Problem 1. Hohmann Transfer and Plane Changes. Geostationary orbits are circular with a period equal to the length of a sidereal day (23 hr and 56 m), and an inclination of i = 0o. The minimum inclination that can be achieved for a spacecraft launched from the surface of the earth is equal to the latitude of the launch site. For American satellites, which are typically launched from Cape Canaveral (latitude = 28.50N), this means it is impossible to directly launch into an equatorial orbit i = 00. With this in mind, in order to send a spacecraft into a geostationary orbit from Florida, it is first launched into a circular Low Earth Orbit (LEO) with an altitude of 200 km and inclination i = 28.50. It then performs a series of orbital maneuvers to change the orbital plane and transfer to GEO. Consider the following four sets of possible maneuvers: I. A plane change burn is made to change the LEO orbit to a circular orbit with the same altitude but an inclination of i = 0o. In other words, the magnitude of v remains the same but its direction is changed. Now in the correct plane, the satellite is allowed to make one complete orbit, and then a Hohmann transfer is used to send the orbit out to GEO. II. In the original inclination of i = 28.50, a Hohmann transfer is performed to send the satellite to GEO. After the orbit has been circularized at GEO, the spacecraft is allowed to complete one orbit. A plane change is then performed to change the orbit to an inclination of i = 0o. III. At LEO, a plane change is implemented that simultaneously changes the inclination of the orbit to i = 0o and sets it on course for a Hohmann transfer in the new plane. In other words, both the magnitude and direction of v change. At GEO altitude, a burn is applied to circularize the orbit. IV. At LEO, a burn is applied to send the satellite on a Hohmann transfer in the same plane (i = 28.50). Once at GEO altitude, a burn is applied that simultaneously changes the inclination of the orbit to i = 0o and circularizes it. The first two maneuvers (I and II) require three burns (denoted 1, 2, and 3), and the maneuvers in the second set (III and IV) each require two burns (denoted 1 and 2). In all cases, let the initial circular orbit at LEO have a longitude of ascending nodes at 22 = 0o and assume the satellite has just enough fuel to complete the specified maneuvers.