This horizontal acceleration is also used to move from one orbit to another, using an elliptical transfer orbit. When the Space Shuttle has to rendezvous from a low orbit, r0V §
and with, for instance, the Space Station, it accelerates along its orbit. Instead of travelling faster around the orbit, the orbit rises; it has in fact become a slightly elliptical orbit, the perigee being where the thrust was turned on. This elliptical orbit intersects with the Space Station orbit at the point where the Station is located when the Shuttle arrives there—if the burn was actuated at the correct time. Once at the Station, the Shuttle has to accelerate, again horizontally, in order to match its speed to that of the Station. This acceleration effectively puts the Shuttle into a circular orbit at the new altitude.
It should be clarified that the velocity of a spacecraft in a circular orbit decreases with the square root of the radius. So the Space Station is moving more slowly in its orbit than is the Shuttle in its lower orbit. However, for an elliptical orbit the velocity at apogee is lower than the intersecting circular orbit, because of the exchange of kinetic energy for potential energy as the spacecraft rises. At the same time, the velocity of the elliptical orbit is greater than that of the intersecting circular orbit at perigee. So to transfer from a low orbit to a higher orbit, two velocity increments are necessary: one at the perigee of the elliptical transfer orbit, and another at the apogee.
For convenient reference, some of the equations for the velocity of tangential circular and elliptic orbits are presented here.
where r0 is the perigee radius, r2 is the apogee radius, Vi is the elliptic orbit velocity at perigee, and V2 is the elliptic orbit velocity at apogee. These are shown in Figure 1.9.
It will perhaps be obvious from the foregoing that the manoeuvre to 'catch up' with a spacecraft in the same orbit is quite complicated. Simply accelerating will not cause the Shuttle to move faster round the orbit; it will put it into an elliptical orbit, which will pass above the target. The Shuttle needs to decelerate and drop to a lower and faster orbit; and then, at the correct point in the lower orbit it should accelerate again to bring it up to the target. Both of these manoeuvres are elliptical orbit transfers. A further acceleration is then needed to match the target speed; that is, to circularise the orbit of the shuttle. These orbits need only be separated from one another by a few tens of kilometres, and the velocity changes are small.
A decrease in velocity when a spacecraft is in a circular orbit causes it to enter an elliptical orbit. The apogee is at the same altitude as the circular orbit, and the perigee is determined by the velocity decrease. At perigee, the spacecraft velocity is greater than the corresponding new circular orbit, so a further decrease is needed. This is the initial manoeuvre when a spacecraft returns to Earth: atmospheric drag for a circular orbit for a parabolic escape orbit for a parabolic escape orbit for elliptic orbits for elliptic orbits
Orbit altitude, tq (km)
Figure 1.9. Injection velocity and altitude.
Orbit altitude, tq (km)
takes over before perigee is reached, and the resultant deceleration causes the trajectory to steepen continuously until it intersects with the Earth's surface.
On an airless body like the Moon, the elliptical orbit taken up should intersect with the surface at the desired landing point, and the vehicle has to be brought to rest using thrust from the motors. The Apollo 11 descent ellipse did not pass through the correct landing point, because the non-spherical components of the Moon's gravitational field were not known accurately. Armstrong had to take over control, and by using lateral thrust from the motors he guided it to the correct point.
Having discussed how a spacecraft can move from one orbit to a higher orbit, it is possible to see how a spacecraft can leave the surface of a planet and enter into a low orbit around that planet. For the Moon it is a reversal of the descent just described. For minimum energy expenditure, it would, in theory, be better to launch in a horizontal direction, regarding the launch point as being on a circular orbit at zero altitude. The velocity of the spacecraft, at rest on the surface, is wrong for the circular orbit, otherwise it would be hovering above the ground; so, during the burn, a significant vertical component of velocity is needed, while horizontal velocity is gained. There is also the question of mountains. The Apollo lunar module took off with a short vertical segment, before moving into a horizontal trajectory. Once in an elliptical orbit with an apogee at the required altitude, the spacecraft coasts towards it. At apogee, a further burn circularises the orbit.
For launch from Earth, the atmosphere is a significant problem. Although the density of air drops rapidly with height, the velocity of a spacecraft is large, and drag is proportional to the square of the velocity. Below 200 km there is sufficient drag to make an orbit unstable. In fact, to have a lifetime measured in years, an orbit needs to be above 500 km. This means that a significant proportion of the stored chemical energy in a rocket has to be used to raise the spacecraft above the atmosphere.
Atmospheric drag also slows the rocket in the lower atmosphere. For this reason, spacecraft are launched vertically; height is gained, and the drag in the dense lower atmosphere is minimised, because the rocket is not yet moving very fast. Once the densest part of the atmosphere is passed—at about 30 km—a more inclined trajectory can be followed. It cannot be horizontal because the velocity is not yet high enough. Ultimately, sufficient velocity is reached for an elliptical trajectory to the desired altitude to be followed, and the spacecraft can then coast. At apogee a further horizontal acceleration is needed to enter the circular orbit.
During a launch, the velocity is never high enough to attain a circular orbit at an intermediate altitude; this occurs only after the final injection has taken place. Intermediate orbits would require a still larger horizontal velocity, because of the inverse dependence of velocity on radius. Thus the motion of the rocket before final injection is along a trajectory which is always steeper than the free space ellipse, and it intersects the Earth's surface. When a launch fails, this becomes obvious. The nonoptimum nature of the launch trajectory, compared with a transfer ellipse, means that a good deal of the energy of the rocket is lost to the Earth's gravitational field. This gravity loss is dealt with in Chapter 5.
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