Changing Orbits to M288

How do we get from any arbitrary orbit to M288? The following techniques may not be optimal for minimal \Delta V , but they set an upper bound. We will assume Kepler orbits and a circular M288 destination orbit, and heavy objects unaffected by light pressure. Actual thinsat orbits are elliptical and precess over a year with J2 and light pressure.

http://upload.wikimedia.org/wikipedia/commons/thumb/e/eb/Orbit1.svg/2000px-Orbit1.svg.png

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.

Maneuvers use the least \Delta V farther out, so we will use different strategies depending on whether the apogee of the origin orbit is higher or lower than the M288 orbit r_288 .

Describing an arbitrary Kepler Orbit

attachment:orb01.png

Blue curve:

An orbit in the horizontal X,Y plane

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

attachment:orb02.png

Red curve:

A similar orbit with a 45 degree argument of perigee

(rotate by \omega around vertical Z axis):

attachment:orb04.png

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

gnuplot source and you will need to crop and convert and resize using gimp to reproduce the above images.

\mu

earth gravitational constant

3.986004418e14 m3/s2

T

sidereal orbital period (s)

{ 2 \pi } \over \omega

a

semimajor axis (m)

0.5 ( r_a + r_p )

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

e

eccentricity (unitless)

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

{ { r_a - r_p } \over { r_a + r_[ } }

r_p

perigee, periapsis (m)

a ( 1 - e )

r_a

apogee, apoapsis (m)

a ( 1 + e )

v_o

orbit velocity (m/s)

\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)

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

v_{\perp}

velocity perpendicular to radius (m/s)

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

v_r

radial velocity (m/s)

e v_0 \sin( \theta )

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