Vis-viva Equation
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In
astrodynamics Orbital mechanics or astrodynamics is the application of ballistics and celestial mechanics to rockets, satellites, and other spacecraft. The motion of these objects is usually calculated from Newton's laws of motion and the Newton's law of univ ...
, the ''vis-viva'' equation 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 which applies when the only force acting on an object is its own weight which is the gravitational force determined by the product of the mass of the object and the strength of the surrounding
gravitational field In physics, a gravitational field or gravitational acceleration field is a vector field used to explain the influences that a body extends into the space around itself. A gravitational field is used to explain gravitational phenomena, such as ...
. '' Vis viva'' (Latin for "living force") is a term from the history of mechanics and this name is given to the orbital equation originally derived by
Isaac Newton Sir Isaac Newton () was an English polymath active as a mathematician, physicist, astronomer, alchemist, theologian, and author. Newton was a key figure in the Scientific Revolution and the Age of Enlightenment, Enlightenment that followed ...
. 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 ani ...
of the accelerating
force In physics, a force is an influence that can cause an Physical object, object to change its velocity unless counterbalanced by other forces. In mechanics, force makes ideas like 'pushing' or 'pulling' mathematically precise. Because the Magnitu ...
s of a
system A system is a group of interacting or interrelated elements that act according to a set of rules to form a unified whole. A system, surrounded and influenced by its open system (systems theory), environment, is described by its boundaries, str ...
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.


Formulation

For any Keplerian orbit ( elliptic, parabolic, hyperbolic, or radial), the ''vis-viva'' equation is as follows: v^2 = GM \left( - \right) where: * is the relative speed of the two bodies * is the distance between the two bodies' centers of mass * 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 longe ...
( 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, or for
parabola In mathematics, a parabola is a plane curve which is Reflection symmetry, mirror-symmetrical and is approximately U-shaped. It fits several superficially different Mathematics, mathematical descriptions, which can all be proved to define exactl ...
s, and for
hyperbola In mathematics, a hyperbola is a type of smooth function, smooth plane curve, 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, called connected component ( ...
s) * is the
gravitational constant The gravitational constant is an empirical physical constant involved in the calculation of gravitational effects in Sir Isaac Newton's law of universal gravitation and in Albert Einstein's general relativity, theory of general relativity. It ...
* is the mass of the central body The product of can also be expressed as the
standard gravitational parameter The standard gravitational parameter ''μ'' of a celestial body is the product of the gravitational constant ''G'' and the mass ''M'' of that body. For two bodies, the parameter may be expressed as , or as when one body is much larger than the ...
using the Greek letter .


Practical applications

Given the total mass and the scalars and at a single point of the orbit, one can compute: * and at any other point in the orbit; and * the
specific orbital energy In the gravitational two-body problem, the specific orbital energy \varepsilon (or specific ''vis-viva'' energy) of two orbiting bodies is the constant quotient of their mechanical energy (the sum of their mutual potential energy, \varepsilon ...
\varepsilon\,\!, allowing an object orbiting a larger object to be classified as having not enough energy to remain in orbit, hence being " suborbital" (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 an object to escape from contact with or orbit of a primary body, assuming: * Ballistic trajectory – no other forces are acting on the object, such as ...
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 For a given orbital radius, the escape velocity will be \sqrt times the orbital velocity.


Derivation for elliptic orbits (0 ≤ eccentricity < 1)

Specific total energy is constant throughout the orbit. Thus, using the subscripts and 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, \begin \fracv_a^2 &= \left( \frac - \frac\right) \cdot \frac \\ \fracv_a^2 &= GM \left( \frac \right) \frac \\ \fracv_a^2 &= GM \frac \end 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 can be derived using r_a + r_p = 2a and r_a r_p = b^2, where is semi-major axis and 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


References

{{DEFAULTSORT:Vis-Viva Equation Orbits Conservation laws Equations of astronomy