A New Paradigm for Lunar Orbits
November 30, 2006: It's 2015. You're NASA's chief engineer designing a moonbase for Shackleton Crater at the Moon's south pole. You're also designing a com-system that will allow astronauts constant radio contact with Earth.
But you know that direct transmissions won't work--not always. As seen from Shackleton Crater, Earth is below the horizon for two to three weeks each month (depending on the base's location). This blocks all radio signals, which travel line of sight.
Right: Artist Pat Rawling's concept of a manned lunar base. [More]
The solution seems obvious. Simply place a satellite in a high, circular orbit going almost over the Moon's poles. Better yet, place three satellites into the same orbit 120 degrees apart. Two would always be above the lunar horizon to relay messages to and from Earth.
There's just one problem.
"High-altitude circular orbits around the Moon are unstable," says Todd A. Ely, senior engineer for guidance, navigation, and control at NASA's Jet Propulsion Laboratory. "Put a satellite into a circular lunar orbit above an altitude of about 750 miles (1200 km) and it'll either crash into the lunar surface or it'll be flung away from the Moon altogether in a hyperbolic orbit." Depending on the specific orbit, this can happen fast: within tens of days.
Satellites in Earth orbit don't experience this sort of interference from the Moon. The Moon has just 1/80th Earth's mass—scarcely more than 1%. Relatively speaking, the Moon is a gravitational pipsqueak. Indeed, to any satellite in Earth orbit, the gravitational pull of the Sun is 160 times stronger than any lunar influence.
Any satellite in orbit around the Moon higher than about 750 miles, however, finds itself in a kind of celestial tug-of-war between Moon and Earth. Earth's pull can actually change the shape of an orbit from a circle to an elongated ellipse.
Stable circular lunar orbits do exist below an inclination of 39.6º, says Ely, but they spend so much time near the equator that "they are terrible orbits for covering the poles."
NASA wants to explore the Moon's polar regions for many reasons--not least is that deep polar craters may contain ice, which astronauts could harvest and melt for drinking or split into hydrogen and oxygen for rocket fuel and other uses. The instability of polar orbits poses a real problem for exploration.
Now for the good news. Ely and several colleagues have discovered a whole new class of "frozen" or stable high-altitude lunar orbits. Pictured right, they are all inclined at steep angles to the Moon's equatorial plane so they get far above the horizon at the lunar poles, and--surprise--they are all also quite elliptical.
"For better South Pole coverage, you want an ellipse with an eccentricity of about 0.6, which is pretty oval," Ely says. An eccentricity of 0 is a circle, along which a satellite travels at a constant speed around a primary body (say, the Moon) at its center. With Earth nearby, that's out of the question: "An inclined circular orbit is kind of a blank canvas where Earth can quickly work its will," Ely says.
In contrast, an eccentricity of 0.6 is an ellipse about as oval as an American football minus the pointed ends; the Moon would be at one focus of the ellipse. "The ellipse effectively 'locks in' the satellite's behavior to make it tougher for Earth to change," Ely explains. [See the appendix below for details.] How stable are they? Ely and his colleagues calculate that certain elliptical, high-inclination, high-altitude lunar orbits may remain stable for periods of at least a century. Indeed, Ely hypothesizes the orbits could last indefinitely.
For lunar communications and navigation, Ely recommends spacing three satellites 120º apart in the same elliptical orbit at an inclination of 51º. Each satellite in turn would go screaming down past periapsis (closest approach to the lunar surface) only 450 miles (700 km) above the north lunar pole, but would each linger fully 8 hours of its 12-hour orbit at 5,000 miles (8,000 km) above the horizon over the south lunar pole. In this configuration, two of the three satellites would always be in radio line-of-sight from a South Pole moonbase.
High-inclination, highly elliptical orbits being cheapest and most stable for communications satellites around the Moon? To Earth-centered satellite engineers used to thinking in terms of circular equatorial orbits, "it's a new paradigm," Ely declares.
Editor's note: This story describes problems keeping satellites in high orbit around the Moon. Low-orbiting satellites have problems, too. Lunar "mascons" tug on them and cause them to crash into the ground. Earth affects high orbits, mascons affect low orbits. For more information read Science@NASA's Bizarre Lunar Orbits.
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Bizarre Lunar Orbits (Science@NASA) -- Mysterious concentrations of mass in the Moon's ancient lava seas disturb the orbits of Moon-circling spacecraft.
Two recent papers by Ely describe stable high-altitude lunar orbits and their challenges:
APPENDIX: THE STABILITY OF HIGH LUNAR ORBITS
The stability of high lunar orbits (as well as of stars and black holes) is all about angular momentum—the force that keeps a top or gyroscope or ice skater spinning upright, even if perturbed slightly from the side.
For anything that's spinning, physicists use the right hand rule. Curl the fingers of your right hand to point in the direction of the spin. Then your thumb will point along the axis of spin. More importantly, it will point in the direction of what physicists call the angular momentum vector, which has one absolute direction in space.
You can actually feel the angular momentum vector. Try this. Take off the front wheel of a bicycle. Hold it horizontally by the axle, with one arm above and one arm below, leaving the wheel free to spin. Have a friend get the wheel spinning as fast as possible. Once the wheel is spinning, try to tilt it at a different angle or even to turn it over. You will find that the spinning wheel resists you with surprising force. Indeed, the angular momentum of bicycle wheels is what makes it easier to balance a bicycle when riding fast than when riding slowly.
One last brief primer before turning to orbits: The magnitude of the angular momentum vector depends on three quantities: the rate of spin, the mass of the spinning object, and the distance of the mass from the axis (the lever arm distance). Moreover, angular momentum is conserved—that is, absent any losses such as friction or applied external torques (twisting motions), the angular momentum vector will remain constant. Thus, if rate of spin, mass, or lever arm changes, then the remaining quantities must change in some compensating way to keep angular momentum constant. Example: if a spinning skater brings her arms in close to her body (shortens the lever arm distance), she starts spinning faster. The constancy of the angular momentum vector is also why gyroscopes are used to stabilize the orientation of spacecraft (such as the Hubble Space Telescope) in space.
What does all this have to do with lunar orbits? Every orbit has angular momentum. Satellites can be fairly massive (kilograms), and the lever arms can be hundreds or thousands of kilometers long. Now, if a lunar orbit is circular with the Moon at the center, the satellite travels with constant velocity—a situation that makes it vulnerable to Earth's gravitational pull.
The effect of all these fascinating dynamics is not to speed the satellite up in its orbit, but to apply a torque (twisting motion) that alters the inclination (tilt) of the plane of the satellite’s orbit. Such a change in tilt is resisted by the angular momentum vector, just as the spinning bicycle wheel resisted your attempts to change its tilt. The only way the orbit can compensate to conserve angular momentum is to change its shape or eccentricity: specifically, to become less circular (eccentricity = 0) and more elliptical (eccentricity > 0 but < 1). If the original circular orbit was steeply inclined, however, the change in shape can be so radical that the satellite is thrown into a hyperbola (eccentricity > 1) and flies completely away from the Moon.
Below a critical inclination of 39.6º, the orbital plane of a lunar satellite wobbles up and down with the line joining the ellipse’s apoapsis (farthest point from the Moon) and periapsis (closest point to the Moon) being dragged around by Earth as if it were attached by a leash. Such low-inclination elliptical orbits circulate around the Moon. Above that critical inclination of 39.6º, the line joining the orbit's periapsis and apoapsis stays relatively fixed in space, providing a stable orbit for communications and navigation satellites with minimum fuel needed for periodic course corrections.