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{\Large ASTR450 Homework \# 8 -- Programming\\
Due Tuesday, April 15}
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1. {\bf Two-Body Problem.} Write a two part computer program that
translates i) from orbital elements $a,e,i,\Omega,\omega,\nu$ to
positions and velocities ($x,y,z,v_x,v_y,v_z$) and ii) from positions
and velocities back to orbital elements. Devise your own algorithms,
or use the ones given in Danby, Section 6.15. Your routines only need
to work for elliptical orbits.  For Danby's algorithm, note that Eq.
6.15.4 comes from 6.2.5 and that {\bf P} is a vector pointing from the
mass-occupied focus to pericenter with magnitude $e$. If you don't
feel inspired, you can write program ii) to work for 2D orbits
$i=\Omega=0$ only, but extra credit will be given for a fully
functional program that can handle 3D orbits.

Test your program by translating $(x,y,z,v_x,v_y,v_z) \rightarrow
(a,e,i,\Omega,\omega,\nu) \rightarrow (x,y,z,v_x,v_y,v_z$) and
$(a,e,i,\Omega,\omega,\nu) \rightarrow (x,y,z,v_x,v_y,v_z) \rightarrow
(a,e,i,\Omega,\omega,\nu)$ for a number of
test cases. Also test your programs against results from the 2D and 3D
orbit viewers and against your intuition (e.g. circular orbits ought
to have ${\bf r} \perp {\bf v}$).

2.{\bf Perturbations.} Using the paper by Burns et al. (1979), fill in
the missing steps in the derivations of Eqs 24 and 28 for $da/dt$ and
$de/dt$. Show enough work that I can see that you understand the
derivations!

3. Do your taxes.



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