In article S503f.428336$_o.76593@attbi_s71,
Sam Wormley wrote:

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.


The answer lies in mathematics of Kepler, and especially Newton.

Johannes Kepler -- more than 900 pages of calculation in about four
years attempted to measure the orbit of Mars

o from a moving platform
o who's orbit was not centered on the Sun
o rotating on its axis
o of a planet varying in its orbital speed around the Sun.

Kepler's calculations were immensely difficult...

No, it was actually very simple - so simple that it could be
performed by the mathematics which was avalable at Kepler's time (no
calculus, only arithmetic and geometry). Today a motivated schoolkid
would be able to reproduce Kepler's calculations.

Kepler performed his calculations like this: he compared Mars' position
as seen from Earth at one day with Mars' position as seen from Earth
exactly one "Mars year" later. Mars was then in the same position in
its orbit, while the Earth was at two different positions in its orbit.
By assuming a circular orbit of the Earth and performing triangulation,
Kepler was able to deduce the position in space of Mars.

Kepler then repeated this process for Mars in many different
positions in its orbit. From that, Kepler deduced the elongated
orbit of Mars.

Kepler was lucky that he selected Mars for this. If Kepler instead
had selected e.g. Venus, then the orbit of Venus would to Kepler
be indistinguishable from a circle.

but he eventually realized that an ellipse

....and it was with great agony that Kepler dared think that Mars'
orbit might be an ellipse - why? Because an ellipse has two foci: at
one focus the Sun was situated, but the other focus is .... empty!
Since an empty focus "served no purpose", Kepler avoided the ellipse
as long as he could until he first had tried numerous other orbital
shapes with only one focus, such as ovals and "egg lines" - they
didn't match the observations. Finally Kepler tried the ellipse ....
then everything matched beautifully and Kepler felt as if he had woken
up from a long nightmare.

with an eccentricity of about nine percent agreed with
Tycho Brahe's observational data. Kepler found that the orbits of planets
in our solar system follow ellipses, sweeping out equal areas in equal
times (Newton would later show this as the conservation of angular
momentum as you point out).

Newton discovered (and showed mathematically) that objects in free fall
(such as planets influenced by a central force like the Sun's gravity)
follow the paths of conic sections.

Ref: http://learning.physics.iastate.edu/DemoRoom/MU.htm#22

The combination of Newton's law of gravity and F=ma . The task of
deducing all three of Kepler's laws from Newton's universal law of
gravitation is known as the Kepler problem. Its solution is one of the
crowning achievements of Western thought.

A model for Gravitation was essential.
http://scienceworld.wolfram.com/physics/Gravity.html

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