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.
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.
Under standard assumptions, no other forces acting except two spherically symmetrical bodies m1 and m2,[1] the orbital speed (
v
v=\sqrt{\mu\left({2\over{r}}-{1\over{a}}\right)}
\mu
r
a
{1\over{a}}
Under standard assumptions the orbital period (
T
T=2\pi\sqrt{a3\over{\mu}}
\mu
a
a
Under standard assumptions, the specific orbital energy (
\epsilon
{v2\over{2}}-{\mu\over{r}}=-{\mu\over{2a}}=\epsilon<0
v
r
a
\mu
Using the virial theorem to find:
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
| Mm | |
| 2a |
where a is the semi major axis.
Since gravity is a central force, the angular momentum is constant:
| L |
=r x F=r x F(r)\hat{r
At the closest and furthest approaches, the angular momentum is perpendicular to the distance from the mass orbited, therefore:
L=rp=rmv
The total energy of the orbit is given by[5]
E=
| 1 | |
| 2 |
mv2-G
| Mm | |
| r |
Substituting for v, the equation becomes
E=
| 1 | |
| 2 |
| L2 | |
| mr2 |
-G
| Mm | |
| r |
This is true for r being the closest / furthest distance so two simultaneous equations are made, which when solved for E:
E=-G
| Mm | |
| r1+r2 |
Since and
r2=a-a\epsilon
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
\phi
h=rv\cos\phi
h
v
r
\phi
\psi
\nu
\phi=\nu+
| \pi | |
| 2 |
-\psi
\cos\phi=\sin(\psi-\nu)=\sin\psi\cos\nu-\cos\psi\sin\nu=
| 1+e\cos\nu | |
| \sqrt{1+e2+2e\cos\nu |
\tan\phi=
| e\sin\nu | |
| 1+e\cos\nu |
where
e
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
See main article: article and orbit equation.
m2
m1
m1
M=E-e\sinE
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 (
r
v
For this case it is convenient to use the following assumptions which differ somewhat from the standard assumptions above:
F1
r
v
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:
F2=\left(fx,fy\right)
The general equation of an ellipse under these assumptions using vectors is:
|F2-p|+|p|=2a \midz=0
where:
a
F2=\left(fx,fy\right)
p=\left(x,y\right)
The semi-major axis length (a) can be calculated as:
a=
| \mu|r| | |
| 2\mu-|r|v2 |
where
\mu =Gm1
The empty focus (
F2=\left(fx,fy\right)
e=
| r | |
| |r| |
-
| v x h | |
| \mu |
Where
h
h=r x v
Then
F2=-2ae
This can be done in cartesian coordinates using the following procedure:
The general equation of an ellipse under the assumptions above is:
\sqrt{\left(fx-x\right)2+\left(fy-y\right)2}+\sqrt{x2+y2}=2a \midz=0
Given:
rx,ry
vx,vy
and
\mu=Gm1
Then:
h=rxvy-ryvx
r=
| 2 | |
| \sqrt{r | |
| x |
+
| 2} | |
| r | |
| y |
a=
| \mur | ||||||||||||||
|
ex=
| rx | |
| r |
-
| hvy | |
| \mu |
ey=
| ry | |
| r |
+
| hvx | |
| \mu |
Finally, the empty focus coordinates
fx=-2aex
fy=-2aey
Now the result values fx, fy and a 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-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.
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.
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).
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.
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.