 m.

#06

Object A is hotter. Wien's Law is that $\lambda_{\rm peak} \propto 1/T$, so the ratio of temperatures is $T_A/T_B = \lambda_B/ \lambda_A = 650/200 = 3.25$. Stefan's Law is that the rate of energy emitted from a unit of area goes as $T^4$, and so object A radiates $(3.25)^4=111$ times more energy per second than object B.

#09

This is a proportions problem, similar to the previous problem. Using Wien's Law, we have that $T_*/T_\odot = \lambda_\odot/ \lambda_*$. Solving for the protostar peak wavelength $\lambda_*$ by cross-multiplying, we have that $\lambda_* = (T_\odot/T_*)\times \lambda_\odot = (5800/1000)\times 500 = 2900$ nm, or 29,000 Å, far in the infrared.

#13

A Doppler shift problem. The signal is 100 MHz, but you want to receive it at 99 MHz. That is a drop in frequency by 1 MHz out of 100 MHz, or 1%. This is normally called a redshift (recall, a drop in frequency is an increase in wavelength). The spacecraft would have to be moving away from the Earth. The speed $v$ is given by $v/c = \Delta \nu/\nu_0 = (\nu-\nu_0)/\nu_0$, where $c$ is the speed of light, $\nu$ is the measured frequency, and $\nu_0$ is the original emitted frequency. Keeping frequencies in MHz, and using $c=300,000$ km/s, the spacecraft must move away from the Earth at $v=(1/100)\times c = 3,000$ km/s. (That converts to almost 11 million km per hr!)