# Elliptic orbit

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In astrodynamics or celestial mechanics, an elliptic orbit or elliptical orbit is a Kepler orbit with an eccentricity 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 orbit with negative energy. This includes the radial elliptic orbit, with eccentricity equal to 1.

## Contents

In a gravitational two-body problem with negative energy, both bodies follow similar elliptic orbits with the same orbital period around their common barycenter. Also the relative position of one body with respect to the other follows an elliptic orbit.

Examples of elliptic orbits include Hohmann transfer orbits, Molniya orbits, and tundra orbits.

## Velocity

Under standard assumptions, no other forces acting except two spherically symmetrical bodies m1 and m2, [1] the orbital speed (${\displaystyle v\,}$) of one body traveling along an elliptic orbit can be computed from the vis-viva equation as: [2]

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

where:

• ${\displaystyle \mu \,}$ is the standard gravitational parameter, G(m1+m2), often expressed as GM when one body is much larger than the other.
• ${\displaystyle r\,}$ is the distance between the orbiting body and center of mass.
• ${\displaystyle a\,\!}$ is the length of the semi-major axis.

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

## Orbital period

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

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

where:

Conclusions:

• The orbital period is equal to that for a circular 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).

## Energy

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

${\displaystyle {v^{2} \over {2}}-{\mu \over {r}}=-{\mu \over {2a}}=\epsilon <0}$

where:

• ${\displaystyle v\,}$ is the orbital speed of the orbiting body,
• ${\displaystyle r\,}$ is the distance of the orbiting body from the central body,
• ${\displaystyle a\,}$ is the length of the semi-major axis,
• ${\displaystyle \mu \,}$ 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 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

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

where a is the semi major axis.

#### Derivation

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

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

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]

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

We may substitute for v and obtain

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

This is true for r being the closest / furthest distance so we get two simultaneous equations which we solve for E:

${\displaystyle E=-G{\frac {Mm}{r_{1}+r_{2}}}}$

Since ${\textstyle r_{1}=a+a\epsilon }$ and ${\displaystyle r_{2}=a-a\epsilon }$, 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 ${\displaystyle \phi }$ satisfies the equation: [6]

${\displaystyle h\,=r\,v\,\cos \phi }$

where:

• ${\displaystyle h\,}$ is the specific relative angular momentum of the orbit,
• ${\displaystyle v\,}$ is the orbital speed of the orbiting body,
• ${\displaystyle r\,}$ is the radial distance of the orbiting body from the central body,
• ${\displaystyle \phi \,}$ is the flight path angle

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

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

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 ${\displaystyle \phi }$ 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 defines the path of an orbiting body ${\displaystyle m_{2}\,\!}$ around central 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. Because Kepler's equation ${\displaystyle M=E-e\sin E}$ has no general closed-form solution for the Eccentric anomaly (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 (${\displaystyle \mathbf {r} }$) 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 (${\displaystyle \mathbf {F1} }$) 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: ${\displaystyle \mathbf {F2} =\left(f_{x},f_{y}\right)}$ .

#### Using vectors

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

${\displaystyle |\mathbf {F2} -\mathbf {p} |+|\mathbf {p} |=2a\qquad \mid z=0}$

where:

• ${\displaystyle a\,\!}$ is the length of the semi-major axis.
• ${\displaystyle \mathbf {F2} =\left(f_{x},f_{y}\right)}$ 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 ${\displaystyle \mu \ =Gm_{1}}$ is the standard gravitational parameter.

The empty focus (${\displaystyle \mathbf {F2} =\left(f_{x},f_{y}\right)}$) can be found by first determining the Eccentricity 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

${\displaystyle \mathbf {F2} =-2a\mathbf {e} }$

#### 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:

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

Given:

${\displaystyle r_{x},r_{y}\quad }$ the initial position coordinates
${\displaystyle v_{x},v_{y}\quad }$ the initial velocity coordinates

and

${\displaystyle \mu =Gm_{1}\quad }$ the gravitational parameter

Then:

${\displaystyle h=r_{x}v_{y}-r_{y}v_{x}\quad }$ specific angular momentum
${\displaystyle r={\sqrt {r_{x}^{2}+r_{y}^{2}}}\quad }$ initial distance from F1 (at the origin)
${\displaystyle a={\frac {\mu r}{2\mu -r\left(v_{x}^{2}+v_{y}^{2}\right)}}\quad }$ the semi-major axis length

${\displaystyle e_{x}={\frac {r_{x}}{r}}-{\frac {hv_{y}}{\mu }}\quad }$ the Eccentricity vector coordinates
${\displaystyle e_{y}={\frac {r_{y}}{r}}+{\frac {hv_{x}}{\mu }}\quad }$

Finally, the empty focus coordinates

${\displaystyle f_{x}=-2ae_{x}\quad }$
${\displaystyle f_{y}=-2ae_{y}\quad }$

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, planets, asteroids, most comets, 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 of the planets, dwarf planets, and Halley'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 and Eris.

Distances of selected bodies of the Solar System from the Sun. The left and right edges of each bar correspond to the perihelion and aphelion of the body, respectively, hence long bars denote high orbital 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.

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 were the first to realize that the Sun's motion along the ecliptic 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 and moving slower when it is farther away at aphelion. [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 his first law of planetary motion. Later, Isaac Newton explained this as a corollary of his law of universal gravitation.

## Related Research Articles

In celestial mechanics, an orbit is the curved trajectory of an object such as the trajectory of a planet around a star, or of a natural satellite around a planet, or of an artificial satellite around an object or position in space such as a planet, moon, asteroid, or Lagrange point. Normally, orbit refers to a regularly repeating trajectory, although it may also refer to a non-repeating trajectory. To a close approximation, planets and satellites follow elliptic orbits, with the center of mass being orbited at a focal point of the ellipse, as described by Kepler's laws of planetary motion.

An ellipsoid is a surface that can be obtained from a sphere by deforming it by means of directional scalings, or more generally, of an affine transformation.

Orbital mechanics or astrodynamics is the application of ballistics and celestial mechanics to the practical problems concerning the motion of rockets, satellites, and other spacecraft. The motion of these objects is usually calculated from Newton's laws of motion and the law of universal gravitation. Orbital mechanics is a core discipline within space-mission design and control.

In celestial mechanics, true anomaly is an angular parameter that defines the position of a body moving along a Keplerian orbit. It is the angle between the direction of periapsis and the current position of the body, as seen from the main focus of the ellipse.

In astrodynamics or celestial mechanics a parabolic trajectory is a Kepler orbit with the eccentricity equal to 1 and is an unbound orbit that is exactly on the border between elliptical and hyperbolic. When moving away from the source it is called an escape orbit, otherwise a capture orbit. It is also sometimes referred to as a C3 = 0 orbit (see Characteristic energy).

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 fact that according to Newtonian theory such an orbit has the shape of a hyperbola. In more technical terms this can be expressed by the condition that the orbital eccentricity is greater than one.

In celestial mechanics, the specific relative angular momentum 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 position and relative linear momentum, divided by the mass of the body in question.

In astrodynamics, an orbit equation defines the path of orbiting body around central body relative to , without specifying position as a function of time. Under standard assumptions, a body moving under the influence of a force, directed to a central body, with a magnitude inversely proportional to the square of the distance, has an orbit that is a conic section with the central body located at one of the two foci, or the focus.

In orbital mechanics, mean motion is the angular speed required for a body to complete one orbit, assuming constant speed in a circular orbit which completes in the same time as the variable speed, elliptical orbit of the actual body. The concept applies equally well to a small body revolving about a large, massive primary body or to two relatively same-sized bodies revolving about a common center of mass. While nominally a mean, and theoretically so in the case of two-body motion, in practice the mean motion is not typically an average over time for the orbits of real bodies, which only approximate the two-body assumption. It is rather the instantaneous value which satisfies the above conditions as calculated from the current gravitational and geometric circumstances of the body's constantly-changing, perturbed orbit.

Spacecraft flight dynamics is the application of mechanical dynamics to model how the external forces acting on a space vehicle or spacecraft determine its flight path. These forces are primarily of three types: propulsive force provided by the vehicle's engines; gravitational force exerted by the Earth and other celestial bodies; and aerodynamic lift and drag.

A theoretical motivation for general relativity, including the motivation for the geodesic equation and the Einstein field equation, can be obtained from special relativity by examining the dynamics of particles in circular orbits about the Earth. A key advantage in examining circular orbits is that it is possible to know the solution of the Einstein Field Equation a priori. This provides a means to inform and verify the formalism.

In geometry, the elliptic coordinate system is a two-dimensional orthogonal coordinate system in which the coordinate lines are confocal ellipses and hyperbolae. The two foci and are generally taken to be fixed at and , respectively, on the -axis of the Cartesian coordinate system.

Elliptic cylindrical coordinates are a three-dimensional orthogonal coordinate system that results from projecting the two-dimensional elliptic coordinate system in the perpendicular -direction. Hence, the coordinate surfaces are prisms of confocal ellipses and hyperbolae. The two foci and are generally taken to be fixed at and , respectively, on the -axis of the Cartesian coordinate system.

Prolate spheroidal coordinates are a three-dimensional orthogonal coordinate system that results from rotating the two-dimensional elliptic coordinate system about the focal axis of the ellipse, i.e., the symmetry axis on which the foci are located. Rotation about the other axis produces oblate spheroidal coordinates. Prolate spheroidal coordinates can also be considered as a limiting case of ellipsoidal coordinates in which the two smallest principal axes are equal in length.

Oblate spheroidal coordinates are a three-dimensional orthogonal coordinate system that results from rotating the two-dimensional elliptic coordinate system about the non-focal axis of the ellipse, i.e., the symmetry axis that separates the foci. Thus, the two foci are transformed into a ring of radius in the x-y plane. Oblate spheroidal coordinates can also be considered as a limiting case of ellipsoidal coordinates in which the two largest semi-axes are equal in length.

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In celestial mechanics, a Kepler orbit is the motion of one body relative to another, as an ellipse, parabola, or hyperbola, which forms a two-dimensional orbital plane in three-dimensional space. A Kepler orbit can also form a straight line. It considers only the point-like gravitational attraction of two bodies, neglecting perturbations due to gravitational interactions with other objects, atmospheric drag, solar radiation pressure, a non-spherical central body, and so on. It is thus said to be a solution of a special case of the two-body problem, known as the Kepler problem. As a theory in classical mechanics, it also does not take into account the effects of general relativity. Keplerian orbits can be parametrized into six orbital elements in various ways.

In celestial mechanics, Lambert's problem is concerned with the determination of an orbit from two position vectors and the time of flight, posed in the 18th century by Johann Heinrich Lambert and formally solved with mathematical proof by Joseph-Louis Lagrange. It has important applications in the areas of rendezvous, targeting, guidance, and preliminary orbit determination.

In geometry, the major axis of an ellipse is its longest diameter: a line segment that runs through the center and both foci, with ends at the two most widely separated points of the perimeter. The semi-major axis is the longest semidiameter or one half of the major axis, and thus runs from the centre, through a focus, and to the perimeter. The semi-minor axis of an ellipse or hyperbola is a line segment that is at right angles with the semi-major axis and has one end at the center of the conic section. For the special case of a circle, the lengths of the semi-axes are both equal to the radius of the circle.

In astrodynamics and celestial mechanics a radial trajectory is a Kepler orbit with zero angular momentum. Two objects in a radial trajectory move directly towards or away from each other in a straight line.

## References

1. Bate, Mueller, White (1971). Fundamentals Of Astrodynamics (First ed.). New York: Dover. pp. 11–12. ISBN   0-486-60061-0.{{cite book}}: CS1 maint: multiple names: authors list (link)
2. Lissauer, Jack J.; de Pater, Imke (2019). Fundamental Planetary Sciences: physics, chemistry, and habitability. New York, NY, USA: Cambridge University Press. pp. 29–31. ISBN   9781108411981.
3. Bate, Mueller, White (1971). Fundamentals Of Astrodynamics (First ed.). New York: Dover. p. 33. ISBN   0-486-60061-0.{{cite book}}: CS1 maint: multiple names: authors list (link)
4. Bate, Mueller, White (1971). Fundamentals Of Astrodynamics (First ed.). New York: Dover. pp. 27–28. ISBN   0-486-60061-0.{{cite book}}: CS1 maint: multiple names: authors list (link)
5. Bate, Mueller, White (1971). Fundamentals Of Astrodynamics (First ed.). New York: Dover. p. 15. ISBN   0-486-60061-0.{{cite book}}: CS1 maint: multiple names: authors list (link)
6. Bate, Mueller, White (1971). Fundamentals Of Astrodynamics (First ed.). New York: Dover. p. 18. ISBN   0-486-60061-0.{{cite book}}: CS1 maint: multiple names: authors list (link)
7. Bate, Mueller, White (1971). Fundamentals Of Astrodynamics (First ed.). New York: Dover. p. 17. ISBN   0-486-60061-0.{{cite book}}: CS1 maint: multiple names: authors list (link)
8. David Leverington (2003), Babylon to Voyager and beyond: a history of planetary astronomy, Cambridge University Press, pp. 6–7, ISBN   0-521-80840-5