The Vis-viva equation is a fundamental equation used in orbital mechanics which relates the velocity of an object in an orbit around a planet, to its distance from the center of the planet and the semi-major axis of its orbit. The beauty of this equation is that a lot of deductions can be made for different relations between the semi major axis and the positional distance of the object.
Let’s see the equation first.
- v is the relative speed of the two bodies
- r is the distance between the two bodies
- a is the length of the semi-major axis (a > 0 for ellipses, a = ∞ or 1/a = 0 for parabolas, and a < 0 for hyperbolas)
- G is the gravitational constant
- M is the mass of the central body
The derivation of this equation uses simple conservation of momentum and geometrical properties of an orbit. If you are interested in the derivation, please check this.
Simple deductions from Vis-viva equation —
Orbital velocity: To determine the orbital velocity of an object in a circular orbit of radius r, just substitute a= r in the above equation and you get —
Escape velocity: To determine escape velocity the object will have to reach infinity starting from it’s position. So a = infinity and we get from vis viva equation —
Delta-v calculations: Vis-viva makes it easy to calculate the Delta-v required for orbit maneuvers. If you have find how Delta-v has to be imparted to an object, simply find the final and initial values of “V” for the final and initial orbit parameters. And subtracting Vinitial from Vfinal will give you the required Delta-v.