Wednesday, November 19, 2014

a quantum of sunshine

the Planck-spectrum is dominated by Boltzmann-statistics at high energies and by Bose-statistics at low energies. could you estimate how much solar power we receive per unit area at Earth from the quantum-mechanical part of the Sun's spectrum?

Wednesday, November 12, 2014

tidal forces in the gravity train

earlier this year there was a question about the gravity train to New Zealand. a passenger would feel two kinds of tidal forces: firstly, there's a differential gravitational field between head and feet, and secondly, the shoulders would each follow trajectories pointing towards the Earth's centre. which of the two effects is larger?

Wednesday, November 5, 2014

anti-Earth

imagine there would be a planet identical in mass as our Earth circling the Sun in our orbit with a lag of exactly 6 months, i.e. it would always be behind the Sun and not directly visible for us. would there be nevertheless possibilities of detecting it?

Wednesday, October 29, 2014

correlation coefficient

can you provide a proof of the Cauchy-Schwarz inequality and show with it that the correlation coefficient
\begin{equation}
r = \frac{\langle xy\rangle}{\sqrt{\langle x^2\rangle\:\langle y^2\rangle}}
\end{equation}
always ranges between $-1\leq r\leq +1$?

bonus question: can you compute the normalisation, variance and kurtosis of $p(x)\mathrm{d}x\propto\exp(-x^4)\mathrm{d}x$?

Wednesday, October 22, 2014

Eddington temperature

Astrophysical objects are limited in their luminosity by the Eddington luminosity, where radiation pressure exceeds gravity. Would it be sensible to define a limiting Eddington-temperature for Planckian spectra?