elliptic orbits
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In
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 ...
or
celestial mechanics Celestial mechanics is the branch of astronomy that deals with the motions of objects in outer space. Historically, celestial mechanics applies principles of physics (classical mechanics) to astronomical objects, such as stars and planets, to ...
, an elliptic orbit or elliptical orbit is a
Kepler orbit Johannes Kepler (; ; 27 December 1571 – 15 November 1630) was a German astronomer, mathematician, astrologer, natural philosopher and writer on music. He is a key figure in the 17th-century Scientific Revolution, best known for his laws ...
with an
eccentricity Eccentricity or eccentric may refer to: * Eccentricity (behavior), odd behavior on the part of a person, as opposed to being "normal" Mathematics, science and technology Mathematics * Off-Centre (geometry), center, in geometry * Eccentricity (g ...
of less than 1; this includes the special case of a circular orbit, with eccentricity equal to 0. In a stricter sense, it is a Kepler orbit with the eccentricity greater than 0 and less than 1 (thus excluding the circular orbit). In a wider sense, it is a Kepler's orbit with negative
energy In physics, energy (from Ancient Greek: ἐνέργεια, ''enérgeia'', “activity”) is the quantitative property that is transferred to a body or to a physical system, recognizable in the performance of work and in the form of hea ...
. This includes the radial elliptic orbit, with eccentricity equal to 1. In a gravitational two-body problem with
negative energy Negative energy is a concept used in physics to explain the nature of certain fields, including the gravitational field and various quantum field effects. Gravitational potential energy Gravitational potential energy can be defined as being n ...
, both bodies follow similar elliptic orbits with the same
orbital period The orbital period (also revolution period) is the amount of time a given astronomical object takes to complete one orbit around another object. In astronomy, it usually applies to planets or asteroids orbiting the Sun, moons orbiting planets ...
around their common
barycenter In astronomy, the barycenter (or barycentre; ) is the center of mass of two or more bodies that orbit one another and is the point about which the bodies orbit. A barycenter is a dynamical point, not a physical object. It is an important con ...
. Also the relative position of one body with respect to the other follows an elliptic orbit. Examples of elliptic orbits include:
Hohmann transfer orbit In astronautics, the Hohmann transfer orbit () is an orbital maneuver used to transfer a spacecraft between two orbits of different altitudes around a central body. Examples would be used for travel between low Earth orbit and the Moon, or ...
,
Molniya orbit A Molniya orbit ( rus, Молния, p=ˈmolnʲɪjə, a=Ru-молния.ogg, "Lightning") is a type of satellite orbit designed to provide communications and remote sensing coverage over high latitudes. It is a highly elliptical orbit with an ...
, and
tundra orbit A Tundra orbit (russian: орбита «Тундра») is a highly elliptical geosynchronous orbit with a high inclination (approximately 63.4°), an orbital period of one sidereal day, and a typical eccentricity between 0.2 and 0.3. A satelli ...
.


Velocity

Under standard assumptions, no other forces acting except two spherically symmetrical bodies m1 and m2, the
orbital speed In gravitationally bound systems, the orbital speed of an astronomical body or object (e.g. planet, moon, artificial satellite, spacecraft, or star) is the speed at which it orbits around either the barycenter or, if one body is much more mas ...
(v\,) of one body traveling along an elliptic orbit can be computed from the
vis-viva equation In astrodynamics, the ''vis-viva'' equation, also referred to as orbital-energy-invariance law, is one of the equations that model the motion of orbiting bodies. It is the direct result of the principle of conservation of mechanical energy which ...
as: :v = \sqrt where: *\mu\, is the standard gravitational parameter, G(m1+m2), often expressed as GM when one body is much larger than the other. *r\, is the distance between the orbiting body and center of mass. *a\,\! is the length of the semi-major axis. The velocity equation for a
hyperbolic trajectory In astrodynamics or celestial mechanics, a hyperbolic trajectory or hyperbolic orbit is the trajectory of any object around a central body with more than enough speed to escape the central object's gravitational pull. The name derives from the f ...
has either + , or it is the same with the convention that in that case ''a'' is negative.


Orbital period

Under standard assumptions the orbital period(T\,\!) of a body travelling along an elliptic orbit can be computed as: :T=2\pi\sqrt where: *\mu is the standard gravitational parameter. *a\,\! is the length of the semi-major axis. Conclusions: *The orbital period is equal to that for a circular orbit with the orbital radius equal to the semi-major axis (a\,\!), *For a given semi-major axis the orbital period does not depend on the eccentricity (See also:
Kepler's third law In astronomy, Kepler's laws of planetary motion, published by Johannes Kepler between 1609 and 1619, describe the orbits of planets around the Sun. The laws modified the heliocentric theory of Nicolaus Copernicus, replacing its circular orbi ...
).


Energy

Under standard assumptions, the
specific orbital energy In the gravitational two-body problem, the specific orbital energy \varepsilon (or vis-viva energy) of two orbiting bodies is the constant sum of their mutual potential energy (\varepsilon_p) and their total kinetic energy (\varepsilon_k), divide ...
(\epsilon) of an elliptic orbit is negative and the orbital energy conservation equation (the
Vis-viva equation In astrodynamics, the ''vis-viva'' equation, also referred to as orbital-energy-invariance law, is one of the equations that model the motion of orbiting bodies. It is the direct result of the principle of conservation of mechanical energy which ...
) for this orbit can take the form: :-=-=\epsilon<0 where: *v\, is the
orbital speed In gravitationally bound systems, the orbital speed of an astronomical body or object (e.g. planet, moon, artificial satellite, spacecraft, or star) is the speed at which it orbits around either the barycenter or, if one body is much more mas ...
of the orbiting body, *r\, is the distance of the orbiting body from the
central body A primary (also called a gravitational primary, primary body, or central body) is the main physical body of a gravitationally bound, multi-object system. This object constitutes most of that system's mass and will generally be located near the syst ...
, *a\, is the length of the semi-major axis, *\mu\, is the standard gravitational parameter. Conclusions: *For a given semi-major axis the specific orbital energy is independent of the eccentricity. Using the
virial theorem In mechanics, the virial theorem provides a general equation that relates the average over time of the total kinetic energy of a stable system of discrete particles, bound by potential forces, with that of the total potential energy of the system. ...
we find: *the time-average of the specific potential energy is equal to −2ε **the time-average of ''r''−1 is ''a''−1 *the time-average of the specific kinetic energy is equal to ε


Energy in terms of semi major axis

It can be helpful to know the energy in terms of the semi major axis (and the involved masses). The total energy of the orbit is given by :E = - G \frac, where a is the semi major axis.


Derivation

Since gravity is a central force, the angular momentum is constant: :\dot = \mathbf \times \mathbf = \mathbf \times F(r)\mathbf = 0 At the closest and furthest approaches, the angular momentum is perpendicular to the distance from the mass orbited, therefore: :L = r p = r m v. The total energy of the orbit is given by :E = \fracm v^2 - G \frac. We may substitute for v and obtain :E = \frac\frac - G \frac. This is true for r being the closest / furthest distance so we get two simultaneous equations which we solve for E: :E = - G \frac Since r_1 = a + a \epsilon and r_2 = a - a \epsilon, where epsilon is the eccentricity of the orbit, we finally have the stated result.


Flight path angle

The flight path angle is the angle between the orbiting body's velocity vector (= the vector tangent to the instantaneous orbit) and the local horizontal. Under standard assumptions of the conservation of angular momentum the flight path angle \phi satisfies the equation: :h\, = r\, v\, \cos \phi where: * h\, is the
specific relative angular momentum In celestial mechanics, the specific relative angular momentum (often denoted \vec or \mathbf) of a body is the angular momentum of that body divided by its mass. In the case of two orbiting bodies it is the vector product of their relative positi ...
of the orbit, * v\, is the
orbital speed In gravitationally bound systems, the orbital speed of an astronomical body or object (e.g. planet, moon, artificial satellite, spacecraft, or star) is the speed at which it orbits around either the barycenter or, if one body is much more mas ...
of the orbiting body, * r\, is the radial distance of the orbiting body from the
central body A primary (also called a gravitational primary, primary body, or central body) is the main physical body of a gravitationally bound, multi-object system. This object constitutes most of that system's mass and will generally be located near the syst ...
, * \phi \, is the flight path angle \psi is the angle between the orbital velocity vector and the semi-major axis. \nu is the local true anomaly. \phi = \nu + \frac - \psi, therefore, :\cos \phi = \sin(\psi - \nu) = \sin\psi\cos\nu - \cos\psi\sin\nu = \frac :\tan \phi = \frac where e is the eccentricity. The angular momentum is related to the vector cross product of position and velocity, which is proportional to the sine of the angle between these two vectors. Here \phi is defined as the angle which differs by 90 degrees from this, so the cosine appears in place of the sine.


Equation of motion


From initial position and velocity

An
orbit equation In astrodynamics, an orbit equation defines the path of orbiting body m_2\,\! around central body m_1\,\! relative to m_1\,\!, without specifying position as a function of time. Under standard assumptions, a body moving under the influence of a f ...
defines the path of an
orbiting body In astrodynamics, an orbiting body is any physical body that orbits a more massive one, called the primary body. The orbiting body is properly referred to as the secondary body (m_2), which is less massive than the primary body (m_1). Thus, m_2 ...
m_2\,\! around
central body A primary (also called a gravitational primary, primary body, or central body) is the main physical body of a gravitationally bound, multi-object system. This object constitutes most of that system's mass and will generally be located near the syst ...
m_1\,\! relative to m_1\,\!, without specifying position as a function of time. If the eccentricity is less than 1 then the equation of motion describes an elliptical orbit. Because
Kepler's equation In orbital mechanics, Kepler's equation relates various geometric properties of the orbit of a body subject to a central force. It was first derived by Johannes Kepler in 1609 in Chapter 60 of his ''Astronomia nova'', and in book V of his '' Epi ...
M = E - e \sin E has no general
closed-form solution In mathematics, a closed-form expression is a mathematical expression that uses a finite number of standard operations. It may contain constants, variables, certain well-known operations (e.g., + − × ÷), and functions (e.g., ''n''th roo ...
for the
Eccentric anomaly In orbital mechanics, the eccentric anomaly is an angular parameter that defines the position of a body that is moving along an elliptic Kepler orbit. The eccentric anomaly is one of three angular parameters ("anomalies") that define a position ...
(E) in terms of the Mean anomaly (M), equations of motion as a function of time also have no closed-form solution (although numerical solutions exist for both). However, closed-form time-independent path equations of an elliptic orbit with respect to a central body can be determined from just an initial position (\mathbf) and velocity (\mathbf). For this case it is convenient to use the following assumptions which differ somewhat from the standard assumptions above: :# The central body's position is at the origin and is the primary focus (\mathbf) of the ellipse (alternatively, the center of mass may be used instead if the orbiting body has a significant mass) :# The central body's mass (m1) is known :# The orbiting body's initial position(\mathbf) and velocity(\mathbf) are known :# The ellipse lies within the XY-plane The fourth assumption can be made without loss of generality because any three points (or vectors) must lie within a common plane. Under these assumptions the second focus (sometimes called the "empty" focus) must also lie within the XY-plane: \mathbf = \left(f_x,f_y\right) .


Using vectors

The general equation of an ellipse under these assumptions using vectors is: : , \mathbf - \mathbf, + , \mathbf, = 2a \qquad\mid z=0 where: *a\,\! is the length of the semi-major axis. *\mathbf = \left(f_x,f_y\right) is the second (“empty”) focus. *\mathbf = \left(x,y\right) is any (x,y) value satisfying the equation. The semi-major axis length (a) can be calculated as: :a = \frac where \mu\ = Gm_1 is the standard gravitational parameter. The empty focus (\mathbf = \left(f_x,f_y\right)) can be found by first determining the
Eccentricity vector In celestial mechanics, the eccentricity vector of a Kepler orbit is the dimensionless vector with direction pointing from apoapsis to periapsis and with magnitude equal to the orbit's scalar eccentricity. For Kepler orbits the eccentricity vec ...
: :\mathbf = \frac - \frac Where \mathbf is the specific angular momentum of the orbiting body: :\mathbf = \mathbf \times \mathbf Then :\mathbf = -2a\mathbf


Using XY Coordinates

This can be done in cartesian coordinates using the following procedure: The general equation of an ellipse under the assumptions above is: : \sqrt + \sqrt = 2a \qquad\mid z=0 Given: :r_x, r_y \quad the initial position coordinates :v_x, v_y \quad the initial velocity coordinates and :\mu = Gm_1 \quad the gravitational parameter Then: :h = r_x v_y - r_y v_x \quad specific angular momentum :r = \sqrt \quad initial distance from F1 (at the origin) :a = \frac \quad the semi-major axis length :e_x = \frac - \frac \quad the
Eccentricity vector In celestial mechanics, the eccentricity vector of a Kepler orbit is the dimensionless vector with direction pointing from apoapsis to periapsis and with magnitude equal to the orbit's scalar eccentricity. For Kepler orbits the eccentricity vec ...
coordinates :e_y = \frac + \frac \quad Finally, the empty focus coordinates :f_x = - 2 a e_x \quad :f_y = - 2 a e_y \quad Now the result values ''fx, fy'' and ''a'' can be applied to the general ellipse equation above.


Orbital parameters

The state of an orbiting body at any given time is defined by the orbiting body's position and velocity with respect to the central body, which can be represented by the three-dimensional Cartesian coordinates (position of the orbiting body represented by x, y, and z) and the similar Cartesian components of the orbiting body's velocity. This set of six variables, together with time, are called the orbital state vectors. Given the masses of the two bodies they determine the full orbit. The two most general cases with these 6 degrees of freedom are the elliptic and the hyperbolic orbit. Special cases with fewer degrees of freedom are the circular and parabolic orbit. Because at least six variables are absolutely required to completely represent an elliptic orbit with this set of parameters, then six variables are required to represent an orbit with any set of parameters. Another set of six parameters that are commonly used are the orbital elements.


Solar System

In the
Solar System The Solar System Capitalization of the name varies. The International Astronomical Union, the authoritative body regarding astronomical nomenclature, specifies capitalizing the names of all individual astronomical objects but uses mixed "Solar ...
,
planet A planet is a large, rounded astronomical body that is neither a star nor its remnant. The best available theory of planet formation is the nebular hypothesis, which posits that an interstellar cloud collapses out of a nebula to create a you ...
s, asteroids, most
comet A comet is an icy, small Solar System body that, when passing close to the Sun, warms and begins to release gases, a process that is called outgassing. This produces a visible atmosphere or coma, and sometimes also a tail. These phenomena ...
s and some pieces of space debris have approximately elliptical orbits around the Sun. Strictly speaking, both bodies revolve around the same focus of the ellipse, the one closer to the more massive body, but when one body is significantly more massive, such as the sun in relation to the earth, the focus may be contained within the larger massing body, and thus the smaller is said to revolve around it. The following chart of the
perihelion and aphelion An apsis (; ) is the farthest or nearest point in the orbit of a planetary body about its primary body. For example, the apsides of the Earth are called the aphelion and perihelion. General description There are two apsides in any elli ...
of the
planet A planet is a large, rounded astronomical body that is neither a star nor its remnant. The best available theory of planet formation is the nebular hypothesis, which posits that an interstellar cloud collapses out of a nebula to create a you ...
s,
dwarf planet A dwarf planet is a small planetary-mass object that is in direct orbit of the Sun, smaller than any of the eight classical planets but still a world in its own right. The prototypical dwarf planet is Pluto. The interest of dwarf planets to ...
s and
Halley's Comet Halley's Comet or Comet Halley, officially designated 1P/Halley, is a short-period comet visible from Earth every 75–79 years. Halley is the only known short-period comet that is regularly visible to the naked eye from Earth, and thus the on ...
demonstrates the variation of the eccentricity of their elliptical orbits. For similar distances from the sun, wider bars denote greater eccentricity. Note the almost-zero eccentricity of Earth and Venus compared to the enormous eccentricity of Halley's Comet and Eris.


Radial elliptic trajectory

A radial trajectory can be a double line segment, which is a degenerate ellipse with semi-minor axis = 0 and eccentricity = 1. Although the eccentricity is 1, this is not a parabolic orbit. Most properties and formulas of elliptic orbits apply. However, the orbit cannot be closed. It is an open orbit corresponding to the part of the degenerate ellipse from the moment the bodies touch each other and move away from each other until they touch each other again. In the case of point masses one full orbit is possible, starting and ending with a singularity. The velocities at the start and end are infinite in opposite directions and the potential energy is equal to minus infinity. The radial elliptic trajectory is the solution of a two-body problem with at some instant zero speed, as in the case of dropping an object (neglecting air resistance).


History

The
Babylonians Babylonia (; Akkadian: , ''māt Akkadī'') was an ancient Akkadian-speaking state and cultural area based in the city of Babylon in central-southern Mesopotamia (present-day Iraq and parts of Syria). It emerged as an Amorite-ruled state c. 1 ...
were the first to realize that the Sun's motion along the
ecliptic The ecliptic or ecliptic plane is the orbital plane of the Earth around the Sun. From the perspective of an observer on Earth, the Sun's movement around the celestial sphere over the course of a year traces out a path along the ecliptic agains ...
was not uniform, though they were unaware of why this was; it is today known that this is due to the Earth moving in an elliptic orbit around the Sun, with the Earth moving faster when it is nearer to the Sun at
perihelion An apsis (; ) is the farthest or nearest point in the orbit of a planetary body about its primary body. For example, the apsides of the Earth are called the aphelion and perihelion. General description There are two apsides in any elli ...
and moving slower when it is farther away at
aphelion An apsis (; ) is the farthest or nearest point in the orbit of a planetary body about its primary body. For example, the apsides of the Earth are called the aphelion and perihelion. General description There are two apsides in any ell ...
. In the 17th century, Johannes Kepler discovered that the orbits along which the planets travel around the Sun are ellipses with the Sun at one focus, and described this in his first law of planetary motion. Later,
Isaac Newton Sir Isaac Newton (25 December 1642 – 20 March 1726/27) was an English mathematician, physicist, astronomer, alchemist, theologian, and author (described in his time as a " natural philosopher"), widely recognised as one of the grea ...
explained this as a corollary of his law of universal gravitation.


See also

*
Apsis An apsis (; ) is the farthest or nearest point in the orbit of a planetary body about its primary body. For example, the apsides of the Earth are called the aphelion and perihelion. General description There are two apsides in any ell ...
*
Characteristic energy In astrodynamics, the characteristic energy (C_3) is a measure of the excess specific energy over that required to just barely escape from a massive body. The units are length2 time−2, i.e. velocity squared, or energy per mass. Every object in ...
* Ellipse *
List of orbits Summary A simple list of just the common orbit abbreviations. List of abbreviations of common Earth orbits List of abbreviations of other orbits Classifications The following is a list of types of orbits: Centric classifications * Gal ...
* Orbital eccentricity *
Orbit equation In astrodynamics, an orbit equation defines the path of orbiting body m_2\,\! around central body m_1\,\! relative to m_1\,\!, without specifying position as a function of time. Under standard assumptions, a body moving under the influence of a f ...
* Parabolic trajectory


References


Sources

* * *


External links


Java applet animating the orbit of a satellite
in an elliptic Kepler orbit around the Earth with any value for semi-major axis and eccentricity.

Lunar photographic comparison

Solar photographic comparison * http://www.castor2.ca {{Portal bar, Astronomy, Stars, Spaceflight, Outer space, Solar System Orbits