Orbit Circularization

What is the ΔV needed for apogee insertion into a circular equatorial orbit from a launch loop transfer orbit, and for perigee insertion from a space elevator transfer orbit?

The destination orbit has a radius of r_d and a velocity of v_d = \sqrt{ \mu / r_d } were \mu = 398600.4418 km³ / s².

The launch loop calculation is fairly simple.

An 80 kilometer breech altitude launch loop defines a transfer orbit with a perigee r_p = 6378 + 80 km = 6458 km . The semimajor axis is a = 0.5 * ( r_p + r_d ) , the eccentricity e = ( r_d - r_p ) / ( r_d + r_p ) , the characteristic velocity is v_0 = \sqrt{ \mu / ( a * ( 1 - e^2 ) ) } , and the apogee velocity is v_a = ( 1 - e ) v_0 . Combining and simplifying:

{v_a}^2 = ( 1 - e )^2 {v_0}^2 = ( \mu / a ) ( 1 - e )^2 / ( 1 - e^2 )

{v_a}^2 = ( 2 \mu r_p ) / ( r_d ( r_d + r_p ) )

v_a = \sqrt{ ( 2 \mu r_p ) / ( r_d ( r_d + r_p ) ) }

\Delta V = v_d - v_a

The space elevator calculation is more complicated.

For transfer orbits below geosynchronous, the apogee of the orbit is the release radius r_a , and the angular velocity is \omega_E = 2 \pi / P_{sidereal} , where P_{sidereal} is the sidereal day, 86164.0905 seconds. We want to find r_a and the perigee velocity v_p . We will have to work backwards.

v_a = \omega_E r_a

{v_a}^2 / 2 \mu = 1 / r_a - 1 / ( r_a + r_p )

Solve for r_p , perigee radius, the candidate r_d

r_p = \Large { r_a \over { \Large { { \huge { 2 \mu } \over { {r_a}^3 {\omega_E}^2 } } } - 1 } }

Testing empirically with a spreadsheet:

So, a 20 iteration binary search of r_a between 29700 km and 53000 km will converge r_p on our target r_d