Thursday, April 29, 2010

SIDEBAR: LaGrange and the Points

In considering a three body problem, Lagrange considered two large Solar System bodies, the Sun and a planet, and a smaller object which would be influenced by gravity of both. After considerable work, Lagrange proposed a frame of reference that rotates with the larger bodies., and he found five specific fixed points where the third, smaller body experiences zero net force.

L4 and L5 are sometimes called triangular points.
The general triangular configuration was discovered by Lagrange.
Examples
The Sun–Earth L4 and L5 points lie 60° ahead of and 60° behind the Earth as it orbits the Sun. They contain interplanetary dust.
The Earth–Moon L4 and L5 points lie 60° ahead of and 60° behind the Moon as it orbits the Earth. They may contain interplanetary dust in what is called Kordylewski clouds.
The Sun–Jupiter L4 and L5 points are occupied by the Trojan asteroids.
Neptune also has Trojan objects at its L4 and L5 points.

Joseph-Louis Lagrange

Though history considers him a French mathematician, Lagrange was born Italian in Turin, Italy (1736). Lagrange's interest in mathematics was sparked by Halley's work on "the use of algebra in optics".
Born into a prominent family, his father's financial difficulties set the environment for Lagrange mathematical inclination. Largely self taught, he quickly mastered all he math books that he could read. At age 19, Lagrange became the professor of mathematics at the Royal Artillery School in Turin. Lagrange later claimed: “If I had been rich, I probably would not have devoted myself to mathematics."
On 12 August 1755, Lagrange sent Euler his results on the tautochrone containing his method of maxima and minima. Euler replied on 6 September saying how impressed he was with Lagrange's new ideas.
He applied sophisticated mathematical methods to many problems including those of astronomy. For example, he applied his methods to the study the orbits of Jupiter and Saturn.
After entering a mathematical method for determining orbits of Jovian moons for the Académie des Sciences prize of 1766, d'Alembert
encouraged Lagrange to accept a post in Berlin. A generous offer was sent by Frederick II in April, and Lagrange accepted. Leaving Turin in August, he visited d'Alembert in Paris, then Caraccioli in London before arriving in Berlin in October. Lagrange succeeded Euler as Director of Mathematics at the Berlin Academy on 6 November 1766.
For 20 years. Lagrange worked at Berlin, producing a steady stream of top quality papers and regularly winning the prize from the Académie des Sciences of Paris. He shared the 1772 prize on the three body problem with Euler, won the prize for 1774, another one on the motion of the moon, and he won the 1780 prize on perturbations of the orbits of comets by the planets.
On 18 May 1787 he left Berlin to become a member of the Académie des Sciences in Paris, where he remained for the rest of his career. Lagrange survived the French Revolution while others did not and this may to some extent be due to his attitude which he had expressed many years before when he wrote: “I believe that, in general, one of the first principles of every wise man is to conform strictly to the laws of the country in which he is living, even when they are unreasonable.”
Lagrange was made a member of the committee of the Académie des Sciences to standardise weights and measures in May 1790. In 1793 the Reign of Terror commenced and the Académie des Sciences, along with the other learned societies, was suppressed on 8 August. The weights and measures commission was the only one allowed to continue and Lagrange became its chairman when others such as the chemist Lavoisier, Borda, Laplace, Coulomb, Brisson and Delambre were thrown off the commission.
In September 1793, a new law required the arrest of all foreigners born in enemy countries and all their property to be confiscated. While Lagrange certainly fell under the terms of the law, Lavoisier intervened to grant him an exception. Paradoxically, Lavoisier was himself condemned to death by guillotine in May 1794. Lagrange said on the day of Lavoisier’s execution: “It took only a moment to cause this head to fall and a hundred years will not suffice to produce its like.”
Napoleon named Lagrange to the Legion of Honour and Count of the Empire in 1808. On 3 April 1813 he was awarded the Grand Croix of the Ordre Impérial de la Réunion. He died a week later. The “Lagrangian points” are named in Lagrange's honor.

Five Lagrangian Points

Lagrange points are locations in space where gravitational forces and the orbital motion of a body balance each other. Over a hundred years after the mathematical theory was formulated, it was confirmed with the discovery of Trojan asteroids at the the Sun–Jupiter Lagrange points, L4 and L5, in 1906.
L4 orbits Sol 60° ahead of Earth.
L1 stays slightly inside Earth's Solar orbit.
L3 shares same orbit as the Earth.
Sun and Earth both orbit around the two bodies' barycenter, well inside the Sun.
INTUITIVE. For Earth's gravity to cancel Sol's, L1 must stay closer to the smaller Earth.
L3 orbits Sol directly opposite Earth.
L1, L2, L3 are on the M1-M2 Line.
Farthest from Earth, the Sun–Earth L3 is L-point most influenced by nonTerran gravitational forces.
L2 stays slightly outside Earth's Solar orbit.
Pulp science fiction often put a "Counter-Earth" in L3; however, space based observations now show no such object.
L5 is 60° behind Earth in it's orbit about Sol.
Reference "L5 Society".
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Of the five Lagrangian points in the Sun-Earth system. L1, L2, and L3 are unstable, but L4 and L5 resist gravitational perturbations. Thus, spacecraft would be truly stable at L points 4 and 5; like a ball in a bowl: when gently pushed away, it continues to orbit the Lagrange point without frequent rocket firings. These positions have been studied as possible sites for artificial space stations in the distant future.
L4 and L5 lie at the third corners of the two equilateral triangles in the plane of orbit whose common base is the line between the centers of the two masses, such that the point lies behind (L5) or ahead of (L4) the smaller mass with regard to its orbit around the larger mass.
The reason these points are in balance is that, at L4 and L5, the distances to the two masses are equal. Accordingly, the gravitational forces from the two massive bodies are in the same ratio as the masses of the two bodies, and so the resultant force acts through the barycenter of the system; additionally, the geometry of the triangle ensures that the resultant acceleration is to the distance from the barycenter in the same ratio as for the two massive bodies. The barycenter being both the center of mass and center of rotation of the system, this resultant force is exactly that required to keep a body at the Lagrange point in orbital equilibrium with the rest of the system.

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