Celestial Mechanics

In the last 30 years high performance computers have been used to study the n-body problem (n = 3 to n = 10 or more) by stepwise integration of the orbits of the gravitationally interacting bodies. Earlier computers were incapable of performing such calculations over sufficiently long time intervals. The study of the stability of Pluto's orbits over the last 10,000,000 years mentioned above was made for n = 5 (the Sun and the Jovian planets) perturbing Pluto's orbit. Some other studies have treated the solar system as a n = 9 system (the Sun and the eight major planets) over time intervals of several million years.

However, the finite increments of space and time used in stepwise integrations introduce small uncertainties in the predicted positions of solar system objects; these uncertainties increase as the time interval covered by the calculations increases. This has led to the application to celestial mechanics of a new concept in science, chaos, which started to develop in the 1970s. Chaos studies indicate that, due to increasing inaccuracy of prediction from integration calculations and also due to incompleteness of the mathematical models integrated, meaningful predictions about the state or position of a system cannot be made beyond some finite time. One result is that Pluto's orbit is chaotic over times of about 800 million years, so that its orbit and position in the early solar system or billions of years from now cannot be specified. Also the rotation of Saturn's satellite Hyperion appears to be chaotic. Chaos is now being applied to studies of the stability of the solar system, a problem which celestial mechanics has considered for centuries without finding a definite answer.

Chaos has also been able to show how certain orbits of main belt asteroids can, over billions of years, evolve into orbits which cross the orbits of Mars and Earth, producing near-Earth asteroids (NEA), of which about 100 are now known. Computer predictions of NEA orbits are now being made to identify NEA which may collide with Earth in the future; such collisions would threaten the very existence of our civilization. The prediction of such Earth-impacting asteroids may allow them to be dejected past Earth or to be destroyed; the space technology to do this may be available soon.

High performance computers and the concept of chaos are now also being used to study the satellite systems of the Jovian planets. They have also been used to study the orbits of stars in multiple star systems and the trajectories of stars in star clusters and galaxies.

The search for planets around other stars is also a recent development. It uses the theory of the two-body problem, starting from earlier work on astrometric double stars. These are stars whose proper motions on the sky are not straight lines as are the case for single stars, but are wavelike curves with periods of some years. This indicates that they are actually double stars with the visible star moving around the system's center of mass (which has straight-line proper motion) with an unseen companion. The stellar companions of Airius A (Gliese 244A), Procyon A (Gliese 280A), Ross 614 A (Gliese 234A), and Mu Cassiopeiae (Gliese 53A) were first detected as astrometric double stars before being observed optically. Small departures of the proper motions of stars from straight lines have been used since 1940 to predict the presence of companions of substellar mass (less than 0.07 solar mass) around nearby stars.

Action of a star around a double star system's center of mass produces periodic variations of the Doppler shift of the star's spectral lines as the star first approaches Earth, then recedes from it as seen from the system's center of mass. Since 1980, very precise spectroscopic observations have allowed searches for companions of substellar mass of visible stars to be made at several observatories.

These methods have allowed several dozen companions of substellar mass (so-called "brown dwarfs" and bodies of Jovian planet mass) to be suspected near stars other than the Sun. Unfortunately, as of late 1994 none of the suspected bodies of planetary mass associated with other stars has been confirmed by consistent observations at two or more observatories. Surprisingly, the two or three most reliably established planets have been detected orbiting a pulsar, which is a neutron star, a star that has used up its nuclear energy sources and has almost completed its evolution. The planets have been detected by apparent periodic variations in the period of the radio pulses from this neutronstar pulsar PSR 1257 + 12, and moreover, they seem to have masses on the order of Earth's mass or less. The search for planets orbiting normal stars continues; this is closely associated with the Search for Extra-Terrestrial Intelligence (the SETI Project).

Since 1957, the Space Age has accelerated the development of the branch of celestial mechanics called astrodynamics, which is becoming increasingly important. In addition to the traditional gravitational interactions between celestial bodies, astrodynamics must also consider (rocket) propulsion effects that are necessary for inserting artificial satellites and other spacecraft into their necessary orbits and trajectories. Aerodynamic effects must sometimes be considered for planets and satellites with appreciable atmospheres (Venus, Earth, Mars, the Jovian planets, Io, Titan, Neptune's satellite Britons and Pluto). Trajectory building is a new part of astrodynamics; it consists of combining different conic section orbits and propulsion segments along with planet and planetary satellite flybys to increase spacecraft payload on missions requiring very large propellant expenditures. The spacecraft Voyagers 1 and 2, Magellan, and Galileo have all used trajectory building, and future spacecraft such as the Cassini/Huygens mission to Saturn and Titan plan to use it to reach their destinations. Minor perturbations due to light pressure, the Poynting-Robertson Effect, and electromagnetic effects sometimes must also be considered. The solar sail is now being studied in spacecraft design as a way of using the light pressure from sunlight on solar sails to maneuver spacecraft and propel them through interplanetary space. Finally, the development of astrodynamics has increased the importance of hyperbolic orbits, since so far all flybys of planets and planetary satellites by spacecraft have occurred along hyperbolic orbits. The spacecraft Pioneers 10 and 11 Voyagers 1 and 2 are leaving the solar system along hyperbolic orbits with respect to the Sun that will take them into interstellar trajectories around the center of our Milky Way galaxy. Their hyperbolic orbits are being checked by intermittent radio signals from their transmitters as they leave the solar system for perturbations that could be produced by the gravitational attractions of undiscovered trans-Neptunian planets.