How to Calculate Orbital Velocity for Elliptical Orbits Using the Vis-Viva Equation

Help support this channel on Patreon:
/ astrophil
Join this channel to get access to perks:
/ @astrophil2000
Welcome, my name is Phil, and in this video, I will show you how to calculate the orbital velocity for an elliptical orbit using the vis-viva equation.
The Vis Viva equation is a key concept in orbital mechanics, used to calculate the speed of an object as it moves in orbit around a planet or star. It helps explain how an object’s velocity changes depending on its position in its orbit and is based on energy conservation.
An object moves faster when it's closer to the body it's orbiting, like a satellite speeding up as it approaches Earth or a planet moving faster when it's closer to the Sun. This happens because of gravity's stronger pull at shorter distances. When the object is farther away, its speed slows down.
The Vis Viva equation tells us the exact speed at specific points in the orbit and ties into Kepler's Laws, which describe the motion of planets. For example, Kepler's Second Law explains why planets move faster when they’re closer to the Sun and slower when they’re farther away.

[@AstroPhil2000]

Here is is the derivation for the vis-viva equation: ua-cam.com/video/RMekZWsOTn4/v-deo.html

[@karhukivi]

Interesting! If M1>>M2 and r=a, the the two formulae are identical. Is this an empirical formula or is it derived from the equation of an ellipse?

[@AstroPhil2000]

Yes, it becomes the circular velocity equation when r =a. It is derived from the specific orbital energy and the fact that it is a balance between the kinetic and potential energy.