Its measured value is 6. 67408×10−11 m3⋅kg−1⋅s−2. The constant of proportionality, G, is the gravitational constant, the gravitational constant is called Big G, for disambiguation with small g, which is the local gravitational field of Earth. The two quantities are related by g = GME/r2 E. In the general theory of relativity, the Einstein field equations, R μ ν −12 R g μ ν =8 π G c 4 T μ ν, the scaled gravitational constant κ = 8π/c4G ≈2. 071×10−43 s2·m−1·kg−1 is known as Einsteins constant. The gravitational constant is a constant that is difficult to measure with high accuracy. This is because the force is extremely weak compared with other fundamental forces. In SI units, the 2014 CODATA-recommended value of the constant is. In cgs, G can be written as G ≈6. 674×10−8 cm3·g−1·s−2, in other words, in Planck units, G has the numerical value of 1. In astrophysics, it is convenient to measure distances in parsecs, velocities in kilometers per second, in these units, the gravitational constant is, G ≈4.302 ×10 −3 p c M ⊙ −12.
In orbital mechanics, the period P of an object in orbit around a spherical object obeys G M =3 π V P2 where V is the volume inside the radius of the orbit. It follows that P2 =3 π G V M ≈10.896 h r 2 g c m −3 V M. This way of expressing G shows the relationship between the density of a planet and the period of a satellite orbiting just above its surface. Cavendish measured G implicitly, using a torsion balance invented by the geologist Rev. John Michell and he used a horizontal torsion beam with lead balls whose inertia he could tell by timing the beams oscillation. Their faint attraction to other balls placed alongside the beam was detectable by the deflection it caused, cavendishs aim was not actually to measure the gravitational constant, but rather to measure Earths density relative to water, through the precise knowledge of the gravitational interaction. In modern units, the density that Cavendish calculated implied a value for G of 6. 754×10−11 m3·kg−1·s−2, the accuracy of the measured value of G has increased only modestly since the original Cavendish experiment. G is quite difficult to measure, because gravity is weaker than other fundamental forces.
Published values of G have varied rather broadly, and some recent measurements of precision are, in fact. This led to the 2010 CODATA value by NIST having 20% increased uncertainty than in 2006, for the 2014 update, CODATA reduced the uncertainty to less than half the 2010 value
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 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, 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
This speed can be measured in the SI unit of angular velocity, radians per second, or in terms of degrees per second, degrees per hour, etc. Angular velocity is usually represented by the symbol omega, the direction of the angular velocity vector is perpendicular to the plane of rotation, in a direction that is usually specified by the right-hand rule. The angular velocity of a particle is measured around or relative to a point, called the origin. As shown in the diagram, if a line is drawn from the origin to the particle, the velocity of the particle has a component along the radius, if there is no radial component, the particle moves in a circle. On the other hand, if there is no cross-radial component, a radial motion produces no change in the direction of the particle relative to the origin, so, for the purpose of finding the angular velocity, the radial component can be ignored. Therefore, the rotation is completely produced by the perpendicular motion around the origin, the angular velocity in two dimensions is a pseudoscalar, a quantity that changes its sign under a parity inversion.
The positive direction of rotation is taken, by convention, to be in the direction towards the y axis from the x axis, if the parity is inverted, but the orientation of a rotation is not, the sign of the angular velocity changes. There are three types of angular velocity involved in the movement on an ellipse corresponding to the three anomalies, in three dimensions, the angular velocity becomes a bit more complicated. The angular velocity in case is generally thought of as a vector, or more precisely. It now has not only a magnitude, but a direction as well, the magnitude is the angular speed, and the direction describes the axis of rotation that Eulers rotation theorem guarantees must exist. The right-hand rule indicates the direction of the angular velocity pseudovector. Let u be a vector along the instantaneous rotation axis. This is the definition of a vector space, the only property that presents difficulties to prove is the commutativity of the addition. This can be proven from the fact that the velocity tensor W is skew-symmetric, therefore, R = e W t is a rotation matrix and in a time dt is an infinitesimal rotation matrix.
Therefore, it can be expanded as R = I + W ⋅ d t +122 +, in such a frame, each vector is a particular case of the previous case, in which the module of the vector is constant. Though it just a case of a moving particle, this is a very important one for its relationship with the rigid body study. There are two ways to describe the angular velocity of a rotating frame, the angular velocity vector. Both entities are related and they can be calculated from each other, in a consistent way with the general definition, the angular velocity of a frame is defined as the angular velocity of each of the three vectors of the frame
In physics, circular motion is a movement of an object along the circumference of a circle or rotation along a circular path. It can be uniform, with constant angular rate of rotation and constant speed, the rotation around a fixed axis of a three-dimensional body involves circular motion of its parts. The equations of motion describe the movement of the center of mass of a body, since the objects velocity vector is constantly changing direction, the moving object is undergoing acceleration by a centripetal force in the direction of the center of rotation. Without this acceleration, the object would move in a straight line, in physics, uniform circular motion describes the motion of a body traversing a circular path at constant speed. Since the body describes circular motion, its distance from the axis of rotation remains constant at all times, though the bodys speed is constant, its velocity is not constant, velocity, a vector quantity, depends on both the bodys speed and its direction of travel.
This changing velocity indicates the presence of an acceleration, this acceleration is of constant magnitude. This acceleration is, in turn, produced by a force which is constant in magnitude. For motion in a circle of radius r, the circumference of the circle is C = 2π r, the axis of rotation is shown as a vector ω perpendicular to the plane of the orbit and with a magnitude ω = dθ / dt. The direction of ω is chosen using the right-hand rule, the acceleration is given by a = ω × v = ω ×, which is a vector perpendicular to both ω and v of magnitude ω |v| = ω2 r and directed exactly opposite to r. In the simplest case the speed and radius are constant, consider a body of one kilogram, moving in a circle of radius one metre, with an angular velocity of one radian per second. The speed is one metre per second, the inward acceleration is one metre per square second, v2/r. It is subject to a force of one kilogram metre per square second. The momentum of the body is one kg·m·s−1, the moment of inertia is one kg·m2.
The angular momentum is one kg·m2·s−1, the kinetic energy is 1/2 joule. The circumference of the orbit is 2π metres, the period of the motion is 2π seconds per turn. It is convenient to introduce the unit vector orthogonal to u ^ R as well and it is customary to orient u ^ θ to point in the direction of travel along the orbit. The velocity is the derivative of the displacement, v → = d d t r → = d R d t u ^ R + R d u ^ R d t. Because the radius of the circle is constant, the component of the velocity is zero
In elementary geometry, the property of being perpendicular is the relationship between two lines which meet at a right angle. The property extends to other related geometric objects, a line is said to be perpendicular to another line if the two lines intersect at a right angle. Perpendicularity can be shown to be symmetric, meaning if a first line is perpendicular to a second line, for this reason, we may speak of two lines as being perpendicular without specifying an order. Perpendicularity easily extends to segments and rays, in symbols, A B ¯ ⊥ C D ¯ means line segment AB is perpendicular to line segment CD. A line is said to be perpendicular to an if it is perpendicular to every line in the plane that it intersects. This definition depends on the definition of perpendicularity between lines, two planes in space are said to be perpendicular if the dihedral angle at which they meet is a right angle. Perpendicularity is one instance of the more general mathematical concept of orthogonality, perpendicularity is the orthogonality of classical geometric objects.
Thus, in advanced mathematics, the perpendicular is sometimes used to describe much more complicated geometric orthogonality conditions. The word foot is used in connection with perpendiculars. This usage is exemplified in the top diagram, the diagram can be in any orientation. The foot is not necessarily at the bottom, step 2, construct circles centered at A and B having equal radius. Let Q and R be the points of intersection of two circles. Step 3, connect Q and R to construct the desired perpendicular PQ, to prove that the PQ is perpendicular to AB, use the SSS congruence theorem for and QPB to conclude that angles OPA and OPB are equal. Then use the SAS congruence theorem for triangles OPA and OPB to conclude that angles POA, to make the perpendicular to the line g at or through the point P using Thales theorem, see the animation at right. The Pythagorean Theorem can be used as the basis of methods of constructing right angles, for example, by counting links, three pieces of chain can be made with lengths in the ratio 3,4,5.
These can be out to form a triangle, which will have a right angle opposite its longest side. This method is useful for laying out gardens and fields, where the dimensions are large, the chains can be used repeatedly whenever required. If two lines are perpendicular to a third line, all of the angles formed along the third line are right angles
The solution is a useful approximation for describing slowly rotating astronomical objects such as many stars and planets, including Earth and the Sun. The solution is named after Karl Schwarzschild, who first published the solution in 1916, according to Birkhoffs theorem, the Schwarzschild metric is the most general spherically symmetric, vacuum solution of the Einstein field equations. A Schwarzschild black hole or static black hole is a hole that has no electric charge or angular momentum. A Schwarzschild black hole is described by the Schwarzschild metric, the Schwarzschild black hole is characterized by a surrounding spherical boundary, called the event horizon, which is situated at the Schwarzschild radius, often called the radius of a black hole. Any non-rotating and non-charged mass that is smaller than its Schwarzschild radius forms a black hole, the analogue of this solution in classical Newtonian theory of gravity corresponds to the gravitational field around a point particle. In practice, the ratio rs/r is almost always extremely small, for example, the Schwarzschild radius rs of the Earth is roughly 8.9 mm, while the Sun, which is 3.
3×105 times as massive has a Schwarzschild radius of approximately 3.0 km. Even at the surface of the Earth, the corrections to Newtonian gravity are only one part in a billion, the ratio only becomes large close to black holes and other ultra-dense objects such as neutron stars. The Schwarzschild metric is a solution of Einsteins field equations in empty space and that is, for a spherical body of radius R the solution is valid for r > R. It was the first exact solution of the Einstein field equations other than the flat space solution. Schwarzschild died shortly after his paper was published, as a result of a disease he contracted while serving in the German army during World War I, johannes Droste in 1916 independently produced the same solution as Schwarzschild, using a simpler, more direct derivation. In the early years of general relativity there was a lot of confusion about the nature of the found in the Schwarzschild. In Schwarzschilds original paper, he put what we now call the event horizon at the origin of his coordinate system, in this paper he introduced what is now known as the Schwarzschild radial coordinate, as an auxiliary variable.
In his equations, Schwarzschild was using a different radial coordinate that was zero at the Schwarzschild radius, a more complete analysis of the singularity structure was given by David Hilbert in the following year, identifying the singularities both at r =0 and r = rs. Although there was consensus that the singularity at r =0 was a genuine physical singularity. They, did not recognize that their solutions were just coordinate transforms, later, in 1932, Georges Lemaître gave a different coordinate transformation to the same effect and was the first to recognize that this implied that the singularity at r = rs was not physical. A similar result was rediscovered by George Szekeres, and independently Martin Kruskal, the new coordinates nowadays known as Kruskal-Szekeres coordinates were much simpler than Synges but both provided a single set of coordinates that covered the entire spacetime. This led to identification of the r = rs singularity in the Schwarzschild metric as an event horizon.
The Schwarzschild solution appears to have singularities at r =0 and r = rs, since the Schwarzschild metric is only expected to be valid for radii larger than the radius R of the gravitating body, there is no problem as long as R > rs
The escape velocity from Earth is about 11.186 km/s at the surface. More generally, escape velocity is the speed at which the sum of a kinetic energy. With escape velocity in a direction pointing away from the ground of a massive body, once escape velocity is achieved, no further impulse need be applied for it to continue in its escape. When given a speed V greater than the speed v e. In these equations atmospheric friction is not taken into account, escape velocity is only required to send a ballistic object on a trajectory that will allow the object to escape the gravity well of the mass M. The existence of escape velocity is a consequence of conservation of energy, by adding speed to the object it expands the possible places that can be reached until with enough energy they become infinite. For a given gravitational potential energy at a position, the escape velocity is the minimum speed an object without propulsion needs to be able to escape from the gravity. Escape velocity is actually a speed because it does not specify a direction, no matter what the direction of travel is, the simplest way of deriving the formula for escape velocity is to use conservation of energy.
Imagine that a spaceship of mass m is at a distance r from the center of mass of the planet and its initial speed is equal to its escape velocity, v e. At its final state, it will be a distance away from the planet. The same result is obtained by a calculation, in which case the variable r represents the radial coordinate or reduced circumference of the Schwarzschild metric. All speeds and velocities measured with respect to the field, the escape velocity at a point in space is equal to the speed that an object would have if it started at rest from an infinite distance and was pulled by gravity to that point. In common usage, the point is on the surface of a planet or moon. On the surface of the Earth, the velocity is about 11.2 km/s. However, at 9,000 km altitude in space, it is less than 7.1 km/s. The escape velocity is independent of the mass of the escaping object and it does not matter if the mass is 1 kg or 1,000 kg, what differs is the amount of energy required. For an object of mass m the energy required to escape the Earths gravitational field is GMm / r, a related quantity is the specific orbital energy which is essentially the sum of the kinetic and potential energy divided by the mass.
An object has reached escape velocity when the orbital energy is greater or equal to zero
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, 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 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, 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, with related inventions such as gears, in mathematics, the study of the circle has helped inspire the development of geometry 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
In physics, mass is a property of a physical body. It is the measure of a resistance to acceleration when a net force is applied. It determines the strength of its gravitational attraction to other bodies. The basic SI unit of mass is the kilogram, Mass is not the same as weight, even though mass is often determined by measuring the objects weight using a spring scale, rather than comparing it directly with known masses. An object on the Moon would weigh less than it does on Earth because of the lower gravity and this is because weight is a force, while mass is the property that determines the strength of this force. In Newtonian physics, mass can be generalized as the amount of matter in an object, however, at very high speeds, special relativity postulates that energy is an additional source of mass. Thus, any body having mass has an equivalent amount of energy. In addition, matter is a defined term in science. There are several distinct phenomena which can be used to measure mass, active gravitational mass measures the gravitational force exerted by an object.
Passive gravitational mass measures the force exerted on an object in a known gravitational field. The mass of an object determines its acceleration in the presence of an applied force, according to Newtons second law of motion, if a body of fixed mass m is subjected to a single force F, its acceleration a is given by F/m. A bodys mass determines the degree to which it generates or is affected by a gravitational field and this is sometimes referred to as gravitational mass. The standard International System of Units unit of mass is the kilogram, the kilogram is 1000 grams, first defined in 1795 as one cubic decimeter of water at the melting point of ice. Then in 1889, the kilogram was redefined as the mass of the prototype kilogram. As of January 2013, there are proposals for redefining the kilogram yet again. In this context, the mass has units of eV/c2, the electronvolt and its multiples, such as the MeV, are commonly used in particle physics. The atomic mass unit is 1/12 of the mass of a carbon-12 atom, the atomic mass unit is convenient for expressing the masses of atoms and molecules.
Outside the SI system, other units of mass include, the slug is an Imperial unit of mass, the pound is a unit of both mass and force, used mainly in the United States
In stellar dynamics a box orbit refers to a particular type of orbit that can be seen in triaxial systems, i. e. systems that do not possess a symmetry around any of its axes. They contrast with the orbits that are observed in spherically symmetric or axisymmetric systems. In a box orbit, the star oscillates independently along the three different axes as it moves through the system, as a result of this motion, it fills in a box-shaped region of space. Unlike loop orbits, the stars on box orbits can come close to the center of the system. As a special case, if the frequencies of oscillation in different directions are commensurate, such orbits are sometimes called boxlets. Horseshoe orbit Lissajous curve List of orbits