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(Originally posted by Noctavis, created by SiliconDream and Cyberbob)

All calculations and simulations which were used to derive the information herein were performed on a TI-86 and my brain. As such, there are almost certainly mistakes, some big, some small, some obvious, some not. If you find any, please mention them and I'll fix it. Thanks.



Lagrange Points--Overview
     The Lagrange points of a two-body system are moving points in space, following a circular path around the system's center of mass, with the following property: if an object's position and velocity are matched to that of such a point (in other words, the object is placed at the Lagrange point and given a push so that it's moving with the same speed and in the same direction as the point), it will experience gravitational forces from the two bodies such that it's made to follow the Lagrange point indefinitely; that is, its position at any time in the future will be that of the Lagrange point. That is, the path of each Lagrange point represents a convenient periodic, circular orbit you can drop an object into, and know where it'll be in, say, a thousand years. This is a definite rarity, which is why Lagrange points are so important; as a rule, it's simply impossible to figure out (analytically-a computer can calculate it for the near future, to a specified accuracy) what the trajectory will be of an object orbiting two massive bodies. The Lagrange points represent one of the very few analytic solutions of the infamous three-body-problem. Well, infamous if you're a physicist. :-) And they're obviously useful for space exploration; you can park a satellite or station or probe-or a ringworld, for that matter-at a Lagrange point and, provided you know how to do long division, you can figure out exactly where it'll be in the future.

     Or almost. There's one more property of the points that you have to take into account, and that's the stability of their equilibrium. As long as the object is undisturbed by forces other than the two bodies' gravity, it'll be perfectly well behaved; but what if something disturbs it? What if a meteoroid bounces off it, or it runs through a dust cloud, or another body passes by and gives it a gravitational kick? If it's moved slightly away from the Lagrange point, will it be pulled back to it or will it continue to drift away? The answer depends on which point you're looking at. Some points are unstable; even a miniscule force on an object will cause it to gradually drift away, lose its nice circular orbit, and follow a completely unpredictable trajectory. Other points are stable; if an object's kicked away from such a point, it'll simply oscillate back and forth across it, or orbit closely around it. This latter behavior is what most artificial objects placed at a Lagrange point are made to do; they follow a very small orbit around the point as it orbits the system's center of mass in turn. Such an orbit is called a - gasp - Halo orbit, I guess because there's no visible object at the orbit's center.

     There are 5 Lagrange points in a system, imaginatively labeled L1-L5. L1, L2 and L3 lie on the line running through both bodies; L3 and L2 are on either side of the pair, while L1 is in between them. All three of these points are unstable. L4 and L5 are in the system's orbital plane, but not on the line between the bodies; instead, they're placed such that if you draw lines between the two bodies and either point, you get an equilateral triangle. These are the only two stable points.



The Halo System
     The Halo is a ringworld, approximately 10,000 miles across, which orbits a Lagrange point of the gas giant Threshold and its moon, Basis. We've been told little about the composition and structure of these bodies, but from the fact that the Lagrange point concept applies, we can infer that Threshold is over 24 times as massive as Basis and that Basis's orbit around Threshold is almost circular, rather than strongly elliptical. No official name has been given for the primary star in the Halo system-and as far as we know, it is a one-star system-but many fans have taken to calling it Soell, after Matt.

     There's a bit of confusion about precisely which point the Halo orbits. The holographic representation at the beginning of the Halo trailer, as well as the story synopsis given by Joe Staaten, imply that it's L1; however, Nathan Bitner's forum posts suggest that the hologram's wrong and it's L4 or L5. This seems more likely, as only L4 and L5 have stable orbits.


Halo's Spin
     The Halo spins around its axis of symmetry, thus exerting centrifugal force on objects on its inner surface. This force is directed outward and simulates gravity. (Yes, I know centrifugal force is generally considered fictional. So is gravity. Shut up.) It's not clear precisely how strong the Halo's pseudogravity actually is, or even if Bungie has decided on a particular strength, but it seems to be something like Earth-normal. From this, and the approximate diameter of the ring, we can calculate that the Halo revolves roughly once every 1 hour and 35 minutes.

     The fact that the Halo's apparent gravity stems from its spin has some semi-significant consequences for beings living on the surface. First, an object moving spinward or counterspinward (relative to the ground) has a different angular velocity than a stationary object, and so feels a different amount of gravity. The pseudogravity's strength can be calculated as 1 g, multiplied by (1 + V/19870)^2, where V is the object's spinward speed relative to the ground in miles per hour (if V is counterspinward, make it negative.) So, for instance, an object moving spinward at 8200 m.p.h. feels double normal gravity; an object moving counterspinward at 5800 m.p.h. feels half normal gravity; an object moving counterspinward at 19870 m.p.h. feels no gravity at all. Second, an object moving up or down (again, relative to the ground of the Halo) is compelled by the Coriolis force to change its angular velocity to conserve angular momentum, just as (in the example used by textbooks around the world), a spinning ice skater revolves faster when she pulls in her arms and slower when she extends them. An object moving upwards feels a spinward acceleration equal in strength to .0011*S per second, where S is the magnitude of the object's downward velocity; an object moving down feels an acceleration of the same magnitude but counterspinward. So if you fire a rifle bullet upward at, say, Mach 1, its path will gradually bend spinward and after 1 second it'll be drifting that way at about .8 m.p.h. (in addition to its vertical velocity.)

     Now, how does this affect everyday Halo life? Well, you won't feel much when you're just moving around under your own steam. The gravitational changes from walking in the spinward or counterspinward directions will be negligible, and you'd have to fall or jump from a lethal height to notice any Coriolis-caused drift. High-speed vehicles like jets would probably be noticeably affected, but it's pretty unlikely that a player will ever have a chance to pilot a supersonic vehicle. The main thing that may be influenced by these effects is gunplay. Bullets move fast enough to experience powerful forces, and travel far enough (especially if you're sniping) that they have a chance to be significantly displaced by those forces. If you snipe at an enemy 1 km spinward with a Mach 1 bullet, the slug will drop 5 inches farther during flight than it would if you were sniping counterspinward; not very much, but enough to mess up that perfect headshot you were hoping for. And the effect increases for faster bullets and longer distances. FPS veterans, be prepared to revamp your sniping skillz for a different physics model. That is, assuming Halo's physics will be this realistic.


Day and Night
     It's clear from the position of the sun in the sky in the screenshots, and from our knowledge of the Halo's spin, that it'll have a day/night cycle which takes about an hour and a half to complete. The Halo's sunward half experience night, since its inner surface is in shadow; the spaceward half experiences day, since its inner surface is illuminated. It's not precisely half and half, though. The ends of the spaceward half are shadowed by the sunward half, to a degree that depends upon the angle between the Halo's axis of rotation and its orbital plane (here I'm talking about the primary orbit, that of Threshold around Soell). When this angle is close to 0, there's very little shadowing, and days are almost as long as nights; when the angle is larger, the shadowing is greater and the nights are significantly longer; when the angle is 90 degrees, the entire spaceward side is shadowed by the sunward side and there is no day whatsoever.

     It's worthwhile to take a moment to imagine how day and night would actually look to an inhabitant of the Halo. If you stand facing the sunward edge, with the ring climbing into the sky on either side, the sun will appear to follow a circular path which has as its center a fixed point on the horizon directly in front of you. The radius of this circle, traced out once a day, depends on the axial angle. When the angle is 0, the circle is simply a point; the sun doesn't appear to move at all through the course of a day, but just sits on the horizon (or just below it if the Halo has high walls on its edges). When the angle is small but nonzero, the sun traces out a small circle low in the sky, night occurring when it's on the half-circle below the horizon. For a larger angle, the sun climbs farther into the sky in the course of a day, but days are shorter; because the sun rises and sets farther along the Halo from either side of you, it rises later and sets earlier (because the Halo, climbing into the sky, meets the sun before it's descended to your level). For an angle almost at 90 degrees, the sun almost reaches the zenith at noon-but the days are only a few minutes long, as the sun only comes out from behind the opposite side of the ring overhead when it's near its highest point in the sky. And for a 90-degree axial angle, the sun never comes out; its entire path is hidden behind the Halo, even when it's directly overhead. It's possible, though, that at noon, the opposite of the Halo might be narrow enough that you could see the sun's corona or edges behind it, as we do on Earth during an eclipse.


     Any other heavenly bodies which are also sufficiently far away/slow-moving to appear fixed--stars, distant planets, possibly Basis and Threshold--will also follow circular paths through the sky centered on that same point on the horizon that the sun circles. So the entire night starscape will seem to rotate around this point, just as the Earth's starscape appears (for those of us in the northern hemisphere) to rotate around the Pole Star. On any given night, certain stars very close to the central point will appear to remain in roughly the same place all night and so might be used as navigation beacons (again, just as the Pole Star is on Earth); however, as the seasons shift and the Halo's axial angle changes, these stars will move away from the central point and their immobility (and hence their usefulness) will be reduced to some degree.



Text by SiliconDream, graphics by CyberBob, HTMLized by Noctavis, Rock Lobster by the B-52s. Originally posted December 16, 2000.