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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 ...
, 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 In physical sciences, mechanical energy is the sum of potential energy and kinetic energy. The principle of conservation of mechanical energy states that if an isolated system is subject only to conservative forces, then the mechanical energy is ...
which applies when the only force acting on an object is its own weight. ''
Vis viva ''Vis viva'' (from the Latin for "living force") is a historical term used for the first recorded description of what we now call kinetic energy in an early formulation of the principle of conservation of energy. Overview Proposed by Gottfried L ...
'' (Latin for "living force") is a term from the history of mechanics, and it survives in this sole context. It represents the principle that the difference between the total
work Work may refer to: * Work (human activity), intentional activity people perform to support themselves, others, or the community ** Manual labour, physical work done by humans ** House work, housework, or homemaking ** Working animal, an animal t ...
of the
accelerating In mechanics, acceleration is the rate of change of the velocity of an object with respect to time. Accelerations are vector quantities (in that they have magnitude and direction). The orientation of an object's acceleration is given by th ...
force In physics, a force is an influence that can change the motion of an object. A force can cause an object with mass to change its velocity (e.g. moving from a state of rest), i.e., to accelerate. Force can also be described intuitively as a p ...
s of a
system A system is a group of Interaction, interacting or interrelated elements that act according to a set of rules to form a unified whole. A system, surrounded and influenced by its environment (systems), environment, is described by its boundaries, ...
and that of the retarding forces is equal to one half the ''vis viva'' accumulated or lost in the system while the work is being done.


Equation

For any
Keplerian orbit Johannes Kepler (; ; 27 December 1571 – 15 November 1630) was a German astronomer, German mathematician, mathematician, astrologer, Natural philosophy, natural philosopher and writer on music. He is a key figure in the 17th-century Scienti ...
(
elliptic In mathematics, an ellipse is a plane curve surrounding two focal points, such that for all points on the curve, the sum of the two distances to the focal points is a constant. It generalizes a circle, which is the special type of ellipse in ...
, parabolic,
hyperbolic Hyperbolic is an adjective describing something that resembles or pertains to a hyperbola (a curve), to hyperbole (an overstatement or exaggeration), or to hyperbolic geometry. The following phenomena are described as ''hyperbolic'' because they ...
, or
radial Radial is a geometric term of location which may refer to: Mathematics and Direction * Vector (geometric), a line * Radius, adjective form of * Radial distance, a directional coordinate in a polar coordinate system * Radial set * A bearing f ...
), the ''vis-viva'' equation is as follows: :v^2 = GM \left( - \right) where: * ''v'' is the relative speed of the two bodies * ''r'' is the distance between the two bodies centers of mass * ''a'' is the length of the
semi-major axis 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 (major semiaxis) is the long ...
(''a'' > 0 for
ellipse In mathematics, an ellipse is a plane curve surrounding two focus (geometry), focal points, such that for all points on the curve, the sum of the two distances to the focal points is a constant. It generalizes a circle, which is the special ty ...
s, ''a'' = ∞ or 1/''a'' = 0 for
parabola In mathematics, a parabola is a plane curve which is mirror-symmetrical and is approximately U-shaped. It fits several superficially different mathematical descriptions, which can all be proved to define exactly the same curves. One descript ...
s, and ''a'' < 0 for
hyperbola In mathematics, a hyperbola (; pl. hyperbolas or hyperbolae ; adj. hyperbolic ) is a type of smooth curve lying in a plane, defined by its geometric properties or by equations for which it is the solution set. A hyperbola has two pieces, cal ...
s) * ''G'' is the
gravitational constant The gravitational constant (also known as the universal gravitational constant, the Newtonian constant of gravitation, or the Cavendish gravitational constant), denoted by the capital letter , is an empirical physical constant involved in ...
* ''M'' is the mass of the central body The product of GM can also be expressed as the
standard gravitational parameter In celestial mechanics, the standard gravitational parameter ''μ'' of a celestial body is the product of the gravitational constant ''G'' and the mass ''M'' of the bodies. For two bodies the parameter may be expressed as G(m1+m2), or as GM when ...
using the Greek letter μ.


Derivation for elliptic orbits (0 ≤ eccentricity < 1)

In the vis-viva equation the mass ''m'' of the orbiting body (e.g., a spacecraft) is taken to be negligible in comparison to the mass ''M'' of the central body (e.g., the Earth). The central body and orbiting body are also often referred to as the primary and a particle respectively. In the specific cases of an elliptical or circular orbit, the vis-viva equation may be readily derived from conservation of energy and momentum. Specific total energy is constant throughout the orbit. Thus, using the subscripts ''a'' and ''p'' to denote apoapsis (apogee) and periapsis (perigee), respectively, : \varepsilon = \frac - \frac = \frac - \frac Rearranging, : \frac - \frac = \frac - \frac Recalling that for an elliptical orbit (and hence also a circular orbit) the velocity and radius vectors are perpendicular at apoapsis and periapsis, conservation of angular momentum requires specific angular momentum h = r_pv_p = r_av_a = \text, thus v_p = \fracv_a: : \frac \left( 1-\frac \right) v_a^2 = \frac - \frac : \frac \left( \frac \right) v_a^2 = \frac - \frac Isolating the kinetic energy at apoapsis and simplifying, : \fracv_a^2 = \left( \frac - \frac\right) \cdot \frac : \fracv_a^2 = GM \left( \frac \right) \frac : \fracv_a^2 = GM \frac From the geometry of an ellipse, 2a=r_p+r_a where ''a'' is the length of the semimajor axis. Thus, : \frac v_a^2 = GM \frac = GM \left( \frac - \frac \right) = \frac - \frac Substituting this into our original expression for specific orbital energy, : \varepsilon = \frac - \frac = \frac - \frac = \frac - \frac = - \frac Thus, \varepsilon = - \frac and the vis-viva equation may be written : \frac - \frac = -\frac or : v^2 = GM \left( \frac - \frac \right) Therefore, the conserved angular momentum L = mh can be derived using r_a + r_p = 2a and r_a r_p = b^2, where a is semi-major axis and b is semi-minor axis of the elliptical orbit, as follows - :v_a^2 = GM \left( \frac - \frac \right) = \frac \left( \frac \right) = \frac \left( \frac \right) = \frac \left( \frac \right)^2 and alternately, :v_p^2 = GM \left( \frac - \frac \right) = \frac \left( \frac \right) = \frac \left( \frac \right) = \frac \left( \frac \right)^2 Therefore, specific angular momentum h = r_p v_p = r_a v_a = b \sqrt, and Total angular momentum L = mh = mb \sqrt


Practical applications

Given the total mass and the scalars ''r'' and ''v'' at a single point of the orbit, one can compute ''r'' and ''v'' at any other point in the orbit.For the
three-body problem In physics and classical mechanics, the three-body problem is the problem of taking the initial positions and velocities (or momenta) of three point masses and solving for their subsequent motion according to Newton's laws of motion and Newton's ...
there is hardly a comparable vis-viva equation: conservation of energy reduces the larger number of degrees of freedom by only one.
Given the total mass and the scalars ''r'' and ''v'' at a single point of the orbit, one can compute 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), divided ...
\varepsilon\,\!, allowing an object orbiting a larger object to be classified as having not enough energy to remain in orbit, hence being "
suborbital A sub-orbital spaceflight is a spaceflight in which the spacecraft reaches outer space, but its trajectory intersects the atmosphere or surface of the gravitating body from which it was launched, so that it will not complete one orbital r ...
" (a ballistic missile, for example), having enough energy to be "orbital", but without the possibility to complete a full orbit anyway because it eventually collides with the other body, or having enough energy to come from and/or go to infinity (as a meteor, for example). The formula for
escape velocity In celestial mechanics, escape velocity or escape speed is the minimum speed needed for a free, non- propelled object to escape from the gravitational influence of a primary body, thus reaching an infinite distance from it. It is typically ...
can be obtained from the Vis-viva equation by taking the limit as a approaches \infty: :v_e^2=GM \left(\frac-0 \right)\rightarrow v_e=\sqrt


Notes


References


External links


Vis-viva equation calculators
{{DEFAULTSORT:Vis-Viva Equation Orbits Conservation laws