Assignment 6

46. Assignment 6#

Upload a pdf and a notebook for the following problems:

Planck’s law as a function of wavelength:

(46.1)#\[ B_\lambda(\lambda, T)=\frac{2 h c^2}{\lambda^5} \frac{1}{e^{h c /\left(\lambda k_{\mathrm{B}} T\right)}-1} \]

Use change of variables to rewrite (46.1) in terms of wavenumber \(\tilde{\nu} = 1/\lambda\) and show that it is:

(46.2)#\[ B_{\tilde{\nu}}(\tilde{\nu}, T)=2 h c^2 \tilde{\nu}^3 \frac{1}{e^{h c \tilde{\nu} /\left(k_{\mathrm{B}} T\right)}-1} \]

Integrate (46.2) to find the Stefan-Boltzman equation given that

(46.3)#\[ \int_0^\infty \frac{u^3}{(e^u -1 )} du = \frac{\pi^4}{15} \]

i.e. show that:

\[ \int_0^\infty B_{\nu} d\nu = \frac{\sigma}{\pi} T^4 \]

where

\[ \sigma=\frac{2 \pi^4 k^4}{15 c^2 h^3} \]

Show that the maximum value for (46.1) occurs at:

\[ \lambda_{max} \propto \frac{1}{T} \]

Integrate \(Z=\int D^6 n(D) dD\) on paper, assuming a Marshall Palmer size distribution and show that it integrates to:

\[ Z \approx 300 RR^{1.5} \]

with Z in \(mm^6\,m^{-3}\) and RR in mm/hr. It’s helpful to know that:

\[ \int^\infty_0 x^n \exp( -a x) dx = n! / a^{n+1} \]

Repeat using numerical integration in python (i.e. np.diff and np.sum) and show that the result agrees.