In astrodynamics, the orbital-energy-invariance law or vis-viva equation, is one of the equations that model the motion of orbiting bodies.

It represents the principle that the difference between the aggregate work of the accelerating forces of a system and that of the retarding forces is equal to one half the vis viva (Latin meaning – living force) accumulated or lost in the system while the work is being done.

The formula for orbital-energy-invariance law or Vis-viva equation is given as,

v = the relative speed of the two bodies

G = the gravitational constant

M = mass of the central body

r = the distance between the orbiting bodies

a = the length of the semi-major axis(a > 0 for ellipses, a = ∞ or 1/a = 0 for parabolas,

                                                            and a < 0 for hyperbolas)

The specific energy (energy per unit mass) of a space vehicle is composed of two components, the specific potential energy and the specific kinetic energy.

The specific orbital energy of two orbiting bodies is the constant sum of their mutual potential energy and their total kinetic energy, divided by the reduced mass.

Under standard assumptions, specific orbital energy (ϵ) of elliptic orbit is negative and the orbital energy conservation equation, also known as the Vis-viva equation, for this orbit can take the form as,

ϵ = -μ /2a

ϵ = <0

v =  Orbital speed of body traveling along an elliptical orbit.

μ  = is the standard gravitational parameter

r = the distance between the orbiting bodies

a = the length of the semi-major axis

In astrodynamics, the orbital speed of a body traveling along an elliptic orbit can be computed from the orbital-energy-invariance law. It is one of the equations that model the motion of orbiting bodies.

Under standard assumptions the orbital speed (v) of a body traveling along an elliptic orbit is computed from the orbital-energy-invariance law as,

v =  orbital speed of a body traveling along an elliptic orbit

μ = is the standard gravitational parameter

r = the distance between the orbiting bodies

a = the length of the semi-major axis

The orbital period is the time taken for a given object to make one complete orbit around another object. It is applicable in astronomy.

The orbital period T (in seconds) of two bodies orbiting each other in a circular or elliptic orbit is given as,

Where,

μ = G × M

G is the gravitational constant,

M is the mass of the more massive body.

a = the orbit’s semi-major axis in meters

μ = the standard gravitational parameter in m³/s²

All bounded orbits where the gravity of a central body is strong are elliptical in nature. An example 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 centre of gravity of mass M is given as,

To properly use this formula, the units must be consistent – M must be in kilograms, and r must be in meters

v = velocity

G = gravitational constant

M = mass of the body

r = distance from the centre of gravity of mass M