Of the important components of space fightthe ascent, the longdistance travel through outer space, and the return to Earth (the landing)we want to address only the most critical component at this point: the ascent. The ascent represents by far the greatest demands placed on the performance of the propulsion system and is also, therefore, of critical importance for the structure of the entire vehicle.
Figure 19. Vertical ascent"steep ascent"of a space rocket.
Key: 1. Climbing velocity=0; 2. Climbing altitude that is supposed to be reached; 3. Free ascent (without power as a "hurl upwards"): the climbing velocity decreases gradually as a result of the decelerating effect of the Earth's gravity; 4. Measure for the climbing velocity at various altitudes; 5. Climbing velocity="highest velocity of climbing"; 6. Power ascent: the climbing velocity increases continuously thanks to the accelerating effect of the propulsion system; 7. Launch.
For implementing the ascent, two fundamental possibilities, the "steep ascent" and "flat ascent," present themselves as the ones mentioned at the beginning in the section about movement in the gravity fields of outer space. In the case of the steep ascent, the vehicle is lifted in at least an approximately vertical direction. During the ascent, the climbing velocity, starting at zero, initially increases continuously thanks to the thrusting force of the reaction propulsion system (Figure 19); more specifically, it increases until a high climbing velocity is attainedwe will designate it as the "maximum velocity of climbing"such that now the power can be shut off and the continued ascent, as a "hurl upward," can continually proceed up to the desired altitude only under the effect of the kinetic energy that has been stored in the vehicle.
In the case of the flat ascent, on the other hand, the vehicle is not lifted vertically, but in an inclined (sloped) direction, and it is a matter not so much of attaining an altitude but rather, more importantly, of gaining horizontal velocity and increasing it until the orbiting velocity necessary for free orbital motion and consequently the "stable state of suspension" are attained (Figures 5 and 20). We will examine this type of ascent in more detail later.
Figure 20. "Flat ascent" of a space rocket. The expenditure of energy for the ascent is the lowest in this case.
Key: 1. Free circular orbit; 2. Earth; 3. Earth rotation; 4. Vertical direction; 5. Inclined direction of launch; 6. This altitude should be as low as possible!; 7. Ascent curve (an ellipse or parabola)
First, however, we want to examine some other points, including the question: How is efficiency varying during the ascent? For regardless how the ascent takes place, the required final velocity can only gradually be attained in any case, leading to the consequence that the travel (climbing) velocity of the space rocket will be lower in the beginning and greater later on (depending on the altitude of the final velocity) than the velocity of expulsion. Accordingly, the efficiency of the propulsion system must also be constantly changing during the power ascent, because the efficiency, in accordance with our previous definitions, is a function of the ratio of the values of the velocities of travel and expulsion (see Table 1, page 29). Accordingly in the beginning, it will only be low, increasing gradually with an increasing climbing velocity, and will finally exceed its maximum (if the final velocity to be attained is correspondingly large) and will then drop again.
In order to be able to visualize the magnitude of the efficiency under these conditions, the "average efficiency of the propulsion system" hrm resulting during the duration of the propulsion must be taken into consideration. As can be easily seen, this efficiency is a function, on the one hand, of the velocity of expulsion c, which we want to assume as constant for the entire propulsion phase, and, on the other hand, of the final velocity v' attained at the end of the propulsion period.
The following formula provides an explanation on this point:
Table 4 was prepared using this formula.
Ratio of the final Average efficiency of
velocity v' to the the propulsion system
velocity of expulsion hrm during the
c: acceleration phase
hrm hrm in percentages
0 0 0
0.2 0.18 18
0.6 0.44 44
1 0.58 58
1.2 0.62 62
1.4 0.64 64
1.591.8 0.65 65
2 0.64 64
2.2 0.63 63
2.6 0.61 61
3 0.54 54
4 0.47 47
5 0.30 30
6 0.17 17
7 0.09 9
The table shows the average efficiency of the propulsion system as a function of the ratio of the final velocity v' attained at the end of the propulsion phase to the velocity of expulsion c existing during the propulsion phase, that is, a function of v'/c. Accordingly by way of example at a velocity of expulsion of c=3,000 meters per second and for a propulsion phase at the end of which the final velocity of v=3,000 meters per second is attained (that is, for v'/c=1), the average efficiency of the propulsion system would be 58 percent. It would be 30 percent for the final velocity of v=12,000 meters per second (that is, v'/c=4), and so on. In the best case (that is, for v'/c=1.59) in our example, the efficiency would even attain 65 percent for a propulsion phase at a final velocity of v'=4,770 meters per second.
In any case it can be seen that even during the ascent, the efficiency is generally still not unfavorable despite the fluctuations in the ratio of the velocities of travel and expulsion.
Figure 21. As long as the vehicle has to be supported (carried) by the propulsion system during the ascent, the forward thrust of the vehicle is decreased by its weight.
Key: 1. Direction of flight (ascent); 2. Total reactive force; 3. Remaining propulsive force available for acceleration; 4. Weight of the vehicle; 5. Direction of expulsion (exhaust).
Besides the efficiency problem being of interest in all cases, a second issue of extreme importance exists especially for the ascent. As soon as the launch has taken place and, thus, the vehicle has lifted off its support (solid base or suspension, watersurface, launch balloon, etc.), it is carried only by the propulsion system (Figure 21), somethingaccording to the nature of the reactive forcethat depends on to a continual expenditure of energy (fuel consumption). As a result, that amount of propellants required for the liftoff is increased by a further, not insignificant value. This condition lasts only untildepending on the type of ascent, steep or flateither the necessary highest climbing velocity or the required horizontal orbiting velocity is attained. The sooner this happens, the shorter the time during which the vehicle must be supported by the propulsion system and the lower the related propellant consumption will be. We see then that a high velocity must be attained as rapidly as possible during the ascent.
Figure 22. During the duration of propulsion, forces of inertia are activated in the vehicle due to the acceleration of the vehicle (increase in velocity) caused by propulsion; the forces manifest themselves for the vehicle like an increase in gravity.
Key: 1. Actual acceleration of climb; 2. Reaction; 3. Normal weight; 4. Force of inertia; 5. Total increased effect of gravity (equals the total reactive force of the propulsion system).
However, a limit is soon set in this regard for space ships that are supposed to be suitable for transporting people. Because the related acceleration always results in the release of inertial forces during a forced velocity increase (as in this case for the propulsion system) and not caused solely by the free interaction of the inertial forces. These forces are manifested for the vehicle during the ascent like an increase in gravity (Figure 22) and may not exceed a certain level, thus ensuring that the passengers do not suffer any injuries. Comparison studies carried out by Oberth as well as by Hohmann and previous experiences in aviation (e.g., during spiral flights) indicate that an actual acceleration of climb up to 30 m/sec2 may be acceptable during a vertical ascent. In this case during the duration of propulsion, the vehicle and its contents would be subjected to the effect of the force of gravity of four times the strength of the Earth's normal gravity. Do not underestimate what this means! It means nothing less than that the feet would have to support almost four times the customary body weight. Therefore, this ascent phase, lasting only a few minutes, can be spent by the passengers only in a prone position, for which purpose Oberth anticipated hammocks.
Taking into account the limitations in the magnitude of the acceleration, the highest climbing velocity that would be required for the total separation from the Earth can be attained only at an altitude of approximately 1,600 km with space ships occupied by humans during a vertical ascent. The rate of climb is then around 10,000 meters per second and is attained after somewhat more than 5 minutes. The propulsion system must be active that long. In accordance with what was stated previously, the vehicle is supported (carried) by the propulsion system during this time, and furthermore the resistance of the Earth's atmosphere still has to be overcome. Both conditions cause, however, an increase of the energy consumption such that the entire energy expenditure necessary for the ascent up to the total separation from the Earth finally becomes just as large as if an ideal highest velocity of around 13,000 meters per second would have to be imparted in total to the vehicle. Now this velocity (not the actual maximum climbing velocity of 10,000 meters per second) is critical for the amount of the propellants required.
Somewhat more favorable is the case when the ascent does not take place vertically, but on an inclined trajectory; in particular, when during the ascent the vehicle in addition strives to attain free orbital motion around the Earth as close to its surface as practical, taking the air drag into account (perhaps at an altitude of 60 to 100 km above sea level). And only thenthrough a further increase of the orbiting velocitythe vehicle works its way up to the highest velocity necessary for attaining the desired altitude or for the total separation from the Earth ("flat ascent," Figure 20).
The inclined direction of ascent has the advantage that the Earth's gravity does not work at full strength against the propulsion system (Figure 23), resulting, therefore, in a greater actual acceleration in the case of a uniform ideal acceleration (uniform propulsion)which, according to what has been previously stated, is restricted when taking the wellbeing of the passengers into account. The greater acceleration results in the highest velocity necessary for the ascent being attained earlier.
However, the transition into the free orbital motion as soon as possible causes the vehicle to escape the Earth's gravity more rapidly than otherwise (because of the larger effect of the centrifugal force). Both conditions now cause the duration to be shortened during which the vehicle must be carried by the propulsion system, saving on the expenditure of energy as a result. Consequently, the ideal highest velocity to be imparted to the vehicle for totally separating from the Earth is only around 12,000 meters per second when employing this ascent maneuver, according to Oberth. In my opinion, however, we should come closest to the actually attainable velocity in practice when assuming an ideal highest velocity of approximately 12,500 meters per second.
Figure 23. Acceleration polygon for: 1.) vertical ascent, 2.) inclined ascent, 3.) flat ascent. It can clearly be seen that the actual acceleration from 1.) to 3.) becomes greater and greater, despite a constant ideal acceleration (force of the propulsion system). (The acceleration polygon for 2.) is emphasized by hatched lines.)
Key: 1. Direction of the effect of the propulsion system; 2. Direction of the actual ascent; 3. Acceleration of gravity; 4. Ideal acceleration; 5. Actual acceleration.
Regardless of how the ascent proceeds, it requires in every case very significant accelerations, such that the vehicle attains a velocity of a projectile at an altitude of several kilometers. This conditionbecause of the thick density of the deepest layers of air closest to the surface of the Earthresults in the air drag reaching undesirably high values in the very initial phases of the ascent, something that is particularly true for space rockets without people on board. Considerably greater accelerations of climb can be employed in unmanned vehicles than in manned ones because health is not a consideration for the former.
To come to grips with this disadvantage, the launch will take place from a point on the Earth's surface as high as possible, e.g., from a launch balloon or another air vehicle or from a correspondingly high mountain. For very large space ships, however, only the latter option is possible due to their weight, even though in this case the launch would preferably be carried out at a normal altitude.