1.
Kepler orbit
–
In celestial mechanics, a Kepler orbit is the motion of one body relative to another, as an ellipse, parabola, or hyperbola, which forms a two-dimensional orbital plane in three-dimensional space. It is thus said to be a solution of a case of the two-body problem. As a theory in classical mechanics, it also does not take account the effects of general relativity. Keplerian orbits can be parametrized into six orbital elements in various ways, in most applications, there is a large central body, the center of mass of which is assumed to be the center of mass of the entire system. By decomposition, the orbits of two objects of mass can be described as Kepler orbits around their common center of mass. Variations in the motions of the planets were explained by smaller circular paths overlaid on the larger path, as measurements of the planets became increasingly accurate, revisions to the theory were proposed. In 1543, Nicolaus Copernicus published a model of the solar system. In 1601, Johannes Kepler acquired the extensive, meticulous observations of the planets made by Tycho Brahe, Kepler would spend the next five years trying to fit the observations of the planet Mars to various curves. In 1609, Kepler published the first two of his three laws of planetary motion, the first law states, The orbit of every planet is an ellipse with the sun at a focus. More generally, the path of an object undergoing Keplerian motion may also follow a parabola or a hyperbola, alternately, the equation can be expressed as, r = p 1 + e cos Where p is called the semi-latus rectum of the curve. This form of the equation is useful when dealing with parabolic trajectories. Despite developing these laws from observations, Kepler was never able to develop a theory to explain these motions, between 1665 and 1666, Isaac Newton developed several concepts related to motion, gravitation and differential calculus. However, these concepts were not published until 1687 in the Principia, in which he outlined his laws of motion, newtons law of gravitation states, Every point mass attracts every other point mass by a force pointing along the line intersecting both points. The laws of Kepler and Newton formed the basis of modern celestial mechanics until Albert Einstein introduced the concepts of special and general relativity in the early 20th century. For most applications, Keplerian motion approximates the motions of planets and satellites to relatively high degrees of accuracy and is used extensively in astronomy and astrodynamics. See also Orbit Analysis To solve for the motion of an object in a two body system, two simplifying assumptions can be made,1, the bodies are spherically symmetric and can be treated as point masses. There are no external or internal forces acting upon the other than their mutual gravitation. The shapes of large bodies are close to spheres
2.
Circle
–
A circle is a simple closed shape in Euclidean geometry. The distance between any of the points and the centre is called the radius, a circle is a simple closed curve which divides the plane into two regions, an interior and an exterior. Annulus, the object, the region bounded by two concentric circles. Arc, any connected part of the circle, centre, the point equidistant from the points on the circle. Chord, a segment whose endpoints lie on the circle. Circumference, the length of one circuit along the circle, or the distance around the circle and it is a special case of a chord, namely the longest chord, and it is twice the radius. Disc, the region of the bounded by a circle. Lens, the intersection of two discs, passant, a coplanar straight line that does not touch the circle. Radius, a line segment joining the centre of the circle to any point on the circle itself, or the length of such a segment, sector, a region bounded by two radii and an arc lying between the radii. Segment, a region, not containing the centre, bounded by a chord, secant, an extended chord, a coplanar straight line cutting the circle at two points. Semicircle, an arc that extends from one of a diameters endpoints to the other, in non-technical common usage it may mean the diameter, arc, and its interior, a two dimensional region, that is technically called a half-disc. A half-disc is a case of a segment, namely the largest one. Tangent, a straight line that touches the circle at a single point. The word circle derives from the Greek κίρκος/κύκλος, itself a metathesis of the Homeric Greek κρίκος, the origins of the words circus and circuit are closely related. The circle has been known since before the beginning of recorded history, natural circles would have been observed, such as the Moon, Sun, and 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, which, with related inventions such as gears, in mathematics, the study of the circle has helped inspire the development of geometry, astronomy and calculus. 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 a value of π.300 BCE – Book 3 of Euclids Elements deals with the properties of circles
3.
Ellipse
–
In mathematics, an ellipse is a curve in a plane surrounding two focal points such that the sum of the distances to the two focal points is constant for every point on the curve. As such, it is a generalization of a circle, which is a type of an ellipse having both focal points at the same location. The shape of an ellipse is represented by its eccentricity, which for an ellipse can be any number from 0 to arbitrarily close to, ellipses are the closed type of conic section, a plane curve resulting from the intersection of a cone by a plane. Ellipses have many similarities with the two forms of conic sections, parabolas and hyperbolas, both of which are open and unbounded. The cross section of a cylinder is an ellipse, unless the section is parallel to the axis of the cylinder and this ratio is called the eccentricity of the ellipse. Ellipses are common in physics, astronomy and engineering, for example, the orbit of each planet in our solar system is approximately an ellipse with the barycenter of the planet–Sun pair at one of the focal points. The same is true for moons orbiting planets and all other systems having two astronomical bodies, the shapes of planets and stars are often well described by ellipsoids. It is also the simplest Lissajous figure formed when the horizontal and vertical motions are sinusoids with the same frequency, a similar effect leads to elliptical polarization of light in optics. The name, ἔλλειψις, was given by Apollonius of Perga in his Conics, in order to omit the special case of a line segment, one presumes 2 a > | F1 F2 |, E =. The midpoint C of the segment joining the foci is called the center of the ellipse. The line through the foci is called the major axis and it contains the vertices V1, V2, which have distance a to the center. The distance c of the foci to the center is called the distance or linear eccentricity. The quotient c a is the eccentricity e, the case F1 = F2 yields a circle and is included. C2 is called the circle of the ellipse. This property should not be confused with the definition of an ellipse with help of a directrix below, for an arbitrary point the distance to the focus is 2 + y 2 and to the second focus 2 + y 2. Hence the point is on the ellipse if the condition is fulfilled 2 + y 2 +2 + y 2 =2 a. The shape parameters a, b are called the major axis. The points V3 =, V4 = are the co-vertices and it follows from the equation that the ellipse is symmetric with respect to both of the coordinate axes and hence symmetric with respect to the origin
4.
Parabola
–
A parabola is a two-dimensional, mirror-symmetrical curve, which is approximately U-shaped when oriented as shown in the diagram below, but which can be in any orientation in its plane. It fits any of several different mathematical descriptions which can all be proved to define curves of exactly the same shape. One description of a parabola involves a point and a line, the focus does not lie on the directrix. The parabola is the locus of points in that plane that are equidistant from both the directrix and the focus, a parabola is a graph of a quadratic function, y = x2, for example. The line perpendicular to the directrix and passing through the focus is called the axis of symmetry, the point on the parabola that intersects the axis of symmetry is called the vertex, and is the point where the parabola is most sharply curved. The distance between the vertex and the focus, measured along the axis of symmetry, is the focal length, the latus rectum is the chord of the parabola which is parallel to the directrix and passes through the focus. Parabolas can open up, down, left, right, or in some arbitrary direction. Any parabola can be repositioned and rescaled to fit exactly on any other parabola — that is, conversely, light that originates from a point source at the focus is reflected into a parallel beam, leaving the parabola parallel to the axis of symmetry. The same effects occur with sound and other forms of energy and this reflective property is the basis of many practical uses of parabolas. The parabola has many important applications, from an antenna or parabolic microphone to automobile headlight reflectors to the design of ballistic missiles. They are frequently used in physics, engineering, and many other areas, the earliest known work on conic sections was by Menaechmus in the fourth century BC. He discovered a way to solve the problem of doubling the cube using parabolas, the name parabola is due to Apollonius who discovered many properties of conic sections. It means application, referring to application of concept, that has a connection with this curve. The focus–directrix property of the parabola and other conics is due to Pappus, Galileo showed that the path of a projectile follows a parabola, a consequence of uniform acceleration due to gravity. The idea that a reflector could produce an image was already well known before the invention of the reflecting telescope. Designs were proposed in the early to mid seventeenth century by many mathematicians including René Descartes, Marin Mersenne, when Isaac Newton built the first reflecting telescope in 1668, he skipped using a parabolic mirror because of the difficulty of fabrication, opting for a spherical mirror. Parabolic mirrors are used in most modern reflecting telescopes and in satellite dishes, solving for y yields y =14 f x 2. The length of the chord through the focus is called latus rectum, one half of it semi latus rectum
5.
Escape orbit
–
In astrodynamics or celestial mechanics a parabolic trajectory is a Kepler orbit with the eccentricity equal to 1. When moving away from the source it is called an escape orbit and it is also sometimes referred to as a C3 =0 orbit. Parabolic trajectories are escape trajectories, separating positive-energy hyperbolic trajectories from negative-energy elliptic orbits. At any position the body has the escape velocity for that position. This is entirely equivalent to the energy being 0, C3 =0 Barkers equation relates the time of flight to the true anomaly of a parabolic trajectory. There are two cases, the move away from each other or towards each other. At any time the speed from t =0 is 1.5 times the current speed
6.
Hyperbola
–
In mathematics, a hyperbola is a type of smooth curve lying in a plane, defined by its geometric properties or by equations for which it is the solution set. A hyperbola has two pieces, called connected components or branches, that are mirror images of each other, the hyperbola is one of the three kinds of conic section, formed by the intersection of a plane and a double cone. If the plane intersects both halves of the double cone but does not pass through the apex of the cones, each branch of the hyperbola has two arms which become straighter further out from the center of the hyperbola. Diagonally opposite arms, one from each branch, tend in the limit to a common line, called the asymptote of those two arms. So there are two asymptotes, whose intersection is at the center of symmetry of the hyperbola, which can be thought of as the point about which each branch reflects to form the other branch. In the case of the curve f =1 / x the asymptotes are the two coordinate axes, hyperbolas share many of the ellipses analytical properties such as eccentricity, focus, and directrix. Typically the correspondence can be made with nothing more than a change of sign in some term, many other mathematical objects have their origin in the hyperbola, such as hyperbolic paraboloids, hyperboloids, hyperbolic geometry, hyperbolic functions, and gyrovector spaces. The word hyperbola derives from the Greek ὑπερβολή, meaning over-thrown or excessive, hyperbolae were discovered by Menaechmus in his investigations of the problem of doubling the cube, but were then called sections of obtuse cones. The term hyperbola is believed to have coined by Apollonius of Perga in his definitive work on the conic sections. The rectangle could be applied to the segment, be shorter than the segment or exceed the segment, the midpoint M of the line segment joining the foci is called the center of the hyperbola. The line through the foci is called the major axis and it contains the vertices V1, V2, which have distance a to the center. The distance c of the foci to the center is called the distance or linear eccentricity. The quotient c a is the eccentricity e, C2 is called the director circle of the hyperbola. In order to get the branch of the hyperbola, one has to use the director circle related to F1. This property should not be confused with the definition of a hyperbola with help of a directrix below, for an arbitrary point the distance to the focus is 2 + y 2 and to the second focus 2 + y 2. Hence the point is on the hyperbola if the condition is fulfilled 2 + y 2 −2 + y 2 = ±2 a. Remove the square roots by suitable squarings and use the relation b 2 = c 2 − a 2 to obtain the equation of the hyperbola, the shape parameters a, b are called the semi major axis and semi minor axis or conjugate axis. As opposed to an ellipse, a hyperbola has two vertices
7.
Conic section
–
In mathematics, a conic section is a curve obtained as the intersection of the surface of a cone with a plane. The three types of conic section are the hyperbola, the parabola, and the ellipse, the circle is a special case of the ellipse, and is of sufficient interest in its own right that it was sometimes called a fourth type of conic section. The conic sections have been studied by the ancient Greek mathematicians with this work culminating around 200 BC, the conic sections of the Euclidean plane have various distinguishing properties. Many of these have used as the basis for a definition of the conic sections. The type of conic is determined by the value of the eccentricity, in analytic geometry, a conic may be defined as a plane algebraic curve of degree 2, that is, as the set of points whose coordinates satisfy a quadratic equation in two variables. This equation may be written in form, and some geometric properties can be studied as algebraic conditions. In the Euclidean plane, the conic sections appear to be different from one another. By extending the geometry to a projective plane this apparent difference vanishes, further extension, by expanding the real coordinates to admit complex coordinates, provides the means to see this unification algebraically. The conic sections have been studied for thousands of years and have provided a source of interesting. A conic is the curve obtained as the intersection of a plane, called the cutting plane and we shall assume that the cone is a right circular cone for the purpose of easy description, but this is not required, any double cone with some circular cross-section will suffice. Planes that pass through the vertex of the cone will intersect the cone in a point and these are called degenerate conics and some authors do not consider them to be conics at all. Unless otherwise stated, we assume that conic refers to a non-degenerate conic. There are three types of conics, the ellipse, parabola, and hyperbola, the circle is a special kind of ellipse, although historically it had been considered as a fourth type. The circle and the ellipse arise when the intersection of the cone and plane is a closed curve, if the cutting plane is parallel to exactly one generating line of the cone, then the conic is unbounded and is called a parabola. In the remaining case, the figure is a hyperbola, in this case, the plane will intersect both halves of the cone, producing two separate unbounded curves. A property that the conic sections share is often presented as the following definition, a conic section is the locus of all points P whose distance to a fixed point F is a constant multiple of the distance from P to a fixed line L. For 0 < e <1 we obtain an ellipse, for e =1 a parabola, a circle is a limiting case and is not defined by a focus and directrix, in the plane. The eccentricity of a circle is defined to be zero and its focus is the center of the circle, an ellipse and a hyperbola each have two foci and distinct directrices for each of them
8.
Two-body problem
–
In classical mechanics, the two-body problem is to determine the motion of two point particles that interact only with each other. Common examples include a satellite orbiting a planet, a planet orbiting a star, the two-body problem can be re-formulated as two one-body problems, a trivial one and one that involves solving for the motion of one particle in an external potential. Since many one-body problems can be solved exactly, the corresponding two-body problem can also be solved, by contrast, the three-body problem cannot be solved in terms of first integrals, except in special cases. Let x1 and x2 be the positions of the two bodies, and m1 and m2 be their masses. The goal is to determine the trajectories x1 and x2 for all t, given the initial positions x1 and x2. The two dots on top of the x position vectors denote their second derivative with respect to time, adding and subtracting these two equations decouples them into two one-body problems, which can be solved independently. Adding equations and results in an equation describing the center of mass motion, by contrast, subtracting equation from equation results in an equation that describes how the vector r = x1 − x2 between the masses changes with time. The solutions of these independent one-body problems can be combined to obtain the solutions for the trajectories x1 and x2. The resulting equation, R ¨ =0 shows that the velocity V = dR/dt of the center of mass is constant, hence, the position R of the center of mass can be determined at all times from the initial positions and velocities. The motion of two bodies with respect to each other always lies in a plane, introducing the assumption that the force between two particles acts along the line between their positions, it follows that r × F =0 and the angular momentum vector L is constant. We now have, μ r ¨ = F r ^, Kepler orbit Energy drift Equation of the center Eulers three-body problem Gravitational two-body problem Kepler problem n-body problem Virial theorem Two-body problem Landau LD, Lifshitz EM. Two-body problem at Eric Weissteins World of Physics
9.
Klemperer rosette
–
A Klemperer rosette is a gravitational system of heavier and lighter bodies orbiting in a regular repeating pattern around a common barycenter. It was first described by W. B, the number of mass types can be increased, so long as the arrangement pattern is cylic, e. g. etc. Klemperer also mentioned octagonal and rhombic rosettes, while all Klemperer rosettes are vulnerable to destabilization, the hexagonal rosette should have extra stability due to the planets sitting in each others L4 and L5 Lagrangian points. Klemperer does indeed mention this configuration at the start of his article, in Larry Nivens novel Ringworld, the Puppeteers Fleet of Worlds is arranged in such a configuration which Niven calls a Kemplerer rosette, this misspelling is one possible source of this confusion. Another is the similarity between Klemperers name and that of Johannes Kepler, who described certain laws of motion in the 17th century. It is notable that these planets were maintained in position by large engines in addition to gravitational force. This is the case whether the center of the Rosette is in free space, the short-form reason is that any perturbation destroys the symmetry, which increases the perturbation, which further damages the symmetry, and so on. Rosette simulations using Java applets Kemplerer Rosette by Larry Niven from Ringworld
10.
Orbit
–
In physics, an orbit is the gravitationally curved path of an object around a point in space, for example the orbit of a planet about a star or a natural satellite around a planet. Normally, orbit refers to a regularly repeating path around a body, to a close approximation, planets and satellites follow elliptical orbits, with the central mass being orbited at a focal point of the ellipse, as described by Keplers laws of planetary motion. For ease of calculation, in most situations orbital motion is adequately approximated by Newtonian Mechanics, historically, 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 it assumed the heavens were fixed apart from the motion of the spheres, and was developed without any understanding of gravity. After the planets motions were accurately measured, theoretical mechanisms such as deferent. Originally 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 understanding of orbits was first formulated by Johannes Kepler whose results are summarised in his three laws of planetary motion. Second, he found that the speed of each planet is not constant, as had previously been thought. Third, Kepler found a 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 respectively about 5.2 and 0.723 AU distant from the Sun, their orbital periods respectively 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. Idealised orbits meeting these rules are known as Kepler orbits, isaac Newton demonstrated that Keplers 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 sizes are in inverse proportion to their masses. Where one body is 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. Lagrange developed a new approach to Newtonian mechanics emphasizing energy more than force, in a dramatic vindication of classical mechanics, in 1846 le Verrier was able to predict the position of Neptune based on unexplained perturbations in the orbit of Uranus. 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 usually approximated very well by the Newtonian predictions but the differences are measurable. Essentially all the evidence that can distinguish between the theories agrees with relativity theory to within experimental measurement accuracy
11.
Eccentricity (mathematics)
–
In mathematics, the eccentricity, denoted e or ε, is a parameter associated with every conic section. It can be thought of as a measure of how much the conic section deviates from being circular, in particular, The eccentricity of a circle is zero. The eccentricity of an ellipse which is not a circle is greater than zero, the eccentricity of a parabola is 1. The eccentricity of a hyperbola is greater than 1, the eccentricity of a line is infinite. Furthermore, two sections are similar if and only if they have the same eccentricity. Any conic section can be defined as the locus of points whose distances to a point and that ratio is called eccentricity, commonly denoted as e. The eccentricity can also be defined in terms of the intersection of a plane, for β =0 the plane section is a circle, for β = α a parabola. The linear eccentricity of an ellipse or hyperbola, denoted c, is the distance between its center and either of its two foci, the eccentricity can be defined as the ratio of the linear eccentricity to the semimajor axis a, that is, e = c a. The eccentricity is sometimes called first eccentricity to distinguish it from the second eccentricity, the eccentricity is also sometimes called numerical eccentricity. In the case of ellipses and hyperbolas the linear eccentricity is sometimes called half-focal separation, three notational conventions are in common use, e for the eccentricity and c for the linear eccentricity. ε for the eccentricity and e for the linear eccentricity, E or ϵ for the eccentricity and f for the linear eccentricity. This article uses the first notation, where, for the ellipse and the hyperbola, a is the length of the semi-major axis and b is the length of the semi-minor axis. The eccentricity of an ellipse is strictly less than 1, for any ellipse, let a be the length of its semi-major axis and b be the length of its semi-minor axis. The eccentricity is also the ratio of the axis a to the distance d from the center to the directrix. The eccentricity can be expressed in terms of the g, e = g. Define the maximum and minimum radii r max and r min as the maximum and minimum distances from either focus to the ellipse, the eccentricity of a hyperbola can be any real number greater than 1, with no upper bound. The eccentricity of a hyperbola is 2. The eccentricity of a quadric is the eccentricity of a designated section of it
12.
Circular orbit
–
A circular orbit is the orbit at a fixed distance around any point by an object rotating around a fixed axis. Below we consider a circular orbit in astrodynamics or celestial mechanics under standard assumptions, here the centripetal force is the gravitational force, and the axis mentioned above is the line through the center of the central mass perpendicular to the plane of motion. In this case, not only the distance, but also the speed, angular speed, potential, there is no periapsis or apoapsis. This orbit has no radial version, transverse acceleration causes change in direction. If it is constant in magnitude and changing in direction with the velocity, we get a circular motion. For this centripetal acceleration we have a = v 2 r = ω2 r where, v is velocity of orbiting body. The formula is dimensionless, describing a ratio true for all units of measure applied uniformly across the formula. If the numerical value of a is measured in meters per second per second, then the values for v will be in meters per second, r in meters. μ = G M is the standard gravitational parameter, the orbit equation in polar coordinates, which in general gives r in terms of θ, reduces to, r = h 2 μ where, h = r v is specific angular momentum of the orbiting body. Maneuvering into a circular orbit, e. g. It is also a matter of maneuvering into the orbit, for the sake of convenience, the derivation will be written in units in which c = G =1. The four-velocity of a body on an orbit is given by. The dot above a variable denotes derivation with respect to proper time τ
13.
Elliptic orbit
–
In astrodynamics or celestial mechanics an elliptic orbit is a Kepler orbit with the eccentricity less than 1, this includes the special case of a circular orbit, with eccentricity equal to zero. In a stricter sense, it is a Kepler orbit with the eccentricity greater than 0, in a wider sense it is a Kepler orbit with negative energy. This includes the radial elliptic orbit, with eccentricity equal to 1, in a gravitational two-body problem with negative energy both bodies follow similar elliptic orbits with the same orbital period around their common barycenter. Also the relative position of one body with respect to the other follows an elliptic orbit, examples of elliptic orbits include, Hohmann transfer orbit, Molniya orbit and tundra orbit. A is the length of the semi-major axis, the velocity equation for a hyperbolic trajectory has either +1 a, or it is the same with the convention that in that case a is negative. Conclusions, For a given semi-major axis the orbital energy is independent of the eccentricity. ν is the true anomaly. The angular momentum is related to the cross product of position and velocity. Here ϕ is defined as the angle which differs by 90 degrees from this and this set of six variables, together with time, are called the orbital state vectors. Given the masses of the two bodies they determine the full orbit, the two most general cases with these 6 degrees of freedom are the elliptic and the hyperbolic orbit. Special cases with fewer degrees of freedom are the circular and parabolic orbit, another set of six parameters that are commonly used are the orbital elements. In the Solar System, planets, asteroids, most comets, the following chart of the perihelion and aphelion of the planets, dwarf planets and Halleys Comet demonstrates the variation of the eccentricity of their elliptical orbits. For similar distances from the sun, wider bars denote greater eccentricity, note the almost-zero eccentricity of Earth and Venus compared to the enormous eccentricity of Halleys Comet and Eris. A radial trajectory can be a line segment, which is a degenerate ellipse with semi-minor axis =0. Although the eccentricity is 1, this is not a parabolic orbit, most properties and formulas of elliptic orbits apply. However, the orbit cannot be closed and it is an open orbit corresponding to the part of the degenerate ellipse from the moment the bodies touch each other and move away from each other until they touch each other again. In the case of point masses one full orbit is possible, the velocities at the start and end are infinite in opposite directions and the potential energy is equal to minus infinity. The radial elliptic trajectory is the solution of a problem with at some instant zero speed
14.
Parabolic trajectory
–
In astrodynamics or celestial mechanics a parabolic trajectory is a Kepler orbit with the eccentricity equal to 1. When moving away from the source it is called an escape orbit and it is also sometimes referred to as a C3 =0 orbit. Parabolic trajectories are escape trajectories, separating positive-energy hyperbolic trajectories from negative-energy elliptic orbits. At any position the body has the escape velocity for that position. This is entirely equivalent to the energy being 0, C3 =0 Barkers equation relates the time of flight to the true anomaly of a parabolic trajectory. There are two cases, the move away from each other or towards each other. At any time the speed from t =0 is 1.5 times the current speed
15.
Hyperbolic trajectory
–
In astrodynamics or celestial mechanics, a hyperbolic trajectory is the trajectory of any object around a central body with more than enough speed to escape the central objects gravitational pull. The name derives from the fact that according to Newtonian theory such an orbit has the shape of a hyperbola, in more technical terms this can be expressed by the condition that the orbital eccentricity is greater than one. Under standard assumptions a body traveling along this trajectory will coast to infinity, similarly to parabolic trajectory all hyperbolic trajectories are also escape trajectories. The specific energy of a hyperbolic trajectory orbit is positive, planetary flybys, used for gravitational slingshots, can be described within the planets sphere of influence using hyperbolic trajectories. Like an elliptical orbit, a trajectory for a given system can be defined by its semi major axis. However, with a hyperbolic orbit other parameters may more useful in understanding a bodys motion, the following table lists the main parameters describing the path of body following a hyperbolic trajectory around another under standard assumptions and the formula connecting them. The semi major axis is not immediately visible with an hyperbolic trajectory, usually, by convention, it is negative, to keep various equations are consistent with elliptical orbits. With a hyperbolic trajectory the orbital eccentricity is greater than 1, the eccentricity is directly related to the angle between the asymptotes. With eccentricity just over 1 the hyperbola is a v shape. At e =2 the asymptotes are at right angles, with e >2 the asymptotes are more that 120° apart, and the periapsis distance is greater than the semi major axis. As eccentricity increases further the motion approaches a straight line, with bodies experiencing gravitational forces and following hyperbolic trajectories it is equal to the semi-minor axis of the hyperbola. In the situation of a spacecraft or comet approaching a planet, if the central body is known the trajectory can now be found, including how close the approaching body will be at periapsis. If this is less than planets radius an impact should be expected, a body approaching Jupiter from the outer solar system with a speed of 5.5 km/h, will need the impact parameter to be at least 770, 000km or 11 times Jupiter radius to avoid collision. As, typically, all variables can be determined accurately. μ = b v ∞2 tan δ /2 where δ =2 θ ∞ − π is the angle the body is deflected from a straight line in its course. Where μ is a parameter w and a is the semi-major axis of the orbit. The flight path angle is the angle between the direction of velocity and the perpendicular to the direction, so it is zero at periapsis. For example, at a place where escape speed is 11.2 km/s,11.62 −11.22 =3.02 This is an example of the Oberth effect
16.
Orbital energy
–
It is expressed in J/kg = m2·s−2 or MJ/kg = km2·s−2. For an elliptical orbit the specific energy is the negative of the additional energy required to accelerate a mass of one kilogram to escape velocity. For a hyperbolic orbit, it is equal to the energy compared to that of a parabolic orbit. In this case the orbital energy is also referred to as characteristic energy. For a hyperbolic trajectory this specific orbital energy is given by ϵ = μ2 a. or the same as for an ellipse. In this case the orbital energy is also referred to as characteristic energy and is equal to the excess specific energy compared to that for a parabolic orbit. It is related to the excess velocity v ∞ by 2 ϵ = C3 = v ∞2. It is relevant for interplanetary missions, thus, if orbital position vector and orbital velocity vector are known at one position, and μ is known, then the energy can be computed and from that, for any other position, the orbital speed. In the case of circular orbits, this rate is one half of the gravity at the orbit and this corresponds to the fact that for such orbits the total energy is one half of the potential energy, because the kinetic energy is minus one half of the potential energy. If the central body has radius R, then the energy of an elliptic orbit compared to being stationary at the surface is − μ2 a + μ R = μ2 a R. The International Space Station has a period of 91.74 minutes. The energy is −29.6 MJ/kg, the energy is −59.2 MJ/kg. Compare with the energy at the surface, which is −62.6 MJ/kg. The extra potential energy is 3.4 MJ/kg, the extra energy is 33.0 MJ/kg. The average speed is 7.7 km/s, the net delta-v to reach this orbit is 8.1 km/s, the increase per meter would be 4.4 J/kg, this rate corresponds to one half of the local gravity of 8.8 m/s². For an altitude of 100 km, The energy is −30.8 MJ/kg, the energy is −61.6 MJ/kg. Compare with the energy at the surface, which is −62.6 MJ/kg. The extra potential energy is 1.0 MJ/kg, the extra energy is 31.8 MJ/kg
17.
Angular momentum
–
In physics, angular momentum is the rotational analog of linear momentum. It is an important quantity in physics because it is a conserved quantity – the angular momentum of a system remains constant unless acted on by an external torque. The definition of momentum for a point particle is a pseudovector r×p. This definition can be applied to each point in continua like solids or fluids, unlike momentum, angular momentum does depend on where the origin is chosen, since the particles position is measured from it. The angular momentum of an object can also be connected to the angular velocity ω of the object via the moment of inertia I. However, while ω always points in the direction of the rotation axis, Angular momentum is additive, the total angular momentum of a system is the vector sum of the angular momenta. For continua or fields one uses integration, torque can be defined as the rate of change of angular momentum, analogous to force. Applications include the gyrocompass, control moment gyroscope, inertial systems, reaction wheels, flying discs or Frisbees. In general, conservation does limit the motion of a system. In quantum mechanics, angular momentum is an operator with quantized eigenvalues, Angular momentum is subject to the Heisenberg uncertainty principle, meaning only one component can be measured with definite precision, the other two cannot. Also, the spin of elementary particles does not correspond to literal spinning motion, Angular momentum is a vector quantity that represents the product of a bodys rotational inertia and rotational velocity about a particular axis. Angular momentum can be considered an analog of linear momentum. Thus, where momentum is proportional to mass m and linear speed v, p = m v, angular momentum is proportional to moment of inertia I. Unlike mass, which only on amount of matter, moment of inertia is also dependent on the position of the axis of rotation. Unlike linear speed, which occurs in a line, angular speed occurs about a center of rotation. Therefore, strictly speaking, L should be referred to as the angular momentum relative to that center and this simple analysis can also apply to non-circular motion if only the component of the motion which is perpendicular to the radius vector is considered. In that case, L = r m v ⊥, where v ⊥ = v sin θ is the component of the motion. It is this definition, × to which the moment of momentum refers
18.
Gravity
–
Gravity, or gravitation, is a natural phenomenon by which all things with mass are brought toward one another, including planets, stars and galaxies. Since energy and mass are equivalent, all forms of energy, including light, on Earth, gravity gives weight to physical objects and causes the ocean tides. Gravity has a range, although its effects become increasingly weaker on farther objects. The most extreme example of this curvature of spacetime is a hole, from which nothing can escape once past its event horizon. More gravity results in time dilation, where time lapses more slowly at a lower gravitational potential. Gravity is the weakest of the four fundamental interactions of nature, the gravitational attraction is approximately 1038 times weaker than the strong force,1036 times weaker than the electromagnetic force and 1029 times weaker than the weak force. As a consequence, gravity has an influence on the behavior of subatomic particles. On the other hand, gravity is the dominant interaction at the macroscopic scale, for this reason, in part, pursuit of a theory of everything, the merging of the general theory of relativity and quantum mechanics into quantum gravity, has become an area of research. While the modern European thinkers are credited with development of gravitational theory, some of the earliest descriptions came from early mathematician-astronomers, such as Aryabhata, who had identified the force of gravity to explain why objects do not fall out when the Earth rotates. Later, the works of Brahmagupta referred to the presence of force, described it as an attractive force. Modern work on gravitational theory began with the work of Galileo Galilei in the late 16th and this was a major departure from Aristotles belief that heavier objects have a higher gravitational acceleration. Galileo postulated air resistance as the reason that objects with less mass may fall slower in an atmosphere, galileos work set the stage for the formulation of Newtons theory of gravity. In 1687, English mathematician Sir Isaac Newton published Principia, which hypothesizes the inverse-square law of universal gravitation. Newtons theory enjoyed its greatest success when it was used to predict the existence of Neptune based on motions of Uranus that could not be accounted for by the actions of the other planets. Calculations by both John Couch Adams and Urbain Le Verrier predicted the position of the planet. A discrepancy in Mercurys orbit pointed out flaws in Newtons theory, the issue was resolved in 1915 by Albert Einsteins new theory of general relativity, which accounted for the small discrepancy in Mercurys orbit. The simplest way to test the equivalence principle is to drop two objects of different masses or compositions in a vacuum and see whether they hit the ground at the same time. Such experiments demonstrate that all objects fall at the rate when other forces are negligible
19.
Electrostatics
–
Electrostatics is a branch of physics that deals with the phenomena and properties of stationary or slow-moving electric charges. Since classical physics, it has known that some materials such as amber attract lightweight particles after rubbing. The Greek word for amber, ήλεκτρον, or electron, was the source of the word electricity, Electrostatic phenomena arise from the forces that electric charges exert on each other. Such forces are described by Coulombs law, Electrostatics involves the buildup of charge on the surface of objects due to contact with other surfaces. This is because the charges that transfer are trapped there for a long enough for their effects to be observed. We begin with the magnitude of the force between two point charges q and Q. It is convenient to one of these charges, q, as a test charge. As we develop the theory, more source charges will be added.854187817 ×10 −12 C2 N −1 m −2, the SI units of ε0 are equivalently A2s4 kg−1m−3 or C2N−1m−2 or F m−1. Coulombs constant is, k e ≈14 π ε0 ≈8.987551787 ×109 N m 2 C −2. A single proton has a charge of e, and the electron has a charge of −e and these physical constants are currently defined so that ε0 and k0 are exactly defined, and e is a measured quantity. Electric field lines are useful for visualizing the electric field, field lines begin on positive charge and terminate on negative charge. Electric field lines are parallel to the direction of the field. The electric field, E →, is a field that can be defined everywhere. It is convenient to place a hypothetical test charge at a point, by Coulombs Law, this test charge will experience a force that can be used to define the electric field as follow F → = q E →. For a single point charge at the origin, the magnitude of electric field is E = k e Q / R2. The fact that the force can be calculated by summing all the contributions due to individual source particles is an example of the superposition principle. If the charge is distributed over a surface or along a line, the Divergence Theorem allows Gausss Law to be written in differential form, ∇ → ⋅ E → = ρ ε0. Where ∇ → ⋅ is the divergence operator, the definition of electrostatic potential, combined with the differential form of Gausss law, provides a relationship between the potential Φ and the charge density ρ, ∇2 ϕ = − ρ ε0
20.
Classical physics
–
Classical physics refers to theories of physics that predate modern, more complete, or more widely applicable theories. As such, the definition of a classical theory depends on context, classical physical concepts are often used when modern theories are unnecessarily complex for a particular situation. Classical theory has at least two meanings in physics. In the context of mechanics, classical theory refers to theories of physics that do not use the quantisation paradigm. Likewise, classical field theories, such as general relativity and classical electromagnetism, are those that do not use quantum mechanics, in the context of general and special relativity, classical theories are those that obey Galilean relativity. Modern physics includes quantum theory and relativity, when applicable, a physical system can be described by classical physics when it satisfies conditions such that the laws of classical physics are approximately valid. In practice, physical objects ranging from larger than atoms and molecules, to objects in the macroscopic and astronomical realm. Beginning at the level and lower, the laws of classical physics break down. Electromagnetic fields and forces can be described well by classical electrodynamics at length scales, unlike quantum physics, classical physics is generally characterized by the principle of complete determinism, although deterministic interpretations of quantum mechanics do exist. Mathematically, classical physics equations are those in which Plancks constant does not appear, according to the correspondence principle and Ehrenfests theorem, as a system becomes larger or more massive the classical dynamics tends to emerge, with some exceptions, such as superfluidity. This is why we can usually ignore quantum mechanics when dealing with everyday objects, however, one of the most vigorous on-going fields of research in physics is classical-quantum correspondence. This field of research is concerned with the discovery of how the laws of physics give rise to classical physics found at the limit of the large scales of the classical level. Computer modeling is essential for quantum and relativistic physics, classic physics is considered the limit of quantum mechanics for large number of particles. On the other hand, classic mechanics is derived from relativistic mechanics, for example, in many formulations from special relativity, a correction factor 2 appears, where v is the velocity of the object and c is the speed of light. For velocities much smaller than that of light, one can neglect the terms with c2 and these formulas then reduce to the standard definitions of Newtonian kinetic energy and momentum. This is as it should be, for special relativity must agree with Newtonian mechanics at low velocities, computer modeling has to be as real as possible. Classical physics would introduce an error as in the superfluidity case, in order to produce reliable models of the world, we can not use classic physics. It is true that quantum theories consume time and computer resources, and the equations of physics could be resorted to provide a quick solution
21.
Kepler problem
–
The Kepler conjecture, named after the 17th-century mathematician and astronomer Johannes Kepler, is a mathematical conjecture about sphere packing in three-dimensional Euclidean space. It says that no arrangement of equally sized spheres filling space has an average density than that of the cubic close packing. The density of these arrangements is around 74. 05%, in 1998 Thomas Hales, following an approach suggested by Fejes Tóth, announced that he had a proof of the Kepler conjecture. Hales proof is a proof by exhaustion involving the checking of many cases using complex computer calculations. Referees have said that they are 99% certain of the correctness of Hales proof, in 2014, the Flyspeck project team, headed by Hales, announced the completion of a formal proof of the Kepler conjecture using a combination of the Isabelle and HOL Light proof assistants. Imagine filling a container with small equal-sized spheres. The density of the arrangement is equal to the volume of the spheres divided by the volume of the container. To maximize the number of spheres in the means to create an arrangement with the highest possible density. Experiment shows that dropping the spheres in randomly will achieve a density of around 65%, however, a higher density can be achieved by carefully arranging the spheres as follows. Start with a layer of spheres in a lattice, then put the next layer of spheres in the lowest points you can find above the first layer. The conjecture was first stated by Johannes Kepler in his paper On the six-cornered snowflake and he had started to study arrangements of spheres as a result of his correspondence with the English mathematician and astronomer Thomas Harriot in 1606. Harriot was a friend and assistant of Sir Walter Raleigh, who had set Harriot the problem of determining how best to stack cannonballs on the decks of his ships. Harriot published a study of various stacking patterns in 1591, and this meant that any packing arrangement that disproved the Kepler conjecture would have to be an irregular one. But eliminating all possible arrangements is very difficult, and this is what made the Kepler conjecture so hard to prove. After Gauss, no progress was made towards proving the Kepler conjecture in the nineteenth century. In 1900 David Hilbert included it in his list of twenty three unsolved problems of mathematics—it forms part of Hilberts eighteenth problem, the next step toward a solution was taken by László Fejes Tóth. Fejes Tóth showed that the problem of determining the density of all arrangements could be reduced to a finite number of calculations. This meant that a proof by exhaustion was, in principle, possible, as Fejes Tóth realised, a fast enough computer could turn this theoretical result into a practical approach to the problem
22.
Specific orbital energy
–
It is expressed in J/kg = m2·s−2 or MJ/kg = km2·s−2. For an elliptical orbit the specific energy is the negative of the additional energy required to accelerate a mass of one kilogram to escape velocity. For a hyperbolic orbit, it is equal to the energy compared to that of a parabolic orbit. In this case the orbital energy is also referred to as characteristic energy. For a hyperbolic trajectory this specific orbital energy is given by ϵ = μ2 a. or the same as for an ellipse. In this case the orbital energy is also referred to as characteristic energy and is equal to the excess specific energy compared to that for a parabolic orbit. It is related to the excess velocity v ∞ by 2 ϵ = C3 = v ∞2. It is relevant for interplanetary missions, thus, if orbital position vector and orbital velocity vector are known at one position, and μ is known, then the energy can be computed and from that, for any other position, the orbital speed. In the case of circular orbits, this rate is one half of the gravity at the orbit and this corresponds to the fact that for such orbits the total energy is one half of the potential energy, because the kinetic energy is minus one half of the potential energy. If the central body has radius R, then the energy of an elliptic orbit compared to being stationary at the surface is − μ2 a + μ R = μ2 a R. The International Space Station has a period of 91.74 minutes. The energy is −29.6 MJ/kg, the energy is −59.2 MJ/kg. Compare with the energy at the surface, which is −62.6 MJ/kg. The extra potential energy is 3.4 MJ/kg, the extra energy is 33.0 MJ/kg. The average speed is 7.7 km/s, the net delta-v to reach this orbit is 8.1 km/s, the increase per meter would be 4.4 J/kg, this rate corresponds to one half of the local gravity of 8.8 m/s². For an altitude of 100 km, The energy is −30.8 MJ/kg, the energy is −61.6 MJ/kg. Compare with the energy at the surface, which is −62.6 MJ/kg. The extra potential energy is 1.0 MJ/kg, the extra energy is 31.8 MJ/kg
23.
Standard gravitational parameter
–
In celestial mechanics, the standard gravitational parameter μ of a celestial body is the product of the gravitational constant G and the mass M of the body. μ = G M For several objects in the Solar System, the SI units of the standard gravitational parameter are m3 s−2. However, units of km3 s−2 are frequently used in the scientific literature, the central body in an orbital system can be defined as the one whose mass is much larger than the mass of the orbiting body, or M ≫ m. This approximation is standard for planets orbiting the Sun or most moons, conversely, measurements of the smaller bodys orbit only provide information on the product, μ, not G and M separately. This can be generalized for elliptic orbits, μ =4 π2 a 3 / T2 where a is the semi-major axis, for parabolic trajectories rv2 is constant and equal to 2μ. For elliptic and hyperbolic orbits μ = 2a| ε |, where ε is the orbital energy. The value for the Earth is called the gravitational constant. However, the M can be out only by dividing the MG by G. The uncertainty of those measures is 1 to 7000, so M will have the same uncertainty, the value for the Sun is called the heliocentric gravitational constant or geopotential of the Sun and equals 1. 32712440018×1020 m3 s−2. Note that the mass is also denoted by μ. Astronomical system of units Planetary mass
24.
Arccsc
–
In mathematics, the inverse trigonometric functions are the inverse functions of the trigonometric functions. Specifically, they are the inverses of the sine, cosine, tangent, cotangent, secant, and cosecant functions, inverse trigonometric functions are widely used in engineering, navigation, physics, and geometry. There are several notations used for the trigonometric functions. The most common convention is to name inverse trigonometric functions using a prefix, e. g. arcsin, arccos, arctan. This convention is used throughout the article, when measuring in radians, an angle of θ radians will correspond to an arc whose length is rθ, where r is the radius of the circle. Similarly, in programming languages the inverse trigonometric functions are usually called asin, acos. The notations sin−1, cos−1, tan−1, etc, the confusion is somewhat ameliorated by the fact that each of the reciprocal trigonometric functions has its own name—for example, −1 = sec. Nevertheless, certain authors advise against using it for its ambiguity, since none of the six trigonometric functions are one-to-one, they are restricted in order to have inverse functions. There are multiple numbers y such that sin = x, for example, sin =0, when only one value is desired, the function may be restricted to its principal branch. With this restriction, for x in the domain the expression arcsin will evaluate only to a single value. These properties apply to all the trigonometric functions. The principal inverses are listed in the following table, if x is allowed to be a complex number, then the range of y applies only to its real part. Trigonometric functions of trigonometric functions are tabulated below. This is derived from the tangent addition formula tan = tan + tan 1 − tan tan , like the sine and cosine functions, the inverse trigonometric functions can be calculated using power series, as follows. For arcsine, the series can be derived by expanding its derivative,11 − z 2, as a binomial series, the series for arctangent can similarly be derived by expanding its derivative 11 + z 2 in a geometric series and applying the integral definition above. Arcsin = z + z 33 + z 55 + z 77 + ⋯ = ∑ n =0 ∞, for example, arccos x = π /2 − arcsin x, arccsc x = arcsin , and so on. Alternatively, this can be expressed, arctan z = ∑ n =0 ∞22 n 2. There are two cuts, from −i to the point at infinity, going down the imaginary axis and it works best for real numbers running from −1 to 1
25.
Inverse trigonometric functions
–
In mathematics, the inverse trigonometric functions are the inverse functions of the trigonometric functions. Specifically, they are the inverses of the sine, cosine, tangent, cotangent, secant, and cosecant functions, inverse trigonometric functions are widely used in engineering, navigation, physics, and geometry. There are several notations used for the trigonometric functions. The most common convention is to name inverse trigonometric functions using a prefix, e. g. arcsin, arccos, arctan. This convention is used throughout the article, when measuring in radians, an angle of θ radians will correspond to an arc whose length is rθ, where r is the radius of the circle. Similarly, in programming languages the inverse trigonometric functions are usually called asin, acos. The notations sin−1, cos−1, tan−1, etc, the confusion is somewhat ameliorated by the fact that each of the reciprocal trigonometric functions has its own name—for example, −1 = sec. Nevertheless, certain authors advise against using it for its ambiguity, since none of the six trigonometric functions are one-to-one, they are restricted in order to have inverse functions. There are multiple numbers y such that sin = x, for example, sin =0, when only one value is desired, the function may be restricted to its principal branch. With this restriction, for x in the domain the expression arcsin will evaluate only to a single value. These properties apply to all the trigonometric functions. The principal inverses are listed in the following table, if x is allowed to be a complex number, then the range of y applies only to its real part. Trigonometric functions of trigonometric functions are tabulated below. This is derived from the tangent addition formula tan = tan + tan 1 − tan tan , like the sine and cosine functions, the inverse trigonometric functions can be calculated using power series, as follows. For arcsine, the series can be derived by expanding its derivative,11 − z 2, as a binomial series, the series for arctangent can similarly be derived by expanding its derivative 11 + z 2 in a geometric series and applying the integral definition above. Arcsin = z + z 33 + z 55 + z 77 + ⋯ = ∑ n =0 ∞, for example, arccos x = π /2 − arcsin x, arccsc x = arcsin , and so on. Alternatively, this can be expressed, arctan z = ∑ n =0 ∞22 n 2. There are two cuts, from −i to the point at infinity, going down the imaginary axis and it works best for real numbers running from −1 to 1
26.
Medieval Latin
–
Despite the clerical origin of many of its authors, medieval Latin should not be confused with Ecclesiastical Latin. There is no consensus on the exact boundary where Late Latin ends. Medieval Latin had a vocabulary, which freely borrowed from other sources. Greek provided much of the vocabulary of Christianity. The various Germanic languages spoken by the Germanic tribes, who invaded southern Europe, were major sources of new words. Germanic leaders became the rulers of parts of the Roman Empire that they conquered, other more ordinary words were replaced by coinages from Vulgar Latin or Germanic sources because the classical words had fallen into disuse. Latin was also spread to such as Ireland and Germany. Works written in the lands, where Latin was a language with no relation to the local vernacular, also influenced the vocabulary. English words like abstract, subject, communicate, matter, probable, the high point of the development of medieval Latin as a literary language came with the Carolingian renaissance, a rebirth of learning kindled under the patronage of Charlemagne, king of the Franks. On the other hand, strictly speaking there was no form of medieval Latin. Every Latin author in the period spoke Latin as a second language, with varying degrees of fluency, and syntax, grammar. For instance, rather than following the classical Latin practice of placing the verb at the end. Unlike classical Latin, where esse was the auxiliary verb, medieval Latin writers might use habere as an auxiliary, similar to constructions in Germanic. The accusative and infinitive construction in classical Latin was often replaced by a clause introduced by quod or quia. This is almost identical, for example, to the use of que in similar constructions in French. In every age from the late 8th century onwards, there were learned writers who were familiar enough with classical syntax to be aware that these forms and usages were wrong, however the use of quod to introduce subordinate clauses was especially pervasive and is found at all levels. That resulted in two features of Medieval Latin compared with Classical Latin. First, many attempted to show off their knowledge of Classical Latin by using rare or archaic constructions
27.
Ancient Greek language
–
Ancient Greek includes the forms of Greek used in ancient Greece and the ancient world from around the 9th century BC to the 6th century AD. It is often divided into the Archaic period, Classical period. It is antedated in the second millennium BC by Mycenaean Greek, the language of the Hellenistic phase is known as Koine. Koine is regarded as a historical stage of its own, although in its earliest form it closely resembled Attic Greek. Prior to the Koine period, Greek of the classic and earlier periods included several regional dialects, Ancient Greek was the language of Homer and of fifth-century Athenian historians, playwrights, and philosophers. It has contributed many words to English vocabulary and has been a subject of study in educational institutions of the Western world since the Renaissance. This article primarily contains information about the Epic and Classical phases of the language, Ancient Greek was a pluricentric language, divided into many dialects. The main dialect groups are Attic and Ionic, Aeolic, Arcadocypriot, some dialects are found in standardized literary forms used in literature, while others are attested only in inscriptions. There are also several historical forms, homeric Greek is a literary form of Archaic Greek used in the epic poems, the Iliad and Odyssey, and in later poems by other authors. Homeric Greek had significant differences in grammar and pronunciation from Classical Attic, the origins, early form and development of the Hellenic language family are not well understood because of a lack of contemporaneous evidence. Several theories exist about what Hellenic dialect groups may have existed between the divergence of early Greek-like speech from the common Proto-Indo-European language and the Classical period and they have the same general outline, but differ in some of the detail. The invasion would not be Dorian unless the invaders had some relationship to the historical Dorians. The invasion is known to have displaced population to the later Attic-Ionic regions, the Greeks of this period believed there were three major divisions of all Greek people—Dorians, Aeolians, and Ionians, each with their own defining and distinctive dialects. Often non-west is called East Greek, Arcadocypriot apparently descended more closely from the Mycenaean Greek of the Bronze Age. Boeotian had come under a strong Northwest Greek influence, and can in some respects be considered a transitional dialect, thessalian likewise had come under Northwest Greek influence, though to a lesser degree. Most of the dialect sub-groups listed above had further subdivisions, generally equivalent to a city-state and its surrounding territory, Doric notably had several intermediate divisions as well, into Island Doric, Southern Peloponnesus Doric, and Northern Peloponnesus Doric. The Lesbian dialect was Aeolic Greek and this dialect slowly replaced most of the older dialects, although Doric dialect has survived in the Tsakonian language, which is spoken in the region of modern Sparta. Doric has also passed down its aorist terminations into most verbs of Demotic Greek, by about the 6th century AD, the Koine had slowly metamorphosized into Medieval Greek
28.
Orbital state vectors
–
State vectors are defined with respect to some frame of reference, usually but not always an inertial reference frame. The position vector r describes the position of the body in the frame of reference. Together, these two vectors and the time at which they are valid uniquely describe the bodys trajectory. The body does not actually have to be in orbit for its state vector to determine its trajectory, it only has to move ballistically, i. e. solely under the effects of its own inertia and gravity. For example, it could be a spacecraft or missile in a suborbital trajectory, if other forces such as drag or thrust are significant, they must be added vectorially to those of gravity when performing the integration to determine future position and velocity. For any object moving through space, the velocity vector is tangent to the trajectory, the state vectors can be easily used to compute the angular momentum vector as h = r × v. Because even satellites in low Earth orbit experience significant perturbations, the Keplerian elements computed from the vector at any moment are only valid at that time. Such element sets are known as osculating elements because they coincide with the actual orbit only at that moment
29.
Elliptical orbit
–
In astrodynamics or celestial mechanics an elliptic orbit is a Kepler orbit with the eccentricity less than 1, this includes the special case of a circular orbit, with eccentricity equal to zero. In a stricter sense, it is a Kepler orbit with the eccentricity greater than 0, in a wider sense it is a Kepler orbit with negative energy. This includes the radial elliptic orbit, with eccentricity equal to 1, in a gravitational two-body problem with negative energy both bodies follow similar elliptic orbits with the same orbital period around their common barycenter. Also the relative position of one body with respect to the other follows an elliptic orbit, examples of elliptic orbits include, Hohmann transfer orbit, Molniya orbit and tundra orbit. A is the length of the semi-major axis, the velocity equation for a hyperbolic trajectory has either +1 a, or it is the same with the convention that in that case a is negative. Conclusions, For a given semi-major axis the orbital energy is independent of the eccentricity. ν is the true anomaly. The angular momentum is related to the cross product of position and velocity. Here ϕ is defined as the angle which differs by 90 degrees from this and this set of six variables, together with time, are called the orbital state vectors. Given the masses of the two bodies they determine the full orbit, the two most general cases with these 6 degrees of freedom are the elliptic and the hyperbolic orbit. Special cases with fewer degrees of freedom are the circular and parabolic orbit, another set of six parameters that are commonly used are the orbital elements. In the Solar System, planets, asteroids, most comets, the following chart of the perihelion and aphelion of the planets, dwarf planets and Halleys Comet demonstrates the variation of the eccentricity of their elliptical orbits. For similar distances from the sun, wider bars denote greater eccentricity, note the almost-zero eccentricity of Earth and Venus compared to the enormous eccentricity of Halleys Comet and Eris. A radial trajectory can be a line segment, which is a degenerate ellipse with semi-minor axis =0. Although the eccentricity is 1, this is not a parabolic orbit, most properties and formulas of elliptic orbits apply. However, the orbit cannot be closed and it is an open orbit corresponding to the part of the degenerate ellipse from the moment the bodies touch each other and move away from each other until they touch each other again. In the case of point masses one full orbit is possible, the velocities at the start and end are infinite in opposite directions and the potential energy is equal to minus infinity. The radial elliptic trajectory is the solution of a problem with at some instant zero speed
30.
Periapsis
–
An apsis is an extreme point in an objects orbit. The word comes via Latin from Greek and is cognate with apse, for elliptic orbits about a larger body, there are two apsides, named with the prefixes peri- and ap-, or apo- added to a reference to the thing being orbited. For a body orbiting the Sun, the point of least distance is the perihelion, the terms become periastron and apastron when discussing orbits around other stars. For any satellite of Earth including the Moon the point of least distance is the perigee, for objects in Lunar orbit, the point of least distance is the pericynthion and the greatest distance the apocynthion. For any orbits around a center of mass, there are the terms pericenter and apocenter, periapsis and apoapsis are equivalent alternatives. A straight line connecting the pericenter and apocenter is the line of apsides and this is the major axis of the ellipse, its greatest diameter. For a two-body system the center of mass of the lies on this line at one of the two foci of the ellipse. When one body is larger than the other it may be taken to be at this focus. Historically, in systems, apsides were measured from the center of the Earth. In orbital mechanics, the apsis technically refers to the distance measured between the centers of mass of the central and orbiting body. However, in the case of spacecraft, the family of terms are used to refer to the orbital altitude of the spacecraft from the surface of the central body. The arithmetic mean of the two limiting distances is the length of the axis a. The geometric mean of the two distances is the length of the semi-minor axis b, the geometric mean of the two limiting speeds is −2 ε = μ a which is the speed of a body in a circular orbit whose radius is a. The words pericenter and apocenter are often seen, although periapsis/apoapsis are preferred in technical usage, various related terms are used for other celestial objects. The -gee, -helion and -astron and -galacticon forms are used in the astronomical literature when referring to the Earth, Sun, stars. The suffix -jove is occasionally used for Jupiter, while -saturnium has very rarely used in the last 50 years for Saturn. The -gee form is used as a generic closest approach to planet term instead of specifically applying to the Earth. During the Apollo program, the terms pericynthion and apocynthion were used when referring to the Moon, regarding black holes, the term peri/apomelasma was used by physicist Geoffrey A. Landis in 1998 before peri/aponigricon appeared in the scientific literature in 2002
31.
Apoapsis
–
An apsis is an extreme point in an objects orbit. The word comes via Latin from Greek and is cognate with apse, for elliptic orbits about a larger body, there are two apsides, named with the prefixes peri- and ap-, or apo- added to a reference to the thing being orbited. For a body orbiting the Sun, the point of least distance is the perihelion, the terms become periastron and apastron when discussing orbits around other stars. For any satellite of Earth including the Moon the point of least distance is the perigee, for objects in Lunar orbit, the point of least distance is the pericynthion and the greatest distance the apocynthion. For any orbits around a center of mass, there are the terms pericenter and apocenter, periapsis and apoapsis are equivalent alternatives. A straight line connecting the pericenter and apocenter is the line of apsides and this is the major axis of the ellipse, its greatest diameter. For a two-body system the center of mass of the lies on this line at one of the two foci of the ellipse. When one body is larger than the other it may be taken to be at this focus. Historically, in systems, apsides were measured from the center of the Earth. In orbital mechanics, the apsis technically refers to the distance measured between the centers of mass of the central and orbiting body. However, in the case of spacecraft, the family of terms are used to refer to the orbital altitude of the spacecraft from the surface of the central body. The arithmetic mean of the two limiting distances is the length of the axis a. The geometric mean of the two distances is the length of the semi-minor axis b, the geometric mean of the two limiting speeds is −2 ε = μ a which is the speed of a body in a circular orbit whose radius is a. The words pericenter and apocenter are often seen, although periapsis/apoapsis are preferred in technical usage, various related terms are used for other celestial objects. The -gee, -helion and -astron and -galacticon forms are used in the astronomical literature when referring to the Earth, Sun, stars. The suffix -jove is occasionally used for Jupiter, while -saturnium has very rarely used in the last 50 years for Saturn. The -gee form is used as a generic closest approach to planet term instead of specifically applying to the Earth. During the Apollo program, the terms pericynthion and apocynthion were used when referring to the Moon, regarding black holes, the term peri/apomelasma was used by physicist Geoffrey A. Landis in 1998 before peri/aponigricon appeared in the scientific literature in 2002
32.
Semimajor axis
–
In geometry, the major axis of an ellipse is its longest diameter, a line segment that runs through the center and both foci, with ends at the widest points of the perimeter. The semi-major axis is one half of the axis, and thus runs from the centre, through a focus. Essentially, it is the radius of an orbit at the two most distant points. For the special case of a circle, the axis is the radius. One can think of the axis as an ellipses long radius. The semi-major axis of a hyperbola is, depending on the convention, thus it is the distance from the center to either vertex of the hyperbola. A parabola can be obtained as the limit of a sequence of ellipses where one focus is fixed as the other is allowed to move arbitrarily far away in one direction. Thus a and b tend to infinity, a faster than b, the semi-minor axis is a line segment associated with most conic sections that is at right angles with the semi-major axis and has one end at the center of the conic section. It is one of the axes of symmetry for the curve, in an ellipse, the one, in a hyperbola. The semi-major axis is the value of the maximum and minimum distances r max and r min of the ellipse from a focus — that is. In astronomy these extreme points are called apsis, the semi-minor axis of an ellipse is the geometric mean of these distances, b = r max r min. The eccentricity of an ellipse is defined as e =1 − b 2 a 2 so r min = a, r max = a. Now consider the equation in polar coordinates, with one focus at the origin, the mean value of r = ℓ / and r = ℓ /, for θ = π and θ =0 is a = ℓ1 − e 2. In an ellipse, the axis is the geometric mean of the distance from the center to either focus. The semi-minor axis of an ellipse runs from the center of the ellipse to the edge of the ellipse, the semi-minor axis is half of the minor axis. The minor axis is the longest line segment perpendicular to the axis that connects two points on the ellipses edge. The semi-minor axis b is related to the axis a through the eccentricity e. A parabola can be obtained as the limit of a sequence of ellipses where one focus is fixed as the other is allowed to move arbitrarily far away in one direction
33.
Center of mass
–
The distribution of mass is balanced around the center of mass and the average of the weighted position coordinates of the distributed mass defines its coordinates. Calculations in mechanics are simplified when formulated with respect to the center of mass. It is a point where entire mass of an object may be assumed to be concentrated to visualise its motion. In other words, the center of mass is the equivalent of a given object for application of Newtons laws of motion. In the case of a rigid body, the center of mass is fixed in relation to the body. The center of mass may be located outside the body, as is sometimes the case for hollow or open-shaped objects. In the case of a distribution of separate bodies, such as the planets of the Solar System, in orbital mechanics, the equations of motion of planets are formulated as point masses located at the centers of mass. The center of mass frame is a frame in which the center of mass of a system is at rest with respect to the origin of the coordinate system. The concept of center of mass in the form of the center of gravity was first introduced by the ancient Greek physicist, mathematician, and engineer Archimedes of Syracuse. He worked with simplified assumptions about gravity that amount to a uniform field, in work on floating bodies he demonstrated that the orientation of a floating object is the one that makes its center of mass as low as possible. He developed mathematical techniques for finding the centers of mass of objects of uniform density of various well-defined shapes, Newtons second law is reformulated with respect to the center of mass in Eulers first law. The center of mass is the point at the center of a distribution of mass in space that has the property that the weighted position vectors relative to this point sum to zero. In analogy to statistics, the center of mass is the location of a distribution of mass in space. Solving this equation for R yields the formula R =1 M ∑ i =1 n m i r i, solve this equation for the coordinates R to obtain R =1 M ∭ Q ρ r d V, where M is the total mass in the volume. If a continuous mass distribution has density, which means ρ is constant. The center of mass is not generally the point at which a plane separates the distribution of mass into two equal halves, in analogy with statistics, the median is not the same as the mean. The coordinates R of the center of mass of a system, P1 and P2, with masses m1. The percentages of mass at each point can be viewed as projective coordinates of the point R on this line, another way of interpreting the process here is the mechanical balancing of moments about an arbitrary point
34.
Focus (geometry)
–
In geometry, focuses or foci, singular focus, are special points with reference to which any of a variety of curves is constructed. For example, one or two foci can be used in defining conic sections, the four types of which are the circle, ellipse, parabola, and hyperbola. In addition, two foci are used to define the Cassini oval and the Cartesian oval, and more than two foci are used in defining an n-ellipse. An ellipse can be defined as the locus of points for each of which the sum of the distances to two given foci is a constant, a circle is the special case of an ellipse in which the two foci coincide with each other. Thus, a circle can be simply defined as the locus of points each of which is a fixed distance from a single given focus. A circle can also be defined as the circle of Apollonius, a parabola is a limiting case of an ellipse in which one of the foci is a point at infinity. A hyperbola can be defined as the locus of points for each of which the value of the difference between the distances to two given foci is a constant. It is also possible to describe all conic sections in terms of a focus and a single directrix. A conic is defined as the locus of points for each of which the distance to the focus divided by the distance to the directrix is a positive constant. If e is zero and one the conic is an ellipse, if e=1 the conic is a parabola. If the distance to the focus is fixed and the directrix is a line at infinity, so the eccentricity is zero and it is also possible to describe all the conic sections as loci of points that are equidistant from a single focus and a single, circular directrix. The ellipse thus generated has its focus at the center of the directrix circle. For the parabola, the center of the moves to the point at infinity. The directrix circle becomes a curve with zero curvature, indistinguishable from a straight line. To generate a hyperbola, the radius of the circle is chosen to be less than the distance between the center of this circle and the focus, thus, the focus is outside the directrix circle. The two branches of a hyperbola are thus the two halves of a curve closed over infinity, in projective geometry, all conics are equivalent in the sense that every theorem that can be stated for one can be stated for the others. Plutos ellipse is entirely inside Charons ellipse, as shown in animation of the system. The barycenter is about three-quarters of the distance from Earths center to its surface, moreover, the Pluto-Charon system moves in an ellipse around its barycenter with the Sun, as does the Earth-Moon system
35.
Mercury (planet)
–
Mercury is the smallest and innermost planet in the Solar System. Its orbital period around the Sun of 88 days is the shortest of all the planets in the Solar System and it is named after the Roman deity Mercury, the messenger to the gods. Like Venus, Mercury orbits the Sun within Earths orbit as a planet, so it can only be seen visually in the morning or the evening sky. Also, like Venus and the Moon, the displays the complete range of phases as it moves around its orbit relative to Earth. Seen from Earth, this cycle of phases reoccurs approximately every 116 days, although Mercury can appear as a bright star-like object when viewed from Earth, its proximity to the Sun often makes it more difficult to see than Venus. Mercury is tidally or gravitationally locked with the Sun in a 3,2 resonance, as seen relative to the fixed stars, it rotates on its axis exactly three times for every two revolutions it makes around the Sun. As seen from the Sun, in a frame of reference that rotates with the orbital motion, an observer on Mercury would therefore see only one day every two years. Mercurys axis has the smallest tilt of any of the Solar Systems planets, at aphelion, Mercury is about 1.5 times as far from the Sun as it is at perihelion. Mercurys surface appears heavily cratered and is similar in appearance to the Moons, the polar regions are constantly below 180 K. The planet has no natural satellites. Mercury is one of four planets in the Solar System. It is the smallest planet in the Solar System, with a radius of 2,439.7 kilometres. Mercury is also smaller—albeit more massive—than the largest natural satellites in the Solar System, Ganymede, Mercury consists of approximately 70% metallic and 30% silicate material. Mercurys density is the second highest in the Solar System at 5.427 g/cm3, Mercurys density can be used to infer details of its inner structure. Although Earths high density results appreciably from gravitational compression, particularly at the core, Mercury is much smaller, therefore, for it to have such a high density, its core must be large and rich in iron. Geologists estimate that Mercurys core occupies about 55% of its volume, Research published in 2007 suggests that Mercury has a molten core. Surrounding the core is a 500–700 km mantle consisting of silicates, based on data from the Mariner 10 mission and Earth-based observation, Mercurys crust is estimated to be 35 km thick. One distinctive feature of Mercurys surface is the presence of narrow ridges
36.
Earth
–
Earth, otherwise known as the World, or the Globe, is the third planet from the Sun and the only object in the Universe known to harbor life. It is the densest planet in the Solar System and the largest of the four terrestrial planets, according to radiometric dating and other sources of evidence, Earth formed about 4.54 billion years ago. Earths gravity interacts with objects in space, especially the Sun. During one orbit around the Sun, Earth rotates about its axis over 365 times, thus, Earths axis of rotation is tilted, producing seasonal variations on the planets surface. The gravitational interaction between the Earth and Moon causes ocean tides, stabilizes the Earths orientation on its axis, Earths lithosphere is divided into several rigid tectonic plates that migrate across the surface over periods of many millions of years. About 71% of Earths surface is covered with water, mostly by its oceans, the remaining 29% is land consisting of continents and islands that together have many lakes, rivers and other sources of water that contribute to the hydrosphere. The majority of Earths polar regions are covered in ice, including the Antarctic ice sheet, Earths interior remains active with a solid iron inner core, a liquid outer core that generates the Earths magnetic field, and a convecting mantle that drives plate tectonics. Within the first billion years of Earths history, life appeared in the oceans and began to affect the Earths atmosphere and surface, some geological evidence indicates that life may have arisen as much as 4.1 billion years ago. Since then, the combination of Earths distance from the Sun, physical properties, in the history of the Earth, biodiversity has gone through long periods of expansion, occasionally punctuated by mass extinction events. Over 99% of all species that lived on Earth are extinct. Estimates of the number of species on Earth today vary widely, over 7.4 billion humans live on Earth and depend on its biosphere and minerals for their survival. Humans have developed diverse societies and cultures, politically, the world has about 200 sovereign states, the modern English word Earth developed from a wide variety of Middle English forms, which derived from an Old English noun most often spelled eorðe. It has cognates in every Germanic language, and their proto-Germanic root has been reconstructed as *erþō, originally, earth was written in lowercase, and from early Middle English, its definite sense as the globe was expressed as the earth. By early Modern English, many nouns were capitalized, and the became the Earth. More recently, the name is simply given as Earth. House styles now vary, Oxford spelling recognizes the lowercase form as the most common, another convention capitalizes Earth when appearing as a name but writes it in lowercase when preceded by the. It almost always appears in lowercase in colloquial expressions such as what on earth are you doing, the oldest material found in the Solar System is dated to 4. 5672±0.0006 billion years ago. By 4. 54±0.04 Gya the primordial Earth had formed, the formation and evolution of Solar System bodies occurred along with the Sun
37.
Triton (moon)
–
Triton is the largest natural satellite of the planet Neptune. It was discovered on October 10,1846, by English astronomer William Lassell and it is the only large moon in the Solar System with a retrograde orbit, an orbit in the opposite direction to its planets rotation. At 2,700 kilometres in diameter, it is the seventh-largest moon in the Solar System, because of its retrograde orbit and composition similar to Plutos, Triton is thought to have been a dwarf planet captured from the Kuiper belt. Triton has a surface of frozen nitrogen, a mostly water-ice crust, an icy mantle. The core makes up two-thirds of its total mass, Triton has a mean density of 2.061 g/cm3 and is composed of approximately 15–35% water ice. Triton is one of the few moons in the Solar System known to be geologically active, as a consequence, its surface is relatively young with sparse impact craters, and a complex geological history revealed in intricate cryovolcanic and tectonic terrains. Part of its surface has geysers erupting sublimated nitrogen gas, contributing to a nitrogen atmosphere less than 1/70,000 the pressure of Earths atmosphere at sea level. Triton was discovered by British astronomer William Lassell on October 10,1846, a brewer by trade, Lassell began making mirrors for his amateur telescope in 1820. When John Herschel received news of Neptunes discovery, he wrote to Lassell suggesting he search for possible moons, Lassell did so and discovered Triton eight days later. Lassell also claimed to have discovered rings, although Neptune was later confirmed to have rings, they are so faint and dark that it is doubtful that he actually saw them. Triton is named after the Greek sea god Triton, the son of Poseidon, the name was first proposed by Camille Flammarion in his 1880 book Astronomie Populaire, and was officially adopted many decades later. Until the discovery of the second moon Nereid in 1949, Triton was commonly referred to as the satellite of Neptune. Lassell did not name his own discovery, he successfully suggested the name Hyperion, previously chosen by John Herschel. Triton is unique among all large moons in the Solar System for its orbit around its planet. Most of the irregular moons of Jupiter and Saturn also have retrograde orbits. However, these moons are all much more distant from their primaries, and are small in comparison, Tritons orbit is associated with two tilts, the inclination of Neptunes spin to Neptunes orbit, 30°, and the inclination of Tritons orbit to Neptunes spin, 157°. Tritons orbit precesses forward relative to Neptunes spin with a period of about 678 Earth years and that inclination is currently 130°, Tritons orbit is now near its maximum departure from coplanarity with Neptunes. Tritons rotation is locked to be synchronous with its orbit around Neptune
38.
Neptune
–
Neptune is the eighth and farthest known planet from the Sun in the Solar System. In the Solar System, it is the fourth-largest planet by diameter, the planet. Neptune is 17 times the mass of Earth and is more massive than its near-twin Uranus. Neptune orbits the Sun once every 164.8 years at a distance of 30.1 astronomical units. It is named after the Roman god of the sea and has the astronomical symbol ♆, Neptune is not visible to the unaided eye and is the only planet in the Solar System found by mathematical prediction rather than by empirical observation. Unexpected changes in the orbit of Uranus led Alexis Bouvard to deduce that its orbit was subject to perturbation by an unknown planet. Neptune was subsequently observed with a telescope on 23 September 1846 by Johann Galle within a degree of the predicted by Urbain Le Verrier. Its largest moon, Triton, was discovered shortly thereafter, though none of the remaining known 14 moons were located telescopically until the 20th century. The planets distance from Earth gives it a small apparent size. Neptune was visited by Voyager 2, when it flew by the planet on 25 August 1989, the advent of the Hubble Space Telescope and large ground-based telescopes with adaptive optics has recently allowed for additional detailed observations from afar. Neptunes composition can be compared and contrasted with the Solar Systems other giant planets, however, its interior, like that of Uranus, is primarily composed of ices and rock, which is why Uranus and Neptune are normally considered ice giants to emphasise this distinction. Traces of methane in the outermost regions in part account for the blue appearance. In contrast to the hazy, relatively featureless atmosphere of Uranus, Neptunes atmosphere has active, for example, at the time of the Voyager 2 flyby in 1989, the planets southern hemisphere had a Great Dark Spot comparable to the Great Red Spot on Jupiter. These weather patterns are driven by the strongest sustained winds of any planet in the Solar System, because of its great distance from the Sun, Neptunes outer atmosphere is one of the coldest places in the Solar System, with temperatures at its cloud tops approaching 55 K. Temperatures at the centre are approximately 5,400 K. Neptune has a faint and fragmented ring system. On both occasions, Galileo seems to have mistaken Neptune for a star when it appeared close—in conjunction—to Jupiter in the night sky, hence. At his first observation in December 1612, Neptune was almost stationary in the sky because it had just turned retrograde that day and this apparent backward motion is created when Earths orbit takes it past an outer planet. Because Neptune was only beginning its yearly cycle, the motion of the planet was far too slight to be detected with Galileos small telescope
39.
Titan (moon)
–
Titan is the largest moon of Saturn. It is the only known to have a dense atmosphere. Titan is the sixth ellipsoidal moon from Saturn, frequently described as a planet-like moon, Titan is 50% larger than Earths Moon, and it is 80% more massive. It is the second-largest moon in the Solar System, after Jupiters moon Ganymede, and is larger than the smallest planet, Mercury, discovered in 1655 by the Dutch astronomer Christiaan Huygens, Titan was the first known moon of Saturn, and the sixth known planetary satellite. Titan orbits Saturn at 20 Saturn radii, from Titans surface, Saturn subtends an arc of 5.09 degrees and would appear 11.4 times larger in the sky than the Moon from Earth. Titan is primarily composed of ice and rocky material. The geologically young surface is smooth, with few impact craters, although mountains. The atmosphere of Titan is largely nitrogen, minor components lead to the formation of methane and ethane clouds and nitrogen-rich organic smog. The climate—including wind and rain—creates surface features similar to those of Earth, such as dunes, rivers, lakes, seas, and deltas, Huygens was inspired by Galileos discovery of Jupiters four largest moons in 1610 and his improvements in telescope technology. Christiaan, with the help of his brother Constantijn Huygens, Jr. began building telescopes around 1650 and it was the sixth moon to be discovered. He named it Saturni Luna, publishing in the 1655 tract De Saturni Luna Observatio Nova, after Giovanni Domenico Cassini published his discoveries of four more moons of Saturn between 1673 and 1686, astronomers fell into the habit of referring to these and Titan as Saturn I through V. Other early epithets for Titan include Saturns ordinary satellite, Titan is officially numbered Saturn VI because after the 1789 discoveries the numbering scheme was frozen to avoid causing any more confusion. Numerous small moons have been discovered closer to Saturn since then and he suggested the names of the mythological Titans, brothers and sisters of Cronus, the Greek Saturn. In Greek mythology, the Titans were a race of powerful deities, descendants of Gaia and Uranus, Titan orbits Saturn once every 15 days and 22 hours. Because of this, there is a point on its surface. Longitudes on Titan are measured westward, starting from the passing through this point. Its orbital eccentricity is 0.0288, and the plane is inclined 0.348 degrees relative to the Saturnian equator. Viewed from Earth, Titan reaches a distance of about 20 Saturn radii from Saturn
40.
Uranus
–
Uranus is the seventh planet from the Sun. It has the third-largest planetary radius and fourth-largest planetary mass in the Solar System, Uranus is similar in composition to Neptune, and both have different bulk chemical composition from that of the larger gas giants Jupiter and Saturn. For this reason, scientists often classify Uranus and Neptune as ice giants to distinguish them from the gas giants, the interior of Uranus is mainly composed of ices and rock. Uranus is the planet whose name is derived from a figure from Greek mythology. Like the other giant planets, Uranus has a system, a magnetosphere. The Uranian system has a unique configuration among those of the planets because its axis of rotation is tilted sideways and its north and south poles, therefore, lie where most other planets have their equators. In 1986, images from Voyager 2 showed Uranus as an almost featureless planet in visible light, observations from Earth have shown seasonal change and increased weather activity as Uranus approached its equinox in 2007. Wind speeds can reach 250 metres per second, like the classical planets, Uranus is visible to the naked eye, but it was never recognised as a planet by ancient observers because of its dimness and slow orbit. Uranus had been observed on many occasions before its recognition as a planet, possibly the earliest known observation was by Hipparchos, who in 128 BCE might have recorded it as a star for his star catalogue that was later incorporated into Ptolemys Almagest. The earliest definite sighting was in 1690 when John Flamsteed observed it at least six times, the French astronomer Pierre Lemonnier observed Uranus at least twelve times between 1750 and 1769, including on four consecutive nights. Sir William Herschel observed Uranus on March 13,1781 from the garden of his house at 19 New King Street in Bath, Somerset, England, Herschel engaged in a series of observations on the parallax of the fixed stars, using a telescope of his own design. Herschel recorded in his journal, In the quartile near ζ Tauri, either Nebulous star or perhaps a comet. On March 17 he noted, I looked for the Comet or Nebulous Star and found that it is a Comet, the sequel has shown that my surmises were well-founded, this proving to be the Comet we have lately observed. Herschel notified the Astronomer Royal, Nevil Maskelyne, of his discovery and received this flummoxed reply from him on April 23,1781, I dont know what to call it. It is as likely to be a planet moving in an orbit nearly circular to the sun as a Comet moving in a very eccentric ellipsis. I have not yet seen any coma or tail to it, although Herschel continued to describe his new object as a comet, other astronomers had already begun to suspect otherwise. Finnish-Swedish astronomer Anders Johan Lexell, working in Russia, was the first to compute the orbit of the new object and its nearly circular orbit led him to a conclusion that it was a planet rather than a comet. Berlin astronomer Johann Elert Bode described Herschels discovery as a star that can be deemed a hitherto unknown planet-like object circulating beyond the orbit of Saturn
41.
Jupiter
–
Jupiter is the fifth planet from the Sun and the largest in the Solar System. It is a giant planet with a mass one-thousandth that of the Sun, Jupiter and Saturn are gas giants, the other two giant planets, Uranus and Neptune are ice giants. Jupiter has been known to astronomers since antiquity, the Romans named it after their god Jupiter. Jupiter is primarily composed of hydrogen with a quarter of its mass being helium and it may also have a rocky core of heavier elements, but like the other giant planets, Jupiter lacks a well-defined solid surface. Because of its rotation, the planets shape is that of an oblate spheroid. The outer atmosphere is visibly segregated into several bands at different latitudes, resulting in turbulence, a prominent result is the Great Red Spot, a giant storm that is known to have existed since at least the 17th century when it was first seen by telescope. Surrounding Jupiter is a faint planetary ring system and a powerful magnetosphere, Jupiter has at least 67 moons, including the four large Galilean moons discovered by Galileo Galilei in 1610. Ganymede, the largest of these, has a greater than that of the planet Mercury. Jupiter has been explored on several occasions by robotic spacecraft, most notably during the early Pioneer and Voyager flyby missions and later by the Galileo orbiter. In late February 2007, Jupiter was visited by the New Horizons probe, the latest probe to visit the planet is Juno, which entered into orbit around Jupiter on July 4,2016. Future targets for exploration in the Jupiter system include the probable ice-covered liquid ocean of its moon Europa, Earth and its neighbor planets may have formed from fragments of planets after collisions with Jupiter destroyed those super-Earths near the Sun. Astronomers have discovered nearly 500 planetary systems with multiple planets, Jupiter moving out of the inner Solar System would have allowed the formation of inner planets, including Earth. Jupiter is composed primarily of gaseous and liquid matter and it is the largest of the four giant planets in the Solar System and hence its largest planet. It has a diameter of 142,984 km at its equator, the average density of Jupiter,1.326 g/cm3, is the second highest of the giant planets, but lower than those of the four terrestrial planets. Jupiters upper atmosphere is about 88–92% hydrogen and 8–12% helium by percent volume of gas molecules, a helium atom has about four times as much mass as a hydrogen atom, so the composition changes when described as the proportion of mass contributed by different atoms. Thus, Jupiters atmosphere is approximately 75% hydrogen and 24% helium by mass, the atmosphere contains trace amounts of methane, water vapor, ammonia, and silicon-based compounds. There are also traces of carbon, ethane, hydrogen sulfide, neon, oxygen, phosphine, the outermost layer of the atmosphere contains crystals of frozen ammonia. The interior contains denser materials - by mass it is roughly 71% hydrogen, 24% helium, through infrared and ultraviolet measurements, trace amounts of benzene and other hydrocarbons have also been found
42.
Saturn
–
Saturn is the sixth planet from the Sun and the second-largest in the Solar System, after Jupiter. It is a gas giant with a radius about nine times that of Earth. Although it has only one-eighth the average density of Earth, with its larger volume Saturn is just over 95 times more massive, Saturn is named after the Roman god of agriculture, its astronomical symbol represents the gods sickle. Saturns interior is composed of a core of iron–nickel and rock. This core is surrounded by a layer of metallic hydrogen, an intermediate layer of liquid hydrogen and liquid helium. Saturn has a yellow hue due to ammonia crystals in its upper atmosphere. Saturns magnetic field strength is around one-twentieth of Jupiters, the outer atmosphere is generally bland and lacking in contrast, although long-lived features can appear. Wind speeds on Saturn can reach 1,800 km/h, higher than on Jupiter, sixty-two moons are known to orbit Saturn, of which fifty-three are officially named. This does not include the hundreds of moonlets comprising the rings, Saturn is a gas giant because it is predominantly composed of hydrogen and helium. It lacks a definite surface, though it may have a solid core, Saturns rotation causes it to have the shape of an oblate spheroid, that is, it is flattened at the poles and bulges at its equator. Its equatorial and polar radii differ by almost 10%,60,268 km versus 54,364 km, Jupiter, Uranus, and Neptune, the other giant planets in the Solar System, are also oblate but to a lesser extent. Saturn is the planet of the Solar System that is less dense than water—about 30% less. Although Saturns core is considerably denser than water, the specific density of the planet is 0.69 g/cm3 due to the atmosphere. Jupiter has 318 times the Earths mass, while Saturn is 95 times the mass of the Earth, together, Jupiter and Saturn hold 92% of the total planetary mass in the Solar System. On 8 January 2015, NASA reported determining the center of the planet Saturn, the temperature, pressure, and density inside Saturn all rise steadily toward the core, which causes hydrogen to transition into a metal in the deeper layers. Standard planetary models suggest that the interior of Saturn is similar to that of Jupiter, having a rocky core surrounded by hydrogen. This core is similar in composition to the Earth, but more dense, in 2004, they estimated that the core must be 9–22 times the mass of the Earth, which corresponds to a diameter of about 25,000 km. This is surrounded by a liquid metallic hydrogen layer, followed by a liquid layer of helium-saturated molecular hydrogen that gradually transitions to a gas with increasing altitude