Escape velocity
In physics, escape velocity is the minimum speed needed for a free object to escape from the gravitational influence of a massive body. It is slower the further away from the body an object is, slower for less massive bodies; the escape velocity from Earth is about 11.186 km/s at the surface. More escape velocity is the speed at which the sum of an object's kinetic energy and its gravitational potential energy is equal to zero. With escape velocity in a direction pointing away from the ground of a massive body, the object will move away from the body, slowing forever and approaching, but never reaching, zero speed. Once escape velocity is achieved, no further impulse need to be applied for it to continue in its escape. In other words, if given escape velocity, the object will move away from the other body, continually slowing, will asymptotically approach zero speed as the object's distance approaches infinity, never to come back. Speeds higher than escape velocity have a positive speed at infinity.
Note that the minimum escape velocity assumes that there is no friction, which would increase the required instantaneous velocity to escape the gravitational influence, that there will be no future acceleration or deceleration, which would change the required instantaneous velocity. For a spherically symmetric, massive body such as a star, or planet, the escape velocity for that body, at a given distance, is calculated by the formula v e = 2 G M r, where G is the universal gravitational constant, M the mass of the body to be escaped from, r the distance from the center of mass of the body to the object; the relationship is independent of the mass of the object escaping the massive body. Conversely, a body that falls under the force of gravitational attraction of mass M, from infinity, starting with zero velocity, will strike the massive object with a velocity equal to its escape velocity given by the same formula; when given an initial speed V greater than the escape speed v e, the object will asymptotically approach the hyperbolic excess speed v ∞, satisfying the equation: v ∞ 2 = V 2 − v e 2.
In these equations atmospheric friction is not taken into account. A rocket moving out of a gravity well does not need to attain escape velocity to escape, but could achieve the same result at any speed with a suitable mode of propulsion and sufficient propellant to provide the accelerating force on the object to escape. Escape velocity is only required to send a ballistic object on a trajectory that will allow the object to escape the gravity well of the mass M; the existence of escape velocity is a consequence of conservation of energy and an energy field of finite depth. For an object with a given total energy, moving subject to conservative forces it is only possible for the object to reach combinations of locations and speeds which have that total energy. By adding speed to the object it expands the possible locations that can be reached, with enough energy, they become infinite. For a given gravitational potential energy at a given position, the escape velocity is the minimum speed an object without propulsion needs to be able to "escape" from the gravity.
Escape velocity is a speed because it does not specify a direction: no matter what the direction of travel is, the object can escape the gravitational field. The simplest way of deriving the formula for escape velocity is to use conservation of energy. For the sake of simplicity, unless stated otherwise, we assume that an object is attempting to escape from a uniform spherical planet by moving away from it and that the only significant force acting on the moving object is the planet's gravity. In its initial state, i, imagine that a spaceship of mass m is at a distance r from the center of mass of the planet, whose mass is M, its initial speed is equal to v e. At its final state, f, it will be an infinite distance away from the planet, its speed will be negligibly small and assumed to be 0. Kinetic energy K and gravitational potential energy Ug are the only types of energy that we will deal with, so by the conservation of energy, i = f Kƒ = 0 because final velocity is zero, Ugƒ = 0 because its final distance is infinity, so ⇒ 1 2 m v e 2 + − G M m r
Kepler's equation
In orbital mechanics, Kepler's equation relates various geometric properties of the orbit of a body subject to a central force. It was first derived by Johannes Kepler in 1609 in Chapter 60 of his Astronomia nova, in book V of his Epitome of Copernican Astronomy Kepler proposed an iterative solution to the equation; the equation has played an important role in the history of both physics and mathematics classical celestial mechanics. Kepler's equation is where M is the mean anomaly, E is the eccentric anomaly, e is the eccentricity. The'eccentric anomaly' E is useful to compute the position of a point moving in a Keplerian orbit; as for instance, if the body passes the periastron at coordinates x = a, y = 0, at time t = t0 to find out the position of the body at any time, you first calculate the mean anomaly M from the time and the mean motion n by the formula M = n solve the Kepler equation above to get E get the coordinates from: where a is the semi-major axis, b the semi-minor axis. Kepler's equation is a transcendental equation because sine is a transcendental function, meaning it cannot be solved for E algebraically.
Numerical analysis and series expansions are required to evaluate E. There are several forms of Kepler's equation; each form is associated with a specific type of orbit. The standard Kepler equation is used for elliptic orbits; the hyperbolic Kepler equation is used for hyperbolic trajectories. The radial Kepler equation is used for linear trajectories. Barker's equation is used for parabolic trajectories; when e = 0, the orbit is circular. Increasing e causes the circle to become elliptical; when e = 1, there are three possibilities: a parabolic trajectory, a trajectory going in or out along an infinite ray emanating from the centre of attraction, or a trajectory that goes back and forth along a line segment from the centre of attraction to a point at some distance away. A slight increase in e above 1 results in a hyperbolic orbit with a turning angle of just under 180 degrees. Further increases reduce the turning angle, as e goes to infinity, the orbit becomes a straight line of infinite length.
The Hyperbolic Kepler equation is:. This equation is derived by redefining M to be the square root of −1 times the right-hand side of the elliptical equation: M = i and replacing E by iH; the Radial Kepler equation is: where t is proportional to time and x is proportional to the distance from the centre of attraction along the ray. This equation is derived by multiplying Kepler's equation by 1/2 and setting e to 1: t = 1 2. and making the substitution E = 2 sin − 1 . Calculating M for a given value of E is straightforward. However, solving for E when M is given can be more challenging. There is no closed-form solution. One can write an infinite series expression for the solution to Kepler's equation using Lagrange inversion, but the series does not converge for all combinations of e and M. Confusion over the solvability of Kepler's equation has persisted in the literature for four centuries. Kepler himself expressed doubt at the possibility of finding a general solution; the inverse Kepler equation is the solution of Kepler's equation for all real values of e: E = { ∑ n = 1 ∞ M n 3 n! lim θ → 0 +, e = 1 ∑ n = 1 ∞ M n n! lim θ → 0 + ( d n − 1 d θ n
Aerospace engineering
Aerospace engineering is the primary field of engineering concerned with the development of aircraft and spacecraft. It has two major and overlapping branches: astronautical engineering. Avionics engineering deals with the electronics side of aerospace engineering. Aeronautical engineering was the original term for the field; as flight technology advanced to include craft operating in outer space, the broader term "aerospace engineering" has come into common use. Aerospace engineering the astronautics branch is colloquially referred to as "rocket science". Flight vehicles are subjected to demanding conditions such as those caused by changes in atmospheric pressure and temperature, with structural loads applied upon vehicle components, they are the products of various technological and engineering disciplines including aerodynamics, avionics, materials science, structural analysis and manufacturing. The interaction between these technologies is known as aerospace engineering; because of the complexity and number of disciplines involved, aerospace engineering is carried out by teams of engineers, each having their own specialized area of expertise.
The origin of aerospace engineering can be traced back to the aviation pioneers around the late 19th to early 20th centuries, although the work of Sir George Cayley dates from the last decade of the 18th to mid-19th century. One of the most important people in the history of aeronautics, Cayley was a pioneer in aeronautical engineering and is credited as the first person to separate the forces of lift and drag, which are in effect on any flight vehicle. Early knowledge of aeronautical engineering was empirical with some concepts and skills imported from other branches of engineering. Scientists understood some key elements of aerospace engineering, like fluid dynamics, in the 18th century. Many years after the successful flights by the Wright brothers, the 1910s saw the development of aeronautical engineering through the design of World War I military aircraft. Between World Wars I and II, great leaps were made in Aeronautical Engineering; the advent of mainstream civil aviation accelerated this process.
Notable airplanes of this era include the Curtiss JN 4, the Farman F.60 Goliath, Fokker trimotor. Notable military airplanes of this period include the Mitsubishi A6M Zero, the Supermarine Spitfire and the Messerschmitt Bf 109 from Japan, Great Britain, Germany respectively. A significant development in Aerospace engineering came with the first Jet engine-powered airplane, the Messerschmitt Me 262 which entered service in 1944 towards the end of the second World War; the first definition of aerospace engineering appeared in February 1958. The definition considered the Earth's atmosphere and the outer space as a single realm, thereby encompassing both aircraft and spacecraft under a newly coined word aerospace. In response to the USSR launching the first satellite, Sputnik into space on October 4, 1957, U. S. aerospace engineers launched the first American satellite on January 31, 1958. The National Aeronautics and Space Administration was founded in 1958 as a response to the Cold War. In 1969, Apollo 11, the first manned space mission to the moon took place.
It saw three astronauts enter orbit around the Moon, with two, Neil Armstrong and Buzz Aldrin, visiting the lunar surface. The third astronaut, Michael Collins, stayed in orbit to rendezvous with Armstrong and Aldrin after their visit to the lunar surface; some of the elements of aerospace engineering are: Radar cross-section – the study of vehicle signature apparent to Radar remote sensing. Fluid mechanics – the study of fluid flow around objects. Aerodynamics concerning the flow of air over bodies such as wings or through objects such as wind tunnels. Astrodynamics – the study of orbital mechanics including prediction of orbital elements when given a select few variables. While few schools in the United States teach this at the undergraduate level, several have graduate programs covering this topic. Statics and Dynamics – the study of movement, moments in mechanical systems. Mathematics – in particular, differential equations, linear algebra. Electrotechnology – the study of electronics within engineering.
Propulsion – the energy to move a vehicle through the air is provided by internal combustion engines, jet engines and turbomachinery, or rockets. A more recent addition to this module is ion propulsion. Control engineering – the study of mathematical modeling of the dynamic behavior of systems and designing them using feedback signals, so that their dynamic behavior is desirable; this applies to the dynamic behavior of aircraft, propulsion systems, subsystems that exist on aerospace vehicles. Aircraft structures – design of the physical configuration of the craft to withstand the forces encountered during flight. Aerospace engineering aims to keep structures lightweight and low-cost while maintaining structural integrity. Materials science – related to structures, aerospace engineering studies the materials of which the aerospace structures are to be built. New materials with specific properties are invented, or existing ones are modified to improve their performance. Solid mechanics – Closely related to material science is solid mechanics which deals with stress and strain analysis of the components of the vehicle.
Nowadays there are several Finite Element programs such as MSC
Orbit
In physics, an orbit is the gravitationally curved trajectory of an object, such as the trajectory of a planet around a star or a natural satellite around a planet. Orbit refers to a repeating trajectory, although it may refer to a non-repeating trajectory. To a close approximation and satellites follow elliptic orbits, with the central mass being orbited at a focal point of the ellipse, as described by Kepler's laws of planetary motion. For most situations, orbital motion is adequately approximated by Newtonian mechanics, which explains gravity as a force obeying an inverse-square law. However, Albert Einstein's general theory of relativity, which accounts for gravity as due to curvature of spacetime, with orbits following geodesics, provides a more accurate calculation and understanding of the exact mechanics of orbital motion; the apparent motions of the planets were described by European and Arabic philosophers using the idea of celestial spheres. This model posited the existence of perfect moving spheres or rings to which the stars and planets were attached.
It assumed the heavens were fixed apart from the motion of the spheres, was developed without any understanding of gravity. After the planets' motions were more measured, theoretical mechanisms such as deferent and epicycles were added. Although the model was capable of reasonably predicting the planets' positions in the sky and more epicycles were required as the measurements became more accurate, hence the model became unwieldy. Geocentric it was modified by Copernicus to place the Sun at the centre to help simplify the model; the model was further challenged during the 16th century, as comets were observed traversing the spheres. The basis for the modern understanding of orbits was first formulated by Johannes Kepler whose results are summarised in his three laws of planetary motion. First, he found that the orbits of the planets in our Solar System are elliptical, not circular, as had been believed, that the Sun is not located at the center of the orbits, but rather at one focus. Second, he found that the orbital speed of each planet is not constant, as had been thought, but rather that the speed depends on the planet's distance from the Sun.
Third, Kepler found a universal relationship between the orbital properties of all the planets orbiting the Sun. For the planets, the cubes of their distances from the Sun are proportional to the squares of their orbital periods. Jupiter and Venus, for example, are about 5.2 and 0.723 AU distant from the Sun, their orbital periods about 11.86 and 0.615 years. The proportionality is seen by the fact that the ratio for Jupiter, 5.23/11.862, is equal to that for Venus, 0.7233/0.6152, in accord with the relationship. Idealised orbits meeting these rules are known as Kepler orbits. Isaac Newton demonstrated that Kepler's laws were derivable from his theory of gravitation and that, in general, the orbits of bodies subject to gravity were conic sections. Newton showed that, for a pair of bodies, the orbits' sizes are in inverse proportion to their masses, that those bodies orbit their common center of mass. Where one body is much more massive than the other, it is a convenient approximation to take the center of mass as coinciding with the center of the more massive body.
Advances in Newtonian mechanics were used to explore variations from the simple assumptions behind Kepler orbits, such as the perturbations due to other bodies, or the impact of spheroidal rather than spherical bodies. Lagrange developed a new approach to Newtonian mechanics emphasizing energy more than force, made progress on the three body problem, discovering the Lagrangian points. In a dramatic vindication of classical mechanics, in 1846 Urbain Le Verrier was able to predict the position of Neptune based on unexplained perturbations in the orbit of Uranus. Albert Einstein in his 1916 paper The Foundation of the General Theory of Relativity explained that gravity was due to curvature of space-time and removed Newton's assumption that changes propagate instantaneously; this led astronomers to recognize that Newtonian mechanics did not provide the highest accuracy in understanding orbits. In relativity theory, orbits follow geodesic trajectories which are approximated well by the Newtonian predictions but the differences are measurable.
All the experimental evidence that can distinguish between the theories agrees with relativity theory to within experimental measurement accuracy. The original vindication of general relativity is that it was able to account for the remaining unexplained amount in precession of Mercury's perihelion first noted by Le Verrier. However, Newton's solution is still used for most short term purposes since it is easier to use and sufficiently accurate. Within a planetary system, dwarf planets and other minor planets and space debris orbit the system's barycenter in elliptical orbits. A comet in a parabolic or hyperbolic orbit about a barycenter is not gravitationally bound to the star and therefore is not considered part of the star's planetary system. Bodies which are gravitationally bound to one of the planets in a planetary system, either natural or artificial satellites, follow orbits about a barycenter near or within that planet. Owing to mutual gravitational perturbations, the eccentricities of the planetary orbits vary over time.
Mercury, the smallest planet in the Solar System, has the most eccentric orbit
Sphere of influence (astrodynamics)
A sphere of influence in astrodynamics and astronomy is the oblate-spheroid-shaped region around a celestial body where the primary gravitational influence on an orbiting object is that body. This is used to describe the areas in the Solar System where planets dominate the orbits of surrounding objects such as moons, despite the presence of the much more massive but distant Sun. In the patched conic approximation, used in estimating the trajectories of bodies moving between the neighbourhoods of different masses using a two body approximation and hyperbolae, the SOI is taken as the boundary where the trajectory switches which mass field it is influenced by; the general equation describing the radius of the sphere r S O I of a planet: r S O I ≈ a 2 / 5 where a is the semimajor axis of the smaller object's orbit around the larger body. M and M are the masses of the larger object, respectively. In the patched conic approximation, once an object leaves the planet's SOI, the primary/only gravitational influence is the Sun.
Because the definition of rSOI relies on the presence of the Sun and a planet, the term is only applicable in a three-body or greater system and requires the mass of the primary body to be much greater than the mass of the secondary body. This changes the three-body problem into a restricted two-body problem; the table shows the values of the sphere of gravity of the bodies of the solar system in relation to the Sun.: The Sphere of influence is, in fact, not quite a sphere. The distance to the SOI depends on the angular distance θ from the massive body. A more accurate formula is given by r S O I ≈ a 2 / 5 1 1 + 3 cos 2 10 Averaging over all possible directions we get r S O I ¯ = 0.9431 a 2 / 5 Consider two point masses A and B at locations r A and r B, with mass m A and m B respectively. The distance R = | r B − r A | separates the two objects. Given a massless third point C at location r C, one can ask whether to use a frame centered on A or on B to analyse the dynamics of C. Let's consider a frame centered on A.
The gravity of B is denoted as g B and will be treated as a perturbation to the dynamics of C due to the gravity g A of body A. Due their gravitational interactions, point A is attracted to point B with acceleration a A = G m B R 3, this frame is therefore non-inertial. To quantify the effects of the perturbations in this frame, one should consider the ratio of the perturbations to the main body gravity i.e. χ A = | g B − a A | | g A |. The perturbation g B − a A is known as the tidal forces due to body B, it is possible to construct the perturbation ratio χ B for the frame centered on B by interchanging A ↔ B. As C gets close to A, χ A →
Circle
A circle is a simple closed shape. It is the set of all points in a plane; the distance between any of the points and the centre is called the radius. This article is about circles in Euclidean geometry, and, in particular, the Euclidean plane, except where otherwise noted. A circle is a simple closed curve that divides the plane into two regions: an interior and an exterior. In everyday use, the term "circle" may be used interchangeably to refer to either the boundary of the figure, or to the whole figure including its interior. A circle may be defined as a special kind of ellipse in which the two foci are coincident and the eccentricity is 0, or the two-dimensional shape enclosing the most area per unit perimeter squared, using calculus of variations. A circle is a plane figure bounded by one line, such that all right lines drawn from a certain point within it to the bounding line, are equal; the bounding line is called the point, its centre. Annulus: a ring-shaped object, the region bounded by two concentric circles.
Arc: any connected part of a circle. Specifying two end points of an arc and a center allows for two arcs that together make up a full circle. Centre: the point equidistant from all points on the circle. Chord: a line segment whose endpoints lie on the circle, thus dividing a circle in two sements. Circumference: the length of one circuit along the circle, or the distance around the circle. Diameter: a line segment whose endpoints lie on the circle and that passes through the centre; this is the largest distance between any two points on the circle. It is a special case of a chord, namely the longest chord for a given circle, its length is twice the length of a radius. Disc: the region of the plane bounded by a circle. Lens: the region common to two overlapping discs. Passant: a coplanar straight line that has no point in common with the circle. Radius: a line segment joining the centre of a circle with any single point on the circle itself. Sector: a region bounded by two radii of equal length with a common center and either of the two possible arcs, determined by this center and the endpoints of the radii.
Segment: a region bounded by a chord and one of the arcs connecting the chord's endpoints. The length of the chord imposes a lower boundary on the diameter of possible arcs. Sometimes the term segment is used only for regions not containing the center of the circle to which their arc belongs to. Secant: an extended chord, a coplanar straight line, intersecting a circle in two points. Semicircle: one of the two possible arcs determined by the endpoints of a diameter, taking its midpoint as center. In non-technical common usage it may mean the interior of the two dimensional region bounded by a diameter and one of its arcs, technically called a half-disc. A half-disc is a special case of a segment, namely the largest one. Tangent: a coplanar straight line that has one single point in common with a circle. All of the specified regions may be considered as open, that is, not containing their boundaries, or as closed, including their respective boundaries; the word circle derives from the Greek κίρκος/κύκλος, itself a metathesis of the Homeric Greek κρίκος, meaning "hoop" or "ring".
The origins of the words circus and circuit are related. The circle has been known since before the beginning of recorded history. Natural circles would have been observed, such as the Moon, a short plant stalk blowing in the wind on sand, which forms a circle shape in the sand; the circle is the basis for the wheel, with related inventions such as gears, makes much of modern machinery possible. In mathematics, the study of the circle has helped inspire the development of geometry and calculus. Early science geometry and astrology and astronomy, was connected to the divine for most medieval scholars, many believed that there was something intrinsically "divine" or "perfect" that could be found in circles; some highlights in the history of the circle are: 1700 BCE – The Rhind papyrus gives a method to find the area of a circular field. The result corresponds to 256/81 as an approximate value of π. 300 BCE – Book 3 of Euclid's Elements deals with the properties of circles. In Plato's Seventh Letter there is a detailed explanation of the circle.
Plato explains the perfect circle, how it is different from any drawing, definition or explanation. 1880 CE – Lindemann proves that π is transcendental settling the millennia-old problem of squaring the circle. The ratio of a circle's circumference to its diameter is π, an irrational constant equal to 3.141592654. Thus the length of the circumference C is related to the radius r and diameter d by: C = 2 π r = π d; as proved by Archimedes, in his Measurement of a Circle, the area enclosed by a circle is equal to that of a triangle whose base has the length of the circle's circumference and whose height equals the circle's radius, which comes to π multiplied by the radius squared: A r e a = π r 2. Equivalently, denoting diameter by d, A r e
Lagrangian point
In celestial mechanics, the Lagrangian points are the points near two large bodies in orbit where a smaller object will maintain its position relative to the large orbiting bodies. At other locations, a small object would go into its own orbit around one of the large bodies, but at the Lagrangian points the gravitational forces of the two large bodies, the centripetal force of orbital motion, the Coriolis acceleration all match up in a way that cause the small object to maintain a stable or nearly stable position relative to the large bodies. There are five such points, labeled L1 to L5, all in the orbital plane of the two large bodies, for each given combination of two orbital bodies. For instance, there are five Lagrangian points L1 to L5 for the Sun-Earth system, in a similar way there are five different Lagrangian points for the Earth-Moon system. L1, L2, L3 are on the line through the centers of the two large bodies. L4 and L5 each form an equilateral triangle with the centers of the large bodies.
L4 and L5 are stable, which implies that objects can orbit around them in a rotating coordinate system tied to the two large bodies. Several planets have trojan satellites near their L5 points with respect to the Sun. Jupiter has more than a million of these trojans. Artificial satellites have been placed at L1 and L2 with respect to the Sun and Earth, with respect to the Earth and the Moon; the Lagrangian points have been proposed for uses in space exploration. The three collinear Lagrange points were discovered by Leonhard Euler a few years before Joseph-Louis Lagrange discovered the remaining two. In 1772, Lagrange published an "Essay on the three-body problem". In the first chapter he considered the general three-body problem. From that, in the second chapter, he demonstrated two special constant-pattern solutions, the collinear and the equilateral, for any three masses, with circular orbits; the five Lagrangian points are labeled and defined as follows: The L1 point lies on the line defined by the two large masses M1 and M2, between them.
It is the most intuitively understood of the Lagrangian points: the one where the gravitational attraction of M2 cancels M1's gravitational attraction. Explanation An object that orbits the Sun more than Earth would have a shorter orbital period than Earth, but that ignores the effect of Earth's own gravitational pull. If the object is directly between Earth and the Sun Earth's gravity counteracts some of the Sun's pull on the object, therefore increases the orbital period of the object; the closer to Earth the object is, the greater this effect is. At the L1 point, the orbital period of the object becomes equal to Earth's orbital period. L1 is 0.01 au, 1/100th the distance to the Sun. The L2 point lies on the line beyond the smaller of the two. Here, the gravitational forces of the two large masses balance the centrifugal effect on a body at L2. Explanation On the opposite side of Earth from the Sun, the orbital period of an object would be greater than that of Earth; the extra pull of Earth's gravity decreases the orbital period of the object, at the L2 point that orbital period becomes equal to Earth's.
Like L1, L2 is 0.01 au from Earth. The L3 point lies on the line defined beyond the larger of the two. Explanation Within the Sun-Earth system, the L3 point exists on the opposite side of the Sun, a little outside Earth's orbit and further from the Sun than Earth is; this placement occurs because the Sun is affected by Earth's gravity and so orbits around the two bodies' barycenter, well inside the body of the Sun. At the L3 point, the combined pull of Earth and Sun cause the object to orbit with the same period as Earth; the L4 and L5 points 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 or ahead of the smaller mass with regard to its orbit around the larger mass. The triangular points are stable equilibria, provided that the ratio of M1/M2 is greater than 24.96. This is the case for the Sun–Earth system, the Sun–Jupiter system, and, by a smaller margin, the Earth–Moon system.
When a body at these points is perturbed, it moves away from the point, but the factor opposite of that, increased or decreased by the perturbation will increase or decrease, bending the object's path into a stable, kidney bean-shaped orbit around the point. In contrast to L4 and L5, where stable equilibrium exists, the points L1, L2, L3 are positions of unstable equilibrium. Any object orbiting at L1, L2, or L3 will tend to fall out of orbit, it is common to orbiting the L4 and L5 points of natural orbital systems. These are called "trojans". In the 20th century, asteroids discovered orbiting at the Sun–Jupiter L4 and L5 points were named after characters from Homer's Iliad. Asteroids at the L4 point, which leads Jupiter, are referred to as the "Greek camp", whereas those at the L5 point are referred to as the "Trojan camp". Other examples of natural objects orbiting at Lagrange points: The Sun–Earth L4 and L5 points contain interplanetary dust and at least one asteroid, 2010 TK7, detected in October 2010 by Wide-field Infrared Survey Explorer and announced during July 2011.
The Earth–Moon L4 and