Perturbation (astronomy)

Perturbation is a term used in astronomy in connection with descriptions of the complex motion of a massive body which is subject to appreciable gravitational effects from more than one other massive body.^[1]

Such complex motions of a body can be broken down descriptively into component parts. First, there can be the hypothetical motion that the body would follow, if it moved under the gravitational effect of one other body only. Expressed in other terms, such a motion could be regarded as a solution of a two-body problem, or as an unperturbed Keplerian orbit. Then, the differences between that hypothetical unperturbed motion and the actual motion of the body can be described as perturbations, due to the additional gravitational effects of the additional body or bodies. If there is only one other significant body, then the perturbed motion can be called a solution of a three-body problem: if there are multiple other significant bodies, the motion can represent a higher case of the n-body problem.

Newton at the time of formulating his laws of motion and of gravitation already recognized the existence of perturbations and the complex difficulties of their calculation.^[2] Since Newton's time, several techniques have been developed for the mathematical analysis of perturbations, and they can be divided into two major classes, general perturbations and special perturbations. In methods of analysing general perturbations, general differential equations of motion are solved, usually by series approximations, to give a result which is usually in terms of algebraic and trigonometrical functions, and can be applied generally to many different sets of conditions.^[3] Historically, general perturbations were investigated first. In methods of special perturbations, numerical datasets, representing values for the positions, velocities and accelerative forces on the bodies of interest, are made the basis of numerical integration of the differential equations.^[4]

Most systems that involve multiple gravitational attractions present one primary body which can be regarded as dominant in its gravitational effects (for example, a star, in the case of the star and its planet, or a planet, in the case of the planet and its satellite). Then, the other gravitational effects can be treated as causing perturbations of the hypothetical unperturbed motion of the planet, or the satellite, around its respective primary body.

In the Solar System, many of the perturbations are made up of periodical components, so that the perturbed bodies follow orbits that are periodic or quasi-periodic for long periods of time – such as the Moon in its strongly perturbed orbit, which is the subject of lunar theory.

Planets cause periodical perturbations in the orbits of other planets, a fact which led to the discovery of Neptune in 1846 as a result of its perturbations of the orbit of Uranus.

On-going mutual perturbations of the planets cause long-term quasi-periodic variations in their orbital elements. Venus currently has the orbit with the least eccentricity, i.e. it is the closest to circular, of all the planetary orbits. In 25,000 years' time, Earth will have a more circular (less eccentric) orbit than Venus.

Gravity Simulator plot of the changing orbital eccentricity of Mercury, Venus, Earth, and Mars over the next 50,000 years. The 0 point on this plot is the year 2007.

The orbits of many of the minor bodies of the Solar System, such as comets, are often heavily perturbed, particularly by the gravitational fields of the gas giants. While many of these perturbations are periodic, others are not, and these in particular may represent aspects of chaotic motion. For example, in April 1996, Jupiter's gravitational influence caused the period of Comet Hale-Bopp's orbit to decrease from 4,206 to 2,380 years, a change that will not revert on any periodical basis.^[5]

In astrodynamics and the case of man-made satellites, orbital perturbation may be a consequence of atmospheric drag or solar radiation pressure.

See also

• Nereid one of the outer moons of Neptune with a high orbital eccentricity of ~0.75 and is frequently perturbed
• Osculating orbit

References

1. ^ Roger R. Bate, Donald D. Mueller, Jerry E. White, "Fundamentals of astrodynamics" (Dover Publications, 1971), e.g. at ch.9, p.385.
2. ^ Newton in 1684 wrote: "By reason of the deviation of the Sun from the center of gravity, the centripetal force does not always tend to that immobile center, and hence the planets neither move exactly in ellipses nor revolve twice in the same orbit. Each time a planet revolves it traces a fresh orbit, as in the motion of the Moon, and each orbit depends on the combined motions of all the planets, not to mention the action of all these on each other. But to consider simultaneously all these causes of motion and to define these motions by exact laws admitting of easy calculation exceeds, if I am not mistaken, the force of any human mind." (quoted by Prof G E Smith (Tufts University), in "Three Lectures on the Role of Theory in Science" 1. Closing the loop: Testing Newtonian Gravity, Then and Now); and Prof R F Egerton (Portland State University, Oregon) after quoting the same passage from Newton concluded: "Here, Newton identifies the "many body problem" which remains unsolved analytically."
3. ^ Roger R. Bate, Donald D. Mueller, Jerry E. White, "Fundamentals of astrodynamics" (Dover Publications, 1971), e.g. at p.387 and at section 9.4.3, p.410.
4. ^ Roger R. Bate, Donald D. Mueller, Jerry E. White, "Fundamentals of astrodynamics" (Dover Publications, 1971), pp.387-409.
5. ^ Don Yeomans (1997-04-10). "Comet Hale-Bopp Orbit and Ephemeris Information". JPL/NASA. http://www2.jpl.nasa.gov/comet/ephemjpl8.html. Retrieved 2008-10-23. 

External links

• Solex (by Aldo Vitagliano) predictions for the position/orbit/close approaches of Mars