Orbit
In
For most situations, orbital motion is adequately approximated by
History
Historically, the apparent motions of the planets were described by European and Arabic philosophers using the idea of
The basis for the modern understanding of orbits was first formulated by
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.
Planetary orbits
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Within a
Owing to mutual
As two objects orbit each other, the
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
Principles
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 velocityit 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.
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.
Illustration
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]
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.
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
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 the 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
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:
- The orbit of a planet around the apastron.
- As the planet moves in its orbit, the line from the Sun to the planet sweeps a constant area of the 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."
- For a given orbit, the ratio of the cube of its semi-major axisto 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
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 secular phenomena that have to be dealt with by post-Newtonianmethods.
- 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.[8]
Newtonian analysis of orbital motion
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
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:
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.
The location of the orbiting object at the current time is located in the plane using
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
In order to get an equation for the orbit from equation (1), we need to eliminate time.[9] (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
-
(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:
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
Note that by letting be the semi-major axis and letting so the long axis of the ellipse is along the positive x coordinate we yield:
When the two-body system is under the influence of torque, the angular momentum h is not a constant. After the following calculation:
we will get the Sturm-Liouville equation of two-body system.[10]
-
(5)
Relativistic orbital motion
The above classical (
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
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.
Note that, unless the eccentricity is zero, a is not the average orbital radius. The time-averaged orbital distance is given by:[12]
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
Orbital decay
If an orbit is about a planetary body with a significant atmosphere, its orbit can decay because of
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
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
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
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.[16] 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.[17]
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.[17]
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
- 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."[19]
- Both semi-major axis of 42,164 km (26,199 mi).[20]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).
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 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)
for an elliptical orbit with
Patents
The application of certain orbits or orbital maneuvers to specific useful purposes has been the subject of patents.[21]
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
- Ephemeris is a compilation of positions of naturally occurring astronomical objects as well as artificial satellites in the sky at a given time or times.
- Free drift
- Klemperer rosette
- List of orbits
- Molniya orbit
- Orbit determination
- Orbital spaceflight
- Perifocal coordinate system
- Polar orbit
- Radial trajectory
- Rosetta orbit
- VSOP model
References
- ^ "orbit (astronomy)". Encyclopædia Britannica (Online ed.). Archived from the original on 5 May 2015. Retrieved 28 July 2008.
- ^ "The Space Place :: What's a Barycenter". NASA. Archived from the original on 8 January 2013. Retrieved 26 November 2012.
- ^ Kuhn, The Copernican Revolution, pp. 238, 246–252
- ^ Encyclopædia Britannica, 1968, vol. 2, p. 645
- ^ M Caspar, Kepler (1959, Abelard-Schuman), at pp.131–140; A Koyré, The Astronomical Revolution: Copernicus, Kepler, Borelli (1973, Methuen), pp. 277–279
- about.com. Archivedfrom the original on 18 November 2016. Retrieved 1 June 2008.
- ^ See pages 6 to 8 in Newton's "Treatise of the System of the World" Archived 30 December 2016 at the Wayback Machine (written 1685, translated into English 1728, see Newton's 'Principia' – A preliminary version), for the original version of this 'cannonball' thought-experiment.
- .
- ^ 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.
- .
- ^ Pogge, Richard W.; "Real-World Relativity: The GPS Navigation System" Archived 14 November 2015 at the Wayback Machine. Retrieved 25 January 2008.
- ISBN 9780816524501.
- S2CID 118305813.
- S2CID 120030309.
- S2CID 119537102.
- arXiv:math/0011268.
- ^ a b Peterson, Ivars (23 September 2013). "Strange Orbits". Science News. Archived from the original on 22 November 2015. Retrieved 21 July 2017.
- ^ "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., pp. 37–38 (6-1,6-2); figure 6-1.
- ^ 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.
- ^ Vallado, David A. (2007). Fundamentals of Astrodynamics and Applications. Hawthorne, CA: Microcosm Press. p. 31.
- ^ Ferreira, Becky (19 February 2015). "How Satellite Companies Patent Their Orbits". Motherboard. Vice News. Archived from the original on 18 January 2017. Retrieved 20 September 2018.
Further reading
- 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.
External links
- 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, doi:10.1086/375560., but only the eccentricity data for Earth and Mercuryare available online.
- Understand orbits using direct manipulation Archived 8 November 2017 at the Wayback Machine. Requires JavaScript and Macromedia
- Merrifield, Michael. "Orbits (including the first manned orbit)". Sixty Symbols. Brady Haran for the University of Nottingham.