Editing Geostationary transfer orbit (section)

==Technical description==

GTO is a [[highly elliptical orbit|highly elliptical Earth orbit]] with an [[apogee]] (the point in the orbit of the moon or a satellite at which it is furthest from the earth) of {{convert|42164|km|mi|abbr=on}},<ref>
{{cite book |title=Fundamentals of Astrodynamics and Applications |last=Vallado |first=David A. |year=2007 |publisher=Microcosm Press |location=Hawthorne, CA  |pages=31 }}
</ref> or a height of {{convert|35786|km|mi|abbr=on}} above sea level, which corresponds to the geostationary altitude. The period of a standard geosynchronous transfer orbit is about 10.5 hours.<ref>{{cite book|author=Mark R. Chartrand|title=Satellite Communications for the Nonspecialist|url=https://books.google.com/books?id=MM0d2cMUWbEC&pg=PA164|year=2004|publisher=SPIE Press|isbn=978-0-8194-5185-9|page=164}}</ref> The [[argument of perigee]] is such that apogee occurs on or near the equator. Perigee can be anywhere above the atmosphere, but is usually restricted to a few hundred kilometers above the Earth's surface to reduce launcher delta-V (<math>\Delta V</math>) requirements and to [[Orbital decay|limit the orbital lifetime of the spent booster]] so as to curtail space junk.

If using low-thrust engines such as [[electrical propulsion]] to get from the transfer orbit to geostationary orbit, the transfer orbit can be [[Supersynchronous orbit|supersynchronous]] (having an apogee above the final geosynchronous orbit). However, this method takes much longer to achieve due to the low thrust injected into the orbit.<ref>{{Cite book | last = Spitzer | first = Arnon | title = Optimal Transfer Orbit Trajectory using Electric Propulsion | publisher = [[USPTO]] | date = 1997 | url = https://patents.google.com/patent/US5595360}}</ref><ref>{{Cite book | last = Koppel | first = Christophe R.| title = Method and a system for putting a space vehicle into orbit, using thrusters of high specific impulse | publisher = USPTO | date = 1997 | url = https://patents.google.com/patent/US6213432}}</ref>
The typical launch vehicle injects the satellite to a supersynchronous orbit having the apogee above 42,164&nbsp;km. The satellite's low-thrust engines are thrusted continuously around the geostationary transfer orbits. The thrust direction and magnitude are usually determined to optimize the transfer time and/or duration while satisfying the mission constraints. The out-of-plane component of thrust is used to reduce the initial inclination set by the initial transfer orbit, while the in-plane component simultaneously raises the perigee and lowers the apogee of the intermediate geostationary transfer orbit. In case of using the Hohmann transfer orbit, only a few days are required to reach the geosynchronous orbit. By using low-thrust engines or electrical propulsion, months are required until the satellite reaches its final orbit.

The [[orbital inclination]] of a GTO is the angle between the orbit plane and the Earth's [[equator|equatorial plane]]. It is determined by the [[latitude]] of the launch site and the launch [[azimuth]] (direction). The inclination and eccentricity must both be reduced to zero to obtain a geostationary orbit. If only the [[Orbital eccentricity|eccentricity]] of the orbit is reduced to zero, the result may be a geosynchronous orbit but will not be geostationary. Because the <math>\Delta V</math> required for a plane change is proportional to the instantaneous velocity, the inclination and eccentricity are usually changed together in a single maneuver at apogee, where velocity is lowest.

The required <math>\Delta V</math> for an inclination change at either the ascending or descending [[orbital node|node]] of the orbit is calculated as follows:<ref name="Curtis, H.D. 2010 pp. 356-357">Curtis, H. D. (2010) [[Orbital Mechanics for Engineering Students]], 2nd Ed. Elsevier, Burlington, MA, pp. 356–357.</ref>
:<math>\Delta V = 2 V \sin \frac{\Delta i}{2}.</math>

For a typical GTO with a [[semi-major axis]] of 24,582&nbsp;km, [[perigee]] velocity is 9.88&nbsp;km/s and [[apogee]] velocity is 1.64&nbsp;km/s, clearly making the inclination change far less costly at apogee. In practice, the inclination change is combined with the orbital circularization (or "[[apogee kick]]") burn to reduce the total <math>\Delta V</math> for the two maneuvers. The combined <math>\Delta V</math> is the vector sum of the inclination change <math>\Delta V</math> and the circularization <math>\Delta V</math>, and as the sum of the lengths of two sides of a triangle will always exceed the remaining side's length, total <math>\Delta V</math> in a combined maneuver will always be less than in two maneuvers. The combined <math>\Delta V</math> can be calculated as follows:<ref name="Curtis, H.D. 2010 pp. 356-357"/>
:<math>\Delta V = \sqrt{ V_{t,a}^2 + V_\text{GEO}^2 - 2 V_{t,a} V_\text{GEO} \cos \Delta i},</math>

where <math>V_{t,a}</math> is the velocity magnitude at the apogee of the transfer orbit and <math>V_\text{GEO}</math> is the velocity in GEO.