#format jsmath
= Hitchhiker Orbit Decay =

Hitchhiker, the first server sky experiment, will deploy from a GTO transfer stage into a highly elliptical orbit, using a high-ISP electric engine like the Accion electrospray thruster.  This orbit is purposely designed to decay quickly during high solar activity years, so it will not leave debris in orbit in the long term.

The orbit will decay in two regimes:

 .1 Highly elliptical to near-circular by perigee drag in the denser atmosphere close to Earth.  Perigee around 6,800 kilometers radius, apogee around 42,000 kilometers radius

 .2 Rapidly decaying near-circular orbit, descending quickly through the ISS orbital radius (6792 to 6772 km on 20161004 ) to reentry.

=== Decaying Near-Circular Orbit ===

This is symmetrical and easier to solve.  Presume the thinsat (or group of thinsats) has a "perpendicular" area-to-mass ratio of A/M = 5 m^2^/kg edge on, but that the thinsat is tumbling randomly, for an average A/M of 2.5 m^2^/kg.  Assume the coefficient of drag $ C_D $ is 2.  

For an orbit radius of $ r $, the orbit velocity is  $ \sqrt{ \mu / r } $ and the air density is $ \rho( r ) $.   The drag force is $ 0.5 C_D A \rho v^2 $, and the drag power is $ dE/dt = 0.5 C_D A \rho ( r )  v^3 $.  The circular orbit energy is $ E = M ( v^2 / 2 - \mu / r ) ~ = ~ - M \mu / 2 r $, so $ dE/dr = M \mu / 2 r^2 $ and $ dr/dE = 2 r^2 / M \mu $  Thus:

$ { \Large { dr \over dt } = } { \LARGE { \left( dr \over dE \right) \left( dE \over dt \right) = \left( { 2 r^2 } \over { M \mu } \right) { \large ( 0.5 C_D A \rho( r ) v^3 ) = } \left( { C_D A \rho( r ) } \over { M \mu } \right) \left( { \large r^2 } \left( \mu \over r \right)^{3/2} \right) } } { \Large = C_D (A/M) \rho( r ) \sqrt{ \mu r } } $

At 414 km equatorial altitude ( 6792 km radius ), the average air density is 1.4e-12 kg/m^3^.  $ \mu $ = 3.9860044e14 m^3^/s^2^ amd r = 6.792e6 m, so the orbit descent rate is about 0.36 m/s, or 1.3 kilometers per hour.   At solar max, the air density is about 3.1e-12 kg/m^3^, so the orbit descent rate is about 0.8 m/s or 2.8 km/hr .  To minimize the ISS collision rate, we should end the experiment during a solar maximum.