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Concept in gravitational orbital mechanics
In astrodynamics, 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
Vis-viva_equation
Historical concept in physics
Vis viva (from the Latin for "living force") is a historical term used to describe a quantity similar to kinetic energy in an early formulation of the
Vis_viva
Kepler orbit with an eccentricity of less than one
elliptic orbit is negative and the orbital energy conservation equation (the Vis-viva equation) for this orbit can take the form: v 2 2 − μ r = − μ 2 a =
Elliptic_orbit
Functional equation Functional equation (L-function) Constitutive equation Laws of science Defining equation (physical chemistry) List of equations in classical
List_of_equations
Speed at which a body orbits around the barycenter of a system
semi-major axis of the elliptical orbit. This expression is called the vis-viva equation. For the Earth at perihelion, the value is: 1.327 × 10 20 m 3 s
Orbital_speed
Process that leads to gradual decrease of the distance between two orbiting bodies
potential energies, in an unperturbed two-body orbit. By substituting the vis-viva equation into the kinetic energy component, the orbital energy of a circular
Orbital_decay
Type of orbital maneuver
three required changes in velocity can be obtained directly from the vis-viva equation v 2 = μ ( 2 r − 1 a ) , {\displaystyle v^{2}=\mu \left({\frac {2}{r}}-{\frac
Bi-elliptic_transfer
Orbital mechanics term
In orbital mechanics, Kepler's equation relates various geometric properties of the orbit of a body subject to a central force. It was derived by Johannes
Kepler's_equation
Mathematical equation describing the motion of a rocket
The classical rocket equation, Tsiolkovsky rocket equation, or ideal rocket equation is a mathematical equation that describes the motion of vehicles that
Tsiolkovsky_rocket_equation
Field of classical mechanics concerned with the motion of spacecraft
elliptic orbit is negative, and the orbital energy conservation equation (the vis-viva equation) for this orbit can take the form v 2 2 − μ r = − μ 2 a = ϵ
Orbital_mechanics
Concept in astrodynamics
traveling along a hyperbolic trajectory can be computed from the vis-viva equation as: v = μ ( 2 r + 1 a ) {\displaystyle v={\sqrt {\mu \left({2 \over
Hyperbolic_trajectory
Parameter in the gravitational two-body problem
mass. According to the orbital energy conservation equation (also referred to as vis-viva equation), it does not vary with time: ε = ε k + ε p = v 2 2
Specific_orbital_energy
Type of orbit
Kepler's equation, which is used to solve for true anomalies in elliptical and hyperbolic trajectories, the true anomaly in Barker's equation can be solved
Parabolic_trajectory
Method to calculate trajectory calculations for spacecraft
v_{M}={\sqrt {\frac {\mu _{\odot }}{r_{M}}}}=24.1\ {\text{km/s}}} Using the vis-viva equation, the spacecraft speed on the transfer ellipse at Earth's orbit is
Patched_conic_approximation
Either of two extreme points in a celestial object's orbit
Orbital period Orbital velocity Surface gravity Specific orbital energy Vis-viva equation Celestial mechanics Gravitational influences Barycenter Hill sphere
Apsis
Time an astronomical object takes to complete one orbit around another object
perfect sphere of uniform density, it is possible to rewrite the first equation without measuring the mass as: T = a 3 r 3 3 π G ρ {\displaystyle T={\sqrt
Orbital_period
Spaceflight operation
Orbital period Orbital velocity Surface gravity Specific orbital energy Vis-viva equation Celestial mechanics Gravitational influences Barycenter Hill sphere
Orbit_insertion
Transfer manoeuvre between two orbits
{mv^{2}}{2}}-{\frac {GMm}{r}}={\frac {-GMm}{2a}}.} Solving this equation for velocity results in the vis-viva equation, v 2 = μ ( 2 r − 1 a ) , {\displaystyle v^{2}=\mu
Hohmann_transfer_orbit
Point where an orbit crosses a plane of reference to which it is inclined
Orbital period Orbital velocity Surface gravity Specific orbital energy Vis-viva equation Celestial mechanics Gravitational influences Barycenter Hill sphere
Orbital_node
Specifies the orbit of an object in space
orbital elements is to calculate the mean anomaly by this equation, and then to solve Kepler's equation for the eccentric anomaly. Define ϖ as the longitude
Mean_anomaly
Motion problem in classical mechanics
mass (barycenter) motion. By contrast, subtracting equation (2) from equation (1) results in an equation that describes how the vector r = x1 − x2 between
Two-body_problem
Orbit with a fixed distance from the barycenter
r {\displaystyle r} is the distance from the center of mass. The orbit equation in polar coordinates, which in general gives r in terms of θ, reduces to:[clarification
Circular_orbit
Equilibrium points near two orbiting bodies
to the first two. The location of L1 is the solution to the following equation, gravitation providing the centripetal force: M 1 ( R − r ) 2 − M 2 r 2
Lagrange_point
Branch of engineering
mechanical systems. Mathematics – in particular, calculus, differential equations, and linear algebra. Electrotechnology – the study of electronics within
Aerospace_engineering
Periodic, three-dimensional orbit
Orbital period Orbital velocity Surface gravity Specific orbital energy Vis-viva equation Celestial mechanics Gravitational influences Barycenter Hill sphere
Halo_orbit
Movement during spaceflight
orbit, it is said to be coasting. The Tsiolkovsky rocket equation, or ideal rocket equation, can be useful for analysis of maneuvers by vehicles using
Orbital_maneuver
Region of space gravitationally dominated by a given body
like the patched conic approximation, have been described. The general equation describing the radius of the sphere r SOI {\displaystyle r_{\text{SOI}}}
Sphere of influence (astrodynamics)
Sphere_of_influence_(astrodynamics)
Horizontal angle from north or other reference cardinal direction
Orbital period Orbital velocity Surface gravity Specific orbital energy Vis-viva equation Celestial mechanics Gravitational influences Barycenter Hill sphere
Azimuth
Property of a dynamical system where solutions near an equilibrium point remain so
stability may be discussed for the solutions of differential equations or difference equations describing dynamical systems. The most important type is that
Lyapunov_stability
Type of spacecraft maneuver
every part of the rocket is proportional to its speed and, given this, the equation can be integrated (numerically or otherwise) to calculate the overall increase
Oberth_effect
Concept in celestial mechanics
asymptotically approach the hyperbolic excess speed v∞, satisfying the equation: v ∞ 2 = V 2 − v e 2 . {\displaystyle {v_{\infty }}^{2}=V^{2}-{v_{\text{e}}}^{2}
Escape_velocity
Gravitational loss of momentum and energy by bodies moving through surrounding matter
f(v)\ } is the number density distribution of the stars The result of the equation is the gravitational acceleration produced on the object under consideration
Dynamical_friction
Center of mass of multiple bodies orbiting each other
Orbital period Orbital velocity Surface gravity Specific orbital energy Vis-viva equation Celestial mechanics Gravitational influences Barycenter Hill sphere
Barycenter_(astronomy)
Term in geometry; longest and shortest semidiameters of an ellipse
a hyperbola, it is the one that does not intersect the hyperbola. The equation of an ellipse is ( x − h ) 2 a 2 + ( y − k ) 2 b 2 = 1 , {\displaystyle
Semi-major and semi-minor axes
Semi-major_and_semi-minor_axes
Concept in aerospace engineering
kg}})=0.7998} . The mass fraction plays an important role in the rocket equation: Δ v = − v e ln m f m 0 {\displaystyle \Delta v=-v_{\text{e}}\ln {\frac
Propellant_mass_fraction
Classical approach to the many-body problem of astronomy
primary body. In methods of general perturbations, general differential equations, either of motion or of change in the orbital elements, are solved analytically
Perturbation_(astronomy)
Asteroid
minorplanetcenter.net. Retrieved 12 January 2026. As calculated with the vis-viva-equation : v 2 = G M ( 2 r − 1 a ) {\displaystyle v^{2}=GM\left({2 \over r}-{1
2005_HC4
Problem in physics and celestial mechanics
Newton from astronomer John Flamsteed – Newton was able to produce an equation by straightforward analytical geometry, to predict a planet's motion; i
N-body_problem
Quasi-periodic orbital trajectory
the Three-Body Problem, and Space Mission Design" (PDF). International Conference on Differential Equations. Berlin: World Scientific. pp. 1167–1181.
Lissajous_orbit
Laws describing planetary orbits
System. The inverse-square law is a differential equation. The solutions to this differential equation include the Keplerian motions, as shown, but they
Kepler's laws of planetary motion
Kepler's_laws_of_planetary_motion
Space navigation technique
Orbital period Orbital velocity Surface gravity Specific orbital energy Vis-viva equation Celestial mechanics Gravitational influences Barycenter Hill sphere
Gravity_assist
Measure of the efficiency of a rocket based on its propellant's mass
The fraction on the left-hand side of this equation is the rocket's mass ratio by definition. This equation indicates that a Δv of n {\displaystyle n}
Rocket_mass_ratio
μ2, as a function of the distance and semi-major axis alone using vis-viva equation ( ξ ˙ 2 + η ˙ 2 + ζ ˙ 2 ) = v 2 = μ ( 2 r − 1 a ) {\displaystyle ({\dot
Tisserand's_criterion
Angle between a reference plane and the plane of an orbit
Orbital period Orbital velocity Surface gravity Specific orbital energy Vis-viva equation Celestial mechanics Gravitational influences Barycenter Hill sphere
Orbital_inclination
Measure of aircraft/spacecraft efficiency
and fuel fractions in aviation, see Fuel Fraction. Tsiolkovsky rocket equation "SOYUZ-2 Launch Vehicle / Power Characteristics". JSC SRC Progress. Retrieved
Payload_fraction
Amount by which an orbit deviates from a perfect circle
also helps understand its near-circular orbits and other unique features. Equation of time Tidal circularization While its orbit was initially hyperbolic
Orbital_eccentricity
Specifies the orbit of an object in space
Orbital period Orbital velocity Surface gravity Specific orbital energy Vis-viva equation Celestial mechanics Gravitational influences Barycenter Hill sphere
Argument_of_periapsis
Law of physics and chemistry
conserved so long as the masses did not interact. He called this quantity the vis viva or living force of the system. The principle represents an accurate statement
Conservation_of_energy
Parameter of Keplerian orbits
\over {1-e\,}}}\tan {E \over 2}\,\right)} Alternatively, a form of this equation was derived by R. Broucke and P. Cefola that avoids numerical issues when
True_anomaly
Branch of astronomy
more than force, and developing a method to use a single polar coordinate equation to describe any orbit, even those that are parabolic and hyperbolic. This
Celestial_mechanics
the separation. This equation applies only to radial parabolic trajectories, for general parabolic trajectories see Barker's equation. t ( x , w ) = arcsin
Radial_trajectory
Region in which an astronomical body dominates the attraction of satellites
about the primary (assuming that m ≪ M {\displaystyle m\ll M} ). The above equation can also be written as m r H 2 − M r 2 ( 1 − r H r ) − 2 + M r 2 ( 1 −
Hill_sphere
hole Virtual image Virtual particle Virtual state Virtual work Vis-viva equation Vis viva Viscimetry Viscoelasticity Viscometer Viscosity Viscosity of amorphous
Index_of_physics_articles_(V)
Standard surface gravity
proportional to the radius r {\displaystyle r} . Solving for mass, this equation can be written as g = G ( 4 π ρ 3 ) 2 / 3 M 1 / 3 {\displaystyle g=G\left({\frac
Surface_gravity
Equation describing a state of matter under a given set of conditions
In physics and chemistry, an equation of state is a thermodynamic equation relating state variables, which describe the state of matter under a given
Equation_of_state
Elliptical orbit used to move a spacecraft from one circular orbit to another
Orbital period Orbital velocity Surface gravity Specific orbital energy Vis-viva equation Celestial mechanics Gravitational influences Barycenter Hill sphere
Transfer_orbit
position. The phase angle can be converted in terms of time using Kepler's Equation: t = T 1 2 π ( E − e 1 sin E ) {\displaystyle t={\frac {T_{1}}{2\pi }}(E-e_{1}\sin
Orbit_phasing
Physical quantity
Latin: vis viva, or living force, which defined as the product of the mass of an object and its velocity squared; he believed that total vis viva was conserved
Energy
caloric culture. Bernoulli made a connection with Gottfried Leibniz's vis viva principle, an early formulation of the principle of conservation of energy
History_of_thermodynamics
1738 book on fluid mechanics by Daniel Bernoulli
conservation of energy, as received from Christiaan Huygens's formulation of vis viva (Latin for living forces). The book describes the theory of water flowing
Hydrodynamica
Equation of the state of a hypothetical ideal gas
The ideal gas law, also called the general gas equation, is the equation of state of a hypothetical ideal gas. It is a good approximation of the behavior
Ideal_gas_law
French mathematician, mechanical engineer, and scientist
he prefixed the factor +1⁄2 to Gottfried Wilhelm Leibniz's concept of vis viva, thus specifying today's kinetic energy. Coriolis was born in Paris in
Gaspard-Gustave_de_Coriolis
Equations in thermodynamics
Thermodynamics is expressed by a mathematical framework of thermodynamic equations which relate various thermodynamic quantities and physical properties
Thermodynamic_equations
Relations between flows and forces, or gradients, in thermodynamic systems
difference) coefficients are equal. For many kinetic systems, like the Boltzmann equation or chemical kinetics, the Onsager relations are closely connected to the
Onsager_reciprocal_relations
Common thermodynamic equations and quantities in thermodynamics, using mathematical notation, are as follows: Many of the definitions below are also used
Table of thermodynamic equations
Table_of_thermodynamic_equations
Swiss mathematician and physicist (1700–1782)
results are consequences of a single principle, namely, the conservation of vis viva, an early version of the conservation of energy. This was followed by a
Daniel_Bernoulli
State function whose change relates to the system's maximal work output
‘free heat, combined heat, and heat released’ into ‘vis viva, loss of vis viva, and increase of vis viva.’" In this manner, the total mass of caloric in a
Thermodynamic_free_energy
History of the physical concept
the Soul. The modern concept of kinetic energy emerged from the idea of vis viva (living force), which Gottfried Wilhelm Leibniz defined over the period
History_of_energy
In thermodynamics, Bridgman's thermodynamic equations are a basic set of thermodynamic equations, derived using a method of generating multiple thermodynamic
Bridgman's thermodynamic equations
Bridgman's_thermodynamic_equations
Force distributed over an area
towards the surface element, while the normal vector points outward. The equation has meaning in that, for any surface S in contact with the fluid, the total
Pressure
Type of thermodynamic potential
products are all in their thermodynamic standard states, then the defining equation is written as Δ G ∘ = Δ H ∘ − T Δ S ∘ {\displaystyle \Delta G^{\circ }=\Delta
Gibbs_free_energy
Heat required to raise the temperature of a given unit of mass of a substance
{\displaystyle \rho } is expressed as molar density in the above equation, this equation reduces simply to Mayer's relation, C p , m − C v , m = R {\displaystyle
Specific_heat_capacity
Thermodynamic cycle for spark ignition piston engines
u=(C_{\text{v}})(\delta T)} Inserting the specific heat equation into the thermal efficiency equation (Equation 2) yields. η = 1 − ( C v ( T 4 − T 1 ) C v ( T
Otto_cycle
and connects the equation of state to one or more thermodynamic energy properties. Here we refer to it as a "thermodynamic equation of state." The fundamental
Internal_pressure
Dutch physicist (1837–1923)
physicist who received the Nobel Prize in Physics in 1910 "for his work on the equation of state for gases and liquids." Van der Waals started his career as a
Johannes Diderik van der Waals
Johannes_Diderik_van_der_Waals
Concept in general relativity and quantum field theory
death paradox Loschmidt's paradox Synergetics Theories Caloric theory Vis viva ("living force") Mechanical equivalent of heat Motive power Key publications
Black_hole_thermodynamics
Mathematical model which approximates the behavior of real gases
gas concept is useful because it obeys the ideal gas law, a simplified equation of state, and is amenable to analysis under statistical mechanics. The
Ideal_gas
Non-hypothetical gases whose molecules occupy space and have interactions
Redlich–Kwong equation is another two-parameter equation that is used to model real gases. It is almost always more accurate than the van der Waals equation, and
Real_gas
proving the theorem. Thomson, W. (1849). Notes on hydrodynamics. V. On the vis-viva of a liquid in motion. Camb. Dubl. Math. J, 4, 90-94. Kelvin, W. T. B.
Kelvin's minimum energy theorem
Kelvin's_minimum_energy_theorem
Parameter used to calculate the volume change of a fluid or solid in response to pressure
is known as the equation of state denoted by some function F {\displaystyle F} . The Van der Waals equation is an example of an equation of state for a
Compressibility
Tendency of matter to change volume in response to a change in temperature
reduces intermolecular forces and allows greater atomic displacement. If an equation of state is available, it can be used to predict the values of the thermal
Thermal_expansion
Property of a thermodynamic system
system. To derive a generalised entropy balanced equation, we start with the general balance equation for the change in any extensive quantity θ {\textstyle
Entropy
Thermodynamic cycle
death paradox Loschmidt's paradox Synergetics Theories Caloric theory Vis viva ("living force") Mechanical equivalent of heat Motive power Key publications
Atkinson_cycle
Idealized thermodynamic cycle
efficient. Rearranging the right side of the equation gives what may be a more easily understood form of the equation, namely that the theoretical maximum efficiency
Carnot_cycle
Scalar physical quantities representing system states
be D equations for each potential, resulting in a total of D 2D equations of state because 2D thermodynamic potentials exist. If the D equations of state
Thermodynamic_potential
Thermodynamic potential
induce pressure changes. It is also frequently used to define fundamental equations of state of pure substances. The concept of free energy was developed
Helmholtz_free_energy
Law of thermodynamics establishing the conservation of energy
energy, for example by emphasising Gottfried Wilhelm Leibniz's concept of vis viva, mv2 (mass times speed squared), as distinct from Isaac Newton's momentum
First_law_of_thermodynamics
Physical law for definition of temperature
death paradox Loschmidt's paradox Synergetics Theories Caloric theory Vis viva ("living force") Mechanical equivalent of heat Motive power Key publications
Zeroth_law_of_thermodynamics
Correction factor which describes the deviation of a real gas from ideal gas behavior
values are usually obtained by calculation from equations of state (EOS), such as the virial equation which take compound-specific empirical constants
Compressibility_factor
Physical law for entropy and heat
the universe is in disagreement with Maxwell's equations – then so much the worse for Maxwell's equations. If it is found to be contradicted by observation
Second_law_of_thermodynamics
of material are eliminated, in a recast reduced form of a constitutive equation. The reduced variables are defined in terms of critical variables. The
Theorem of corresponding states
Theorem_of_corresponding_states
Thermodynamic process in which no mass or heat is exchanged with surroundings
Differentiating equation (a3) yields Equation (a4) is often expressed as dU = n CV dT because CV = α R . Now substitute equations (a2) and (a4) into equation (a1)
Adiabatic_process
Volume of fluid which passes per unit time
v = flow velocity, A = cross-sectional vector area/surface. The above equation is only true for uniform or homogeneous flow velocity and a flat or planar
Volumetric_flow_rate
German physicist and physiologist (1821–1894)
of physics. He became interested in electromagnetism, and the Helmholtz equation is named for him. Although he made no major contributions to this field
Hermann_von_Helmholtz
Body of matter in a state of internal equilibrium
contents of the system must be accounted for in an appropriate balance equation. The volume can be the region surrounding a single atom resonating energy
Thermodynamic_system
Model that is used to predict the performance of steam turbine systems
{{\dot {W}}_{\text{turb}}}{{\dot {Q}}_{\text{in}}}}} Each of the next four equations[1] is derived from the energy and mass balance for a control volume. Q
Rankine_cycle
Thermodynamic process
n is the polytropic index, and C is a constant. The polytropic process equation describes expansion and compression processes which include heat transfer
Polytropic_process
Italian mathematician and physicist (1707–1775)
discussed the question of the parallelogram of forces in the context of the vis viva controversy. In 1752, he published the short treatise De usu motus tractorii
Vincenzo_Riccati
Physical property of matter
p}}\right)_{T}dp\right]} For constant pressure ( d p = 0 ) {\displaystyle (dp=0)} the equation simplifies to: C p = d Q d T | p = const = ( ∂ U ∂ T ) p + p ( ∂ V ∂ T
Heat_capacity
Thermodynamic cycle
death paradox Loschmidt's paradox Synergetics Theories Caloric theory Vis viva ("living force") Mechanical equivalent of heat Motive power Key publications
Miller_cycle
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