Orbit problem

"Painius" wrote in message ...
"Odysseus" wrote...
in message ...

Painius wrote:

But why wouldn't an elliptical orbit--if the orbit is truly an ellipse--
be symmetrical about its minor axis? Wouldn't this asymmetry
mean that the orbit is NOT an ellipse and indeed be more egg-
shaped?

Sorry, I was unclear. While the *shape* of an elliptical orbit is
certainly symmetrical about both axes, the *kinematics* concerned are
not. In terms of a 'static' diagram, the asymmetry may be seen in
that the sun (or, more generally, the gravitational centre of the
system) is at one focus of the ellipse while the other focus is 'empty'.

--
Odysseus

Okay, thanks for that. You and Steve have made it much
clearer. Now, i understand that in basic astronomy it's okay
to say, "The Sun is at one of the foci of the ellipse..." Yet is
this precisely true? Is the farthest focus from the planet not
also the center of gravity between the Sun and the planet?

Nope. The center of gravity between two masses depends only on their
(a) relative masses, and (b) current separation. (Halley's Comet has
an eccentricity of .9+, and the other focus of its elliptical orbit is
about at Neptune's orbit. But as the solid form of the comet is
perhaps a hundred miles in diameter, its center of mass with the Sun
is within a few inches of the center of the Sun.)



And as the Earth and Moon revolve around a CG that is
about 1,000 miles beneath the Earth's surface (in the
opposite direction from the Moon), then so do the Sun and
planets revolve around foci that are *near* but not *at* the
center of the Sun?

An Asimov table would make this clearer...

CENTER OF GRAVITY
PLANET (MILES FROM SUN'S CENTER)

Mercury 6
Venus 160
Earth/Moon 300
Mars 50
Jupiter 460,000
Saturn 250,000
Uranus 80,000
Neptune 150,000
Pluto 1,200

...noting that the distance from the center of the Sun to its
surface is 432,000 miles, so the Jupiter/Sun CG is the only
one that is outside the surface of the Sun. And the Sun
goes around this CG once every 11.86 years right in step
with Jupiter (in addition to the other circles brought about
by the other planets).

So some questions are... does each of these CG figures
represent the actual position of the focus that is farthest
from the planet?

Not at all. The distance of the focus is related solely to
eccentricity of orbit, and has nothing to do with relative mass.

The second focus of Pluto's orbit is well outside the orbit of Earth.
Mars's second focus is significantly further than Earth's, because
Mars's eccentricity is substantially higher than Earth's (yet Mars's
mass is substantially smaller).


And would there be any helpful/useful reason to calculate
the position of the other focus? the focus that is nearer to
the planet?

Calculating it requires only that you know the orbital elements. But
there's nothing "there" -- it has no gravitational significance.


Or is the farthest focus actually *at* the center of the Sun
and therefore in a different position than the CG?

The orbit is an ellipse in the reference frame of the Sun. From the
reference frame of the center of mass, both the SUn and the planet
orbit the center of mass in ellipses, both of which have a second
focus (in opposite directions). Neither second focus is significant in
any real way.


And in tangent, how closely does the orbital period of the
Sun/Jupiter system coincide with the sunspot cycle?

Not that closely. One would expect this, since Jupiter's orbit is a
near perfect circle, and the Sun's rotation speed is about 25 days.


happy days and...
starry starry nights!

"Painius" wrote in message ...
"Odysseus" wrote...
in message ...

Painius wrote:

But why wouldn't an elliptical orbit--if the orbit is truly an ellipse--
be symmetrical about its minor axis? Wouldn't this asymmetry
mean that the orbit is NOT an ellipse and indeed be more egg-
shaped?

Sorry, I was unclear. While the *shape* of an elliptical orbit is
certainly symmetrical about both axes, the *kinematics* concerned are
not. In terms of a 'static' diagram, the asymmetry may be seen in
that the sun (or, more generally, the gravitational centre of the
system) is at one focus of the ellipse while the other focus is 'empty'.

--
Odysseus

Okay, thanks for that. You and Steve have made it much
clearer. Now, i understand that in basic astronomy it's okay
to say, "The Sun is at one of the foci of the ellipse..." Yet is
this precisely true? Is the farthest focus from the planet not
also the center of gravity between the Sun and the planet?

Nope. The center of gravity between two masses depends only on their
(a) relative masses, and (b) current separation. (Halley's Comet has
an eccentricity of .9+, and the other focus of its elliptical orbit is
about at Neptune's orbit. But as the solid form of the comet is
perhaps a hundred miles in diameter, its center of mass with the Sun
is within a few inches of the center of the Sun.)



And as the Earth and Moon revolve around a CG that is
about 1,000 miles beneath the Earth's surface (in the
opposite direction from the Moon), then so do the Sun and
planets revolve around foci that are *near* but not *at* the
center of the Sun?

An Asimov table would make this clearer...

CENTER OF GRAVITY
PLANET (MILES FROM SUN'S CENTER)

Mercury 6
Venus 160
Earth/Moon 300
Mars 50
Jupiter 460,000
Saturn 250,000
Uranus 80,000
Neptune 150,000
Pluto 1,200

...noting that the distance from the center of the Sun to its
surface is 432,000 miles, so the Jupiter/Sun CG is the only
one that is outside the surface of the Sun. And the Sun
goes around this CG once every 11.86 years right in step
with Jupiter (in addition to the other circles brought about
by the other planets).

So some questions are... does each of these CG figures
represent the actual position of the focus that is farthest
from the planet?

Not at all. The distance of the focus is related solely to
eccentricity of orbit, and has nothing to do with relative mass.

The second focus of Pluto's orbit is well outside the orbit of Earth.
Mars's second focus is significantly further than Earth's, because
Mars's eccentricity is substantially higher than Earth's (yet Mars's
mass is substantially smaller).


And would there be any helpful/useful reason to calculate
the position of the other focus? the focus that is nearer to
the planet?

Calculating it requires only that you know the orbital elements. But
there's nothing "there" -- it has no gravitational significance.


Or is the farthest focus actually *at* the center of the Sun
and therefore in a different position than the CG?

The orbit is an ellipse in the reference frame of the Sun. From the
reference frame of the center of mass, both the SUn and the planet
orbit the center of mass in ellipses, both of which have a second
focus (in opposite directions). Neither second focus is significant in
any real way.


And in tangent, how closely does the orbital period of the
Sun/Jupiter system coincide with the sunspot cycle?

Not that closely. One would expect this, since Jupiter's orbit is a
near perfect circle, and the Sun's rotation speed is about 25 days.


happy days and...
starry starry nights!

"Painius" wrote in
:


Okay, thanks for that. You and Steve have made it much
clearer. Now, i understand that in basic astronomy it's okay
to say, "The Sun is at one of the foci of the ellipse..." Yet is
this precisely true? Is the farthest focus from the planet not
also the center of gravity between the Sun and the planet?

The "farthest" focus? Which focus is farther from the planet changes as
the planet moves around the sun. This may be clearer if you think of a
comet in a highly elliptical orbit, instead of a planet. One focus is
located inside the sun, very near the center. The other is located way
outside of the sun. At perihelion the comet is nearer the focus inside the
sun; at aphelion it's closer to the other.

However, your point is well taken. The center of the sun is not located
exactly at one of the foci. Neglecting all the other planets, the center
of mass of the sun and the planet in question coincides with one of the
foci. This is near but not exactly at the center of the sun. The actual
presence of the other planets does, of course, complicate things.


--
Steve Gray

"Painius" wrote in
:


Okay, thanks for that. You and Steve have made it much
clearer. Now, i understand that in basic astronomy it's okay
to say, "The Sun is at one of the foci of the ellipse..." Yet is
this precisely true? Is the farthest focus from the planet not
also the center of gravity between the Sun and the planet?

The "farthest" focus? Which focus is farther from the planet changes as
the planet moves around the sun. This may be clearer if you think of a
comet in a highly elliptical orbit, instead of a planet. One focus is
located inside the sun, very near the center. The other is located way
outside of the sun. At perihelion the comet is nearer the focus inside the
sun; at aphelion it's closer to the other.

However, your point is well taken. The center of the sun is not located
exactly at one of the foci. Neglecting all the other planets, the center
of mass of the sun and the planet in question coincides with one of the
foci. This is near but not exactly at the center of the sun. The actual
presence of the other planets does, of course, complicate things.


--
Steve Gray

Jonathan Silverlight wrote:

For instance, the distance of Pluto varies from 2761 to 4589 thousand
million miles, so the empty focus is presumably outside the "inner solar
system" (I can't find the formula).

Where e is the eccentricty, a is the semimajor axis, and c is the
distance from either focus to the centre of an ellipse, c = a*e. So
plugging in the values for a and e given by NASA's Planetary Fact
Sheet at

http://nssdc.gsfc.nasa.gov/planetary/factsheet/plutofact.html,

we have 39.24 AU * 0.2444 = 9.590 AU; doubling this gives a figure of
19.18 AU for the distance between the "empty focus" of Pluto's orbit
and the sun. This is indeed well outside the inner solar system, in
fact very near the orbit of Uranus (for which a = 19.20 AU).

--
Odysseus

Jonathan Silverlight wrote:

For instance, the distance of Pluto varies from 2761 to 4589 thousand
million miles, so the empty focus is presumably outside the "inner solar
system" (I can't find the formula).

Where e is the eccentricty, a is the semimajor axis, and c is the
distance from either focus to the centre of an ellipse, c = a*e. So
plugging in the values for a and e given by NASA's Planetary Fact
Sheet at

http://nssdc.gsfc.nasa.gov/planetary/factsheet/plutofact.html,

we have 39.24 AU * 0.2444 = 9.590 AU; doubling this gives a figure of
19.18 AU for the distance between the "empty focus" of Pluto's orbit
and the sun. This is indeed well outside the inner solar system, in
fact very near the orbit of Uranus (for which a = 19.20 AU).

--
Odysseus