Orbit

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.

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. Joseph-Louis Lagrange 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 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.

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 that 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 the sum of those two energies 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's cannonball, an illustration of how objects can "fall" in a curve

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 and the distance of the object from the barycenter, 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:

No orbit

Suborbital trajectories

Range of interrupted elliptical paths

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

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.

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

${\displaystyle F_{2}=-{\frac {Gm_{1}m_{2}}{r^{2}}}}$

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 body's acceleration:

${\displaystyle F_{2}=m_{2}A_{2}}$

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:

${\displaystyle -{\frac {Gm_{1}m_{2}}{r^{2}}}=m_{2}A_{2}}$

Solving for the acceleration, A₂:

${\displaystyle A_{2}={\frac {F_{2}}{m_{2}}}=-{\frac {1}{m_{2}}}{\frac {Gm_{1}m_{2}}{r^{2}}}=-{\frac {\mu }{r^{2}}}}$

where ${\displaystyle \mu \,}$ is the standard gravitational parameter, in this case ${\displaystyle Gm_{1}}$. 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 ${\displaystyle A=F/m=-kr.}$ Due to the way vectors add, the component of the force in the ${\displaystyle {\hat {\mathbf {x} }}}$ or in the ${\displaystyle {\hat {\mathbf {y} }}}$ directions are also proportionate to the respective components of the distances, ${\displaystyle r''_{x}=A_{x}=-kr_{x}}$. Hence, the entire analysis can be done separately in these dimensions. This results in the harmonic parabolic equations ${\displaystyle x=A\cos(t)}$ and ${\displaystyle y=B\sin(t)}$ of the ellipse. In contrast, with the decreasing relationship ${\displaystyle A=\mu /r^{2}}$, the dimensions cannot be separated.

The location of the orbiting object at the current time ${\displaystyle t}$ 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 ${\displaystyle r}$ be the distance between the object and the center and ${\displaystyle \theta }$ be the angle it has rotated. Let ${\displaystyle {\hat {\mathbf {x} }}}$ and ${\displaystyle {\hat {\mathbf {y} }}}$ be the standard Euclidean bases and let ${\displaystyle {\hat {\mathbf {r} }}=\cos(\theta ){\hat {\mathbf {x} }}+\sin(\theta ){\hat {\mathbf {y} }}}$ and ${\displaystyle {\hat {\boldsymbol {\theta }}}=-\sin(\theta ){\hat {\mathbf {x} }}+\cos(\theta ){\hat {\mathbf {y} }}}$ 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

${\displaystyle {\hat {\mathbf {O} }}=r\cos(\theta ){\hat {\mathbf {x} }}+r\sin(\theta ){\hat {\mathbf {y} }}=r{\hat {\mathbf {r} }}}$

We use ${\displaystyle {\dot {r}}}$ and ${\displaystyle {\dot {\theta }}}$ 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 ${\displaystyle t}$ from that at time ${\displaystyle t+\delta t}$ and dividing by ${\displaystyle \delta t}$. The result is also a vector. Because our basis vector ${\displaystyle {\hat {\mathbf {r} }}}$ moves as the object orbits, we start by differentiating it. From time ${\displaystyle t}$ to ${\displaystyle t+\delta t}$, the vector ${\displaystyle {\hat {\mathbf {r} }}}$ keeps its beginning at the origin and rotates from angle ${\displaystyle \theta }$ to ${\displaystyle \theta +{\dot {\theta }}\ \delta t}$ which moves its head a distance ${\displaystyle {\dot {\theta }}\ \delta t}$ in the perpendicular direction ${\displaystyle {\hat {\boldsymbol {\theta }}}}$ giving a derivative of ${\displaystyle {\dot {\theta }}{\hat {\boldsymbol {\theta }}}}$.

${\displaystyle {\begin{aligned}{\hat {\mathbf {r} }}&=\cos(\theta ){\hat {\mathbf {x} }}+\sin(\theta ){\hat {\mathbf {y} }}\\{\frac {\delta {\hat {\mathbf {r} }}}{\delta t}}={\dot {\mathbf {r} }}&=-\sin(\theta ){\dot {\theta }}{\hat {\mathbf {x} }}+\cos(\theta ){\dot {\theta }}{\hat {\mathbf {y} }}={\dot {\theta }}{\hat {\boldsymbol {\theta }}}\\{\hat {\boldsymbol {\theta }}}&=-\sin(\theta ){\hat {\mathbf {x} }}+\cos(\theta ){\hat {\mathbf {y} }}\\{\frac {\delta {\hat {\boldsymbol {\theta }}}}{\delta t}}={\dot {\boldsymbol {\theta }}}&=-\cos(\theta ){\dot {\theta }}{\hat {\mathbf {x} }}-\sin(\theta ){\dot {\theta }}{\hat {\mathbf {y} }}=-{\dot {\theta }}{\hat {\mathbf {r} }}\end{aligned}}}$

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

${\displaystyle {\begin{aligned}{\hat {\mathbf {O} }}&=r{\hat {\mathbf {r} }}\\{\dot {\mathbf {O} }}&={\frac {\delta r}{\delta t}}{\hat {\mathbf {r} }}+r{\frac {\delta {\hat {\mathbf {r} }}}{\delta t}}={\dot {r}}{\hat {\mathbf {r} }}+r\left[{\dot {\theta }}{\hat {\boldsymbol {\theta }}}\right]\\{\ddot {\mathbf {O} }}&=\left[{\ddot {r}}{\hat {\mathbf {r} }}+{\dot {r}}{\dot {\theta }}{\hat {\boldsymbol {\theta }}}\right]+\left[{\dot {r}}{\dot {\theta }}{\hat {\boldsymbol {\theta }}}+r{\ddot {\theta }}{\hat {\boldsymbol {\theta }}}-r{\dot {\theta }}^{2}{\hat {\mathbf {r} }}\right]\\&=\left[{\ddot {r}}-r{\dot {\theta }}^{2}\right]{\hat {\mathbf {r} }}+\left[r{\ddot {\theta }}+2{\dot {r}}{\dot {\theta }}\right]{\hat {\boldsymbol {\theta }}}\end{aligned}}}$

The coefficients of ${\displaystyle {\hat {\mathbf {r} }}}$ and ${\displaystyle {\hat {\boldsymbol {\theta }}}}$ give the accelerations in the radial and transverse directions. As said, Newton gives this first due to gravity is ${\displaystyle -\mu /r^{2}}$ and the second is zero.

${\displaystyle {\ddot {r}}-r{\dot {\theta }}^{2}=-{\frac {\mu }{r^{2}}}}$

(1)

${\displaystyle r{\ddot {\theta }}+2{\dot {r}}{\dot {\theta }}=0}$

(2)

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

${\displaystyle r{\ddot {\theta }}+2{\dot {r}}{\dot {\theta }}={\frac {1}{r}}{\frac {d}{dt}}\left(r^{2}{\dot {\theta }}\right)=0}$

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

${\displaystyle r^{2}{\dot {\theta }}=h}$

(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 ${\displaystyle r}$ of the orbiting object from the center as a function of its angle ${\displaystyle \theta }$. However, it is easier to introduce the auxiliary variable ${\displaystyle u=1/r}$ and to express ${\displaystyle u}$ as a function of ${\displaystyle \theta }$. Derivatives of ${\displaystyle r}$ with respect to time may be rewritten as derivatives of ${\displaystyle u}$ with respect to angle.

${\displaystyle u={1 \over r}}$

${\displaystyle {\dot {\theta }}={\frac {h}{r^{2}}}=hu^{2}}$ (reworking (3))

${\displaystyle {\begin{aligned}{\frac {\delta u}{\delta \theta }}&={\frac {\delta }{\delta t}}\left({\frac {1}{r}}\right){\frac {\delta t}{\delta \theta }}=-{\frac {\dot {r}}{r^{2}{\dot {\theta }}}}=-{\frac {\dot {r}}{h}}\\{\frac {\delta ^{2}u}{\delta \theta ^{2}}}&=-{\frac {1}{h}}{\frac {\delta {\dot {r}}}{\delta t}}{\frac {\delta t}{\delta \theta }}=-{\frac {\ddot {r}}{h{\dot {\theta }}}}=-{\frac {\ddot {r}}{h^{2}u^{2}}}\ \ \ {\text{ or }}\ \ \ {\ddot {r}}=-h^{2}u^{2}{\frac {\delta ^{2}u}{\delta \theta ^{2}}}\end{aligned}}}$

Plugging these into (1) gives

${\displaystyle {\begin{aligned}{\ddot {r}}-r{\dot {\theta }}^{2}&=-{\frac {\mu }{r^{2}}}\\-h^{2}u^{2}{\frac {\delta ^{2}u}{\delta \theta ^{2}}}-{\frac {1}{u}}\left(hu^{2}\right)^{2}&=-\mu u^{2}\end{aligned}}}$

${\displaystyle {\frac {\delta ^{2}u}{\delta \theta ^{2}}}+u={\frac {\mu }{h^{2}}}}$

(4)

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:

${\displaystyle u(\theta )={\frac {\mu }{h^{2}}}-A\cos(\theta -\theta _{0})}$

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 ${\displaystyle e\equiv h^{2}A/\mu }$ be the eccentricity, letting ${\displaystyle a\equiv h^{2}/\mu \left(1-e^{2}\right)}$ be the semi-major axis. Finally, letting ${\displaystyle \theta _{0}\equiv 0}$ so the long axis of the ellipse is along the positive x coordinate.

${\displaystyle r(\theta )={\frac {a\left(1-e^{2}\right)}{1+e\cos \theta }}}$

When the two-body system is under the influence of torque, the angular momentum h is not a constant. After the following calculation:

${\displaystyle {\begin{aligned}{\frac {\delta r}{\delta \theta }}&=-{\frac {1}{u^{2}}}{\frac {\delta u}{\delta \theta }}=-{\frac {h}{m}}{\frac {\delta u}{\delta \theta }}\\{\frac {\delta ^{2}r}{\delta \theta ^{2}}}&=-{\frac {h^{2}u^{2}}{m^{2}}}{\frac {\delta ^{2}u}{\delta \theta ^{2}}}-{\frac {hu^{2}}{m^{2}}}{\frac {\delta h}{\delta \theta }}{\frac {\delta u}{\delta \theta }}\\\left({\frac {\delta \theta }{\delta t}}\right)^{2}r&={\frac {h^{2}u^{3}}{m^{2}}}\end{aligned}}}$

we will get the Sturm-Liouville equation of two-body system.[9]

${\displaystyle {\frac {\delta }{\delta \theta }}\left(h{\frac {\delta u}{\delta \theta }}\right)+hu={\frac {\mu }{h}}}$

(5)

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.[10]).

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.

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

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:

• Inclination (i)
• Longitude of the ascending node (Ω)
• Argument of periapsis (ω)
• Eccentricity (e)
• Semimajor axis (a)
• Mean anomaly at epoch (M₀).

In principle, once the orbital elements are known for a body, its position can be calculated forward and backward 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.

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.

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.[14] 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.[15]

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.[15]

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).

• Low Earth orbit (LEO): Geocentric orbits with altitudes up to 2,000 km (0–1,240 miles).[16]
• Medium Earth orbit (MEO): Geocentric orbits ranging in altitude from 2,000 km (1,240 miles) to just below geosynchronous orbit at 35,786 kilometers (22,236 mi). Also known as an intermediate circular orbit. These are "most commonly at 20,200 kilometers (12,600 mi), or 20,650 kilometers (12,830 mi), with an orbital period of 12 hours."[17]
• Both geosynchronous orbit (GSO) and geostationary orbit (GEO) are orbits around Earth matching Earth's sidereal rotation period. All geosynchronous and geostationary orbits have a semi-major axis of 42,164 km (26,199 mi).[18] All geostationary orbits are also geosynchronous, but not all geosynchronous orbits are geostationary. A geostationary orbit stays exactly above the equator, whereas a geosynchronous orbit may swing north and south to cover more of the Earth's surface. Both complete one full orbit of Earth per sidereal day (relative to the stars, not the Sun).
• High Earth orbit: Geocentric orbits above the altitude of geosynchronous orbit 35,786 km (22,240 miles).[17]

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 and other travel times related to gravity remain the same. For example, 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 adjusting 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)

${\displaystyle GT^{2}\rho =3\pi \left({\frac {a}{r}}\right)^{3},}$

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.

The application of certain orbits or orbital maneuvers to specific useful purposes has been the subject of patents.[19]

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.

• Abell, George O.; Morrison, David & Wolff, Sidney C. (1987). Exploration of the Universe (Fifth ed.). Saunders College Publishing. ISBN 9780030051432.
• Linton, Christopher (2004). From Eudoxus to Einstein: A History of Mathematical Astronomy. Cambridge University Press. ISBN 978-1-139-45379-0.
• Milani, Andrea; Gronchi, Giovanni F. (2010). Theory of Orbit Determination. Cambridge University Press. Discusses new algorithms for determining the orbits of both natural and artificial celestial bodies.
• Swetz, Frank; Fauvel, John; Johansson, Bengt; Katz, Victor; Bekken, Otto (1995). Learn from the Masters. MAA. ISBN 978-0-88385-703-8.

Look up orbit in Wiktionary, the free dictionary.

Wikimedia Commons has media related to Orbits.

• CalcTool: Orbital period of a planet calculator. Has wide choice of units. Requires JavaScript.
• Java simulation on orbital motion. Requires Java.
• NOAA page on Climate Forcing Data includes (calculated) data on Earth orbit variations over the last 50 million years and for the coming 20 million years
• On-line orbit plotter. Requires JavaScript.
• Orbital Mechanics (Rocket and Space Technology)
• Orbital simulations by Varadi, Ghil and Runnegar (2003) provide another, slightly different series for Earth orbit eccentricity, and also a series for orbital inclination. Orbits for the other planets were also calculated, by F. Varadi; B. Runnegar; M. Ghil (2003). "Successive Refinements in Long-Term Integrations of Planetary Orbits". The Astrophysical Journal. 592 (1): 620–630. Bibcode:2003ApJ...592..620V. doi:10.1086/375560., but only the eccentricity data for Earth and Mercury are available online.
• Understand orbits using direct manipulation. Requires JavaScript and Macromedia
• Merrifield, Michael. "Orbits (including the first manned orbit)". Sixty Symbols. Brady Haran for the University of Nottingham.