Energy requirements altitude change

Referring to Figure 5.6, to initiate the transfer maneuver the spacecraft must be 180 degrees away from the desired point in the GSO orbit. A that point, a rocket burn is required to increase the spacecraft velocity from 25,573 ft/s to 33,643 ft/s, an incremental velocity of 8,070 ft/s (2,460 m/s). The spacecraft is now in an elliptical trajectory towards the 19,323 nautical miles (35,786 km) apogee. When apogee is reached, the elliptical orbital velocity is 5,235 ft/s (1,596 m/s). That is slower than the 10,088 ft/s (3,075 m/s) required for a GSO circular orbit. So at apogee a rocket burn provides a 4,853 ft/s (1,479 m/s) velocity increment necessary to circularize the orbit, otherwise the spacecraft will continue along its elliptical trajectory. The total velocity increment is 12,923 ft/s (3,939 m/s). To return to LEO, the opposite sequence of events is necessary. Again at the orbital location opposite the location point in the LEO orbit, a retro-burn of minus 4,853 ft/s (1,479 m/s) velocity is necessary to slow the spacecraft to the elliptical orbit apogee velocity of 5,235 ft/s (1,596 m/s). When approaching the 100 nautical mile altitude, the elliptical orbit speed is approaching 33,643 ft/s (10,254m/s). In order to achieve a 100 nautical mile circular orbit, a retro-burn of minus 8,070 ft/s (2,460 m/s) is necessary to reach the 100 nautical mile circular orbit speed of 25,573 ft/s (7,795 m/s). For a round trip a total of four rocket firings are required for a total incremental velocity of 25,846 ft/s (7,878 m/s), or greater than the incremental velocity to reach LEO!

So, to change orbital altitude requires the expenditure of energy. The amount of the energy depends on the altitude change desired. The incremental velocity to move from a 100 nautical mile or 200 km orbital altitude is given in Figure 5.7. The incremental velocity curve is very non-linear. A 6,000 ft/s (1,829 m/s) incremental velocity will permit an altitude change of about 3,000 nautical miles (5,556 km). However a burn of twice the velocity increment, 12,000 ft/s (3,658 m/s) will permit an altitude change of about 13,000 nautical miles (24,076 km), or 4.3 times greater.

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