But why an elliptical orbit

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.


Until someone can do the 'why', I'm going to assume that either:

1. it's because the Sun is in a rotation of it's own and all the
planets have to scoot to 'catch' up with it. But since the Sun is
moving too, the now-receding planet's momentum carries it further away
than it expected, so it maps out a stretched circle, longer than it
thought was necessary. So it's stretching the distance on the long-arm
of the orbit and has to scoot back in again, chasing that moving Sun.

or

2. the minute gravitational tug of the planet pulls the Sun closer (out
of its 'fixed' position relative to the planet) which increases the
mutual gravitational attraction so that they are each attracted that
little bit stronger. But since the Plant is still in orbit, it follows
the short arm of the ellipse that little bit faster or a little more
energised. It then maps out the long-arm of the ellipse and
gravitational attraction recedes just a little bit, allowing it to
'stretch that circle'.

or

(this space left blank for the correct answer)

Greg

On 11 Oct 2005 20:27:05 -0700, "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.


Until someone can do the 'why', I'm going to assume that either:

1. it's because the Sun is in a rotation of it's own and all the
planets have to scoot to 'catch' up with it. But since the Sun is
moving too, the now-receding planet's momentum carries it further away
than it expected, so it maps out a stretched circle, longer than it
thought was necessary. So it's stretching the distance on the long-arm
of the orbit and has to scoot back in again, chasing that moving Sun.

or

2. the minute gravitational tug of the planet pulls the Sun closer (out
of its 'fixed' position relative to the planet) which increases the
mutual gravitational attraction so that they are each attracted that
little bit stronger. But since the Plant is still in orbit, it follows
the short arm of the ellipse that little bit faster or a little more
energised. It then maps out the long-arm of the ellipse and
gravitational attraction recedes just a little bit, allowing it to
'stretch that circle'.

The correct answer is found in the mathematics. Any attempt to explain
it in English is a poor approximation at best (your first suggestion is
entirely incorrect; you are sort of on the right track with the second).

In a two body system, with the bodies acting under the force of gravity,
an elliptical path is simply what happens. The math that describes that
is really the answer.

http://en.wikipedia.org/wiki/Kepler%27s_laws
http://astron.berkeley.edu/~converse/Lagrange/Kepler'sFirstLaw.htm

Maybe Brian Tung will chip in. He has a talent for putting the math into
English.

_________________________________________________

Chris L Peterson
Cloudbait Observatory
http://www.cloudbait.com

tt40 wrote:
It then maps out the long-arm of the ellipse and
gravitational attraction recedes just a little bit, allowing it to
'stretch that circle'.

Are you asking why it is an ellipse rather than a circle? If
so, the answer is easy. You can create a perfectly circular
orbit with only two bodies --- the sun and the planet. But
when you introduce everything else, they will not stay
perfectly circular. And, given the chaotic nature of the solar
system's creation, it is too much to ask that it create
perfectly circular orbits at the start.

A circle and an ellipse are the only two possibilities that
work. And to be "perfect" about it, they aren't even perfectly
elliptical. There is always a little tugging and pulling by
other bodies that alter the orbits from perfect. But apart
from those tiny imperfections, they are ellipses.

Hope this helps.

Chuck Taylor
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If you enjoy optics, try
http://groups.yahoo.com/group/ATM_Optics_Software/
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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... but he eventually realized
that an ellipse 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

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

--
----------------------------------------------------------------
Paul Schlyter, Grev Turegatan 40, SE-114 38 Stockholm, SWEDEN
e-mail: pausch at stockholm dot bostream dot se
WWW: http://stjarnhimlen.se/

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.

I'm afraid there *is* no intuitive reason why it's an ellipse. You have
to remember that a mathematical derivation is generally a rather terse
and compact description of "why" something is true. As Chris says, it's
difficult to translate that into English without losing precision. You
can do it, but then the English is much longer than the math.

Even so, the derivation of a simple fact like Newton's law of universal
gravitation leading to an elliptical orbit is fairly involved. If there
really were a simple and intuitive explanation why it's an ellipse, the
mathematics would be pretty short and straightforward. The fact that it
isn't reveals something deep about mathematics and celestial mechanics.
I think it's one of the most beautiful aspects of theoretical astronomy,
and a major reason why Newton's Principia Mathematica is considered the
crowning achievement of science.

--
Brian Tung
The Astronomy Corner at http://astro.isi.edu/
Unofficial C5+ Home Page at http://astro.isi.edu/c5plus/
The PleiadAtlas Home Page at http://astro.isi.edu/pleiadatlas/
My Own Personal FAQ (SAA) at http://astro.isi.edu/reference/faq.txt

In article , Brian Tung 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.

I'm afraid there *is* no intuitive reason why it's an ellipse.

Yes, there is an intuitive reason: the orbits are ellipses with the
Sun at one of the foci because gravity follows an inverse-square law.
If gravity had varied in some other way, then the orbits would (usually)
not have been ellipses.

One example: if gravity would have been directly proportional to
distance (i.e. if gravity had *increased* with distance - we could
call this hypothetical case "rubber band gravity") the orbits would
have been ellipses too, but the Sun would have been at the *center*
of the ellipse, not at one of the foci of the ellipse. This case can
be simulated with a little ball attached to a stick with a very
elastic rubber band.

Another hypothetical case: if gravity would have varied as the inverse
*fifth* power of the distance, all orbits would have been spirals.
I.e. all planets would have either spiralled into the Sun or spiralled
out into space. Needless to say, a solar system would under such
circumstances become very short-lived.


You have
to remember that a mathematical derivation is generally a rather terse
and compact description of "why" something is true. As Chris says, it's
difficult to translate that into English without losing precision. You
can do it, but then the English is much longer than the math.

Even so, the derivation of a simple fact like Newton's law of universal
gravitation leading to an elliptical orbit is fairly involved. If there
really were a simple and intuitive explanation why it's an ellipse, the
mathematics would be pretty short and straightforward. The fact that it
isn't reveals something deep about mathematics and celestial mechanics.
I think it's one of the most beautiful aspects of theoretical astronomy,
and a major reason why Newton's Principia Mathematica is considered the
crowning achievement of science.

--
Brian Tung
The Astronomy Corner at http://astro.isi.edu/
Unofficial C5+ Home Page at http://astro.isi.edu/c5plus/
The PleiadAtlas Home Page at http://astro.isi.edu/pleiadatlas/
My Own Personal FAQ (SAA) at http://astro.isi.edu/reference/faq.txt


--
----------------------------------------------------------------
Paul Schlyter, Grev Turegatan 40, SE-114 38 Stockholm, SWEDEN
e-mail: pausch at stockholm dot bostream dot se
WWW: http://stjarnhimlen.se/

Paul Schlyter wrote:
Yes, there is an intuitive reason: the orbits are ellipses with the
Sun at one of the foci because gravity follows an inverse-square law.
If gravity had varied in some other way, then the orbits would (usually)
not have been ellipses.

That is not a use of the word "intuitive" with which I am familiar.
If I drop a crystal vase from the roof, it will fall with increasing
speed until it strikes the ground and smashes into a thousand pieces.
*That* is intuitive. The fact that it falls at a rate that increases
linearly with time, and so its distance from me increases quadratically
with time (neglecting air resistance for the moment)--well, it's likely
intuitive to you and me, but probably not to a quite a few people.

But to say that an inverse-square law leads to a conic section--*that*
is not intuitive. To paraphrase Tom from another thread, try to explain
that without any mathematics. I don't think you can. The fact that
other forces lead to different shapes doesn't explain to me, clearly
and convincingly, why this particular force leads to an ellipse (or a
parabola, or a hyperbola). You can't explain something like that in
the negative.

That it comes out as an ellipse, rather than some other oval, is
something deep, beautiful and unexpected about celestial mechanics. I
would like to internalize it more than I have previously, but as I say,
I'm not confident that it can be done.

--
Brian Tung
The Astronomy Corner at http://astro.isi.edu/
Unofficial C5+ Home Page at http://astro.isi.edu/c5plus/
The PleiadAtlas Home Page at http://astro.isi.edu/pleiadatlas/
My Own Personal FAQ (SAA) at http://astro.isi.edu/reference/faq.txt

(Paul Schlyter) wrote in :

Another hypothetical case: if gravity would have varied as the inverse
*fifth* power of the distance, all orbits would have been spirals.

Are the orbits stable if gravity varies as the inverse 4th or 6th power of
the distance?

In article 02,
John Schutkeker wrote:

(Paul Schlyter) wrote in :

Another hypothetical case: if gravity would have varied as the inverse
*fifth* power of the distance, all orbits would have been spirals.

Are the orbits stable if gravity varies as the inverse 4th or 6th power of
the distance?

No.

If gravity varies as r^(-n) then circular orbits are stable only if n 3

For other shapes of the orbits the stability criterion get much more complex.
Even under normal r^(-2) gravity a very elongated elliptical orbit with
an eccentricity sufficiently close to one is unstable, since it can easily
be perturbed into a parabolic or hyperbolic orbit.

--
----------------------------------------------------------------
Paul Schlyter, Grev Turegatan 40, SE-114 38 Stockholm, SWEDEN
e-mail: pausch at stockholm dot bostream dot se
WWW: http://stjarnhimlen.se/