Animate a planet's orbit.
A planet's orbit around a star has the shape of an ellipse. The purpose of
this exercise is to make an animation of the movement along the orbit. One
should see a small disk, representing the planet, moving along an elliptic
curve. An evolving solid line shows the development of the planet's orbit as
the planet moves.
The points \((x, y)\) along the ellipse are given by the expressions
$$
x=a \cos (\omega t), \quad y=b \sin (\omega t)
$$
where \(a\) is the semimajor axis of the ellipse, \(b\) is the semiminor axis,
\(\omega\) is an angular velocity of the planet around the star, and \(t\) denotes
time. One complete orbit corresponds to \(t \in[0,2 \pi / \omega] .\) Let us
discretize time into time points \(t_{k}=k \Delta t\), where \(\Delta t=2 \pi
/(\omega n) .\) Each frame in the movie corresponds to \((x, y)\) points along
the curve with \(t\) values \(t_{0}, t_{1}, \ldots, t_{i}, i\) representing the
frame number \((i=1, \ldots, n)\). Let the plot title of each frame display the
planet's instantaneous velocity magnitude. This magnitude is the length of the
velocity vector
$$
\left(\frac{d x}{d t}, \frac{d y}{d t}\right)=(-\omega a \sin (\omega t),
\omega b \cos (\omega t))
$$
which becomes \(\omega \sqrt{a^{2} \sin ^{2}(\omega t)+b^{2} \cos ^{2}(\omega
t)}\)
Implement the visualization of the planet's orbit using the method above. Run
the special case of a circle and verify that the magnitude of the velocity
remains constant as the planet moves. Name of program file: planet_orbit.py.
