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)

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 )

\omega

argument of perigee, radians

i

inclination, radians

Orbits in V-H coordinates

We are interested in delivering an object from an arbitrary starting orbit to M288, and the \Delta V impulses necessary. First we will describe the orbit in radial coordinates ( V, H ), then compute the two impulses:

There may be more optimal strategies, and we may use non-traditional propulsion systems like VASIMR engines or tethers to accomplish the \Delta V , but the calculations are easier in bulk if we treat all 10,000+ objects in the NORAD catalog the same way. Although I can't share the whole catalog here, you can sign up for it, and I will risk sharing some data for the hundreds of SL-6 Molniya third stage rocket bodies, which threaten the M288 orbits and whose aluminum tanks can provide many tons of ballast for server sky.

Let's do the math in Cartesian (x,y,z) coordinates. The z component is in the direction of the earth's north pole, with the x and y components confined to the equatorial plane.

Start with the basic Kepler orbit with the argument of perigee \omega (the angle from the x axis), in the equatorial plane:

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The orbit crosses the equatorial plane where \theta = \omega and \theta = \pi + \omega . Thus:

Because of the inclination i , the perpendicular velocity is out-of-plane, and has a Z (north-south) and an H (in-plane) component, like so:

Our goal is to add a \Delta V_1 , and end up with a velocity vector that is perigee (or apogee) of an elliptical equatorial orbit whose apogee (or perigee) is at M288, where we add \Delta V_2 and end up in a circular M288 orbit. We will choose the orbital pass and time the transfer so we pass through a gap in the M288 constellation. The constellation of light-pressured thinsats will be an elliptical orbit wrapped around the central, circular M288 orbit, so we will likely be able to pass "over or under" the elliptical orbit in the equatorial plane. That orbit is defined by:

The velocity change at r_1 is :

The total \Delta V_1 is the vector sum of those three orthogonal vector components.

The orbit insertion into the final orbit is simple - a thrust along the orbit path at (perigee/apogee), changing velocity from V_{H 3} (the velocity of the incoming orbit) to V_{M288} .

There are TWO possible starting positions and velocities to perform this calculation on, with two possible pairs of \Delta V . We can choose either pair, based on the smallest scalar total \Delta V , the smallest \Delta V_1 , the smallest \Delta V_2 , or some weighted average of the latter.

Molniya Rocket Body Transfer

121 SL-6 third stage rocket bodies remain in orbit after placing Molniya satellites into 12 hour, high eccentricity, high inclination orbits that dwell above northern Asia.

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