$($use python code$)$The three$-$body problem has occupied the minds of great mathematicians for over $300$ years. In

contrast with the case of two point particles, a system of three point particles interacting under

gravitational attraction is in general impossible to solve analytically, and in many cases the solution

is unstable. Until recently the only known stable solutions were provided by making simplifying

assumptions. However, thirteen new stable $($numerical$)$ solutions to the three$-$body problem were

published in February $2013.$ Each solution has parameters given by:

where a and $b$ specify the particular solution. Note that the three planets remain in the plane

$z=0$ at all times.

$(1)$ Write down, in vector form, the six ODEs governing the motion of the three planets interacting

gravitationally. $($If written in component form, these six equations would give $18$ equations.$)$

$(2)$ Adapt your two$\_$body function to solve the three$-$body problem. Name your new function

three$\_$body $($x$),$ where $x$ is now a Numpy array of shape $3,2,3.$

$(3)$ For the parameters $a=0\mathrm{.}347111$ and $b=0\mathrm{.}532728,$ the three planets chase each other in a

figure$-$eight of period $T6\mathrm{.}324449.($This isolated example was first discovered numerically

in $1993$ by Cris Moore.$)$ Calculate the complete orbit of the three planets numerically using

your function rk$4$ and then plot it$.$

$(4)$ Compute and plot the complete orbits of the three planets for the following parameters:

$($a$)$ Butterfly I of period $T6\mathrm{.}235641$ for $a=0\mathrm{.}306893$ and $b=0\mathrm{.}125507.$

$($b$)$ Ying$-$Yang $1$b of period $T10\mathrm{.}962563$ for $a=0\mathrm{.}282699$ and $b=0\mathrm{.}327209.$

$($c$)$ Dragonfly of period $T21\mathrm{.}270975$ for $a=0\mathrm{.}080584$ and $b=0\mathrm{.}588836.$

If the solutions appear to be unstable, you may need to reduce the step size, $h.$ The complete set

of parameters to generate solutions of the three$-$body problem can be found at the Three$-$Body

Gallery $($Institute of Physics Belgrade$).$