Orbital Insertion
Launch loops will be located south of the equator, to avoid storms and hurricanes associated with the Inter Tropical Convergence Zone (ITCZ). Launching due east from a launch loop below the equator places the apogee well north of the equatorial plane; this orbit will cross the equatorial plane at an orbital angle ±90° from perigee.
A previous attempt (Old_LaunchAzimuth) assumed that launching with an azimuth to the north would help orbital insertion by accomplishing this with one rocket firing. While this is possible for lower target orbits, it adds too much \delta V to the firing - the azimuth must be removed, and the transfer orbit is farther from the ideal Hohmann.
Instead, the loop should launch payloads at zero azimuth, due east. The orbit will still be inclined by the launch latitude, which must be corrected. However, angle changes are easier at lower velocities, that is, high in the orbit.
Since launch loop will throw payloads directly into transfer orbits, launch azimuth should be arranged to put the equatorial plane crossing near apogee as far from earth as possible. For logistical convenience and safety, the launch loop west station should be south of the intertropical convergence zone (ITC) in January, and the whole system should be erected when the rain and wind is minimized. A loop with a 2000 kilometer track and 300 kilometer inclines will be 23 degrees west to east.
What is the optimum azimuth for injection into a geostationary circular equatorial orbit (42164 km radius), or into a server sky M288 circular equatorial orbit (12789 km radius)?
Equatorial Launch, Two Impulse Hohmann
If we launch from the equator, latitude \phi = 0 , the best trajectory is a two impulse Hohmann, with the Launch Loop providing launch velocity v_L at an altitude a_L , radius from the center of the Earth r_L = R_E + a_L .
The launch vehicle will follow an elliptical transfer orbit to an apogee with radius r_A = r_D .
A Newtonian orbit with a perigee of r_L and an apogee of r_D has a semimajor axis r_S = { \Large { { r_D + r_L } \over 2 } } and an eccentricity e = { \Large { { r_D - r_L } \over { r_D + r_L } } } .
Let \theta be the angle from perigee; an equatorial elliptical transfer orbit will intersect the destination orbit where \theta = \pi .
The orbit radius is r = { \Large { { ( 1 + e ) r_L } \over { 1 + e \cos( \theta ) } } } .
The orbit's characteristic velocity is \large v_0 = { \Large { \sqrt{ { \mu \over 2 } \left( { 1 \over r_L } + { 1 \over r_P } \right) } } } .
Earth's standard gravitational parameter is \mu = 3.9860044e14 m³/s².
The velocity as a function of angle is v = v_0 ( 1 + e \cos( \theta ) , so perigee velocity (or absolute launch velocity) v_L = v_0 ( 1 + e ) and apogee velocity v_A = v_0 ( 1-e ) = \sqrt{\Large{{\mu \over{2 r_D}}\left({{r_L}\over{r_D}} +1 \right)}}.
The destination circular orbit velocity is v_D = \sqrt{\Large{\mu \over{r_D}}} . The orbital insertion \Delta V or "apogee kick" is v_D - v_A = \sqrt{\Large{\mu \over{r_D}}}-\sqrt{\Large{{\mu \over{2 r_D}}\left( 1 + {{r_L}\over{r_D}}\right)}} .
v_D - v_A = \sqrt{\Large{\mu \over{r_D}}} \left(1-\sqrt{\large{{1\over 2}\left(1+{{r_L}\over{r_D}}\right)}}\right) .
The perigee velocity is the exit velocity relative to the loop vector-summed with the Earth rotation velocity at the exit latitude, 7.292115e-5 radians/sec \times r_L \cos( \phi ) . For a loop exit altitude of 120 km, and an exit latitude of 5 degrees south, r_L = 6498 km and the rotation velocity at that altitude is 472 m/s.
This will get a drawing later
x is the direction of the perigee at the equator, pointing out of the page, in the equatorial plane.
y is the direction 90 degrees east of perigee orthogonal to x, pointing right on the page, in the equatorial plane.
z is north, pointing up on the page, orthogonal to x and y,and the equatorial plane.
The planar equatorial orbit has a radius vector of:
\left[\matrix{ r_x \\ r_y \\ r_z }\right] ~ = ~ \left[\matrix{ \cos( \theta ) \\ \sin( \theta ) \\ 0 }\right] { \Large { { (1 + e) r_L } \over { 1 + e \cos( \theta ) } } }
And a velocity vector of:
\left[\matrix{ v_x \\ v_y \\ v_z }\right] ~ = ~ \left[\matrix{ - \sin( \theta ) \\ \cos( \theta ) \\ 0 }\right] { v_0 ( 1 + e \cos( \theta ) ) }
Rotation by Launch Latitude
The Launch Loop exit will be in the southern hemisphere at latitude \phi , inclined towards the north, to avoid hurricanes and the inter tropical convergence zone (ITCZ). The eastern exit end of the loop can tolerate more weather than the western elevator end. Wind loads determine the area-to-mass scaling and the operating schedule of the loop, so less wind is better.
Since the loop is to the south, the equatorial orbit is inclined by the latitude \phi . That will put the perigee well north of the destination equatorial orbital plane. To compute the new position and velocity vectors, we rotate both components around the y axis by an angle of \phi . Another matrix multiplication in front of the planar rotation:
\left[\matrix{ r_x \\ r_y \\ r_z }\right] ~=~ \left[\matrix{ \cos( \phi ) & 0 & - \sin( \phi ) \\ 0 & 1 & 0 \\ \sin( \phi ) & 0 & \cos( \phi ) }\right] \left[\matrix{ \cos( \theta ) \\ \sin( \theta ) \\ 0 }\right] { \Large { { (1 + e) r_L } \over { 1 + e \cos( \theta ) } } }
\left[\matrix{ v_x \\ v_y \\ v_z }\right] ~=~ \left[\matrix{ \cos( \phi ) & 0 & - \sin( \phi ) \\ 0 & 1 & 0 \\ \sin( \phi ) & 0 & \cos( \phi ) }\right] \left[\matrix{ - \sin( \theta ) \\ \cos( \theta ) \\ 0 }\right] { v_0 ( 1 + e \cos( \theta ) ) }
Reducing the Matrices to a Unit Vector
\left[\matrix{ r_x \\ r_y \\ r_z }\right] ~=~ \left[\matrix{ \cos( \theta )\\ \sin( \theta ) \\ \sin( \phi ) \cos( \theta ) }\right] { \Large { { (1 + e) r_L } \over { 1 + e \cos( \theta ) } } }
\left[\matrix{ v_x \\ v_y \\ v_z }\right] ~=~ \left[\matrix{ -\cos( \phi ) \sin( \theta ) \\ \cos( \theta ) \\ -\sin( \phi ) \sin( \theta ) }\right] { v_0 ( 1 + e \cos( \theta ) ) }
Assuming there are many assets orbiting in the equatorial plane, this orbit is mostly above or below that plane, except where it crosses the plane, \cos( \theta ) = 0 , and \theta = \pm \pi/2 , 90 degrees from perigee and apogee. The radius at the plane crossing is r_{cross} = \left( { \Large {{2 r_P r_L} \over { r_P+ r_L}} } \right) .
Two delta V from the Transfer Orbit to the Destination Orbit
Circularizing the orbit in the equatorial plane involves two steps:
- 1: raise perigee using thrust at apogee
- 2: change the orbital plane with thrust at the equator crossings.
Raising perigee can be done with high specific impulse, high fuel efficiency electric thrusters, or with relatively low specific impulse chemical rockets. The delta V's are on the order of 1500 m/s
Kourou is located approximately 500 kilometres (310 mi) north of the equator, at a latitude of 5°10'. Ariane 5 GTO launch azimuth 80 degrees from north. Is the launch azimuth designed to clear the coast (running NW-SE) for safety reasons? The inclination of the GTO orbit is 0.5°