Editing Orbital mechanics (section)

==Laws of astrodynamics==
{{See also|Laplace–Runge–Lenz vector}}
The fundamental laws of astrodynamics are [[Newton's law of universal gravitation]] and [[Newton's laws of motion]], while the fundamental mathematical tool is [[differential calculus]].

In a Newtonian framework, the laws governing orbits and trajectories are in principle [[T-symmetry|time-symmetric]].

Standard assumptions in astrodynamics include non-interference from outside bodies, negligible mass for one of the bodies, and negligible other forces (such as from the solar wind, atmospheric drag, etc.).  More accurate calculations can be made without these simplifying assumptions, but they are more complicated.  The increased accuracy often does not make enough of a difference in the calculation to be worthwhile.

[[Kepler's laws of planetary motion]] may be derived from Newton's laws, when it is assumed that the orbiting body is subject only to the gravitational force of the central attractor. When an engine thrust or propulsive force is present, Newton's laws still apply, but Kepler's laws are invalidated.  When the thrust stops, the resulting orbit will be different but will once again be described by Kepler's laws which have been set out above. The three laws are:
# The [[orbit]] of every [[planet]] is an [[ellipse]] with the Sun at one of the [[focus (geometry)|foci]].
# A [[line (mathematics)|line]] joining a planet and the Sun sweeps out equal areas during equal intervals of time.
# The [[square (algebra)|square]]s of the [[orbital period]]s of planets are directly [[proportionality (mathematics)|proportional]] to the [[cube (arithmetic)|cube]]s of the [[semi-major axis]] of the orbits.

===Escape velocity===
{{Main|Escape velocity}}
The formula for an [[escape velocity]] is derived as follows. The [[specific energy]] (energy per unit [[mass]]) of any space vehicle is composed of two components, the [[specific potential energy]] and the [[specific kinetic energy]]. The specific potential energy associated with a planet of [[mass]] ''M'' is given by
:<math>\epsilon_p = - \frac{G M}{r} \,</math>

where ''G'' is the [[gravitational constant]] and ''r'' is the distance between the two bodies;

while the [[specific kinetic energy]] of an object is given by
:<math>\epsilon_k = \frac{v^2}{2} \,</math>

where ''v'' is its Velocity;

and so the total [[specific orbital energy]] is
:<math> \epsilon = \epsilon_k+\epsilon_p = \frac{v^2}{2} - \frac{G M}{r} \,</math>

Since [[conservation of energy|energy is conserved]], <math> \epsilon</math> cannot depend on the distance, <math>r</math>, from the center of the central body to the space vehicle in question, i.e. ''v'' must vary with ''r'' to keep the specific orbital energy constant. Therefore, the object can reach infinite <math>r</math> only if this quantity is nonnegative, which implies
:<math>v\geq\sqrt{\frac{2 G M}{r}}.</math>

The escape velocity from the Earth's surface is about 11&nbsp;km/s, but that is insufficient to send the body an infinite distance because of the gravitational pull of the Sun. To escape the Solar System from a location at a distance from the Sun equal to the distance Sun–Earth, but not close to the Earth, requires around 42&nbsp;km/s velocity, but there will be "partial credit" for the Earth's orbital velocity for spacecraft launched from Earth, if their further acceleration (due to the propulsion system) carries them in the same direction as Earth travels in its orbit.

===Formulae for free orbits===
Orbits are [[conic section]]s, so the formula for the distance of a body for a given angle corresponds to the formula for that curve in [[polar coordinates]], which is:
:<math>r = \frac{ p }{1 + e \cos \theta}</math>
:<math>\mu= G(m_1+m_2)\,</math>
:<math>p=h^2/\mu\,</math>

<math>\mu</math> is called the [[gravitational parameter]]. <math>m_1</math> and <math>m_2</math> are the masses of objects 1 and 2, and <math>h</math> is the [[specific angular momentum]] of object 2 with respect to object 1. The parameter <math>\theta</math> is known as the [[true anomaly]], <math>p</math> is the [[conic section#Conic parameters|semi-latus rectum]], while <math>e</math> is the [[orbital eccentricity]], all obtainable from the various forms of the six independent [[orbital elements]].

===Circular orbits===
{{Main|Circular orbit}}
All bounded orbits where the gravity of a central body dominates are elliptical in nature. A special case of this is the circular orbit, which is an ellipse of zero eccentricity. The formula for the velocity of a body in a circular orbit at distance ''r'' from the center of gravity of mass ''M'' can be derived as follows:

Centrifugal acceleration matches the acceleration due to gravity.

So, <math display="block">\frac{v^2}{r} = \frac{GM}{r^2}</math>

Therefore,
:<math>\ v = \sqrt{\frac{GM} {r}\ }</math>

where <math>G</math> is the [[gravitational constant]], equal to
:6.6743 &times; 10<sup>&minus;11</sup> m<sup>3</sup>/(kg·s<sup>2</sup>)

To properly use this formula, the units must be consistent; for example, <math>M</math> must be in kilograms, and <math>r</math> must be in meters. The answer will be in meters per second.

The quantity <math>GM</math> is often termed the [[standard gravitational parameter]], which has a different value for every planet or moon in the [[Solar System]].

Once the circular orbital velocity is known, the [[escape velocity]] is easily found by multiplying by <math>\sqrt{2}</math>:
:<math>\ v = \sqrt 2\sqrt{\frac {GM} {r}\ } = \sqrt{\frac {2GM} {r}\ }.</math>

To escape from gravity, the kinetic energy must at least match the negative potential energy. Therefore, <math display="block">\frac{1}{2}mv^2 = \frac{GMm}{r}</math>
:<math>v = \sqrt{\frac {2GM} {r}\ }.</math>

===Elliptical orbits===
If <math>0 < e < 1</math>, then the denominator of the equation of free orbits varies with the true anomaly <math>\theta</math>, but remains positive, never becoming zero. Therefore, the relative position vector remains bounded, having its smallest magnitude at periapsis <math>r_p</math>, which is given by:
:<math>r_p=\frac{p}{1+e}</math>

The maximum value <math>r</math> is reached when <math>\theta = 180^\circ</math>. This point is called the apoapsis, and its radial coordinate, denoted <math>r_a</math>, is
:<math>r_a=\frac{p}{1-e}</math>

Let <math>2a</math> be the distance measured along the apse line from periapsis <math>P</math> to apoapsis <math>A</math>, as illustrated in the equation below:
:<math>2a=r_p+r_a</math>

Substituting the equations above, we get:
:<math>a=\frac{p}{1-e^2}</math>

a is the semimajor axis of the ellipse. Solving for <math>p</math>, and substituting the result in the conic section curve formula above, we get:
:<math>r=\frac{a(1-e^2)}{1+e\cos\theta}</math>

====Orbital period====
Under standard assumptions the [[orbital period]] (<math>T\,\!</math>) of a body traveling along an elliptic orbit can be computed as:
:<math>T=2\pi\sqrt{a^3\over{\mu}}</math>
where:
*<math>\mu\,</math> is the [[standard gravitational parameter]],
*<math>a\,\!</math> is the length of the [[semi-major axis]].
Conclusions:
*The orbital period is equal to that for a [[circular orbit]] with the orbit radius equal to the [[semi-major axis]] (<math>a\,\!</math>),
*For a given semi-major axis the orbital period does not depend on the eccentricity (See also: [[Kepler's laws of planetary motion#Third law|Kepler's third law]]).

====Velocity====
Under standard assumptions the [[orbital speed]] (<math>v\,</math>) of a body traveling along an '''elliptic orbit''' can be computed from the [[Vis-viva equation]] as:
:<math>v=\sqrt{\mu\left({2\over{r}}-{1\over{a}}\right)}</math>
where:
*<math>\mu\,</math> is the [[standard gravitational parameter]],
*<math>r\,</math> is the distance between the orbiting bodies.
*<math>a\,\!</math> is the length of the [[semi-major axis]].

The velocity equation for a [[hyperbolic trajectory]] is <math>v=\sqrt{\mu\left({2\over{r}}+\left\vert {1\over{a}} \right\vert\right)}</math>.

====Energy====
Under standard assumptions, [[specific orbital energy]] (<math>\epsilon\,</math>) of elliptic orbit is negative and the orbital energy conservation equation (the [[Vis-viva equation]]) for this orbit can take the form:
:<math>{v^2\over{2}}-{\mu\over{r}}=-{\mu\over{2a}}=\epsilon<0</math>
where:
*<math>v\,</math> is the speed of the orbiting body,
*<math>r\,</math> is the distance of the orbiting body from the center of mass of the [[central body]],
*<math>a\,</math> is the [[semi-major axis]],
*<math>\mu\,</math> 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 <math>2\epsilon</math>
*the time-average of <math>r^{-1}</math> is <math>a^{-1}</math>
*the time-average of the specific kinetic energy is equal to <math>-\epsilon</math>

===Parabolic orbits===
If the eccentricity equals 1, then the orbit equation becomes:
:<math>r={{h^2}\over{\mu}}{{1}\over{1+\cos\theta}}</math>
where:
*<math>r\,</math> is the radial distance of the orbiting body from the mass center of the [[central body]],
*<math>h\,</math> is [[specific angular momentum]] of the [[orbiting body]],
*<math>\theta\,</math> is the [[true anomaly]] of the orbiting body,
*<math>\mu\,</math> is the [[standard gravitational parameter]].

As the true anomaly θ approaches 180°, the denominator approaches zero, so that ''r'' tends towards infinity. Hence, the energy of the trajectory for which ''e''=1 is zero, and is given by:
:<math>\epsilon={v^2\over2}-{\mu\over{r}}=0</math>
where:
*<math>v\,</math> is the speed of the orbiting body.

In other words, the speed anywhere on a parabolic path is:
:<math>v=\sqrt{2\mu\over{r}}</math>

===Hyperbolic orbits===
If <math>e>1</math>, the orbit formula,
:<math>r={{h^2}\over{\mu}}{{1}\over{1+e\cos\theta}}</math>

describes the geometry of the hyperbolic orbit. The system consists of two symmetric curves. The orbiting body occupies one of them; the other one is its empty mathematical image. Clearly, the denominator of the equation above goes to zero when <math>\cos\theta = -1/e</math>. we denote this value of true anomaly
:<math>\theta_\infty = \cos^{-1} \left( -\frac1e \right)</math>
since the radial distance approaches infinity as the true anomaly approaches <math>\theta_\infty</math>, known as the ''true anomaly of the asymptote''. Observe that <math>\theta_\infty</math> lies between 90° and 180°. From the trigonometric identity <math>\sin^2\theta+\cos^2\theta=1</math> it follows that:
:<math>\sin\theta_\infty = \frac1e \sqrt{e^2 - 1}</math>

====Energy====
Under standard assumptions, [[specific orbital energy]] (<math>\epsilon\,</math>) of a [[hyperbolic trajectory]] is greater than zero and the [[orbital energy conservation equation]] for this kind of trajectory takes form:
:<math>\epsilon={v^2\over2}-{\mu\over{r}}={\mu\over{-2a}}</math>
where:
*<math>v\,</math> is the [[orbital speed|orbital velocity]] of orbiting body,
*<math>r\,</math> is the radial distance of orbiting body from [[central body]],
*<math>a\,</math> is the negative [[semi-major axis]] of the [[orbit]]'s [[hyperbola]],
*<math>\mu\,</math> is [[standard gravitational parameter]].

====Hyperbolic excess velocity====
{{See also|Characteristic energy}}

Under standard assumptions the body traveling along a hyperbolic trajectory will attain at <math>r =</math> infinity an [[orbital speed|orbital velocity]] called hyperbolic excess velocity (<math>v_\infty\,\!</math>) that can be computed as:
:<math>v_\infty=\sqrt{\mu\over{-a}}\,\!</math>
where:
*<math>\mu\,\!</math> is [[standard gravitational parameter]],
*<math>a\,\!</math> is the negative [[semi-major axis]] of [[orbit]]'s [[hyperbola]].

The hyperbolic excess velocity is related to the [[specific orbital energy]] or characteristic energy by
:<math>2\epsilon=C_3=v_{\infty}^2\,\!</math>