Lagrange pointIn celestial mechanics, the Lagrange points (ləˈɡrɑːndʒ; also Lagrangian points or libration points) are points of equilibrium for small-mass objects under the gravitational influence of two massive orbiting bodies. Mathematically, this involves the solution of the restricted three-body problem. Normally, the two massive bodies exert an unbalanced gravitational force at a point, altering the orbit of whatever is at that point. At the Lagrange points, the gravitational forces of the two large bodies and the centrifugal force balance each other.
Solar SystemThe Solar System is the gravitationally bound system of the Sun and the objects that orbit it. The largest of such objects are the eight planets, in order from the Sun: four terrestrial planets named Mercury, Venus, Earth and Mars, two gas giants named Jupiter and Saturn, and two ice giants named Uranus and Neptune. The terrestrial planets have a definite surface and are mostly made of rock and metal. The gas giants are mostly made of hydrogen and helium, while the ice giants are mostly made of 'volatile' substances such as water, ammonia, and methane.
SunThe Sun is the star at the center of the Solar System. It is a nearly perfect ball of hot plasma, heated to incandescence by nuclear fusion reactions in its core. The Sun radiates this energy mainly as light, ultraviolet, and infrared radiation, and is the most important source of energy for life on Earth. The Sun's radius is about , or 109 times that of Earth. Its mass is about 330,000 times that of Earth, comprising about 99.86% of the total mass of the Solar System.
Formation and evolution of the Solar SystemThe formation of the Solar System began about 4.6 billion years ago with the gravitational collapse of a small part of a giant molecular cloud. Most of the collapsing mass collected in the center, forming the Sun, while the rest flattened into a protoplanetary disk out of which the planets, moons, asteroids, and other small Solar System bodies formed. This model, known as the nebular hypothesis, was first developed in the 18th century by Emanuel Swedenborg, Immanuel Kant, and Pierre-Simon Laplace.
Euler–Lagrange equationIn the calculus of variations and classical mechanics, the Euler–Lagrange equations are a system of second-order ordinary differential equations whose solutions are stationary points of the given action functional. The equations were discovered in the 1750s by Swiss mathematician Leonhard Euler and Italian mathematician Joseph-Louis Lagrange. Because a differentiable functional is stationary at its local extrema, the Euler–Lagrange equation is useful for solving optimization problems in which, given some functional, one seeks the function minimizing or maximizing it.
Joseph-Louis LagrangeJoseph-Louis Lagrange (born Giuseppe Luigi Lagrangia or Giuseppe Ludovico De la Grange Tournier; 25 January 1736 – 10 April 1813), also reported as Giuseppe Luigi Lagrange or Lagrangia, was an Italian mathematician, physicist and astronomer, later naturalized French. He made significant contributions to the fields of analysis, number theory, and both classical and celestial mechanics.
SpacecraftA spacecraft (: spacecraft) is a vehicle that is designed to fly in outer space and operate there. Spacecraft are used for a variety of purposes, including communications, Earth observation, meteorology, navigation, space colonization, planetary exploration, and transportation of humans and cargo. All spacecraft except single-stage-to-orbit vehicles cannot get into space on their own, and require a launch vehicle (carrier rocket).
Three-body problemIn physics and classical mechanics, the three-body problem is the problem of taking the initial positions and velocities (or momenta) of three point masses and solving for their subsequent motion according to Newton's laws of motion and Newton's law of universal gravitation. The three-body problem is a special case of the n-body problem. Unlike two-body problems, no general closed-form solution exists, as the resulting dynamical system is chaotic for most initial conditions, and numerical methods are generally required.
Lagrangian mechanicsIn physics, Lagrangian mechanics is a formulation of classical mechanics founded on the stationary-action principle (also known as the principle of least action). It was introduced by the Italian-French mathematician and astronomer Joseph-Louis Lagrange in his 1788 work, Mécanique analytique. Lagrangian mechanics describes a mechanical system as a pair consisting of a configuration space and a smooth function within that space called a Lagrangian. For many systems, where and are the kinetic and potential energy of the system, respectively.
Earth's orbitEarth orbits the Sun at an average distance of 149.60 million km (92.96 million mi) in a counterclockwise direction as viewed from above the Northern Hemisphere. One complete orbit takes 365.256 days (1 sidereal year), during which time Earth has traveled 940 million km (584 million mi). Ignoring the influence of other Solar System bodies, Earth's orbit, also known as Earth's revolution, is an ellipse with the Earth-Sun barycenter as one focus with a current eccentricity of 0.0167.
EarthEarth is the third planet from the Sun and the only astronomical object known to harbor life. This is enabled by Earth being a water world, the only one in the Solar System sustaining liquid surface water. Almost all of Earth's water is contained in its global ocean, covering 70.8% of Earth's surface. The remaining 29.2% of Earth's surface is land, most of which is located in the form of continental landmasses within one hemisphere, Earth's land hemisphere.
Uncrewed spacecraftRobotic spacecraft or uncrewed spacecraft are spacecraft without people on board. Uncrewed spacecraft may have varying levels of autonomy from human input; they may be remote controlled, remote guided or autonomous: they have a pre-programmed list of operations, which they will execute unless otherwise instructed. A robotic spacecraft for scientific measurements is often called a space probe or space observatory. Many space missions are more suited to telerobotic rather than crewed operation, due to lower cost and risk factors.
Euler's three-body problemIn physics and astronomy, Euler's three-body problem is to solve for the motion of a particle that is acted upon by the gravitational field of two other point masses that are fixed in space. This problem is exactly solvable, and yields an approximate solution for particles moving in the gravitational fields of prolate and oblate spheroids. This problem is named after Leonhard Euler, who discussed it in memoirs published in 1760.
N-body problemIn physics, the n-body problem is the problem of predicting the individual motions of a group of celestial objects interacting with each other gravitationally. Solving this problem has been motivated by the desire to understand the motions of the Sun, Moon, planets, and visible stars. In the 20th century, understanding the dynamics of globular cluster star systems became an important n-body problem. The n-body problem in general relativity is considerably more difficult to solve due to additional factors like time and space distortions.
Docking and berthing of spacecraftDocking and berthing of spacecraft is the joining of two space vehicles. This connection can be temporary, or partially permanent such as for space station modules. Docking specifically refers to joining of two separate free-flying space vehicles. Berthing refers to mating operations where a passive module/vehicle is placed into the mating interface of another space vehicle by using a robotic arm. Because the modern process of un-berthing requires more crew labor and is time-consuming, berthing operations are unsuited for rapid crew evacuations in the event of an emergency.
Earth observation satelliteAn Earth observation satellite or Earth remote sensing satellite is a satellite used or designed for Earth observation (EO) from orbit, including spy satellites and similar ones intended for non-military uses such as environmental monitoring, meteorology, cartography and others. The most common type are Earth imaging satellites, that take s, analogous to aerial photographs; some EO satellites may perform remote sensing without forming pictures, such as in GNSS radio occultation.
Spacecraft attitude controlSpacecraft attitude control is the process of controlling the orientation of a spacecraft (vehicle or satellite) with respect to an inertial frame of reference or another entity such as the celestial sphere, certain fields, and nearby objects, etc. Controlling vehicle attitude requires sensors to measure vehicle orientation, actuators to apply the torques needed to orient the vehicle to a desired attitude, and algorithms to command the actuators based on (1) sensor measurements of the current attitude and (2) specification of a desired attitude.
Stardust (spacecraft)Stardust was a 385-kilogram robotic space probe launched by NASA on 7 February 1999. Its primary mission was to collect dust samples from the coma of comet Wild 2, as well as samples of cosmic dust, and return them to Earth for analysis. It was the first sample return mission of its kind. En route to comet Wild 2, it also flew by and studied the asteroid 5535 Annefrank. The primary mission was successfully completed on 15 January 2006 when the sample return capsule returned to Earth.
Observational astronomyObservational astronomy is a division of astronomy that is concerned with recording data about the observable universe, in contrast with theoretical astronomy, which is mainly concerned with calculating the measurable implications of physical models. It is the practice and study of observing celestial objects with the use of telescopes and other astronomical instruments. As a science, the study of astronomy is somewhat hindered in that direct experiments with the properties of the distant universe are not possible.
Interplanetary Transport NetworkThe Interplanetary Transport Network (ITN) is a collection of gravitationally determined pathways through the Solar System that require very little energy for an object to follow. The ITN makes particular use of Lagrange points as locations where trajectories through space can be redirected using little or no energy. These points have the peculiar property of allowing objects to orbit around them, despite lacking an object to orbit. While it would use little energy, transport along the network would take a long time.
