Orbit

In physics, an orbit is the gravitationally curved trajectory of an object,[1] such as the trajectory of a planet around a star or a natural satellite around a planet. Normally, orbit refers to a regularly repeating trajectory, although it may also refer to a non-repeating trajectory. To a close approximation, planets and satellites follow elliptic orbits, with the central mass being orbited at a focal point of the ellipse,[2] 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.[3] 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.

Orbit2
Two bodies of different masses orbiting a common barycenter. The relative sizes and type of orbit are similar to the PlutoCharon system.

History

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 planets were attached. 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 more accurately measured, theoretical mechanisms such as deferent and epicycles were added. Although the model was capable of reasonably accurately predicting the planets' positions in the sky, more and more epicycles were required as the measurements became more accurate, hence the model became increasingly unwieldy. 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.[4][5]

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 (or epicyclic), as had previously been believed, and that the Sun is not located at the center of the orbits, but rather at one focus.[6] Second, he found that the orbital speed of each planet is not constant, as had previously 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 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 practically equal to that for Venus, 0.7233/0.6152, in accord with the relationship. Idealised orbits meeting these rules are known as Kepler orbits.

Conic sections with plane
The lines traced out by orbits dominated by the gravity of a central source are conic sections: the shapes of the curves of intersection between a plane and a cone. Parabolic (1) and hyperbolic (3) orbits are escape orbits, whereas elliptical and circular orbits (2) are captive.
Gravity Wells Potential Plus Kinetic Energy - Circle-Ellipse-Parabola-Hyperbola
This image shows the four trajectory categories with the gravitational potential well of the central mass's field of potential energy shown in black and the height of the kinetic energy of the moving body shown in red extending above that, correlating to changes in speed as distance changes according to Kepler's laws.

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 (this assumes that the force of gravity propagates instantaneously). Newton showed that, for a pair of bodies, the orbits' sizes are in inverse proportion to their masses, and that those bodies orbit their common center of mass. Where one body is much more massive than the other (as is the case of an artificial satellite orbiting a planet), 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 then 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 (1736–1813) developed a new approach to Newtonian mechanics emphasizing energy more than force, and 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 (1879-1955) 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 usually approximated very well by the Newtonian predictions (except where there are very strong gravity fields and very high speeds) but the differences are measurable. Essentially 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 significantly easier to use and sufficiently accurate.

Planetary orbits

Within a planetary system, planets, dwarf planets, asteroids and other minor planets, comets, 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. At the present epoch, Mars has the next largest eccentricity while the smallest orbital eccentricities are seen with Venus and Neptune.

As two objects orbit each other, the periapsis is that point at which the two objects are closest to each other and the apoapsis is that point at which they are the farthest. (More specific terms are used for specific bodies. For example, perigee and apogee are the lowest and highest parts of an orbit around Earth, while perihelion and aphelion are the closest and farthest points of an orbit around the Sun.)

In the case of planets orbiting a star, the mass of the star and all its satellites are calculated to be at a single point called the barycenter. The paths of all the star's satellites are elliptical orbits about that barycenter. Each satellite in that system will have its own elliptical orbit with the barycenter at one focal point of that ellipse. At any point along its orbit, any satellite will have a certain value of kinetic and potential energy with respect to the barycenter, and that energy is a constant value at every point along its orbit. As a result, as a planet approaches periapsis, the planet will increase in speed as its potential energy decreases; as a planet approaches apoapsis, its velocity will decrease as its potential energy increases.

Understanding orbits

There are a few common ways of understanding orbits:

  • A force, such as gravity, pulls an object into a curved path as it attempts to fly off in a straight line.
  • As the object is pulled toward the massive body, it falls toward that body. However, if it has enough tangential velocity it will not fall into the body but will instead continue to follow the curved trajectory caused by that body indefinitely. The object is then said to be orbiting the body.

As an illustration of an orbit around a planet, the Newton's cannonball model may prove useful (see image below). This is a 'thought experiment', in which a cannon on top of a tall mountain is able to fire a cannonball horizontally at any chosen muzzle speed. The effects of air friction on the cannonball are ignored (or perhaps the mountain is high enough that the cannon is above the Earth's atmosphere, which is the same thing).[7]

Newton Cannon
Newton's cannonball, an illustration of how objects can "fall" in a curve
Conic sections, orbits, and gravitational potential
Conic sections describe the possible orbits (yellow) of small objects around the Earth. A projection of these orbits onto the gravitational potential (blue) of the Earth makes it possible to determine the orbital energy at each point in space.

If the cannon fires its ball with a low initial speed, the trajectory of the ball curves downward and hits the ground (A). As the firing speed is increased, the cannonball hits the ground farther (B) away from the cannon, because while the ball is still falling towards the ground, the ground is increasingly curving away from it (see first point, above). All these motions are actually "orbits" in a technical sense – they are describing a portion of an elliptical path around the center of gravity – but the orbits are interrupted by striking the Earth.

If the cannonball is fired with sufficient speed, the ground curves away from the ball at least as much as the ball falls – so the ball never strikes the ground. It is now in what could be called a non-interrupted, or circumnavigating, orbit. For any specific combination of height above the center of gravity and mass of the planet, there is one specific firing speed (unaffected by the mass of the ball, which is assumed to be very small relative to the Earth's mass) that produces a circular orbit, as shown in (C).

As the firing speed is increased beyond this, non-interrupted elliptic orbits are produced; one is shown in (D). If the initial firing is above the surface of the Earth as shown, there will also be non-interrupted elliptical orbits at slower firing speed; these will come closest to the Earth at the point half an orbit beyond, and directly opposite the firing point, below the circular orbit.

At a specific horizontal firing speed called escape velocity, dependent on the mass of the planet, an open orbit (E) is achieved that has a parabolic path. At even greater speeds the object will follow a range of hyperbolic trajectories. In a practical sense, both of these trajectory types mean the object is "breaking free" of the planet's gravity, and "going off into space" never to return.

The velocity relationship of two moving objects with mass can thus be considered in four practical classes, with subtypes:

  1. No orbit
  2. Suborbital trajectories
    • Range of interrupted elliptical paths
  3. Orbital trajectories (or simply "orbits")
    • Range of elliptical paths with closest point opposite firing point
    • Circular path
    • Range of elliptical paths with closest point at firing point
  4. Open (or escape) trajectories
    • Parabolic paths
    • Hyperbolic paths

It is worth noting that orbital rockets are launched vertically at first to lift the rocket above the atmosphere (which causes frictional drag), and then slowly pitch over and finish firing the rocket engine parallel to the atmosphere to achieve orbit speed.

Once in orbit, their speed keeps them in orbit above the atmosphere. If e.g., an elliptical orbit dips into dense air, the object will lose speed and re-enter (i.e. fall). Occasionally a space craft will intentionally intercept the atmosphere, in an act commonly referred to as an aerobraking maneuver

Orbitalaltitudes

Newton's laws of motion

Newton's law of gravitation and laws of motion for two-body problems

In most situations relativistic effects can be neglected, and Newton's laws give a sufficiently accurate description of motion. The acceleration of a body is equal to the sum of the forces acting on it, divided by its mass, and the gravitational force acting on a body is proportional to the product of the masses of the two attracting bodies and decreases inversely with the square of the distance between them. To this Newtonian approximation, for a system of two point masses or spherical bodies, only influenced by their mutual gravitation (called a two-body problem), their trajectories can be exactly calculated. If the heavier body is much more massive than the smaller, as in the case of a satellite or small moon orbiting a planet or for the Earth orbiting the Sun, it is accurate enough and convenient to describe the motion in terms of a coordinate system that is centered on the heavier body, and we say that the lighter body is in orbit around the heavier. For the case where the masses of two bodies are comparable, an exact Newtonian solution is still sufficient and can be had by placing the coordinate system at the center of mass of the system.

Defining gravitational potential energy

Energy is associated with gravitational fields. A stationary body far from another can do external work if it is pulled towards it, and therefore has gravitational potential energy. Since work is required to separate two bodies against the pull of gravity, their gravitational potential energy increases as they are separated, and decreases as they approach one another. For point masses the gravitational energy decreases to zero as they approach zero separation. It is convenient and conventional to assign the potential energy as having zero value when they are an infinite distance apart, and hence it has a negative value (since it decreases from zero) for smaller finite distances.

Orbital energies and orbit shapes

When only two gravitational bodies interact, their orbits follow a conic section. The orbit can be open (implying the object never returns) or closed (returning). Which it is depends on the total energy (kinetic + potential energy) of the system. In the case of an open orbit, the speed at any position of the orbit is at least the escape velocity for that position, in the case of a closed orbit, the speed is always less than the escape velocity. Since the kinetic energy is never negative, if the common convention is adopted of taking the potential energy as zero at infinite separation, the bound orbits will have negative total energy, the parabolic trajectories zero total energy, and hyperbolic orbits positive total energy.

An open orbit will have a parabolic shape if it has velocity of exactly the escape velocity at that point in its trajectory, and it will have the shape of a hyperbola when its velocity is greater than the escape velocity. When bodies with escape velocity or greater approach each other, they will briefly curve around each other at the time of their closest approach, and then separate, forever.

All closed orbits have the shape of an ellipse. A circular orbit is a special case, wherein the foci of the ellipse coincide. The point where the orbiting body is closest to Earth is called the perigee, and is called the periapsis (less properly, "perifocus" or "pericentron") when the orbit is about a body other than Earth. The point where the satellite is farthest from Earth is called the apogee, apoapsis, or sometimes apifocus or apocentron. A line drawn from periapsis to apoapsis is the line-of-apsides. This is the major axis of the ellipse, the line through its longest part.

Kepler's laws

Bodies following closed orbits repeat their paths with a certain time called the period. This motion is described by the empirical laws of Kepler, which can be mathematically derived from Newton's laws. These can be formulated as follows:

  1. The orbit of a planet around the Sun is an ellipse, with the Sun in one of the focal points of that ellipse. [This focal point is actually the barycenter of the Sun-planet system; for simplicity this explanation assumes the Sun's mass is infinitely larger than that planet's.] The planet's orbit lies in a plane, called the orbital plane. The point on the orbit closest to the attracting body is the periapsis. The point farthest from the attracting body is called the apoapsis. There are also specific terms for orbits about particular bodies; things orbiting the Sun have a perihelion and aphelion, things orbiting the Earth have a perigee and apogee, and things orbiting the Moon have a perilune and apolune (or periselene and aposelene respectively). An orbit around any star, not just the Sun, has a periastron and an apastron.
  2. As the planet moves in its orbit, the line from the Sun to planet sweeps a constant area of the orbital plane for a given period of time, regardless of which part of its orbit the planet traces during that period of time. This means that the planet moves faster near its perihelion than near its aphelion, because at the smaller distance it needs to trace a greater arc to cover the same area. This law is usually stated as "equal areas in equal time."
  3. For a given orbit, the ratio of the cube of its semi-major axis to the square of its period is constant.

Limitations of Newton's law of gravitation

Note that while bound orbits of a point mass or a spherical body with a Newtonian gravitational field are closed ellipses, which repeat the same path exactly and indefinitely, any non-spherical or non-Newtonian effects (such as caused by the slight oblateness of the Earth, or by relativistic effects, thereby changing the gravitational field's behavior with distance) will cause the orbit's shape to depart from the closed ellipses characteristic of Newtonian two-body motion. The two-body solutions were published by Newton in Principia in 1687. In 1912, Karl Fritiof Sundman developed a converging infinite series that solves the three-body problem; however, it converges too slowly to be of much use. Except for special cases like the Lagrangian points, no method is known to solve the equations of motion for a system with four or more bodies.

Approaches to many-body problems

Rather than an exact closed form solution, orbits with many bodies can be approximated with arbitrarily high accuracy. These approximations take two forms:

One form takes the pure elliptic motion as a basis, and adds perturbation terms to account for the gravitational influence of multiple bodies. This is convenient for calculating the positions of astronomical bodies. The equations of motion of the moons, planets and other bodies are known with great accuracy, and are used to generate tables for celestial navigation. Still, there are secular phenomena that have to be dealt with by post-Newtonian methods.
The differential equation form is used for scientific or mission-planning purposes. According to Newton's laws, the sum of all the forces acting on a body will equal the mass of the body times its acceleration (F = ma). Therefore accelerations can be expressed in terms of positions. The perturbation terms are much easier to describe in this form. Predicting subsequent positions and velocities from initial values of position and velocity corresponds to solving an initial value problem. Numerical methods calculate the positions and velocities of the objects a short time in the future, then repeat the calculation ad nauseam. However, tiny arithmetic errors from the limited accuracy of a computer's math are cumulative, which limits the accuracy of this approach.

Differential simulations with large numbers of objects perform the calculations in a hierarchical pairwise fashion between centers of mass. Using this scheme, galaxies, star clusters and other large assemblages of objects have been simulated.

Newtonian analysis of orbital motion

(See also Kepler orbit, orbit equation and Kepler's first law.)

The Earth follows an ellipse round the sun. But unlike the ellipse followed by a pendulum or an object attached to a spring, the sun is at a focal point of the ellipse and not at its centre.

The following derivation applies to such an elliptical orbit. We start only with the Newtonian law of gravitation stating that the gravitational acceleration towards the central body is related to the inverse of the square of the distance between them, namely

eq 1.

where F2 is the force acting on the mass m2 caused by the gravitational attraction mass m1 has for m2, G is the universal gravitational constant, and r is the distance between the two masses centers.

From Newton's Second Law, the summation of the forces acting on m2 related to that bodies acceleration:

eq 2.

where A2 is the acceleration of m2 caused by the force of gravitational attraction F2 of m1 acting on m2.

Combining Eq 1 and 2:

Solving for the acceleration, A2:

where is the standard gravitational parameter, in this case . It is understood that the system being described is m2, hence the subscripts can be dropped.

We assume that the central body is massive enough that it can be considered to be stationary and we ignore the more subtle effects of general relativity.

When a pendulum or an object attached to a spring swings in an ellipse, the inward acceleration/force is proportional to the distance Due to the way vectors add, the component of the force in the or in the directions are also proportionate to the respective components of the distances, . Hence, the entire analysis can be done separately in these dimensions. This results in the harmonic parabolic equations and of the ellipse. In contrast, with the decreasing relationship , the dimensions cannot be separated.

The location of the orbiting object at the current time is located in the plane using Vector calculus in polar coordinates both with the standard Euclidean basis and with the polar basis with the origin coinciding with the center of force. Let be the distance between the object and the center and be the angle it has rotated. Let and be the standard Euclidean bases and let and be the radial and transverse polar basis with the first being the unit vector pointing from the central body to the current location of the orbiting object and the second being the orthogonal unit vector pointing in the direction that the orbiting object would travel if orbiting in a counter clockwise circle. Then the vector to the orbiting object is

We use and to denote the standard derivatives of how this distance and angle change over time. We take the derivative of a vector to see how it changes over time by subtracting its location at time from that at time and dividing by . The result is also a vector. Because our basis vector moves as the object orbits, we start by differentiating it. From time to , the vector keeps its beginning at the origin and rotates from angle to which moves its head a distance in the perpendicular direction giving a derivative of .

We can now find the velocity and acceleration of our orbiting object.

The coefficients of and give the accelerations in the radial and transverse directions. As said, Newton gives this first due to gravity is and the second is zero.

(1)

(2)

Equation (2) can be rearranged using integration by parts.

We can multiply through by because it is not zero unless the orbiting object crashes. Then having the derivative be zero gives that the function is a constant.

(3)

which is actually the theoretical proof of Kepler's second law (A line joining a planet and the Sun sweeps out equal areas during equal intervals of time). The constant of integration, h, is the angular momentum per unit mass.

In order to get an equation for the orbit from equation (1), we need to eliminate time.[8] (See also Binet equation.) In polar coordinates, this would express the distance of the orbiting object from the center as a function of its angle . However, it is easier to introduce the auxiliary variable and to express as a function of . Derivatives of with respect to time may be rewritten as derivatives of with respect to angle.

(reworking (3))

Plugging these into (1) gives

So for the gravitational force – or, more generally, for any inverse square force law – the right hand side of the equation becomes a constant and the equation is seen to be the harmonic equation (up to a shift of origin of the dependent variable). The solution is:

where A and θ0 are arbitrary constants. This resulting equation of the orbit of the object is that of an ellipse in Polar form relative to one of the focal points. This is put into a more standard form by letting be the eccentricity, letting be the semi-major axis. Finally, letting so the long axis of the ellipse is along the positive x coordinate.

Relativistic orbital motion

The above classical (Newtonian) analysis of orbital mechanics assumes that the more subtle effects of general relativity, such as frame dragging and gravitational time dilation are negligible. Relativistic effects cease to be negligible when near very massive bodies (as with the precession of Mercury's orbit about the Sun), or when extreme precision is needed (as with calculations of the orbital elements and time signal references for GPS satellites.[9])

Orbital planes

The analysis so far has been two dimensional; it turns out that an unperturbed orbit is two-dimensional in a plane fixed in space, and thus the extension to three dimensions requires simply rotating the two-dimensional plane into the required angle relative to the poles of the planetary body involved.

The rotation to do this in three dimensions requires three numbers to uniquely determine; traditionally these are expressed as three angles.

Orbital period

The orbital period is simply how long an orbiting body takes to complete one orbit.

Specifying orbits

Six parameters are required to specify a Keplerian orbit about a body. For example, the three numbers that specify the body's initial position, and the three values that specify its velocity will define a unique orbit that can be calculated forwards (or backwards) in time. However, traditionally the parameters used are slightly different.

The traditionally used set of orbital elements is called the set of Keplerian elements, after Johannes Kepler and his laws. The Keplerian elements are six:

In principle once the orbital elements are known for a body, its position can be calculated forward and backwards indefinitely in time. However, in practice, orbits are affected or perturbed, by other forces than simple gravity from an assumed point source (see the next section), and thus the orbital elements change over time.

Orbital perturbations

An orbital perturbation is when a force or impulse which is much smaller than the overall force or average impulse of the main gravitating body and which is external to the two orbiting bodies causes an acceleration, which changes the parameters of the orbit over time.

Radial, prograde and transverse perturbations

A small radial impulse given to a body in orbit changes the eccentricity, but not the orbital period (to first order). A prograde or retrograde impulse (i.e. an impulse applied along the orbital motion) changes both the eccentricity and the orbital period. Notably, a prograde impulse at periapsis raises the altitude at apoapsis, and vice versa, and a retrograde impulse does the opposite. A transverse impulse (out of the orbital plane) causes rotation of the orbital plane without changing the period or eccentricity. In all instances, a closed orbit will still intersect the perturbation point.

Orbital decay

If an orbit is about a planetary body with significant atmosphere, its orbit can decay because of drag. Particularly at each periapsis, the object experiences atmospheric drag, losing energy. Each time, the orbit grows less eccentric (more circular) because the object loses kinetic energy precisely when that energy is at its maximum. This is similar to the effect of slowing a pendulum at its lowest point; the highest point of the pendulum's swing becomes lower. With each successive slowing more of the orbit's path is affected by the atmosphere and the effect becomes more pronounced. Eventually, the effect becomes so great that the maximum kinetic energy is not enough to return the orbit above the limits of the atmospheric drag effect. When this happens the body will rapidly spiral down and intersect the central body.

The bounds of an atmosphere vary wildly. During a solar maximum, the Earth's atmosphere causes drag up to a hundred kilometres higher than during a solar minimum.

Some satellites with long conductive tethers can also experience orbital decay because of electromagnetic drag from the Earth's magnetic field. As the wire cuts the magnetic field it acts as a generator, moving electrons from one end to the other. The orbital energy is converted to heat in the wire.

Orbits can be artificially influenced through the use of rocket engines which change the kinetic energy of the body at some point in its path. This is the conversion of chemical or electrical energy to kinetic energy. In this way changes in the orbit shape or orientation can be facilitated.

Another method of artificially influencing an orbit is through the use of solar sails or magnetic sails. These forms of propulsion require no propellant or energy input other than that of the Sun, and so can be used indefinitely. See statite for one such proposed use.

Orbital decay can occur due to tidal forces for objects below the synchronous orbit for the body they're orbiting. The gravity of the orbiting object raises tidal bulges in the primary, and since below the synchronous orbit the orbiting object is moving faster than the body's surface the bulges lag a short angle behind it. The gravity of the bulges is slightly off of the primary-satellite axis and thus has a component along the satellite's motion. The near bulge slows the object more than the far bulge speeds it up, and as a result the orbit decays. Conversely, the gravity of the satellite on the bulges applies torque on the primary and speeds up its rotation. Artificial satellites are too small to have an appreciable tidal effect on the planets they orbit, but several moons in the Solar System are undergoing orbital decay by this mechanism. Mars' innermost moon Phobos is a prime example, and is expected to either impact Mars' surface or break up into a ring within 50 million years.

Orbits can decay via the emission of gravitational waves. This mechanism is extremely weak for most stellar objects, only becoming significant in cases where there is a combination of extreme mass and extreme acceleration, such as with black holes or neutron stars that are orbiting each other closely.

Oblateness

The standard analysis of orbiting bodies assumes that all bodies consist of uniform spheres, or more generally, concentric shells each of uniform density. It can be shown that such bodies are gravitationally equivalent to point sources.

However, in the real world, many bodies rotate, and this introduces oblateness and distorts the gravity field, and gives a quadrupole moment to the gravitational field which is significant at distances comparable to the radius of the body. In the general case, the gravitational potential of a rotating body such as, e.g., a planet is usually expanded in multipoles accounting for the departures of it from spherical symmetry. From the point of view of satellite dynamics, of particular relevance are the so-called even zonal harmonic coefficients, or even zonals, since they induce secular orbital perturbations which are cumulative over time spans longer than the orbital period.[10][11][12] They do depend on the orientation of the body's symmetry axis in the space, affecting, in general, the whole orbit, with the exception of the semimajor axis.

Multiple gravitating bodies

The effects of other gravitating bodies can be significant. For example, the orbit of the Moon cannot be accurately described without allowing for the action of the Sun's gravity as well as the Earth's. One approximate result is that bodies will usually have reasonably stable orbits around a heavier planet or moon, in spite of these perturbations, provided they are orbiting well within the heavier body's Hill sphere.

When there are more than two gravitating bodies it is referred to as an n-body problem. Most n-body problems have no closed form solution, although some special cases have been formulated.

Light radiation and stellar wind

For smaller bodies particularly, light and stellar wind can cause significant perturbations to the attitude and direction of motion of the body, and over time can be significant. Of the planetary bodies, the motion of asteroids is particularly affected over large periods when the asteroids are rotating relative to the Sun.

Strange orbits

Mathematicians have discovered that it is possible in principle to have multiple bodies in non-elliptical orbits that repeat periodically, although most such orbits are not stable regarding small perturbations in mass, position, or velocity. However, some special stable cases have been identified, including a planar figure-eight orbit occupied by three moving bodies. Further studies have discovered that nonplanar orbits are also possible, including one involving 12 masses moving in 4 roughly circular, interlocking orbits topologically equivalent to the edges of a cuboctahedron.[13]

Finding such orbits naturally occurring in the universe is thought to be extremely unlikely, because of the improbability of the required conditions occurring by chance.[13]

Astrodynamics

Orbital mechanics or astrodynamics is the application of ballistics and celestial mechanics to the practical problems concerning the motion of rockets and other spacecraft. The motion of these objects is usually calculated from Newton's laws of motion and Newton's law of universal gravitation. It is a core discipline within space mission design and control. Celestial mechanics treats more broadly the orbital dynamics of systems under the influence of gravity, including spacecraft and natural astronomical bodies such as star systems, planets, moons, and comets. Orbital mechanics focuses on spacecraft trajectories, including orbital maneuvers, orbit plane changes, and interplanetary transfers, and is used by mission planners to predict the results of propulsive maneuvers. General relativity is a more exact theory than Newton's laws for calculating orbits, and is sometimes necessary for greater accuracy or in high-gravity situations (such as orbits close to the Sun).

Earth orbits

Comparison satellite navigation orbits
Comparison of geostationary Earth orbit with GPS, GLONASS, Galileo and Compass (medium Earth orbit) satellite navigation system orbits with the International Space Station, Hubble Space Telescope and Iridium constellation orbits, and the nominal size of the Earth.[a] The Moon's orbit is around 9 times larger (in radius and length) than geostationary orbit.[b]

Scaling in gravity

The gravitational constant G has been calculated as:

  • (6.6742 ± 0.001) × 10−11 (kg/m3)−1s−2.

Thus the constant has dimension density−1 time−2. This corresponds to the following properties.

Scaling of distances (including sizes of bodies, while keeping the densities the same) gives similar orbits without scaling the time: if for example distances are halved, masses are divided by 8, gravitational forces by 16 and gravitational accelerations by 2. Hence velocities are halved and orbital periods remain the same. Similarly, when an object is dropped from a tower, the time it takes to fall to the ground remains the same with a scale model of the tower on a scale model of the Earth.

Scaling of distances while keeping the masses the same (in the case of point masses, or by reducing the densities) gives similar orbits; if distances are multiplied by 4, gravitational forces and accelerations are divided by 16, velocities are halved and orbital periods are multiplied by 8.

When all densities are multiplied by 4, orbits are the same; gravitational forces are multiplied by 16 and accelerations by 4, velocities are doubled and orbital periods are halved.

When all densities are multiplied by 4, and all sizes are halved, orbits are similar; masses are divided by 2, gravitational forces are the same, gravitational accelerations are doubled. Hence velocities are the same and orbital periods are halved.

In all these cases of scaling. if densities are multiplied by 4, times are halved; if velocities are doubled, forces are multiplied by 16.

These properties are illustrated in the formula (derived from the formula for the orbital period)

for an elliptical orbit with semi-major axis a, of a small body around a spherical body with radius r and average density ρ, where T is the orbital period. See also Kepler's Third Law.

Patents

The application of certain orbits or orbital maneuvers to specific useful purposes have been the subject of patents.[17]

Tidal locking

Some bodies are tidally locked with other bodies, meaning that one side of the celestial body is permanently facing its host object. This is the case for Earth-Moon and Pluto-Charon system.

See also

Notes

  1. ^ Orbital periods and speeds are calculated using the relations 4π2R2 = T2GM and V2R = GM, where R = radius of orbit in metres, T = orbital period in seconds, V = orbital speed in m/s, G = gravitational constant ≈ 6.673×1011 Nm2/kg2, M = mass of Earth ≈ 5.98×1024 kg.
  2. ^ Approximately 8.6 times when the Moon is nearest (363,104 km ÷ 42,164 km) to 9.6 times when the Moon is farthest (405,696 km ÷ 42,164 km).

References

  1. ^ orbit (astronomy) – Britannica Online Encyclopedia
  2. ^ The Space Place :: What's a Barycenter
  3. ^ Kuhn, The Copernican Revolution, pp. 238, 246–252
  4. ^ Encyclopædia Britannica, 1968, vol. 2, p. 645
  5. ^ M Caspar, Kepler (1959, Abelard-Schuman), at pp.131–140; A Koyré, The Astronomical Revolution: Copernicus, Kepler, Borelli (1973, Methuen), pp. 277–279
  6. ^ Jones, Andrew. "Kepler's Laws of Planetary Motion". about.com. Retrieved 1 June 2008.
  7. ^ See pages 6 to 8 in Newton's "Treatise of the System of the World" (written 1685, translated into English 1728, see Newton's 'Principia' – A preliminary version), for the original version of this 'cannonball' thought-experiment.
  8. ^ Fitzpatrick, Richard (2 February 2006). "Planetary orbits". Classical Mechanics – an introductory course. The University of Texas at Austin. Archived from the original on 3 March 2001. Retrieved 14 January 2009.
  9. ^ Pogge, Richard W.; "Real-World Relativity: The GPS Navigation System". Retrieved 25 January 2008.
  10. ^ Iorio, L. (2011). "Perturbed stellar motions around the rotating black hole in Sgr A* for a generic orientation of its spin axis". Physical Review D. 84 (12): 124001. arXiv:1107.2916. Bibcode:2011PhRvD..84l4001I. doi:10.1103/PhysRevD.84.124001.
  11. ^ Renzetti, G. (2013). "Satellite Orbital Precessions Caused by the Octupolar Mass Moment of a Non-Spherical Body Arbitrarily Oriented in Space". Journal of Astrophysics and Astronomy. 34 (4): 341–348. Bibcode:2013JApA...34..341R. doi:10.1007/s12036-013-9186-4.
  12. ^ Renzetti, G. (2014). "Satellite orbital precessions caused by the first odd zonal J3 multipole of a non-spherical body arbitrarily oriented in space". Astrophysics and Space Science. 352 (2): 493–496. Bibcode:2014Ap&SS.352..493R. doi:10.1007/s10509-014-1915-x.
  13. ^ a b Peterson, Ivars (23 September 2013). "Strange Orbits". Science News. Retrieved 21 July 2017.
  14. ^ "NASA Safety Standard 1740.14, Guidelines and Assessment Procedures for Limiting Orbital Debris" (PDF). Office of Safety and Mission Assurance. 1 August 1995. Archived from the original (PDF) on 15 February 2013., pages 37-38 (6-1,6-2); figure 6-1.
  15. ^ a b "Orbit: Definition". Ancillary Description Writer's Guide, 2013. National Aeronautics and Space Administration (NASA) Global Change Master Directory. Archived from the original on 11 May 2013. Retrieved 29 April 2013.
  16. ^ Vallado, David A. (2007). Fundamentals of Astrodynamics and Applications. Hawthorne, CA: Microcosm Press. p. 31.
  17. ^ Ferreira, Becky (19 February 2015). "How Satellite Companies Patent Their Orbits". Motherboard. Vice News. Retrieved 20 September 2018.

Further reading

  • Abell; Morrison & Wolff (1987). Exploration of the Universe (fifth ed.). Saunders College Publishing.
  • Linton, Christopher (2004). From Eudoxus to Einstein. Cambridge: University Press. ISBN 0-521-82750-7
  • Swetz, Frank; et al. (1997). Learn from the Masters!. Mathematical Association of America. ISBN 0-88385-703-0
  • Andrea Milani and Giovanni F. Gronchi. Theory of Orbit Determination (Cambridge University Press; 378 pages; 2010). Discusses new algorithms for determining the orbits of both natural and artificial celestial bodies.

External links

Apollo 11

Apollo 11 was the spaceflight that landed the first two people on the Moon. Commander Neil Armstrong and lunar module pilot Buzz Aldrin, both American, landed the Apollo Lunar Module Eagle on July 20, 1969, at 20:17 UTC. Armstrong became the first person to step onto the lunar surface six hours later on July 21 at 02:56:15 UTC; Aldrin joined him 19 minutes later. They spent about two and a quarter hours together outside the spacecraft, and collected 47.5 pounds (21.5 kg) of lunar material to bring back to Earth. Command module pilot Michael Collins flew the command module Columbia alone in lunar orbit while they were on the Moon's surface. Armstrong and Aldrin spent 21.5 hours on the lunar surface before rejoining Columbia in lunar orbit.

Apollo 11 was launched by a Saturn V rocket from Kennedy Space Center on Merritt Island, Florida, on July 16 at 13:32 UTC, and was the fifth crewed mission of NASA's Apollo program. The Apollo spacecraft had three parts: a command module (CM) with a cabin for the three astronauts, and the only part that returned to Earth; a service module (SM), which supported the command module with propulsion, electrical power, oxygen, and water; and a lunar module (LM) that had two stages – a descent stage for landing on the Moon, and an ascent stage to place the astronauts back into lunar orbit.

After being sent to the Moon by the Saturn V's third stage, the astronauts separated the spacecraft from it and traveled for three days until they entered lunar orbit. Armstrong and Aldrin then moved into Eagle and landed in the Sea of Tranquillity. The astronauts used Eagle's ascent stage to lift off from the lunar surface and rejoin Collins in the command module. They jettisoned Eagle before they performed the maneuvers that blasted them out of lunar orbit on a trajectory back to Earth. They returned to Earth and splashed down in the Pacific Ocean on July 24 after more than eight days in space.

Armstrong's first step onto the lunar surface was broadcast on live TV to a worldwide audience. He described the event as "one small step for [a] man, one giant leap for mankind." Apollo 11 effectively ended the Space Race and fulfilled a national goal proposed in 1961 by President John F. Kennedy: "before this decade is out, of landing a man on the Moon and returning him safely to the Earth."

Apsis

The term apsis (Greek: ἁψίς; plural apsides , Greek: ἁψῖδες; "orbit") refers to an extreme point in the orbit of an object. It denotes either the points on the orbit, or the respective distance of the bodies. The word comes via Latin from Greek, there denoting a whole orbit, and is cognate with apse. Except for a theoretical possibility, there are two apsides in any elliptic orbit, (see discussion below, in article body); they are named with the prefixes peri- (from περί (peri), meaning 'near') and ap-, apo- (from ἀπ(ό) (ap(o)), meaning 'away from'), added in reference to the body being orbited. According to Newton's laws of motion all periodic orbits are ellipses: either the two individual ellipses of both bodies (see the two graphs in the second figure), with the center of mass (or barycenter) of this two-body system at the one common focus of the ellipses, or the orbital ellipses, with one body taken as fixed at one focus, and the other body orbiting this focus (see top figure). All these ellipses share a straight line, the line of apsides, that contains their major axes (the greatest diameter), the foci, and the vertices, and thus also the periapsis and the apoapsis (see both figures). The major axis of the orbital ellipse (top figure) is the distance of the apsides, when taken as points on the orbit, or their sum, when taken as distances.

The major axes of the individual ellipses around the barycenter, respectively the contributions to the major axis of the orbital ellipses are inverse proportional to the masses of the bodies, i.e., a bigger mass implies a smaller axis/contribution. Only when one mass is sufficiently larger than the other, the individual ellipse of the smaller body around the barycenter comprises the individual ellipse of the larger body as shown in the second figure. For remarkable asymmetry, the barycenter of the two bodies may lie well within the bigger body, e.g., the Earth–Moon barycenter is about 75% of the way from Earth's center to its surface. If the smaller mass is negligible compared to the larger, then the orbital parameters are independent of the smaller mass (e.g. for satellites).

For generic situations, the terms pericenter and apocenter are used for naming the extreme points of orbits where the primary is not specified, (see table, top figure); periapsis and apoapsis (or apapsis) are equivalent alternatives, but these terms, frequently, also refer to distances, i.e., the smallest and largest distances between the orbiter and its host body, (see second figure).

For a body orbiting the Sun, the point of least distance is the perihelion (), and the point of greatest distance is the aphelion (); when discussing orbits around other stars the terms become periastron and apastron.

When discussing a satellite of Earth, including the Moon, the point of least distance is the perigee (), and of greatest distance, the apogee, (from Ancient Greek Γῆ (Gē), "land" or "earth").

There are no natural satellites of the Moon: for man-made objects in lunar orbit, the point of least distance may be called the pericynthion () and the greatest distance the apocynthion (); or perilune and apolune are sometimes used.In orbital mechanics, the apsides technically refer to the distance measured between the barycenters of the central body and orbiting body. However, in the case of a spacecraft, the terms are commonly used to refer to the orbital altitude of the spacecraft above the surface of the central body (assuming a constant, standard reference radius).

Communications satellite

A communications satellite is an artificial satellite that relays and amplifies radio telecommunications signals via a transponder; it creates a communication channel between a source transmitter and a receiver at different locations on Earth. Communications satellites are used for television, telephone, radio, internet, and military applications. There are 2,134 communications satellites in Earth’s orbit, used by both private and government organizations. Many are in geostationary orbit 22,200 miles (35,700 km) above the equator, so that the satellite appears stationary at the same point in the sky, so the satellite dish antennas of ground stations can be aimed permanently at that spot and do not have to move to track it.

The high frequency radio waves used for telecommunications links travel by line of sight and so are obstructed by the curve of the Earth. The purpose of communications satellites is to relay the signal around the curve of the Earth allowing communication between widely separated geographical points. Communications satellites use a wide range of radio and microwave frequencies. To avoid signal interference, international organizations have regulations for which frequency ranges or "bands" certain organizations are allowed to use. This allocation of bands minimizes the risk of signal interference.

Geocentric orbit

A geocentric orbit or Earth orbit involves any object orbiting Planet Earth, such as the Moon or artificial satellites. In 1997 NASA estimated there were approximately 2,465 artificial satellite payloads orbiting the Earth and 6,216 pieces of space debris as tracked by the Goddard Space Flight Center. Over 16,291 previously launched objects have decayed into the Earth's atmosphere.A spacecraft enters orbit when its centripetal acceleration due to gravity is less than or equal to the centrifugal acceleration due to the horizontal component of its velocity. For a low Earth orbit, this velocity is about 7,800 m/s (28,100 km/h; 17,400 mph); by contrast, the fastest manned airplane speed ever achieved (excluding speeds achieved by deorbiting spacecraft) was 2,200 m/s (7,900 km/h; 4,900 mph) in 1967 by the North American X-15. The energy required to reach Earth orbital velocity at an altitude of 600 km (370 mi) is about 36 MJ/kg, which is six times the energy needed merely to climb to the corresponding altitude.Spacecraft with a perigee below about 2,000 km (1,200 mi) are subject to drag from the Earth's atmosphere, which decreases the orbital altitude. The rate of orbital decay depends on the satellite's cross-sectional area and mass, as well as variations in the air density of the upper atmosphere. Below about 300 km (190 mi), decay becomes more rapid with lifetimes measured in days. Once a satellite descends to 180 km (110 mi), it has only hours before it vaporizes in the atmosphere. The escape velocity required to pull free of Earth's gravitational field altogether and move into interplanetary space is about 11,200 m/s (40,300 km/h; 25,100 mph).

Geostationary orbit

A geostationary orbit, often referred to as a geosynchronous equatorial orbit (GEO), is a circular geosynchronous orbit 35,786 km (22,236 mi) above Earth's equator and following the direction of Earth's rotation. An object in such an orbit appears motionless, at a fixed position in the sky, to ground observers. Communications satellites and weather satellites are often placed in geostationary orbits, so that the satellite antennae (located on Earth) that communicate with them do not have to rotate to track them, but can be pointed permanently at the position in the sky where the satellites are located. Using this characteristic, ocean-color monitoring satellites with visible and near-infrared light sensors (e.g. GOCI) can also be operated in geostationary orbit in order to monitor sensitive changes of ocean environments.

A geostationary orbit is a particular type of geosynchronous orbit, which has an orbital period equal to Earth's rotational period, or one sidereal day (23 hours, 56 minutes, 4 seconds). Thus, the distinction is that, while an object in geosynchronous orbit returns to the same point in the sky at the same time each day, an object in geostationary orbit never leaves that position. Geosynchronous orbits move around relative to a point on Earth's surface because, while geostationary orbits have an inclination of 0° with respect to the Equator, geosynchronous orbits have varying inclinations and eccentricities.

Geosynchronous orbit

A geosynchronous orbit (sometimes abbreviated GSO) is an orbit around Earth of a satellite with an orbital period that matches Earth's rotation on its axis, which takes one sidereal day (about 23 hours, 56 minutes, and 4 seconds). The synchronization of rotation and orbital period means that, for an observer on Earth's surface, an object in geosynchronous orbit returns to exactly the same position in the sky after a period of one sidereal day. Over the course of a day, the object's position in the sky traces out a path, typically in a figure-8 form, whose precise characteristics depend on the orbit's inclination and eccentricity. Satellites are typically launched in an eastward direction. A circular geosynchronous orbit is 35,786 km (22,236 mi) above the Earth's surface. Those closer to Earth orbit faster than Earth rotates, so from Earth, they appear to move eastward while those that orbit beyond geosynchronous distances appear to move westward.

A special case of geosynchronous orbit is the geostationary orbit, which is a circular geosynchronous orbit in Earth's equatorial plane (that is, directly above the Equator). A satellite in a geostationary orbit appears stationary, always at the same point in the sky, to observers on the surface. Popularly or loosely, the term geosynchronous may be used to mean geostationary. Specifically, geosynchronous Earth orbit (GEO) may be a synonym for geosynchronous equatorial orbit, or geostationary Earth orbit. Communications satellites are often given geostationary or close to geostationary orbits so that the satellite antennas that communicate with them do not have to move, but can be pointed permanently at the fixed location in the sky where the satellite appears.

A semi-synchronous orbit has an orbital period of half a sidereal day (i.e., 11 hours and 58 minutes). Relative to Earth's surface, it has twice this period and hence appears to go around Earth once every day. Examples include the Molniya orbit and the orbits of the satellites in the Global Positioning System.

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, but two-and-a-half times that of all the other planets in the Solar System combined. 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. It is named after the Roman god Jupiter. When viewed from Earth, Jupiter can reach an apparent magnitude of −2.94, bright enough for its reflected light to cast shadows, and making it on average the third-brightest natural object in the night sky after the Moon and Venus.

Jupiter is primarily composed of hydrogen with a quarter of its mass being helium, though helium comprises only about a tenth of the number of molecules. 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 rapid rotation, the planet's shape is that of an oblate spheroid (it has a slight but noticeable bulge around the equator). The outer atmosphere is visibly segregated into several bands at different latitudes, resulting in turbulence and storms along their interacting boundaries. 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 79 known moons, including the four large Galilean moons discovered by Galileo Galilei in 1610. Ganymede, the largest of these, has a diameter 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, which used Jupiter's gravity to increase its speed and bend its trajectory en route to Pluto. 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.

Lagrangian point

In celestial mechanics, the Lagrangian points ( also Lagrange points, L-points, or libration points) are the points near two large bodies in orbit where a smaller object will maintain its position relative to the large orbiting bodies. At other locations, a small object would go into its own orbit around one of the large bodies, but at the Lagrangian points the gravitational forces of the two large bodies, the centripetal force of orbital motion, and (for certain points) the Coriolis acceleration all match up in a way that cause the small object to maintain a stable or nearly stable position relative to the large bodies.

There are five such points, labeled L1 to L5, all in the orbital plane of the two large bodies, for each given combination of two orbital bodies. For instance, there are five Lagrangian points L1 to L5 for the Sun-Earth system, and in a similar way there are five different Lagrangian points for the Earth-Moon system. L1, L2, and L3 are on the line through the centers of the two large bodies. L4 and L5 each form an equilateral triangle with the centers of the large bodies. L4 and L5 are stable, which implies that objects can orbit around them in a rotating coordinate system tied to the two large bodies.

Several planets have trojan satellites near their L4 and L5 points with respect to the Sun. Jupiter has more than a million of these trojans. Artificial satellites have been placed at L1 and L2 with respect to the Sun and Earth, and with respect to the Earth and the Moon. The Lagrangian points have been proposed for uses in space exploration.

Low Earth orbit

A Low Earth Orbit (LEO) is an Earth-centered orbit with an altitude of 2,000 km (1,200 mi) or less (approximately one third of the radius of Earth), or with at least 11.25 periods per day (an orbital period of 128 minutes or less) and an eccentricity less than 0.25. Most of the manmade objects in space are in LEO. A histogram of the mean motion of the cataloged objects shows that the number of objects drops significantly beyond 11.25.There is a large variety of other sources that define LEO in terms of altitude. The altitude of an object in an elliptic orbit can vary significantly along the orbit. Even for circular orbits, the altitude above ground can vary by as much as 30 km (19 mi) (especially for polar orbits) due to the oblateness of Earth's spheroid figure and local topography. While definitions based on altitude are inherently ambiguous, most of them fall within the range specified by an orbit period of 128 minutes because, according to Kepler's third law, this corresponds to a semi-major axis of 8,413 km (5,228 mi). For circular orbits, this in turn corresponds to an altitude of 2,042 km (1,269 mi) above the mean radius of Earth, which is consistent with some of the upper altitude limits in some LEO definitions.

The LEO region is defined by some sources as the region in space that LEO orbits occupy. Some highly elliptical orbits may pass through the LEO region near their lowest altitude (or perigee) but are not in an LEO Orbit because their highest altitude (or apogee) exceeds 2,000 km (1,200 mi). Sub-orbital objects can also reach the LEO region but are not in an LEO orbit because they re-enter the atmosphere. The distinction between LEO orbits and the LEO region is especially important for analysis of possible collisions between objects which may not themselves be in LEO but could collide with satellites or debris in LEO orbits.

The International Space Station conducts operations in LEO. All crewed space stations to date, as well as the majority of satellites, have been in LEO. The altitude record for human spaceflights in LEO was Gemini 11 with an apogee of 1,374.1 km (853.8 mi). Apollo 8 was the first mission to carry humans beyond LEO on December 21–27, 1968. The Apollo program continued during the four-year period spanning 1968 through 1972 with 24 astronauts who flew lunar flights but since then there have been no human spaceflights beyond LEO.

Mercury (planet)

Mercury is the smallest and innermost planet in the Solar System. Its orbital period around the Sun of 87.97 days is the shortest of all the planets in the Solar System. It is named after the Roman deity Mercury, the messenger of the gods.

Like Venus, Mercury orbits the Sun within Earth's orbit as an inferior planet, and never exceeds 28° away from the Sun when viewed from Earth. This proximity to the Sun means the planet can only be seen near the western or eastern horizon during the early evening or early morning. At this time it may appear as a bright star-like object, but is often far more difficult to observe than Venus. The planet telescopically displays the complete range of phases, similar to Venus and the Moon, as it moves in its inner orbit relative to Earth, which reoccurs over the so-called synodic period approximately every 116 days.

Mercury is tidally locked with the Sun in a 3:2 spin-orbit resonance, and rotates in a way that is unique in the Solar System. 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, it appears to rotate only once every two Mercurian years. An observer on Mercury would therefore see only one day every two Mercurian years.

Mercury's axis has the smallest tilt of any of the Solar System's planets (about ​1⁄30 degree). Its orbital eccentricity is the largest of all known planets in the Solar System; at perihelion, Mercury's distance from the Sun is only about two-thirds (or 66%) of its distance at aphelion. Mercury's surface appears heavily cratered and is similar in appearance to the Moon's, indicating that it has been geologically inactive for billions of years. Having almost no atmosphere to retain heat, it has surface temperatures that vary diurnally more than on any other planet in the Solar System, ranging from 100 K (−173 °C; −280 °F) at night to 700 K (427 °C; 800 °F) during the day across the equatorial regions. The polar regions are constantly below 180 K (−93 °C; −136 °F). The planet has no known natural satellites.

Two spacecraft have visited Mercury: Mariner 10 flew by in 1974 and 1975; and MESSENGER, launched in 2004, orbited Mercury over 4,000 times in four years before exhausting its fuel and crashing into the planet's surface on April 30, 2015. The BepiColombo spacecraft is planned to arrive at Mercury in 2025.

Moon landing

A Moon landing is the arrival of a spacecraft on the surface of the Moon. This includes both manned and unmanned (robotic) missions. The first human-made object to reach the surface of the Moon was the Soviet Union's Luna 2 mission, on 13 September 1959.The United States' Apollo 11 was the first manned mission to land on the Moon, on 20 July 1969. There have been six manned U.S. landings (between 1969 and 1972) and numerous unmanned landings, with no soft landings happening from 22 August 1976 until 14 December 2013.

To date, the United States is the only country to have successfully conducted manned missions to the Moon, with the last departing the lunar surface in December 1972. All manned and unmanned soft landings had taken place on the near side of the Moon, until 3 January 2019 when the Chinese Chang'e 4 spacecraft made the first landing on the far side of the Moon.

Orbital eccentricity

The orbital eccentricity of an astronomical object is a parameter that determines the amount by which its orbit around another body deviates from a perfect circle. A value of 0 is a circular orbit, values between 0 and 1 form an elliptic orbit, 1 is a parabolic escape orbit, and greater than 1 is a hyperbola. The term derives its name from the parameters of conic sections, as every Kepler orbit is a conic section. It is normally used for the isolated two-body problem, but extensions exist for objects following a Klemperer rosette orbit through the galaxy.

Orbital inclination

Orbital inclination measures the tilt of an object's orbit around a celestial body. It is expressed as the angle between a reference plane and the orbital plane or axis of direction of the orbiting object.

For a satellite orbiting the Earth directly above the equator, the plane of the satellite's orbit is the same as the Earth's equatorial plane, and the satellite's orbital inclination is 0°. The general case for a circular orbit is that it is tilted, spending half an orbit over the northern hemisphere and half over the southern. If the orbit swung between 20° north latitude and 20° south latitude, then its orbital inclination would be 20°.

Orbital period

The orbital period is the time a given astronomical object takes to complete one orbit around another object, and applies in astronomy usually to planets or asteroids orbiting the Sun, moons orbiting planets, exoplanets orbiting other stars, or binary stars.

For objects in the Solar System, this is often referred to as the sidereal period, determined by a 360° revolution of one celestial body around another, e.g. the Earth orbiting the Sun. The name sidereal is added as it implies that the object returns to the same position relative to the fixed stars projected in the sky. When describing orbits of binary stars, the orbital period is usually referred to as just the period. For example, Jupiter has a sidereal period of 11.86 years while the main binary star Alpha Centauri AB has a period of about 79.91 years.

Another important orbital period definition can refer to the repeated cycles for celestial bodies as observed from the Earth's surface. An example is the so-called synodic period, applying to the elapsed time where planets return to the same kind of phenomena or location. For example, when any planet returns between its consecutive observed conjunctions with or oppositions to the Sun. For example, Jupiter has a synodic period of 398.8 days from Earth; thus, Jupiter's opposition occurs once roughly every 13 months.

Periods in astronomy are conveniently expressed in various units of time, often in hours, days, or years. They can be also defined under different specific astronomical definitions that are mostly caused by small complex eternal gravitational influences by other celestial objects. Such variations also include the true placement of the centre of gravity between two astronomical bodies (barycenter), perturbations by other planets or bodies, orbital resonance, general relativity, etc. Most are investigated by detailed complex astronomical theories using celestial mechanics using precise positional observations of celestial objects via astrometry.

Planet Nine

Planet Nine is a hypothetical planet in the outer region of the Solar System. Its gravitational effects could explain the unusual clustering of orbits for a group of extreme trans-Neptunian objects (eTNOs), bodies beyond Neptune that orbit the Sun at distances averaging more than 250 times that of the Earth. These eTNOs tend to make their closest approaches to the Sun in one sector, and their orbits are similarly tilted. These improbable alignments suggest that an undiscovered planet may be shepherding the orbits of the most distant known Solar System objects.This undiscovered super-Earth-sized planet would have a predicted mass of five to ten times that of the Earth, and an elongated orbit 400 to 800 times as far from the Sun as the Earth. Konstantin Batygin and Michael E. Brown suggest that Planet Nine could be the core of a giant planet that was ejected from its original orbit by Jupiter during the genesis of the Solar System. Others propose that the planet was captured from another star, was once a rogue planet, or that it formed on a distant orbit and was pulled into an eccentric orbit by a passing star.As of the end of 2018, no observation of Planet Nine had been announced. While sky surveys such as Wide-field Infrared Survey Explorer (WISE) and Pan-STARRS did not detect Planet Nine, they have not ruled out the existence of a Neptune-diameter object in the outer Solar System. The ability of these past sky surveys to detect Planet Nine were dependent on its location and characteristics. Further surveys of the remaining regions are ongoing using NEOWISE and the 8-meter Subaru Telescope. Unless Planet Nine is observed, its existence is purely conjectural. Several alternative theories have been proposed to explain the observed clustering of TNOs.

Pluto

Pluto (minor planet designation: 134340 Pluto) is a dwarf planet in the Kuiper belt, a ring of bodies beyond Neptune. It was the first Kuiper belt object to be discovered and is the largest known plutoid (or ice dwarf).

Pluto was discovered by Clyde Tombaugh in 1930 and was originally considered to be the ninth planet from the Sun. After 1992, its status as a planet was questioned following the discovery of several objects of similar size in the Kuiper belt. In 2005, Eris, a dwarf planet in the scattered disc which is 27% more massive than Pluto, was discovered. This led the International Astronomical Union (IAU) to define the term "planet" formally in 2006, during their 26th General Assembly. That definition excluded Pluto and reclassified it as a dwarf planet.

Pluto is the largest and second-most-massive (after Eris) known dwarf planet in the Solar System, and the ninth-largest and tenth-most-massive known object directly orbiting the Sun. It is the largest known trans-Neptunian object by volume but is less massive than Eris. Like other Kuiper belt objects, Pluto is primarily made of ice and rock and is relatively small—about one-sixth the mass of the Moon and one-third its volume. It has a moderately eccentric and inclined orbit during which it ranges from 30 to 49 astronomical units or AU (4.4–7.4 billion km) from the Sun. This means that Pluto periodically comes closer to the Sun than Neptune, but a stable orbital resonance with Neptune prevents them from colliding. Light from the Sun takes about 5.5 hours to reach Pluto at its average distance (39.5 AU).

Pluto has five known moons: Charon (the largest, with a diameter just over half that of Pluto), Styx, Nix, Kerberos, and Hydra. Pluto and Charon are sometimes considered a binary system because the barycenter of their orbits does not lie within either body.

The New Horizons spacecraft performed a flyby of Pluto on July 14, 2015, becoming the first ever spacecraft to do so. During its brief flyby, New Horizons made detailed measurements and observations of Pluto and its moons. In September 2016, astronomers announced that the reddish-brown cap of the north pole of Charon is composed of tholins, organic macromolecules that may be ingredients for the emergence of life, and produced from methane, nitrogen and other gases released from the atmosphere of Pluto and transferred about 19,000 km (12,000 mi) to the orbiting moon.

Satellite

In the context of spaceflight, a satellite is an artificial object which has been intentionally placed into orbit. Such objects are sometimes called artificial satellites to distinguish them from natural satellites such as Earth's Moon.

On 4 October 1957 the Soviet Union launched the world's first artificial satellite, Sputnik 1. Since then, about 8,100 satellites from more than 40 countries have been launched. According to a 2018 estimate, some 4,900 remain in orbit, of those about 1,900 were operational; while the rest have lived out their useful lives and become space debris. Approximately 500 operational satellites are in low-Earth orbit, 50 are in medium-Earth orbit (at 20,000 km), and the rest are in geostationary orbit (at 36,000 km). A few large satellites have been launched in parts and assembled in orbit. Over a dozen space probes have been placed into orbit around other bodies and become artificial satellites to the Moon, Mercury, Venus, Mars, Jupiter, Saturn, a few asteroids, a comet and the Sun. On 18 April 2019 Nepal successfully launched its own satellite with the help of NASA.

Satellites are used for many purposes. Among several other applications, they can be used to make star maps and maps of planetary surfaces, and also take pictures of planets they are launched into. Common types include military and civilian Earth observation satellites, communications satellites, navigation satellites, weather satellites, and space telescopes. Space stations and human spacecraft in orbit are also satellites. Satellite orbits vary greatly, depending on the purpose of the satellite, and are classified in a number of ways. Well-known (overlapping) classes include low Earth orbit, polar orbit, and geostationary orbit.

A launch vehicle is a rocket that places a satellite into orbit. Usually, it lifts off from a launch pad on land. Some are launched at sea from a submarine or a mobile maritime platform, or aboard a plane (see air launch to orbit).

Satellites are usually semi-independent computer-controlled systems. Satellite subsystems attend many tasks, such as power generation, thermal control, telemetry, attitude control and orbit control.

Solar System

The Solar System is the gravitationally bound planetary system of the Sun and the objects that orbit it, either directly or indirectly. Of the objects that orbit the Sun directly, the largest are the eight planets, with the remainder being smaller objects, such as the five dwarf planets and small Solar System bodies. Of the objects that orbit the Sun indirectly—the moons—two are larger than the smallest planet, Mercury.The Solar System formed 4.6 billion years ago from the gravitational collapse of a giant interstellar molecular cloud. The vast majority of the system's mass is in the Sun, with the majority of the remaining mass contained in Jupiter. The four smaller inner planets, Mercury, Venus, Earth and Mars, are terrestrial planets, being primarily composed of rock and metal. The four outer planets are giant planets, being substantially more massive than the terrestrials. The two largest, Jupiter and Saturn, are gas giants, being composed mainly of hydrogen and helium; the two outermost planets, Uranus and Neptune, are ice giants, being composed mostly of substances with relatively high melting points compared with hydrogen and helium, called volatiles, such as water, ammonia and methane. All eight planets have almost circular orbits that lie within a nearly flat disc called the ecliptic.

The Solar System also contains smaller objects. The asteroid belt, which lies between the orbits of Mars and Jupiter, mostly contains objects composed, like the terrestrial planets, of rock and metal. Beyond Neptune's orbit lie the Kuiper belt and scattered disc, which are populations of trans-Neptunian objects composed mostly of ices, and beyond them a newly discovered population of sednoids. Within these populations are several dozen to possibly tens of thousands of objects large enough that they have been rounded by their own gravity. Such objects are categorized as dwarf planets. Identified dwarf planets include the asteroid Ceres and the trans-Neptunian objects Pluto and Eris. In addition to these two regions, various other small-body populations, including comets, centaurs and interplanetary dust clouds, freely travel between regions. Six of the planets, at least four of the dwarf planets, and many of the smaller bodies are orbited by natural satellites, usually termed "moons" after the Moon. Each of the outer planets is encircled by planetary rings of dust and other small objects.

The solar wind, a stream of charged particles flowing outwards from the Sun, creates a bubble-like region in the interstellar medium known as the heliosphere. The heliopause is the point at which pressure from the solar wind is equal to the opposing pressure of the interstellar medium; it extends out to the edge of the scattered disc. The Oort cloud, which is thought to be the source for long-period comets, may also exist at a distance roughly a thousand times further than the heliosphere. The Solar System is located in the Orion Arm, 26,000 light-years from the center of the Milky Way galaxy.

Sun

The Sun is the star at the center of the Solar System. It is a nearly perfect sphere of hot plasma, with internal convective motion that generates a magnetic field via a dynamo process. It is by far the most important source of energy for life on Earth. Its diameter is about 1.39 million kilometers (864,000 miles), or 109 times that of Earth, and its mass is about 330,000 times that of Earth. It accounts for about 99.86% of the total mass of the Solar System.

Roughly three quarters of the Sun's mass consists of hydrogen (~73%); the rest is mostly helium (~25%), with much smaller quantities of heavier elements, including oxygen, carbon, neon, and iron.The Sun is a G-type main-sequence star (G2V) based on its spectral class. As such, it is informally and not completely accurately referred to as a yellow dwarf (its light is closer to white than yellow). It formed approximately 4.6 billion years ago from the gravitational collapse of matter within a region of a large molecular cloud. Most of this matter gathered in the center, whereas the rest flattened into an orbiting disk that became the Solar System. The central mass became so hot and dense that it eventually initiated nuclear fusion in its core. It is thought that almost all stars form by this process.

The Sun is roughly middle-aged; it has not changed dramatically for more than four billion years, and will remain fairly stable for more than another five billion years. It currently fuses about 600 million tons of hydrogen into helium every second, converting 4 million tons of matter into energy every second as a result. This energy, which can take between 10,000 and 170,000 years to escape from its core, is the source of the Sun's light and heat. In about 5 billion years, when hydrogen fusion in its core has diminished to the point at which the Sun is no longer in hydrostatic equilibrium, its core will undergo a marked increase in density and temperature while its outer layers expand to eventually become a red giant. It is calculated that the Sun will become sufficiently large to engulf the current orbits of Mercury and Venus, and render Earth uninhabitable. After this, it will shed its outer layers and become a dense type of cooling star known as a white dwarf, and no longer produce energy by fusion, but still glow and give off heat from its previous fusion.

The enormous effect of the Sun on Earth has been recognized since prehistoric times, and the Sun has been regarded by some cultures as a deity. The synodic rotation of Earth and its orbit around the Sun are the basis of solar calendars, one of which is the predominant calendar in use today.

Gravitational orbits
Types
Parameters
Maneuvers
Orbital mechanics

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