Some spacecraft can be placed directly into their mission orbit by the launch vehicle, and thus have no need to perform orbit transfers. However, other spacecraft have to transfer between orbits to reach their final destination. This process of orbit transfer usually involves the use of a large rocket engine onboard the spacecraft, and such a system is referred to as primary

G. Swinerd, How Spacecraft Fly: Spaceflight Without Formulae, DOI: 10.1007/978-0-387-76572-3_9, © Praxis Publishing, Ltd. 2008

propulsion. If a spacecraft needs such a system, then the mass of the rocket hardware and of the required propellant has a major effect on the overall mass of the spacecraft. A good example of a spacecraft that requires such a primary propulsion system onboard is one of the communication satellites that we talked about in Chapter 2. They can be injected into a low Earth orbit (LEO) by the launch vehicle, but then need to be boosted to the high geostationary Earth orbit (GEO) before they can begin operations. This is usually achieved by using the Hohmann transfer. This strategy, which was invented by Walter Hohmann, is illustrated in Figure 9.1. Hohmann was one of those extraordinary individuals you find in the history of spaceflight. During the early 1900s he was a city architect by profession, in Essen, Germany. But in his free time he devoted his energies to thinking about interplanetary spaceflight. He published his work, including the details of his orbital transfer method, in 1925, at a time when the first Earth satellite was still over a quarter of a century away. His transfer method has been used hundreds of times in placing spacecraft into GEO because it does the job using the minimum amount of rocket fuel, which. means that the overall mass of the spacecraft is minimized and the launch costs can be reduced.

Referring to Figure 9.1, if we assume that the spacecraft is placed initially in LEO by the launcher, how does the Hohmann transfer take us to GEO? To do this, the spacecraft's primary engine is fired twice—once at point 1, to boost it into the geostationary transfer orbit (GTO), and then again at point 2 to push it into the final GEO. The elliptical GTO becomes a sort of bridge, spanning the space between the low orbit and the high mission orbit. Focusing on the first engine firing at point 1, we know from our basic orbits in Chapter 2 that the speed of the spacecraft at this point in the circular LEO

Figure 9.1: The Hohmann transfer between two circular orbits in the same plane. In this case, the spacecraft engine is fired at point 1, to transfer from LEO to GTO. The engine is then fired again at point 2, to take the vehicle into the GEO.

GEO altitude:

i 35,800 km

Figure 9.1: The Hohmann transfer between two circular orbits in the same plane. In this case, the spacecraft engine is fired at point 1, to transfer from LEO to GTO. The engine is then fired again at point 2, to take the vehicle into the GEO.

GEO altitude:

i 35,800 km is about 8 km/sec (5 miles/sec). Recalling Newton's cannon on the mountaintop (Chapter 2), we know that the speed at point 1 at the perigee (the low point of an elliptic orbit) of the GTO is higher. This is very much related to our discussion about what happens if we increase the barrel speed of Newton's cannon beyond the circular orbit speed. With some simple calculations we can determine that the speed at point 1 in the GTO is about 10 km/sec (6.2 miles/sec). Thus if the rocket engine firing at point 1 increases the speed of the spacecraft by 2 km/sec (1.2 mile/sec), then the vehicle will transfer from the LEO to the GTO. A similar calculation shows that the speed increase required at point 2 is about 1.5 km/sec (0.9 miles/sec) to transfer between the GTO and the GEO.

The speed change produced by a rocket burn is referred to as a AV (pronounced "delta vee''), and one of the prime jobs of the mission analysis team is to calculate the total mission AV required to take the spacecraft from launch pad to final mission orbit. Mission analysts spend a lot of time calculating AVs because it is directly related to the amount of rocket fuel required. The first person to realize this was Konstantin Tsiolkovsky, another spaceflight visionary who worked as a high school mathematics and science teacher in Russia around the turn of the 20th century. He published his rocket equation in 1903, in what was arguably the first mathematical treatise on rocket science. This equation calculates the rocket fuel mass for a particular orbit transfer directly from the corresponding AV, which is an important calculation for the spacecraft designer. Mission analysts should also try to minimize the AV, because minimum AV means minimum fuel mass; in turn, minimum fuel mass means maximum payload mass onboard the spacecraft, which means that the overall effectiveness of the spacecraft in achieving its objective is enhanced. You may recall a similar argument in Chapter 5 when we discussed maximizing the payload of a launch vehicle.

We can use Tsiolkovsky's rocket equation to estimate the rocket fuel mass required for the transfer from LEO to GEO. The total A Vof about 3.5 km/sec (2.2 miles/sec) requires that about 70% of the initial mass of the spacecraft in LEO needs to be propellant. This leaves only 30% of the initial mass as hardware (payload and subsystems), which poses a problem for the designer. To help overcome this, it is common practice for the launcher to inject the spacecraft directly into the GTO, so that only the 1.5 km/sec AV at point 2 is handled by the spacecraft's own primary propulsion. In this case, only about 40% of the initial mass needs to be rocket fuel, giving the designer a much more reasonable 60% of hardware mass to play with.

In our discussion of the Hohmann transfer, we used the LEO to GEO journey as an example. But this type of transfer can be used to move a spacecraft between any two circular orbits that share the same plane, for example, between two LEOs a few hundreds of kilometers apart, where the AV might be a few hundred meters per second. The original application that Hohmann had in mind was the transfer between the orbits of two planets, say Earth to Jupiter, where the AV might be on the order of 10 km/sec (6.2 miles/sec).

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