Friday, 5 June 2015

The Second Legrange Point (L2) and the Peculiar Kordean Moon Lina

Coryn was at the controls of the flyer when he and Jaime touched down on the dusty ground beside the large stone pile which was Ferhil Stones.

The sun had set a short while ago, and the full moon Lina had been rising in the opposite quarter of the sky from the sunset, while they had travelled across the countryside. 

 Jaime had stared at it, shaking his head, every now and then.

“What’s wrong?” Coryn had asked him, when he had first noticed the gesture.

“That moon is all wrong,” Jaime had replied.  “I did read about it when I was researching Kordea, but seeing it is a bit different from knowing that it exists.  That moon should not be the way it is, where it is, so consistently.”

“Consistent is about the right description for it,” Coryn had said, glancing at the offending night light.  “It does that every night.  Comes up directly across from where the sun sets, when the sun sets, crosses the sky, and then goes down as the sun rises.  And always full, so it’s handy, if anything.  I was raised on Space Stations so I’ve really no concept of how moons ought to behave.”

“It must be at the L2 point of the sun-planet system,” Jaime had said.

From The Witches’ Stones Book 2: Love and Intrigue under the Seven Moons of Kordea.

Jaime explains the L2 point to Coryn, in the book, but here is a more comprehensive explanation of the L2 point, the moon Lina, Kordea’s sun and their complicated interrelationships.

The L2 point is the second Lagrange point, where the gravitational field of the sun and the planet, as well as the point’s centripetal acceleration interact in such a way that it is a gravitationally stable spot.  In other words, an object located at this spot will tend to remain at this spot.

One can think of this as being like the top of a symmetrical hill, where the gravitational forces from each direction cancel out and the object therefore is not pulled down the hill.  Although the top of a hill is technically a stable point, as we all know, it isn’t easy to balance something at the top of a hill, so in practice some energy must be put into “station keeping”.

In the case of the Sun-Earth system, the L2 point is about 1.5 million km from the Earth, a considerably farther distance than the moon, at about  384 thousand km.  The Sun-Earth-L2 system will be collinear (in a straight line), within the plane of the Earth’s orbit around the sun.

The nice thing about L2 is that an object placed there will follow the Earth in its orbit around the sun, always staying in the same relative place, with the Earth directly between it and the Sun.  Normally, an object 1.15 million km farther out than the Earth’s orbit would move a bit more slowly than the Earth, as orbiting bodies move more slowly as they get farther from the center of the system, and would lag behind the Earth.  But that’s not the case for a small object in the L2 point.  It will seem to follow the Earth in its journey around the sun, staying in the same relative spot.

It’s a good spot for satellites - in fact, the replacement for the Hubble Space Telescope, the James Webb is going to use the L2 point.

In the case of the planet Kordea, the moon Lina is said to be in the sky all night, anywhere on the planet where it happens to be night.  The natural place for this to be would be the L2 point for that system.  A moon at L2 would be seen rising as the sun was setting, on the other side of the sky.  It would continue to rise as the night went on, being highest at midnight, then it would set at dawn, as the sun rises, again on the other side of the sky from the sun.  It would be in a sort of anti-sun position, so whatever the sun did, the moon Lina would do the opposite.

Assuming that Kordea had a tilt similar to the Earth’s, in the Northern Hemisphere summer the moon Lina would rise in the southeast and set in the southwest, similar to what the sun would do in the winter.  In the winter, Lina would rise in the northeast and set in the southwest, like the sun does in summer.  At the equinoxes, it would rise due east and set due west, as does the sun.
So, the Sun-Kordea-Lina system would always have a luminous body in the sky, day or night.  That would be useful on a planet with a sun hotter than our’s, as it would allow human activity to go on during the night, similar to how a full moon allows farmers to harvest through the night in the fall. 

The actual distance to the L2 point can be calculated from the masses of the star and the planet involved.  The calculation assumes that the mass of the body at the L2 point is negligible, compared to the other two masses.  The formula is:

            L2 dist = Star to planet distance X cube root of (planet mass/(3 X star mass)).

Kordea is supposed to be substantially hotter than Earth, so let’s assume the star is bigger and therefore hotter.  We will say that it is 1.2 times the mass of our sun, Kordea is about the same mass as the Earth, and the distance between Kordea and its sun is about the same as the Earth-Sol distance.  Plugging in those figures gives us a distance to the L2 point of about 1.4 million km.
If Lina was about the same size as our moon (which is big, as moons go), then at 1.4 million km it would only subtend about one quarter the angle our full moon does.  That gives only one sixteenth of the brightness of the full moon, all else being equal.

But all else is not equal.  For one thing, the luminosity of a star varies with its mass:

Luminosity = Mass of star to the 3.8 power (approximately)

If we take 1.2 to the 3.8 power, we get almost exactly 2.  So, Kordea’s sun would be twice as luminous as ours.  That would also increase the star’s temperature (stellar mass and temperature are related), and move the peak of the radiation towards the blue end of the spectrum (temperature and the peak of the radiation are related), and would mean more high energy ultra-violet light. Thus, it would make sense for the inhabitants of the planet to avoid the bright sun - not only is it hotter, but it emits more radiation in the dangerous ultra-violet part of the spectrum (which leads to sunburns and skin cancer on Earth).  Kordea would be a lot closer to the hot edge of its star’s habitable zone than the Earth is in the solar system.

Also, our moon has quite a low reflectivity or albedo, only about  0.12 (12% of the light falling on it is reflected).  We can assume that the albedo of Lina is much higher, say 0.8.  Then, when we combine the higher albedo of Lina, the higher luminosity of Kordea’s star, and the distance to the L2 point, it turns out that Lina’s brightness would be almost the same as our full moon.
As noted earlier, two issues arise with putting a “moon” at the L2 point:

  • It only works if the mass of that moon is negligible compared to the mass of the star and planet.  Our moon is less than 1% of the Earth’s mass, but that’s probably still way too big to be considered negligible.  Thus, Jaime considers there must be something very odd about Lina - it must either be hollow, or most of its mass must somehow be gravitationally shielded from the Kordean system.  Either way, it strongly suggests that it is not a natural body.
  • A body at the L2 point is at a stable point, but one from which a small nudge will knock it out of place, the way a small nudge would knock a round boulder down a big hill, if it was balanced at the top.  So, something must be done to ensure that it stays in place, or keeps its station, in satellite jargon.  That is the job of the Kordean witches, of course (perhaps with some extra-dimensional help?).   

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