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Part of a series on
Astrodynamics
Orbital mechanics
Orbital elements Apsis Argument of periapsis Eccentricity Inclination Mean anomaly Orbital nodes Semi-major axis True anomaly
Types oftwo-body orbits by eccentricity Circular orbit Elliptic orbit Transfer orbit (Hohmann transfer orbit Bi-elliptic transfer orbit) Parabolic orbit Hyperbolic orbit Radial orbit Decaying orbit
Equations Dynamical friction Escape velocity Kepler's equation Kepler's laws of planetary motion Orbital period Orbital velocity Surface gravity Specific orbital energy Vis-viva equation
Celestial mechanics
Gravitational influences Barycenter Hill sphere Perturbations Sphere of influence
N-body orbits Lagrangian points (Halo orbits) Lissajous orbits Lyapunov orbits
Engineering and efficiency
Preflight engineering Mass ratio Payload fraction Propellant mass fraction Tsiolkovsky rocket equation
Efficiency measures Gravity assist Oberth effect
Propulsive maneuvers Orbital maneuver Orbit insertion
v t e

Inastrodynamics orcelestial mechanics, anelliptical orbit oreccentric orbit is anorbit with aneccentricity of less than 1;^{[citation needed]} this includes the special case of acircular orbit, with eccentricity equal to 0. Some orbits have been referred to as "elongated orbits" if the eccentricity is "high" but that is not an explanatory term. For the simple two body problem, all orbits are ellipses.

In agravitational two-body problem, both bodies followsimilar elliptical orbits with the sameorbital period around their commonbarycenter. The relative position of one body with respect to the other also follows an elliptic orbit.

Examples of elliptic orbits includeHohmann transfer orbits,Molniya orbits, andtundra orbits.

Under standard assumptions, no other forces acting except two spherically symmetrical bodies${\displaystyle (m_1)}$ and${\displaystyle (m_2)}$,^[1] theorbital speed (${\displaystyle v}$) of one body traveling along anelliptical orbit can be computed from thevis-viva equation as:^[2]

${\displaystyle v=\sqrt{\mu \left(\frac{2}{r}-\frac{1}{a}\right)}}$

where:

• ${\displaystyle \mu }$ is thestandard gravitational parameter,${\displaystyle G(m_1+m_2)}$, often expressed as${\displaystyle GM}$ when one body is much larger than the other.
• ${\displaystyle r}$ is the distance between the centers of mass of both bodies.
• ${\displaystyle a}$ is the length of thesemi-major axis.

The velocity equation for ahyperbolic trajectory has either${\displaystyle (+\frac{1}{a})}$, or it is the same with the convention that in that case${\displaystyle (a)}$ is negative.

Under standard assumptions the orbital period (${\displaystyle T}$) of a body travelling along an elliptic orbit can be computed as:^[3]

${\displaystyle T=2\pi \sqrt{\frac{a^3}{\mu }}}$

where:

• ${\displaystyle \mu }$ is thestandard gravitational parameter.
• ${\displaystyle a}$ is the length of thesemi-major axis.

Conclusions:

• The orbital period is equal to that for acircular orbit with the orbital radius equal to the semi-major axis (${\displaystyle a}$),
• For a given semi-major axis the orbital period does not depend on the eccentricity (See also:Kepler's third law).

Under standard assumptions, thespecific orbital energy (${\displaystyle ϵ}$) of an elliptic orbit is negative and the orbital energy conservation equation (theVis-viva equation) for this orbit can take the form:^[4]

where:

• ${\displaystyle v}$ is theorbital speed of the orbiting body,
• ${\displaystyle r}$ is the distance of the orbiting body from thecentral body,
• ${\displaystyle a}$ is the length of thesemi-major axis,
• ${\displaystyle \mu }$ is thestandard gravitational parameter.

Conclusions:

• For a given semi-major axis the specific orbital energy is independent of the eccentricity.

Using thevirial theorem to find:

• the time-average of the specific potential energy is equal to −2ε
  • the time-average ofr⁻¹ isa⁻¹
• the time-average of the specific kinetic energy is equal to ε

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

${\displaystyle E=-G\frac{Mm}{2a}}$,

where a is the semi major axis.

Since gravity is a central force, the angular momentum is constant:

${\displaystyle \dot{\mathbf{L}}=\mathbf{r}\times \mathbf{F}=\mathbf{r}\times F(r)\hat{\mathbf{r}}=0}$

At the closest and furthest approaches, the angular momentum is perpendicular to the distance from the mass orbited, therefore:

${\displaystyle L=rp=rmv}$.

The total energy of the orbit is given by^[5]

${\displaystyle E=\frac{1}{2}m{v}^2-G\frac{Mm}{r}}$.

Substituting for v, the equation becomes

${\displaystyle E=\frac{1}{2}\frac{L^2}{m{r}^2}-G\frac{Mm}{r}}$.

This is true for r being the closest / furthest distance so two simultaneous equations are made, which when solved for E:

${\displaystyle E=-G\frac{Mm}{r_1+r_2}}$

Since$r_1=a+aϵ$ and${\displaystyle r_2=a-aϵ}$, where epsilon is the eccentricity of the orbit, the stated result is reached.

The flight path angle is the angle between the orbiting body's velocity vector (equal to the vector tangent to the instantaneous orbit) and the local horizontal. Under standard assumptions of the conservation of angular momentum the flight path angle${\displaystyle \varphi }$ satisfies the equation:^[6]

${\displaystyle h=rv\cos \varphi }$

where:

• ${\displaystyle h}$ is thespecific relative angular momentum of the orbit,
• ${\displaystyle v}$ is theorbital speed of the orbiting body,
• ${\displaystyle r}$ is the radial distance of the orbiting body from thecentral body,
• ${\displaystyle \varphi }$ is the flight path angle

${\displaystyle \psi }$ is the angle between the orbital velocity vector and the semi-major axis.${\displaystyle \nu }$ is the localtrue anomaly.${\displaystyle \varphi =\nu +\frac{\pi }{2}-\psi }$, therefore,

${\displaystyle \cos \varphi =\sin (\psi -\nu )=\sin \psi \cos \nu -\cos \psi \sin \nu =\frac{1+e\cos \nu }{\sqrt{1+e^2+2e\cos \nu }}}$

${\displaystyle \tan \varphi =\frac{e\sin \nu }{1+e\cos \nu }}$

where${\displaystyle e}$ 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${\displaystyle \varphi }$ is defined as the angle which differs by 90 degrees from this, so the cosine appears in place of the sine.

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Anorbit equation defines the path of anorbiting body${\displaystyle m_2}$ aroundcentral body${\displaystyle m_1}$ relative to${\displaystyle m_1}$, without specifying position as a function of time. If the eccentricity is less than 1 then the equation of motion describes an elliptical orbit. BecauseKepler's equation${\displaystyle M=E-e\sin E}$ has no generalclosed-form solution for theEccentric anomaly (E)in terms of the Mean anomaly (M), equations of motion as a function of time also have no closed-form solution (althoughnumerical 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 (${\displaystyle \mathbf{r}}$) and velocity (${\displaystyle \mathbf{v}}$).

For this case it is convenient to use the following assumptions which differ somewhat from the standard assumptions above:

1. The central body's position is at the origin and is the primary focus (${\displaystyle \mathbf{F}\mathbf{1}}$) of the ellipse (alternatively, the center of mass may be used instead if the orbiting body has a significant mass)
2. The central body's mass (m1) is known
3. The orbiting body's initial position(${\displaystyle \mathbf{r}}$) and velocity(${\displaystyle \mathbf{v}}$) are known
4. 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:${\displaystyle \mathbf{F}\mathbf{2}=\left(f_x,f_y\right)}$ .

The general equation of an ellipse under these assumptions using vectors is:

${\displaystyle |\mathbf{F}\mathbf{2}-\mathbf{p}|+|\mathbf{p}|=2a\mid z=0}$

where:

• ${\displaystyle a}$ is the length of thesemi-major axis.
• ${\displaystyle \mathbf{F}\mathbf{2}=\left(f_x,f_y\right)}$ is the second ("empty") focus.
• ${\displaystyle \mathbf{p}=\left(x,y\right)}$ is any (x,y) value satisfying the equation.

The semi-major axis length (a) can be calculated as:

${\displaystyle a=\frac{\mu |\mathbf{r}|}{2\mu -|\mathbf{r}|{\mathbf{v}}^2}}$

where${\displaystyle \mu =G{m}_1}$ is thestandard gravitational parameter.

The empty focus (${\displaystyle \mathbf{F}\mathbf{2}=\left(f_x,f_y\right)}$) can be found by first determining theEccentricity vector:

${\displaystyle \mathbf{e}=\frac{\mathbf{r}}{|\mathbf{r}|}-\frac{\mathbf{v}\times \mathbf{h}}{\mu }}$

Where${\displaystyle \mathbf{h}}$ is the specific angular momentum of the orbiting body:^[7]

${\displaystyle \mathbf{h}=\mathbf{r}\times \mathbf{v}}$

Then

${\displaystyle \mathbf{F}\mathbf{2}=-2a\mathbf{e}}$

This can be done in cartesian coordinates using the following procedure:

The general equation of an ellipse under the assumptions above is:

${\displaystyle \sqrt{{\left(f_x-x\right)}^2+{\left(f_y-y\right)}^2}+\sqrt{x^2+y^2}=2a\mid z=0}$

Given:

${\displaystyle r_x,r_y}$ the initial position coordinates

${\displaystyle v_x,v_y}$ the initial velocity coordinates

and

${\displaystyle \mu =G{m}_1}$ the gravitational parameter

Then:

${\displaystyle h=r_x{v}_y-r_y{v}_x}$ specific angular momentum

${\displaystyle r=\sqrt{r_x^2+r_y^2}}$ initial distance from F1 (at the origin)

${\displaystyle a=\frac{\mu r}{2\mu -r\left(v_x^2+v_y^2\right)}}$ the semi-major axis length

${\displaystyle e_x=\frac{r_x}{r}-\frac{h{v}_y}{\mu }}$ theEccentricity vector coordinates

${\displaystyle e_y=\frac{r_y}{r}+\frac{h{v}_x}{\mu }}$

Finally, the empty focus coordinates

${\displaystyle f_x=-2a{e}_x}$

${\displaystyle f_y=-2a{e}_y}$

Now the result valuesfx, fy anda can be applied to the general ellipse equation above.

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-dimensionalCartesian 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 theorbital 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 theorbital elements.

In theSolar System,planets,asteroids, mostcomets, and some pieces ofspace 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 theperihelion and aphelion of theplanets,dwarf planets, andHalley's Comet 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 andEris.

Distances of selected bodies of theSolar System from the Sun. The left and right edges of each bar correspond to theperihelion andaphelion of the body, respectively, hence long bars denote highorbital eccentricity. The radius of the Sun is 0.7 million km, and the radius of Jupiter (the largest planet) is 0.07 million km, both too small to resolve on this image.

Aradial trajectory can be adouble line segment, which is adegenerate 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 ofdropping an object (neglecting air resistance).

TheBabylonians were the first to realize that the Sun's motion along theecliptic 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 atperihelion and moving slower when it is farther away ataphelion.^[8]

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 hisfirst law of planetary motion. Later,Isaac Newton explained this as a corollary of hislaw of universal gravitation.

• Apsis
• Characteristic energy
• Ellipse
• List of orbits
• Orbital eccentricity
• Orbit equation
• Parabolic trajectory

• D'Eliseo, Maurizio M. (2007). "The First-Order Orbital Equation".American Journal of Physics.75 (4):352–355.Bibcode:2007AmJPh..75..352D.doi:10.1119/1.2432126.
• D'Eliseo, Maurizio M.; Mironov, Sergey V. (2009). "The Gravitational Ellipse".Journal of Mathematical Physics.50 (2): 022901.arXiv:0802.2435.Bibcode:2009JMP....50a2901M.doi:10.1063/1.3078419.
• Curtis, Howard D. (2019).Orbital Mechanics for Engineering Students (4th ed.).Butterworth-Heinemann.ISBN 978-0-08-102133-0.

• Apogee - Perigee Lunar photographic comparison
• Aphelion - Perihelion Solar photographic comparison
• Canadian Astronomy, Satellite Tracking and Optical Research (CASTOR)

• Orbits

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