There are five special points where a small mass can orbit in a constant pattern with two larger masses. The Lagrange Points are positions where the gravitational pull of two large masses precisely equals the centripetal force required for a small object to move with them. Of the five Lagrange points, three are unstable and two are stable. The unstable Lagrange points - labeled L1, L2 and L3 - lie along the line connecting the two large masses. The stable Lagrange points - labeled L4 and L5 - form the apex of two equilateral triangles that have the large masses at their vertices.
L4 leads the orbit of earth and L5 follows. L2 is ideal for astronomy because a spacecraft is close enough to readily communicate with Earth, can keep Sun, Earth and Moon behind the spacecraft for solar power and with appropriate shielding provides a clear view of deep space for our telescopes. The L1 and L2 points are unstable on a time scale of approximately 23 days, which requires satellites orbiting these positions to undergo regular course and attitude corrections.
NASA is unlikely to find any use for the L3 point since it remains hidden behind the Sun at all times. The idea of a hidden planet has been a popular topic in science fiction writing.
The L4 and L5 points are home to stable orbits so long as the mass ratio between the two large masses exceeds This condition is satisfied for both the Earth-Sun and Earth-Moon systems, and for many other pairs of bodies in the solar system. Objects found orbiting at the L4 and L5 points are often called Trojans after the three large asteroids Agamemnon, Achilles and Hector that orbit in the L4 and L5 points of the Jupiter-Sun system.
There are hundreds of Trojan Asteroids in the solar system. Most orbit with Jupiter, but others orbit with Mars. In addition, several of Saturn's moons have Trojan companions.
In the Polish astronomer Kordylewski discovered large concentrations of dust at the Trojan points of the Earth-Moon system. The existence of this ring is closely related to the Trojan points, but the story is complicated by the effects of radiation pressure on the dust grains.
The easiest way to understand Lagrange points is to think of them in much the same way that wind speeds can be inferred from a weather map. The forces are strongest when the contours of the effective potential are closest together and weakest when the contours are far apart. This page was originally written with mathematical equations by Neil J. Cornish of the Wikinson Microwave Anistropy Probe team.
What are they? Lagrange Points Lagrange points are positions in space where objects sent there tend to stay put. To create these maps, scientists employed a mathematical process called s Polar Views of Titan's Global Topography.
Hence, the condition for the stability of the Lagrange points which all lie at to small displacements parallel to the -axis is simply see Section 3. Suppose that a Lagrange point is situated in the - plane at coordinates. Let us consider small amplitude - motion in the vicinity of this point by writing Let us search for a solution of the above pair of equations of the form and. We obtain This leave purely imaginary eigenvalues as our only way to have a stable system.
This means that of our four roots, one is positive real, one is negative real, and two are purely imaginary. This means that all the collinear Lagrange points are unstable saddle points. In plain English, if we place any object close, but not perfectly on one of these Lagrange points the object will move away from the Lagrange point.
Whenever I talk about Lagrange points people ask me about space stations. As we just found out, these collinear Lagrange points are unstable. For that we can look at the discriminant,d. If d is negative, then the roots will be a mixture of real, positive, and negative like with the collinear points.
0コメント