The origin elliptical Kepler orbit has four parameters we need to compute two \Delta V burns for our transfer orbit:

* Apogee r_a

* Perigee r_p

* Inclination i

* argument of perigee \omega

We can compute the semimajor axis a and the eccentricity e from the apogee and perigee

The angle of the ascending and descending nodes do not affect the \Delta V burns, though they do affect when we make our burns so we insert into the desired destination region of the M288 orbit.

Blue curve:

An orbit in the horizontal X,Y plane

apogee = apoapsis = 6
perigee= periapsis = 2
semimajor axis = 4
eccentricity = 0.5

Red curve:

A similar orbit with a 45 degree argument of perigee

(rotate by \omega around vertical Z axis):

Green curve:

A similar orbit with a 30 degree inclination

(rotate by i around horizontal X axis):


note that the ascending and descending nodes, where the orbit crosses the equatorial plane, do not have the same radius from the origin

\mu

earth gravitational constant

3.986004418e14 m³/s²

T

sidereal orbital period (s)

\Large { { 2 \pi } \over \omega }

a

semimajor axis (m)

0.5 ( r_a + r_p )

\large \left( \mu \left( T \over { 2 \pi } \right)^2 \right)^{1/3}

e

eccentricity (unitless)

\Large { { r_a } \over { a } } - 1

\Large { { r_a - r_p } \over { r_a + r_p } }

r_p

perigee, periapsis (m)

a ( 1 - e )

r_a

apogee, apoapsis (m)

a ( 1 + e )

v_o

orbit velocity (m/s)

\large \sqrt{ \mu ( 1 - e^2 ) / a }

v_p

perigee velocity (m/s)

( 1 + e ) v_0

v_a

apogee velocity (m/s)

( 1 - e ) v_0

\theta

true anomaly, orbit angle from perigee

radians or degrees

r

radius (m)

\large a ( 1 - e^2 ) / ( 1 + e \cos( \theta ) )

v_{\perp}

velocity perpendicular to radius (m/s)

\large v_0 ( 1 + e \cos( \theta ) )

v_r

radial velocity (m/s)

\large e v_0 \sin( \theta )