tt40 wrote:
In everything I've read about planets and elliptical orbits, I can't
ever recall any author (Feynman, Newton, 'Ask an Astronomer' etc.),
explaining exactly 'why' the orbit is elliptical. Oh sure there's been
lots of mathematics to explain the orbit and how it works, but most of
the explanations don't provide a definitive statement as to why it IS
elliptical.

or

(this space left blank for the correct answer)

It is a direct consequence of conserving angular momentum and total
energy when a body moves in an inverse square law gravitational field.

You can derive the equations for orbital motion this way starting from
first principles. Most decent physics text books should cover this in
some detail. The orbit is elliptical because an ellipse is one solution
of the equations of motion (for a bound orbit).

The other solutions are a parabola for a particle just able to escape,
and a hyperbola for a particle with more than enough kinetic energy to
break free (typically non-periodic comets follow these paths).

Regards,
Martin Brown