# Blog

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## Reaching Orbit

The following code illustrates the effect of the initial velocity on the dynamics of an object released in a gravitational field. A very simple numerical integration of the equation of motion gives the trajectory, which is plotted below for four different initial speeds for a rocket released at 200 km altitude parallel to the Earth's surface. At this altitude the speed needed for a circular orbit is about 7.8 km/s. You can read more about this kind of simulation at the Wikipedia page for Newton's cannonball.

## Moiré patterns in a pair of hexagonal lattices

A Moiré pattern is an interference pattern that occurs when two grids of repeating lines or shapes are rotated by a small amount relative to one another (oblig. xkcd).

## Gyromotion of a charged particle in a magnetic field

A charged particle of mass $m$ and charge $q$ moving with a velocity $\boldsymbol{v}$ in an an electric field $\boldsymbol{E}$ and a magnetic field $\boldsymbol{B}$ is subject to a Lorentz force, $\boldsymbol{F}$, given by $$\boldsymbol{F} = q(\boldsymbol{E} + \boldsymbol{v}\times\boldsymbol{B}).$$ The equation of motion for a single particle is therefore given by Newton's second law as $$\boldsymbol{\ddot{r}} = \frac{q}{m}(\boldsymbol{E} + \boldsymbol{v}\times\boldsymbol{B}).$$ Here we will consider a uniform magnetic field, $\boldsymbol{B} = (0,0,B)$ and zero electric field, $E=0$. In this case, the trajectory of the particle can be obtained by solving the equation of motion analytically, but here we integrate it numerically using SciPy's integrate.odeint method. Assuming the particle starts off with non-zero components of its velocity parallel ($v_\parallel$) and perpendicular ($v_\perp$) to the magnetic field, it moves in a helix, with radius given by $$\rho = \frac{mv_\perp}{|q|B},$$ known as the Larmor or cyclotron radius (or gyroradius).

## Types of plasma

Two important parameters in plasma physics are the electron Debye length, $\lambda_{\mathrm{D}e}$, a measure of the distance over which charge-screening effects occur and deviations from quasi-neutrality are observed, and the number of paricles in a "Debye cube" (of side length $\lambda_{\mathrm{D}e}$), $N_\mathrm{D}$.

## Plotting nuclear fusion cross sections

In a nuclear fusion reaction two atomic nuclei combine to form a single nucleus of lower total mass, the difference in mass, $\Delta m$ being released as energy in accordance with $E = \Delta m c^2$. It is this process which powers stars (in our own sun, hydrogen nuclei are fused into helium), and nuclear fusion has been actively pursued as a potential clean and cheap energy source in reactors on Earth for over 50 years.