But why an elliptical orbit

Chris L Peterson wrote:
On Wed, 12 Oct 2005 12:13:25 GMT, (Paul Schlyter) wrote:
OK, time to answer your question: the orbit is a conic section (the
ellipse being the most common case) because gravity is inversely
proportional to the square of the distance to the gravitating body...
snip Perhaps that is a satisfying answer to somebody already
familiar with the mathematics behind orbital dynamics, but . . . I
doubt most people find it intuitive that an inverse square
gravity law naturally leads to elliptical orbits. . . .

I always thought it is was because gravitational attraction between two
bodies was the result of two force vectors, not one.

The second smaller body has an orbital speed (angular momentum)
combined with its mass. The causes the second smaller body to pull the
larger body slightly off-center. Conversely, the larger body generates
sufficient gravitational force to still hold the smaller orbiting body
in place. As a consequence, a smaller body and larger body always orbit
a common dynamical center, offset from the true gravitational center of
gravity of the larger body.

Although a true circular orbit is theoretically possible assuming an
idealized set of initial conditions, in practice any body perturbing a
two-body orbital system will distort the idealized two-body circular
orbit. Considering the age of solar system and the density of objects
in it, the likelihood of finding any solar system object that has not
be perturbed from a true circular orbit into an elliptical orbit seems
remote.

- Canopus56

canopus56 wrote:
snip

This page may also help the thread top-poster:
http://www-astronomy.mps.ohio-state....t4/orbits.html

These lecture notes by an Ohio State professor note that to sustain a
exactly circular orbit, velocity of the smaller body must be, per
Newtonian gravity:

v_circular = Sqrt( ( G*M ) / r )

where r = radius, G is the gravitational constant, and m is the mass of
the larger first body.

If the initial velocity of the smaller body when captured is slightly
more than v_circular, the orbit will change into an ellipse. The
ellipse will grow larger until escape velocity is reached.


- Canopus56

canopus56 wrote:

http://www-astronomy.mps.ohio-state....t4/orbits.html

These lecture notes by an Ohio State professor note that to sustain a
exactly circular orbit, velocity of the smaller body must be, per
Newtonian gravity:

v_circular = Sqrt( ( G*M ) / r )

where r = radius, G is the gravitational constant, and m is the mass of
the larger first body.

Note that this is just a solution for v in the eccentricity equation I
posted earlier,

e = (rv^2 / GM) - 1

Set e = 0 (the eccentricity of a circle) and you find

rv^2 = GM

which implies that you can find not only the right v for a given r, but
also an r for a given v.

v = sqrt( GM / r )
r = GM / v^2

I'm not a physicist, so I don't know if that's physically right, but I
don't know why it wouldn't be, either.

The equation for escape velocity is yet another solution of the same
relation, but this time with e = 1 (the eccentricity of a parabola):

rv^2 / GM - 1 = 1
rv^2 = 2GM

- Ernie http://home.comcast.net/~erniew

canopus56 wrote:

I always thought it is was because gravitational attraction between
two bodies was the result of two force vectors, not one.

The second smaller body has an orbital speed (angular momentum)
combined with its mass. The causes the second smaller body to pull the
larger body slightly off-center. Conversely, the larger body generates
sufficient gravitational force to still hold the smaller orbiting body
in place. As a consequence, a smaller body and larger body always orbit
a common dynamical center, offset from the true gravitational center of
gravity of the larger body.

It sounds like you're saying that, for example, the sun is pulled to one
focus of an ellipse by the gravity of each planet. That's not right.

Mars's distance from the sun varies by 40 million kilometers, almost 30
solar diameters.

- Ernie http://home.comcast.net/~erniew

Ernie Wright wrote:
It sounds like you're saying that, for example, the sun is pulled to one
focus of an ellipse by the gravity of each planet. That's not right.
Mars's distance from the sun varies by 40 million kilometers, almost 30
solar diameters.

Yes, I was trying to convey that the Sun is pulled from the idealized
center of a prefectly circular orbit to the focus of the Keplerian
ellipse. IMHO, that is correct, but I'm reading the many excellent
posts in this thread and am still learning.

- Canopus56

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 ...

Could you clarify this question? All the responses to date (except for
one that is best left unnamed) have assumed that you were asking
for a simple, intuitive explanation why the orbits are ellipses rather
than some other elongated shape. It sounds to me as though you were
asking why they are elliptical rather than circular.

The simple answer to the latter question is "why not?" A circle is,
in fact, one particular kind of ellipse, but the chances of orbits
being *perfectly* circular are precisely zero. What's really baffling
is why the orbits of all the major planets are so close to being
circular, while the orbits of most comets and extra-solar planets
are extremely eccentric.

If the planets' orbits were grossly elliptical, then nobody would
ever have expected them to be circular, and that particular
confusion would never have existed. The confusion of everybody
before Kepler was caused precisely by the fact that the orbits
*are* circles to a (pretty good) first approximation.

In any case, the Sun's rotation has nothing to do with anything;
the planets can't "feel" this rotation directly. And the shapes
of the orbits also have nothing to do with the fact that the Sun
moves in response to the gravitational tugs of the planets --
although the Sun does, in fact, respond to those tugs.

- Tony Flanders

To Tony

If you had been paying attension you would have realised that the
original Newtonian solution shortcircuits the ability to treat the
Earth's axial and orbital motion in isolation.

For at least a century it is known that the relationship between axial
and orbital motion changes due to the change in Keplerian orbital
geometry from more elliptical to less elliptical while still retaining
what you call Kepler's second law..

What this means is that the cause for elliptical orbits cannot be
attributed solely to any change within the solar system but as Newton
has put all his eggs in one basket using a geocentric/heliocentric
orbital equivalency you and the other clowns get to cut off your nose
to spite your face for playing along with the grandiose sounding
'universal law of gravitation'.

Go ahead and play away with angular momentum ,if you had any
astronomical sense you would drop the claustraphobic Newtonian
quasi-geocentricity (please don't mention relativistic
homocentricity),go back and refer orbital motions to the orbital
motion of the Earth just like they did 400 years ago.Nobody is going to
call you on blustering and bluffing but it is time wasting for an
agenda that was and remains an astronomical fraud.

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 assume you mean, why planetary orbits *aren't circles*. The simplest
answer is that circular orbits require a perfect balance of parameters,
like flipping a coin and having it come up on edge. It's just a lot
more likely that the parameters determining the orbit aren't perfectly
balanced.

The parameters are distance and velocity (velocity is a combination of
speed and direction). Think of them as dials on a control panel. There
are certain combinations that will produce circles, but they have to be
exact. As you turn the dials, you get other conic sections--ellipses,
parabolas, hyperbolas.

But to see why this is so, you can't escape the math that describes how
gravity works. It's not explainable as scooting, wobbling, sloshing, or
anything like that. An ellipse is simply what happens for a broad set
of distance and velocity settings.

Start with Newton's laws of force and gravity,

f = ma
f = GMm/r^2

Set the right side of one equal to the right side of the other and wave
your magic calculus wand, and assuming I didn't screw this up, you get

e = (rv^2 / GM) - 1

where M is the mass of the sun, G is Newton's gravitational constant,
and r and v are distance and speed at perihelion.

The values of r and v are the dials you can twiddle to try to get a
circular orbit. You have to set them so that e, the eccentricity, is
exactly 0. If you plug in the parameters for Mars,

r = 2.07 * 10^11 meters
v = 2.65 * 10^4 meters per second

what you get out should be close to e = 0.093, the eccentricity of
Mars's orbit.

- Ernie http://home.comcast.net/~erniew

Ernie Wright wrote:
I assume you mean, why planetary orbits *aren't circles*.

I couldn't tell. I still can't. I'm hoping the original poster
clarifies the question...

--
Brian Tung
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Brian Tung wrote:

I assume you mean, why planetary orbits *aren't circles*.

I couldn't tell. I still can't. I'm hoping the original poster
clarifies the question...

I drew the inference from his attempted explanations, which couldn't
distinguish between ellipses and other ovally shapes, and from the
context of the recent thread about ancient Greek astronomy. It's a
guess, though.

- Ernie http://home.comcast.net/~erniew