In astrodynamics, orbit phasing is the adjustment of the time-position of spacecraft along its orbit, usually described as adjusting the orbiting spacecraft's true anomaly. Orbital phasing is primarily used in scenarios where a spacecraft in a given orbit must be moved to a different location within the same orbit. The change in position within the orbit is usually defined as the phase angle, ϕ, and is the change in true anomaly required between the spacecraft's current position to the final position. The phase angle can be converted in terms of time using Kepler's Equation:
t = T 1 2 π ( E − e 1 sin E ) {\displaystyle t={\frac {T_{1}}{2\pi }}(E-e_{1}\sin E)}
E = 2 arctan ( 1 − e 1 1 + e 1 tan ϕ 2 ) {\displaystyle E=2\arctan \left({\sqrt {\frac {1-e_{1}}{1+e_{1}}}}\tan {\frac {\phi }{2}}\right)}
where
t is defined as time elapsed to cover phase angle in original orbit T1 is defined as period of original orbit E is defined as change of eccentric anomaly between spacecraft and final position e1 is defined as orbital eccentricity of original orbit φ is defined as change in true anomaly between spacecraft and final position
This time derived from the phase angle is the required time the spacecraft must gain or lose to be located at the final position within the orbit. To gain or lose this time, the spacecraft must be subjected to a simple two-impulse Hohmann transfer which takes the spacecraft away from, and then back to, its original orbit. The first impulse to change the spacecraft's orbit is performed at a specific point in the original orbit (point of impulse, POI), usually performed in the original orbit's periapsis or apoapsis. The impulse creates a new orbit called the “phasing orbit” and is larger or smaller than the original orbit resulting in a different period time than the original orbit. The difference in period time between the original and phasing orbits will be equal to the time converted from the phase angle. Once one period of the phasing orbit is complete, the spacecraft will return to the POI and the spacecraft will once again be subjected to a second impulse, equal and opposite to the first impulse, to return it to the original orbit. When complete, the spacecraft will be in the targeted final position within the original orbit. To find some of the phasing orbital parameters, first one must find the required period time of the phasing orbit using the following equation.
T 2 = T 1 − t {\displaystyle T_{2}=T_{1}-t}
where
T1 is defined as period of original orbit T2 is defined as period of phasing orbit t is defined as time elapsed to cover phase angle in original orbit Once phasing orbit period is determined, the phasing orbit semimajor axis can be derived from the period formula:
a 2 = ( μ T 2 2 π ) 2 / 3 {\displaystyle a_{2}=\left({\frac {{\sqrt {\mu }}T_{2}}{2\pi }}\right)^{2/3}}
where
a2 is defined as semimajor axis of phasing orbit T2 is defined as period of phasing orbit μ is defined as Standard gravitational parameter From the semimajor axis, the phase orbit apogee and perigee can be calculated:
2 a 2 = r a + r p {\displaystyle 2a_{2}=r_{a}+r_{p}}
where
a2 is defined as semimajor axis of phasing orbit ra is defined as apogee of phasing orbit rp is defined as perigee of phasing orbit Finally, the phasing orbit's angular momentum can be found from the equation:
h 2 = 2 μ r a r p r a + r p {\displaystyle h_{2}={\sqrt {2\mu }}{\sqrt {\frac {r_{a}r_{p}}{r_{a}+r_{p}}}}}
where
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