Vis-Viva Orbital Velocity Equation: perigee, apogee & escape speed

Speed in an ideal two-body orbit

Vis-viva follows from conservation of orbital energy in the Newtonian two-body model. It gives speed, not the velocity direction. The example neglects drag, thrust, oblateness and other bodies. The central-body approximation uses mu=GM for a spacecraft of negligible mass.

All radii are measured from the central body's center, not its surface. With mu in km^3/s^2 and radii in km, speed is km/s. For the ellipse specified by perigee and apogee, the selected radius must lie between them.

At perigee and apogee the velocity is transverse, so their radius-speed products agree before rounding. At intermediate points angular momentum uses the transverse component, not total speed. The displayed four-decimal speeds will not give an exact equality when multiplied.

Vis-Viva Elliptical Orbital Velocity v = SQRT(mu * ((2 / r) - (1 / a)))

The semi-major axis is the mean of positive perigee and apogee radii. Perigee cannot exceed apogee; r must lie in that closed interval. Equal apsidal radii describe a circle. These guards do not check collision with a real planet or atmosphere.

=IF(COUNT(B2:B5)<>4,"Invalid inputs",IF(OR(B2<=0,B3<=0,B4<B3,B5<B3,B5>B4),"Invalid inputs",ROUND(SQRT(B2 * ((2 / B5) - (1 / ((B3 + B4) / 2)))), 4)))

Local Parabolic Escape Speed v_esc = SQRT(2 * mu / r)

This is the local zero-energy escape speed in the same isolated gravitational model. It uses only positive mu and radius and does not require the ellipse inputs. It is not a launch delta-v budget.

=IF(COUNT(B2,B5)<>2,"Invalid inputs",IF(OR(B2<=0,B5<=0),"Invalid inputs",ROUND(SQRT(2 * B2 / B5), 4)))

Further reading

• Microsoft Support — SQRT function
• NASA Technical Reports — Vis-Viva Equation