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A small, polar-orbiting moon



 
 
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  #1  
Old October 17th 03, 12:29 PM
Bill Bogen
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Default A small, polar-orbiting moon

I'd like input from all you orbital mechanics out there as to this
concept's feasibility:

What if sometime in prehistory, the Earth had captured a Near Earth
Orbit asteroid, say 10 km in diameter, into a nice circular 20310.8
kilometer polar orbit?

So every 8 hours the Moon2, let's call it 'Cynthia' [derives from
'woman from Kynthos', a reference to Artemis, sister of Apollo and
Greek goddess of the Moon, who was reputed to have been born on the
mountain of Kynthos]
rises from either the Northerly horizon or the Southerly and, in
about 3 hours or less, sinks below the opposite horizon. The ancients
would quickly figure out the regular patterns of its orbit and how to
use it to determine longitude (using only the local time of day; no
accurate clocks synchronized to Greenwich time needed). Accurate maps
would appear early. The apparent size of Cynthia would be only about
1/12 that of Moon1 (Luna) and would vary by about 5%, depending on the
time and place of the observer. The size variation would let the
ancients determine the distance to Cynthia and start giving a sense of
the scale of the solar system. The ancients would probably deduce
that Cynthia was brighter (relative to size) than Luna because it's
closer to Earth. So they'd estimate the distance to Luna and, when
they compared the orbital periods of the two moons, would derive the
gravitational inverse-square law centuries earlier than OTL. Eclipses
(solar transits, really) would happen often, encouraging study of the
Sun's surface (maybe using camera obscura, really big pinhole camera
minus film). Cynthia often being 25 times closer to us than is Luna,
naked-eye observation would show craters and such on Cynthia, an early
intro to Galilean ideas of imperfect heavenly bodies or, better yet,
the idea of other planets and moons being actual places, that is,
destinations.

There is some question as to how bright Cynthia would be. I'd
appreciate any critiques of the following analysis:

… 1) For the non-astronomers out there, the brightness of
celestial bodies/stars/whatever is given as 'visual magnitude'. For
some reason, early astronomers defined that an object of magnitude 1
was 100 times brighter than an object of magnitude 6. So an object of
mag 1 would be 2.512 times brighter than an object of mag 2 because
2.512^(6-1)=100 and 2.512^(2-1)=2.512. Setting magnitudes was an
attempt to compare the brightness of stars but later really bright
objects were assigned magnitudes as well. For example, Luna has a
magnitude of -12.5 (brighter objects have larger negative numbers).

2) Let's wave our hand and make Luna (diameter 3476 km) disappear and
be replaced by Cynthia (diameter 10km). The area, and so the amount
of light reflected, has shrunk to (10/3476)^2 or 0.000008276 of its
former value. Its magnitude has changed by 12.7, meaning it's dimmer
by 2.512^12.7 times. Its magnitude is now 0.2 (-12.5+12.7), like a
really bright star but still not visible in full daylight.

3) Now let's move Cynthia closer to Earth. At closest, Luna is about
378028 km from an observer on Earth, while Cynthia would be about
13938.8 km away, 27.12 times closer. So it's brighter by a factor of
735.5 (27.12^2) due to the inverse square law. This produces a change
in magnitude of -7.2 (2.512^7.2~735.5) so Cynthia, at best, would have
a magnitude of -7.0 (-7.2+0.2).

4) By comparison, Venus can have a magnitude of -4.9 while Luna, as I
mentioned before, has a magnitude of -12.5. So Venus Cynthia
Luna.

5) From sci.astronomy.amateur, "The Great Comet of 1744 reached -7.0
magnitude and was visible 12 degrees from the Sun in broad daylight."

So I think Cynthia would often be visible during the day and available
for use in navigation.

My greatest concern is whether/how long Cynthia might maintain a
stable, circular polar orbit. Would the influence of Luna disrupt her
orbit? I assume the orbit would precess but how fast? Would the
precession be at a constant rate, one that ancients could include in
their tables/calculations? Is it even possible for a Near Earth
Asteroid to take up a circular orbit around Earth?

Thanks in advance for any input.
  #3  
Old October 24th 03, 04:23 AM
Henry Spencer
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Default A small, polar-orbiting moon

In article ,
Mike Miller wrote:
1) Virtually everything in the Inner System (starting at Pluto and
working in) has settled down into a single plane, give or take a few
degrees. It is very improbable that anything in Earth's neighborhood
would be approaching from a steep angle to end up in a polar orbit.


It doesn't have to approach from a steep angle. You can't ignore Earth's
motion around the Sun when thinking about such things. The incoming
object only has to pass Earth a few tens of thousands of kilometers to
(say) the north, and then lose some velocity while there. To pass, say,
50,000km north of Earth, the inclination of its orbit needs to be only
about 0.02deg -- remember, all this is happening 150Mkm from the Sun, so
the angle between the orbits needed to get 50,000km difference is very
small.

It's the "lose some velocity" part that's hard.

The Earth *can* capture objects from heliocentric orbit, as witness the
temporary capture of object J002E3 (which is almost certainly Apollo 12's
S-IVB!) last year. But as witness that case, the resulting orbits tend to
be very large -- well beyond the Moon's -- and rather precarious (J002E3
is gone into solar orbit again).

2) Earth already has a big, jealous companion. I'm not sure Cynthia
could settle down into a circular orbit (it wouldn't be circular to
begin with) before Luna destabilized Cynthia's orbit to intersect
Earth, Luna, or ejected it from the area all together. I'd put my
money on "intersect Earth."


Most likely is to eject it, in fact: both Earth and Moon are rather small
targets, in the celestial scheme of things, and a near-miss that changes
your path into an escape trajectory is much more likely than an actual
collision.

In general, three-body systems which are not "hierarchical" -- one close
pair plus a distant third -- over time have a strong tendency to lose one
body by ejection.
--
MOST launched 30 June; first light, 29 July; 5arcsec | Henry Spencer
pointing, 10 Sept; first science, early Oct; all well. |
  #4  
Old October 25th 03, 02:31 AM
Bill Bogen
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Default A small, polar-orbiting moon

(Henry Spencer) wrote in message ...
In article ,
Mike Miller wrote:
1) Virtually everything in the Inner System (starting at Pluto and
working in) has settled down into a single plane, give or take a few
degrees. It is very improbable that anything in Earth's neighborhood
would be approaching from a steep angle to end up in a polar orbit.


It doesn't have to approach from a steep angle. You can't ignore Earth's
motion around the Sun when thinking about such things. The incoming
object only has to pass Earth a few tens of thousands of kilometers to
(say) the north, and then lose some velocity while there. To pass, say,
50,000km north of Earth, the inclination of its orbit needs to be only
about 0.02deg -- remember, all this is happening 150Mkm from the Sun, so
the angle between the orbits needed to get 50,000km difference is very
small.

It's the "lose some velocity" part that's hard.

The Earth *can* capture objects from heliocentric orbit, as witness the
temporary capture of object J002E3 (which is almost certainly Apollo 12's
S-IVB!) last year. But as witness that case, the resulting orbits tend to
be very large -- well beyond the Moon's -- and rather precarious (J002E3
is gone into solar orbit again).


But an object _could_ (very small chance, I admit) be in heliocentric
orbit and yet pass over the Earth at just the right speed to enter a
circular polar orbit at 20310.8 km radius, could it not? Without
having to shed any velocity at all? (I feel like a cross-examining
attorney;"You admit that my client _could_ have been carrying that
plutonium for perfectly innocent reasons?")

2) Earth already has a big, jealous companion. I'm not sure Cynthia
could settle down into a circular orbit (it wouldn't be circular to
begin with) before Luna destabilized Cynthia's orbit to intersect
Earth, Luna, or ejected it from the area all together. I'd put my
money on "intersect Earth."


Most likely is to eject it, in fact: both Earth and Moon are rather small
targets, in the celestial scheme of things, and a near-miss that changes
your path into an escape trajectory is much more likely than an actual
collision.

In general, three-body systems which are not "hierarchical" -- one close
pair plus a distant third -- over time have a strong tendency to lose one
body by ejection.


And yet Jupiter has a number of moons in pretty stable orbits, over
millenia.
  #5  
Old October 26th 03, 03:57 AM
Joseph Hertzlinger
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Default A small, polar-orbiting moon

On 17 Oct 2003 04:29:24 -0700, Bill Bogen wrote:

The ancients would probably deduce that Cynthia was brighter
(relative to size) than Luna because it's closer to Earth.


I thought the brighness is proportional to the solid angle.

--
http://hertzlinger.blogspot.com
  #7  
Old October 26th 03, 10:53 PM
Henry Spencer
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Default A small, polar-orbiting moon

In article ,
Bill Bogen wrote:
It's the "lose some velocity" part that's hard...


But an object _could_ (very small chance, I admit) be in heliocentric
orbit and yet pass over the Earth at just the right speed to enter a
circular polar orbit at 20310.8 km radius, could it not?


Unfortunately, no, because Earth's gravity will accelerate it as it
approaches. If it arrives from infinity, its speed must be at least
escape velocity (for that distance), which is about 1.4x circular-orbit
velocity.

The only exception to this, which is what got J002E3 captured temporarily,
is if the whole thing is happening at the outer fringes of Earth's sphere
of influence, where the Sun's gravity is quite significant and three-body
complications invalidate simple concepts like "escape velocity". But
that's out around 900000km radius.

(Oh, there's one other exception, if it chances to make a lunar flyby
that robs it of some energy. But that will necessarily leave it in an
orbit that goes out, at least, nearly to the Moon's orbit.)

In general, three-body systems which are not "hierarchical" -- one close
pair plus a distant third -- over time have a strong tendency to lose one
body by ejection.


And yet Jupiter has a number of moons in pretty stable orbits, over
millenia.


Those cases are not really three-body systems, because Jupiter dominates
the situation so overwhelmingly. Interactions between the moons are minor
by comparison. (Sometimes they are non-trivial -- e.g. the resonance with
Europa and Ganymede that maintains the slight eccentricity of Io's orbit
and hence its internal tidal heating -- but not to the extent of actually
altering another moon's orbit substantially.)
--
MOST launched 30 June; first light, 29 July; 5arcsec | Henry Spencer
pointing, 10 Sept; first science, early Oct; all well. |
  #10  
Old October 27th 03, 01:28 PM
Bill Bogen
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Posts: n/a
Default A small, polar-orbiting moon

(Henry Spencer) wrote in message ...
In article ,
Bill Bogen wrote:
It's the "lose some velocity" part that's hard...


But an object _could_ (very small chance, I admit) be in heliocentric
orbit and yet pass over the Earth at just the right speed to enter a
circular polar orbit at 20310.8 km radius, could it not?


Unfortunately, no, because Earth's gravity will accelerate it as it
approaches. If it arrives from infinity, its speed must be at least
escape velocity (for that distance), which is about 1.4x circular-orbit
velocity.


So it doesn't arrive from infinity but rather from an orbit closer to
the Sun so Earth accelerates it to just the right velocity for a
circular polar orbit.

snip
In general, three-body systems which are not "hierarchical" -- one close
pair plus a distant third -- over time have a strong tendency to lose one
body by ejection.


And yet Jupiter has a number of moons in pretty stable orbits, over
millenia.


Those cases are not really three-body systems, because Jupiter dominates
the situation so overwhelmingly. Interactions between the moons are minor
by comparison. (Sometimes they are non-trivial -- e.g. the resonance with
Europa and Ganymede that maintains the slight eccentricity of Io's orbit
and hence its internal tidal heating -- but not to the extent of actually
altering another moon's orbit substantially.)


But if Cynthia is 25 times closer to the Earth than to Luna, wouldn't
Earth's influenece overwhelm Luna's?
 




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