Earth is the third planet from the Sun and the only astronomical object known to harbor life. According to radiometric dating and other sources of evidence, Earth formed over 4.5 billion years ago. Earth's gravity interacts with other objects in space the Sun and the Moon, Earth's only natural satellite. Earth revolves around the Sun in a period known as an Earth year. During this time, Earth rotates about its axis about 366.26 times. Earth's axis of rotation is tilted with respect to its orbital plane; the gravitational interaction between Earth and the Moon causes ocean tides, stabilizes Earth's orientation on its axis, slows its rotation. Earth is the largest of the four terrestrial planets. Earth's lithosphere is divided into several rigid tectonic plates that migrate across the surface over periods of many millions of years. About 71% of Earth's surface is covered with water by oceans; the remaining 29% is land consisting of continents and islands that together have many lakes and other sources of water that contribute to the hydrosphere.
The majority of Earth's polar regions are covered in ice, including the Antarctic ice sheet and the sea ice of the Arctic ice pack. Earth's interior remains active with a solid iron inner core, a liquid outer core that generates the Earth's magnetic field, a convecting mantle that drives plate tectonics. Within the first billion years of Earth's history, life appeared in the oceans and began to affect the Earth's atmosphere and surface, leading to the proliferation of aerobic and anaerobic organisms; some geological evidence indicates. Since the combination of Earth's distance from the Sun, physical properties, geological history have allowed life to evolve and thrive. In the history of the Earth, biodiversity has gone through long periods of expansion 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 7.6 billion humans live on Earth and depend on its biosphere and natural resources for their survival.
Humans have developed diverse cultures. The modern English word Earth developed from a wide variety of Middle English forms, which derived from an Old English noun most spelled eorðe, it has cognates in every Germanic language, their proto-Germanic root has been reconstructed as *erþō. In its earliest appearances, eorðe was being used to translate the many senses of Latin terra and Greek γῆ: the ground, its soil, dry land, the human world, the surface of the world, the globe itself; as with Terra and Gaia, Earth was a personified goddess in Germanic paganism: the Angles were listed by Tacitus as among the devotees of Nerthus, Norse mythology included Jörð, a giantess given as the mother of Thor. Earth was written in lowercase, from early Middle English, its definite sense as "the globe" was expressed as the earth. By Early Modern English, many nouns were capitalized, the earth became the Earth when referenced along with other heavenly bodies. More the name is sometimes given as Earth, by analogy with the names of the other planets.
House styles now vary: Oxford spelling recognizes the lowercase form as the most common, with the capitalized form an acceptable variant. Another convention capitalizes "Earth" when appearing as a name but writes it in lowercase when preceded by the, it 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 Bya the primordial Earth had formed. The bodies in the Solar System evolved with the Sun. In theory, a solar nebula partitions a volume out of a molecular cloud by gravitational collapse, which begins to spin and flatten into a circumstellar disk, the planets grow out of that disk with the Sun. A nebula contains gas, ice grains, dust. According to nebular theory, planetesimals formed by accretion, with the primordial Earth taking 10–20 million years to form. A subject of research is the formation of some 4.53 Bya. A leading hypothesis is that it was formed by accretion from material loosed from Earth after a Mars-sized object, named Theia, hit Earth.
In this view, the mass of Theia was 10 percent of Earth, it hit Earth with a glancing blow and some of its mass merged with Earth. Between 4.1 and 3.8 Bya, numerous asteroid impacts during the Late Heavy Bombardment caused significant changes to the greater surface environment of the Moon and, by inference, to that of Earth. Earth's atmosphere and oceans were formed by volcanic outgassing. Water vapor from these sources condensed into the oceans, augmented by water and ice from asteroids and comets. In this model, atmospheric "greenhouse gases" kept the oceans from freezing when the newly forming Sun had only 70% of its current luminosity. By 3.5 Bya, Earth's magnetic field was established, which helped prevent the atmosphere from being stripped away by the solar wind. A crust formed; the two models that explain land mass propose either a steady growth to the present-day forms or, more a rapid growth early in Earth history followed by a long-term steady continental area. Continents formed by plate tectonics
In physics, angular momentum is the rotational equivalent of linear momentum. It is an important quantity in physics because it is a conserved quantity—the total angular momentum of a closed system remains constant. In three dimensions, the angular momentum for a point particle is a pseudovector r × p, the cross product of the particle's position vector r and its momentum vector p = mv; this definition can be applied to each point in physical fields. Unlike momentum, angular momentum does depend on where the origin is chosen, since the particle's position is measured from it. Just like for angular velocity, there are two special types of angular momentum: the spin angular momentum and the orbital angular momentum; the spin angular momentum of an object is defined as the angular momentum about its centre of mass coordinate. The orbital angular momentum of an object about a chosen origin is defined as the angular momentum of the centre of mass about the origin; the total angular momentum of an object is the sum of orbital angular momenta.
The orbital angular momentum vector of a particle is always parallel and directly proportional to the orbital angular velocity vector ω of the particle, where the constant of proportionality depends on both the mass of the particle and its distance from origin. However, the spin angular momentum of the object is proportional but not always parallel to the spin angular velocity Ω, making the constant of proportionality a second-rank tensor rather than a scalar. Angular momentum is additive. For a continuous rigid body, the total angular momentum is the volume integral of angular momentum density over the entire body. Torque can be defined as the rate of change of angular momentum, analogous to force; the net external torque on any system is always equal to the total torque on the system. Therefore, for a closed system, the total torque on the system must be 0, which means that the total angular momentum of the system is constant; the conservation of angular momentum helps explain many observed phenomena, for example the increase in rotational speed of a spinning figure skater as the skater's arms are contracted, the high rotational rates of neutron stars, the Coriolis effect, the precession of gyroscopes.
In general, conservation does limit the possible motion of a system, but does not uniquely determine what the exact motion is. In quantum mechanics, angular momentum is an operator with quantized eigenvalues. Angular momentum is subject to the Heisenberg uncertainty principle, meaning that at any time, only one component can be measured with definite precision; because of this, it turns out that the notion of an elementary particle "spinning" about an axis does not exist. For technical reasons, elementary particles still possess a spin angular momentum, but this angular momentum does not correspond to spinning motion in the ordinary sense. Angular momentum is a vector quantity that represents the product of a body's rotational inertia and rotational velocity about a particular axis. However, if the particle's trajectory lies in a single plane, it is sufficient to discard the vector nature of angular momentum, treat it as a scalar. Angular momentum can be considered a rotational analog of linear momentum.
Thus, where linear momentum p is proportional to mass m and linear speed v, p = m v, angular momentum L is proportional to moment of inertia I and angular speed ω, L = I ω. Unlike mass, which depends only on amount of matter, moment of inertia is dependent on the position of the axis of rotation and the shape of the matter. Unlike linear speed, which does not depend upon the choice of origin, angular velocity is always measured with respect to a fixed origin; therefore speaking, L should be referred to as the angular momentum relative to that center. Because I = r 2 m for a single particle and ω = v r for circular motion, angular momentum can be expanded, L = r 2 m ⋅ v r, reduced to, L = r m v, the product of the radius of rotation r and the linear momentum of the particle p = m v, where v in this case is the equivalent linear speed at the radius; this simple analysis can apply to non-circular motion if only the component of the motion, perpendicular to the radius vector is considered. In that case, L
In physics, an orbit is the gravitationally curved trajectory of an object, such as the trajectory of a planet around a star or a natural satellite around a planet. Orbit refers to a repeating trajectory, although it may refer to a non-repeating trajectory. To a close approximation and satellites follow elliptic orbits, with the central mass being orbited at a focal point of the ellipse, as described by Kepler's laws of planetary motion. For most situations, orbital motion is adequately approximated by Newtonian mechanics, which explains gravity as a force obeying an inverse-square law. However, Albert Einstein's general theory of relativity, which accounts for gravity as due to curvature of spacetime, with orbits following geodesics, provides a more accurate calculation and understanding of the exact mechanics of orbital motion; the apparent motions of the planets were described by European and Arabic philosophers using the idea of celestial spheres. This model posited the existence of perfect moving spheres or rings to which the stars and planets were attached.
It assumed the heavens were fixed apart from the motion of the spheres, was developed without any understanding of gravity. After the planets' motions were more measured, theoretical mechanisms such as deferent and epicycles were added. Although the model was capable of reasonably predicting the planets' positions in the sky and more epicycles were required as the measurements became more accurate, hence the model became unwieldy. Geocentric it was modified by Copernicus to place the Sun at the centre to help simplify the model; the model was further challenged during the 16th century, as comets were observed traversing the spheres. The basis for the modern understanding of orbits was first formulated by Johannes Kepler whose results are summarised in his three laws of planetary motion. First, he found that the orbits of the planets in our Solar System are elliptical, not circular, as had been believed, that the Sun is not located at the center of the orbits, but rather at one focus. Second, he found that the orbital speed of each planet is not constant, as had been thought, but rather that the speed depends on the planet's distance from the Sun.
Third, Kepler found a universal relationship between the orbital properties of all the planets orbiting the Sun. For the planets, the cubes of their distances from the Sun are proportional to the squares of their orbital periods. Jupiter and Venus, for example, are about 5.2 and 0.723 AU distant from the Sun, their orbital periods about 11.86 and 0.615 years. The proportionality is seen by the fact that the ratio for Jupiter, 5.23/11.862, is equal to that for Venus, 0.7233/0.6152, in accord with the relationship. Idealised orbits meeting these rules are known as Kepler orbits. Isaac Newton demonstrated that Kepler's laws were derivable from his theory of gravitation and that, in general, the orbits of bodies subject to gravity were conic sections. Newton showed that, for a pair of bodies, the orbits' sizes are in inverse proportion to their masses, that those bodies orbit their common center of mass. Where one body is much more massive than the other, it is a convenient approximation to take the center of mass as coinciding with the center of the more massive body.
Advances in Newtonian mechanics were used to explore variations from the simple assumptions behind Kepler orbits, such as the perturbations due to other bodies, or the impact of spheroidal rather than spherical bodies. Lagrange developed a new approach to Newtonian mechanics emphasizing energy more than force, made progress on the three body problem, discovering the Lagrangian points. In a dramatic vindication of classical mechanics, in 1846 Urbain Le Verrier was able to predict the position of Neptune based on unexplained perturbations in the orbit of Uranus. Albert Einstein in his 1916 paper The Foundation of the General Theory of Relativity explained that gravity was due to curvature of space-time and removed Newton's assumption that changes propagate instantaneously; this led astronomers to recognize that Newtonian mechanics did not provide the highest accuracy in understanding orbits. In relativity theory, orbits follow geodesic trajectories which are approximated well by the Newtonian predictions but the differences are measurable.
All the experimental evidence that can distinguish between the theories agrees with relativity theory to within experimental measurement accuracy. The original vindication of general relativity is that it was able to account for the remaining unexplained amount in precession of Mercury's perihelion first noted by Le Verrier. However, Newton's solution is still used for most short term purposes since it is easier to use and sufficiently accurate. Within a planetary system, dwarf planets and other minor planets and space debris orbit the system's barycenter in elliptical orbits. A comet in a parabolic or hyperbolic orbit about a barycenter is not gravitationally bound to the star and therefore is not considered part of the star's planetary system. Bodies which are gravitationally bound to one of the planets in a planetary system, either natural or artificial satellites, follow orbits about a barycenter near or within that planet. Owing to mutual gravitational perturbations, the eccentricities of the planetary orbits vary over time.
Mercury, the smallest planet in the Solar System, has the most eccentric orbit
The velocity of an object is the rate of change of its position with respect to a frame of reference, is a function of time. Velocity is equivalent to a specification of an object's direction of motion. Velocity is a fundamental concept in kinematics, the branch of classical mechanics that describes the motion of bodies. Velocity is a physical vector quantity; the scalar absolute value of velocity is called speed, being a coherent derived unit whose quantity is measured in the SI as metres per second or as the SI base unit of. For example, "5 metres per second" is a scalar. If there is a change in speed, direction or both the object has a changing velocity and is said to be undergoing an acceleration. To have a constant velocity, an object must have a constant speed in a constant direction. Constant direction constrains the object to motion in a straight path thus, a constant velocity means motion in a straight line at a constant speed. For example, a car moving at a constant 20 kilometres per hour in a circular path has a constant speed, but does not have a constant velocity because its direction changes.
Hence, the car is considered to be undergoing an acceleration. Speed describes only how fast an object is moving, whereas velocity gives both how fast it is and in which direction the object is moving. If a car is said to travel at 60 km/h, its speed has been specified. However, if the car is said to move at 60 km/h to the north, its velocity has now been specified; the big difference can be noticed. When something moves in a circular path and returns to its starting point, its average velocity is zero but its average speed is found by dividing the circumference of the circle by the time taken to move around the circle; this is because the average velocity is calculated by only considering the displacement between the starting and the end points while the average speed considers only the total distance traveled. Velocity is defined as the rate of change of position with respect to time, which may be referred to as the instantaneous velocity to emphasize the distinction from the average velocity.
In some applications the "average velocity" of an object might be needed, to say, the constant velocity that would provide the same resultant displacement as a variable velocity in the same time interval, v, over some time period Δt. Average velocity can be calculated as: v ¯ = Δ x Δ t; the average velocity is always equal to the average speed of an object. This can be seen by realizing that while distance is always increasing, displacement can increase or decrease in magnitude as well as change direction. In terms of a displacement-time graph, the instantaneous velocity can be thought of as the slope of the tangent line to the curve at any point, the average velocity as the slope of the secant line between two points with t coordinates equal to the boundaries of the time period for the average velocity; the average velocity is the same as the velocity averaged over time –, to say, its time-weighted average, which may be calculated as the time integral of the velocity: v ¯ = 1 t 1 − t 0 ∫ t 0 t 1 v d t, where we may identify Δ x = ∫ t 0 t 1 v d t and Δ t = t 1 − t 0.
If we consider v as velocity and x as the displacement vector we can express the velocity of a particle or object, at any particular time t, as the derivative of the position with respect to time: v = lim Δ t → 0 Δ x Δ t = d x d t. From this derivative equation, in the one-dimensional case it can be seen that the area under a velocity vs. time is the displacement, x. In calculus terms, the integral of the velocity function v is the displacement function x. In the figure, this corresponds to the yellow area under the curve labeled s. X = ∫ v d t. Since the derivative of the position with respect to time gives the change in position divided by the change in time, velocity is measured in metres per second. Although the concept of an instantaneous velocity might at first seem counter-intuitive, it
A sub-orbital spaceflight is a spaceflight in which the spacecraft reaches outer space, but its trajectory intersects the atmosphere or surface of the gravitating body from which it was launched, so that it will not complete one orbital revolution. For example, the path of an object launched from Earth that reaches the Kármán line, falls back to Earth, is considered a sub-orbital spaceflight; some sub-orbital flights have been undertaken to test spacecraft and launch vehicles intended for orbital spaceflight. Other vehicles are designed only for sub-orbital flight. Flights which attain sufficient velocity to go into low Earth orbit, de-orbit before completing their first full orbit, are not considered sub-orbital. Examples of this include Yuri Gagarin's Vostok 1, flights of the Fractional Orbital Bombardment System. A rocket is used, but experimental sub-orbital spaceflight has been achieved with a space gun. By one definition a sub-orbital spaceflight reaches an altitude higher than 100 km above sea level.
This altitude, known as the Kármán line, was chosen by the Fédération Aéronautique Internationale because it is the point where a vehicle flying fast enough to support itself with aerodynamic lift from the Earth's atmosphere would be flying faster than orbital speed. The US military and NASA award astronaut wings to those flying above 50 mi, although the U. S. State Department appears to not support a distinct boundary between atmospheric flight and spaceflight. During freefall the trajectory is part of an elliptic orbit as given by the orbit equation; the perigee distance is less than the radius of the Earth R including atmosphere, hence the ellipse intersects the Earth, hence the spacecraft will fail to complete an orbit. The major axis is vertical, the semi-major axis a is more than R/2; the specific orbital energy ϵ is given by: ε = − μ 2 a > − μ R where μ is the standard gravitational parameter. Always a < R, corresponding to a lower ϵ than the minimum for a full orbit, − μ 2 R Thus the net extra specific energy needed compared to just raising the spacecraft into space is between 0 and μ 2 R.
To minimize the required delta-v, the high-altitude part of the flight is made with the rockets off. The maximum speed in a flight is attained at the lowest altitude of this free-fall trajectory, both at the start and at the end of it. If one's goal is to "reach space", for example in competing for the Ansari X Prize, horizontal motion is not needed. In this case the lowest required delta-v, to reach 100 km altitude, is about 1.4 km/s. Moving slower, with less free-fall, would require more delta-v. Compare this with orbital spaceflights: a low Earth orbit, with an altitude of about 300 km, needs a speed around 7.7 km/s, requiring a delta-v of about 9.2 km/s. For sub-orbital spaceflights covering a horizontal distance the maximum speed and required delta-v are in between those of a vertical flight and a LEO; the maximum speed at the lower ends of the trajectory are now composed of a horizontal and a vertical component. The higher the horizontal distance covered, the greater the horizontal speed will be.
For the V-2 rocket, just reaching space but with a range of about 330 km, the maximum speed was 1.6 km/s. Scaled Composites SpaceShipTwo, under development will have a similar free-fall orbit but the announced maximum speed is 1.1 km/s. For larger ranges, due to the elliptic orbit the maximum altitude can be much more than for a LEO. On a 10,000-km intercontinental flight, such as that of an intercontinental ballistic missile or possible future commercial spaceflight, the maximum speed is about 7 km/s, the maximum altitude may be more than 1300 km. Any spaceflight that returns to the surface, including sub-orbital ones, will undergo atmospheric reentry; the speed at the start of the reentry is the maximum speed of the flight. The aerodynamic heating caused will vary accordingly: it is much less for a flight with a maximum speed of only 1 km/s than for one with a maximum speed of 7 or 8 km/s. We can calculate the minimum delta-v and the corresponding maximum altitude for a given range, d, assuming a spherical earth of circumference 40 000 km and neglecting the earth's rotation and atmosphere.
Let θ be half the angle that the projectile is to go around the earth, so in degrees it is 45°×d/10 000 km. The minimum-delta-v trajectory corresponds to an ellipse with one focus at the centre of the earth and the other at the point halfway between the launch point and the destination point. (
An equinox is regarded as the instant of time when the plane of Earth's equator passes through the center of the Sun. This occurs 23 September. In other words, it is the moment at which the center of the visible Sun is directly above the Equator; the word is derived from aequus and nox. On the day of an equinox and nighttime are of equal duration all over the planet, they are not equal, due to the angular size of the Sun, atmospheric refraction, the changing duration of the length of day that occurs at most latitudes around the equinoxes. Long before conceiving this equality primitive cultures noted the day when the Sun rises due East and sets due West and indeed this happens on the day closest to the astronomically defined event. In the northern hemisphere, the equinox in March is called the Spring Equinox; the dates are variable, dependent as they are on the leap year cycle. Because the Moon cause the motion of the Earth to vary from a perfect ellipse, the equinox is now defined by the Sun's more regular ecliptic longitude rather than by its declination.
The instants of the equinoxes are defined to be when the longitude of the Sun is 0° and 180°. Systematically observing the sunrise, people discovered that it occurs between two extreme locations at the horizon and noted the midpoint between the two, it was realized that this happens on a day when the durations of the day and the night are equal and the word "equinox" comes from Latin Aequus, meaning "equal", Nox, meaning "night". In the northern hemisphere, the vernal equinox conventionally marks the beginning of spring in most cultures and is considered the start of the New Year in the Assyrian calendar and the Persian calendar or Iranian calendars as Nowruz, while the autumnal equinox marks the beginning of autumn; the equinoxes are the only times. As a result, the northern and southern hemispheres are illuminated. In other words, the equinoxes are the only times when the subsolar point is on the equator, meaning that the Sun is overhead at a point on the equatorial line; the subsolar point crosses the equator moving northward at the March equinox and southward at the September equinox.
When Julius Caesar established the Julian calendar in 45 BC, he set 25 March as the date of the spring equinox. Because the Julian year is longer than the tropical year by about 11.3 minutes on average, the calendar "drifted" with respect to the two equinoxes – so that in AD 300 the spring equinox occurred on about 21 March, by AD 1500 it had drifted backwards to 11 March. This drift induced Pope Gregory XIII to create the modern Gregorian calendar; the Pope wanted to continue to conform with the edicts of the Council of Nicaea in AD 325 concerning the date of Easter, which means he wanted to move the vernal equinox to the date on which it fell at that time, to maintain it at around that date in the future, which he achieved by reducing the number of leap years from 100 to 97 every 400 years. However, there remained a small residual variation in the date and time of the vernal equinox of about ±27 hours from its mean position all because the distribution of 24-hour centurial leap days causes large jumps.
This in turn raised the possibility that it could fall on 22 March, thus Easter Day might theoretically commence before the equinox. The astronomers chose the appropriate number of days to omit so that the equinox would swing from 19 to 21 March but never fall on 22 March; the dates of the equinoxes change progressively during the leap-year cycle, because the Gregorian calendar year is not commensurate with the period of the Earth's revolution about the Sun. It is only after a complete Gregorian leap-year cycle of 400 years that the seasons commence at the same time. In the 21st century the earliest March equinox will be 19 March 2096, while the latest was 21 March 2003; the earliest September equinox will be 21 September 2096 while the latest was 23 September 2003. Vernal equinox and autumnal equinox: these classical names are direct derivatives of Latin; these are the universal and still most used terms for the equinoxes, but are confusing because in the southern hemisphere the vernal equinox does not occur in spring and the autumnal equinox does not occur in autumn.
The equivalent common language English terms spring equinox and autumn equinox are more ambiguous. It has become common for people to refer to the September equinox in the southern hemisphere as the Vernal equinox. March equinox and September equinox: names referring to the months of the year in which they occur, with no ambiguity as to which hemisphere is the context, they are still not universal, however, as not all cultures use a solar-based calendar where the equinoxes occur every year in the same month. Although the terms have become common in the 21st century, they were sometimes used at least as long ago as the mid-20th century. Northward equinox and southward equinox: names referring to the appare
A circular orbit is the orbit with a fixed distance around the barycenter, that is, in the shape of a circle. Below we consider a circular orbit in astrodynamics or celestial mechanics under standard assumptions. Here the centripetal force is the gravitational force, 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 the speed, angular speed and kinetic energy are constant. There is no 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 = ω 2 r where: v is orbital velocity of orbiting body, r is radius of the circle ω is angular speed, measured in radians per unit time; 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 the numerical values for v will be in meters per second, r in meters, ω in radians per second.
The relative velocity is constant: v = G M r = μ r where: G, is the gravitational constant M, is the mass of both orbiting bodies, although in common practice, if the greater mass is larger, the lesser mass is neglected, with minimal change in the result. 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; this is because μ = r v 2 ω 2 r 3 = μ Hence the orbital period can be computed as: T = 2 π r 3 μ Compare two proportional quantities, the free-fall time T f f = π 2 2 r 3 μ and the time to fall to a point mass in a radial parabolic orbit T p a r = 2 3 r 3 μ The fact that the formulas only differ by a constant factor is a priori clear from dimensional analysis. The specific orbital energy is negative, ϵ = − v 2 2 ϵ = − μ 2 r Thus the virial theorem applies without taking a time-average: the kinetic energy of the system is equal to the absolute value of the total energy the potential energy of the system is equal to twice the total energyThe escape velocity from any distance is √2 times the speed in a circular orbit at that distance: the kinetic energy is twice as much, hence the total energy is zero.
Maneuvering into a large circular orbit, e.g. a geostationary orbit, requires a larger delta-v than an escape orbit, although the latter implies getting arbitrarily far away and having more energy than needed for the orbital speed of the circular orbit. It is a matter of maneuvering into the orbit. See Hohmann transfer orbit. In Schwarzschild metric, the orbital velocity for a circular orbit with radius r is given by the following formula: v = G M r − r S where r S = 2 G M c 2 is the Schwarzschild radius of the central body. For the sake of convenience, the derivation will be written in units in which c = G = 1; the four-velocity of a body on a circular orbit is given by: u μ