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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. |
#2
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A small, polar-orbiting moon
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#3
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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. | |
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A small, polar-orbiting moon
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#5
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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 |
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A small, polar-orbiting moon
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#7
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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. | |
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A small, polar-orbiting moon
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#9
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A small, polar-orbiting moon
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#10
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A small, polar-orbiting moon
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