Arthur C. Clarke
An Introduction to Astronautics
First published by Temple Press, 1950.
First American edition by Harper & Brothers, 1951.
Second impression [slightly revised], 1951.
Third Impression, 1952.
This book is intended as a survey of the possibilities and problems of interplanetary flight, as far as they can be foreseen at the present day. Although the science of “astronautics” still belongs to the future, many of its basic conceptions will remain unaltered by the passage of time, and most of the fundamental techniques already exist in embryo. It is, for example, possible to calculate by quite simple methods the velocities and durations required for interplanetary journeys, irrespective of the physical means that may be used to accomplish them.
The attempt has been made throughout this book to keep the treatment quantitative, and to give exact values and magnitudes rather than vague generalities. Nevertheless, almost all mathematics has been relegated to the Appendix, and it is believed that the argument can be followed without undue difficulty even by readers with little mathematical or scientific training.
The approach throughout has been from the astronomical rather than the engineering point of view. The author makes no apologies for this, as there are now several excellent books on rocket technology, but none, at least in English, which develop the theory of astronautics in any detail.
The Earth’s Gravitational Field
Man is still essentially a two-dimensional creature: all his journeys in the vertical direction have so far been of negligible extent. It is, therefore, perhaps not surprising that very curious ideas persist about gravity – one of the commonest being that it ceases, more or less abruptly, at a definite distance from the Earth. The frequently-encountered phrase “beyond Earth’s gravity” is a good example of this survival from pre-Newtonian thinking.
Strictly speaking, no point in the Universe is “beyond Earth’s gravity”, which decreases as the inverse square of the distance and so becomes zero only at infinity. At the greatest heights yet attained by rocket, its value is still nearly 90% of that at sea-level, and one must go to an altitude of 2620 km (1630 miles) before it is even halved.
Over astronomical distances, however, the decrease is extremely rapid, as an inspection of Figure 1 will show. The point beyond which, for practical purposes, the Earth’s gravitation field may be neglected, depends entirely on the particular case being considered. As will be seen later, a body travelling at a very high speed quite close to the Earth will be far less affected by its field than a slow-moving body at a great distance. Thus the Earth is incapable of capturing a meteor skimming just outside the atmosphere at 50 km/sec, while it holds the Moon (moving at 1 km/sec) firmly chained in its orbit a thousand times farther away.
Since the work done in lifting a body of unit mass vertically against the Earth’s field is the product of distance times force, it follows that, for equal distances, this work decreases with height according to the inverse square law or the “g” curve of Figure 1. At ten radii from the Earth’s centre, moving a body through a given vertical distance requires only a hundredth of the energy needed to perform the same feat at sea-level. The total energy, E, required to lift unit mass from the Earth’s surface to “infinity” (or to a point where for all practical purposes gravity can be neglected) is clearly proportional to the area beneath the “g” curve in Figure 1. An integration (see Appendix) gives the surprisingly simple result: –
where g is the value of gravity at the Earth’s surface (981 cm/sec2 or 32.2 ft/sec2) and R is its radius (6360 kms or 3960 miles).
This equation makes possible a striking mental picture of the work involved in lifting or projecting a body completely away from the Earth. The energy expended in climbing a mile is something which can be visualized as not outside the range of normal experience, though only an Alpine guide would consider it a part of the day’s work. […] But as Equation II.2 shows, the escape from Earth is equivalent to a climb of one radius, or almost four thousand miles, under a gravity equal to its sea-level value.
This peculiarly simple law, which we will often invoke, holds for all planets and gravitating bodies. To take a case which, as will be seen later, is not as academic as it sounds, the escape from the Sun (whose radius and surface gravity are 109 and 28 times that of the Earth) is equivalent to a vertical climb of almost 109X28X4000 miles, or approximately 12,000,000 miles (say 20,000,000 kms) against one terrestrial gravity. In the same way, the work required to leave any other body in the Solar System may be easily calculated.
Our position here on the Earth’s surface may best be visualized by an analogy which will play an important part in later discussions. Since the escape from our planet is equivalent to a vertical ascent of four thousand miles against one gravity, we may picture ourselves as being at the bottom of a valley or crater four thousand miles deep, out of which we must climb if we are ever to leave Earth. The walls of this imaginary crater are at first very steep, but as Earth’s gravity weakens they become slowly less vertical and the ascent correspondingly easier. At very great distances (a hundred thousand miles or more) the slope becomes more and more nearly horizontal until at last we have, for all practical purposes, reached the level plain and can move in any direction with no appreciable expenditure of energy.
This imaginary “gravitational pit” has been accurately drawn in Figure 2, which shows the amount of work needed to reach “infinity” from any point within about 300,000 miles (500,000 kms) of the Earth. The figure must of course be regarded as three-dimensional, like the stem of an inverted wine-glass: its section is actually the rectangular hyperbola defined by Equation II.3.
In the same way, all other celestial bodies have their exactly similar gravitation pits. That of the Sun, as we have already seen, is 12,000,000 miles or 20,000,000 kms deep. The Moon’s, on the other hand, is only 170 miles (280 kms) deep, and is represented to scale by the small dimple far up the slope of the Earth’s field in Figure 2. If we imagine this diagram as showing the profiles of two adjacent valleys, it will be seen that the problem of escape from the Moon is enormously simpler than that of leaving the Earth.
There are, in principle, two main ways by which a body can be transferred from the Earth’s surface to infinity. It can be moved at a slow and more or less uniform speed, by the continuous application of some force; but this method, as will be seen later, is excessively wasteful of energy. Alternatively, it can receive the necessary kinetic energy in one instalment, as it were, by being given a velocity sufficient for it to “coast” up the slope of the gravitational crater under its own momentum before coming to rest. The velocity needed to do this is known as the escape or parabolic velocity: it is equal to √(2gR) (Equation II.4) and its numerical value at the Earth’s surface is 11.2 km/sec (7 miles/sec or 25,000 m.p.h.). This is also the velocity which a body would acquire during a fall to the Earth’s surface from a very great distance: it follows therefore that a spaceship leaving the Earth must not only reach this speed on the outward journey but must also neutralize it on the return, if it is to make a safe landing.
Escape velocity, though usually quoted for the Earth’s surface, naturally decreases with distance as a body starting at a considerable altitude would need less initial speed to reach infinity. The rate of decrease is rather slow and is also shown on Figure 1 (the curve being given by Equation II.5).
This curve gives the same sort of information as Figure 2, but in a more useful and immediately understandable form. It shows at a glance the vertical projection velocity needed, at any point, to send a body right away from the Earth – and, conversely, the velocity a body initially at rest would acquire in falling to that point from a great distance.
A planet’s escape velocity is one of its most important characteristics, and not only from the view-point of astronautics. It determines whether that planet can retain an atmosphere, for if the gas molecules have average speeds comparable with the escape velocity, the atmosphere will quickly leak away into space – as has happened in the case of the Moon and is happening for Mars. A table giving this value for the more important bodies in the Solar System will be found in Chapter 10.
Closely related to escape velocity is the conception of orbital or circular velocity, which is the speed at which a body would continue to circle the Earth indefinitely like a second Moon, its outward centrifugal force equalling the inward pull of gravity (just as one may whirl a stone at the end of a piece of string). The necessary speed to maintain a stable orbit at any distance from the Earth is easily calculated (Equation II.8) and near the Earth’s surface is about 7.9 km/s (18,000 m.p.h.) This is less than the corresponding escape velocity in the ratio 1:√2, a proportion which holds universally at all points. It is, therefore, much easier for a body to become a close satellite of Earth than to escape completely, a point which as we shall see later is of great importance. The conception of a rocket or other structure circling round the Earth forever with no expenditure of energy seems peculiarly difficult for the layman to understand – his usual reaction being: “Why doesn’t it fall down?” Perhaps if, like Jupiter, our planet had a dozen or so natural satellites at varying distances, the idea of a few artificial ones would be more readily accepted.
Circular velocity, like escape velocity, decreases slowly with distance according to an inverse square root law, and the values of both for points out to the Moon’s orbit are shown in Figure 1.
We have now considered the two simplest cases of movement possible for a body projected beyond the Earth’s atmosphere at a point where the only force acting is that of gravity. It now remains to consider the more general case, where the motion is neither radial nor circular.
To fix ideas, imagine a point just beyond the atmosphere and consider what happens when a body is given various horizontal speeds. At 7.9 km/sec (5 miles/sec) it will, as we have seen, travel round the Earth forever in a circular orbit. A lesser speed will make it impossible to maintain this orbit and it will eventually fall to Earth – though it may travel half-way round the planet before doing so.
If the original speed is in excess of the orbital velocity, then the body will move outwards. It will recede from the Earth along an elliptical path, gradually losing speed until at the point farthest from the Earth (“apogee”) its motion will again be tangential and it will be travelling at its lowest speed. Thereafter, unable to maintain itself at this distance, it will fall back with increasing velocity to its original point of projection (“perigee”) and will continue to retrace its path indefinitely.
As the initial velocity is increased, the ellipse becomes more and more elongated and the furthest point moves steadily towards infinity. (Figure 4.) When escape velocity is reached, the ellipse changes into a parabola and the body never returns. (This is the reason why escape velocity is often called parabolic velocity.) For speeds greater than this value, the body moves away from Earth along a hyperbola, which at very great speeds indeed (over 100 km/sec) would become almost a straight line.
The rocket motor is unique among prime movers in two respects – its independence of any external medium, and its ability to generate colossal thrusts and powers. Both of these characteristics are required for space-flight, the former for obvious reasons, the second because very large masses fuel are necessary for interplanetary missions.
No detailed discussion of the purely engineering aspects of the rocket will be given in this book, as several excellent works on the subject are now available. (See Bibliography.) But it may be as well to spend some little time considering why the rocket, unlike all other forms of propulsion, can operate in space, which for all practical purposes is a perfect vacuum.
All forms of locomotion depend of reaction. Surface vehicles, through the friction of their wheels, try to thrust the Earth away from them and, to an immeasurably small extent, succeed; but such is the disparity of masses that the effect on the Earth is unnoticeable. Aeroplanes and ships operate by giving momentum to a mass of air or water, thus acquiring equal momentum in the opposite direction. This is most clearly seen in the case of the jet aircraft, the rocket’s closest relative. The jet collects a large quantity of air, which it heats and expels at a very great velocity, thereby obtaining a thrust which is proportional to the product of the jet’s mass and its increase in speed. If it could carry its own oxygen supply, instead of obtaining it from air, a jet aircraft could then operate as a self-contained unit capable of functioning in a vacuum – and would, indeed, then be a type of rocket.
It cannot be too strongly emphasised that neither the rocket nor the jet obtains thrust by “pushing on the air behind”, as a great many people believe. All the “push” occurs inside the combustion chambers and exhaust nozzles, and the subsequent adventures of the burnt gases once they have left the system can have no effect on it whatsoever. It is sometimes helpful to think of the rocket’s still burning and expanding gases as thrusting against the already burnt gas further down the nozzle, so producing a recoil in exactly the same way as the charge in a gun, driving the bullet forwards, forces the gun backwards with equal momentum. The rocket may indeed be regarded as a sort of continuously operating gun firing out a stream of gas instead of solid material.
The velocity which a rocket can attain, after burning all its fuel, is clearly dependent on the speed with which the gases leave the nozzle (the exhaust or jet velocity) and the amount of fuel ejected. These quantities are connected by the simple relation (see Appendix), the most important in the whole of rocketry: –
From these equations, it follows that although the rocket’s speed increases in direct proportion to the exhaust speed, it does so only slowly with increase in mass-ratio. This result is best shown graphically, as in Figure 5.
These curves show that, for a given value of exhaust velocity c, impossibly high values of mass-ratio R would be needed if the rocket is to attain a final speed much greater than c. For R=e=2.718…, the rocket’s final speed would equal its jet speed. This value presents no engineering difficulties: it would mean, for example, building a rocket of empty mass 1 ton, carrying 1.72 tons of fuel. V.2 did considerably better than this, having a mass-ratio of over 3 with its normal one ton warhead, and almost 4 if carrying only light meteorological instruments. But to double the final speed, with the same exhaust velocity, would mean squaring the mass-ratio, i.e. increasing it from 2.72 to 7.4 (e2). This would be a considerable technical feat, though perhaps not an impossible one. A rocket capable of travelling three times as fast as its exhaust would need a mass ratio of 20 (e3), which may be regarded as quite impracticable, since it would require that 95% of the machine’s total mass be fuel and only the remaining 5% be devoted to payload, structure, motors, etc. It appears, therefore, that no simple rocket can be built to travel more than 2 or 3 times as fast as its exhaust. When, as in the next chapter, the inevitable losses to air-resistance and gravitational retardation are considered, it will be seen that the value 2 is more likely to be the upper limit.
The attainment of high exhaust speeds is therefore the first concern of the rocket engineer, and is a problem involving chemistry, thermodynamics, metallurgy, and a great deal of still somewhat empirical mathematics. The absolute maximum of exhaust velocity available from any given fuel is easily calculated by assuming that the motor converts all the propellant’s energy of combustion into kinetic energy at 100% efficiency. The figure obtained in this way, however, has very little relation with reality. In practice, owing to the inevitable losses in any heat engine, no more than about 70% of this theoretical or ideal velocity can ever be achieved: with current designs, the figure is about 55%.
The best-known rocket fuel (that used in V.2) is the alcohol-liquid oxygen combination with an ideal exhaust velocity of about 4.2 km/sec (14,000 f.p.s.). The value realized so far in practice is only some 2.25 km/sec (7,500 f.p.s.).
Considerably more powerful fuels exist, with ideal exhaust velocities up to 6.5 km/sec (21,000 f.p.s.). These involve “combustion”, not with oxygen, but with still more reactive element fluorine. When such propellants are fully developed (which will require many years of research and considerable improvements in metallurgy to permit motor operation at high temperatures) it is possible that exhaust speeds of around 4.5 km/sec (15,000 f.p.s.) may be obtained. It can be shown that, irrespective of the energy contained in any possible propellant, this figure is near the absolute limit which may be achieved with chemical rockets, since much higher values would require impossible temperatures and pressures in the motor.
A table giving some of the more important propellants known at present or likely to be used in the near future is given overleaf, together with their exhaust velocities in km/sec. These values can only be somewhat approximate as they would vary from motor to motor or in the same motor from sea-level to vacuum.
Since we have seen that it is not practicable to build a rocket capable of travelling more than about twice as fast as its exhaust, it would therefore seem that – even when chemical propellants and motors have been developed to the ultimate – we cannot hope to build rockets capable of attaining speeds of over 9 km/sec (20,000 m.p.h.) This would be sufficient to achieve circular velocity, but insufficient for an escape from the Earth. In later chapters, however, we will see that there are various ways of avoiding this difficulty, notably by the principle of “step construction”.
The Problem of Escape by Rocket
The ground covered in the last two chapters now enables us to discuss, in a quantitative manner, the problem of escape from Earth by rocket. We have seen that if a body can attain a speed of more than 11.2 km/sec (or less if it is already at a great height) then it will travel away from the Earth indefinitely with no further expenditure of power. And we have seen how to calculate the final velocity reached by a rocket after combustion of its fuel.
Equation III.2, on which our previous calculations were based, was however derived for the theoretical case of a rocket acted upon by no forces except its own exhaust. A machine rising in the Earth’s atmosphere will experience two retarding forces which may be considerable – air resistance, and the downward pull of gravity. The corrected equation for the rocket’s final velocity after a vertical ascent (at the moment of fuel cut-off) must therefore be written
where t is the time of flight and VD is the total velocity loss due to air-drag. The acceleration of gravity, g, is of course assumed to be constant during the period of powered ascent: this is nearly true in most cases that are likely to occur, for a rocket would burn most of its fuel while still relatively near the Earth.
It is quite impossible to give a general formula for the air-resistance loss: it depends on the shape and size of the rocket, the acceleration characteristics of its path, and the height of take-off. Since a high-velocity rocket spends only a short part of its powered trajectory in the relatively dense lower atmosphere, it does not reach considerable speeds until the air is already very rarefied, and towards the end of the burning period air-drag is quite negligible.
Considerably more important, it will be noticed, is the gravitation loss term, gt. Since this depends directly on the time of operation of the motors, it can be reduced only by short burning times and hence high accelerations. The maximum acceleration which a large rocket can employ is, however, limited by the thrust of its motors. At take-off, when it was fully loaded, V.2 had an acceleration of only 1g and the value for “Viking” is about the same. (The leisurely ascent of a giant rocket invariably surprises those who are only acquainted with the common or back-garden variety, with their accelerations of 50g or more.) When the propellant is nearly exhausted, liquid-fuel rockets may reach accelerations of about 10g unless the motor thrust is reduced. For manned rockets, such “throttling back” might be desirable, though as mentioned in Chapter 9 a properly protected man can tolerate higher linear accelerations than it would be practical to stress a large machine to withstand.
In order to reach escape velocity, therefore, thrust periods of several minutes would be required – and each minute of vertical ascent means a loss to gravity of 0.6 km/sec or 1,300 m.p.h. This would be a very serious matter, but fortunately substantial savings can be effected by using non-vertical departure curves – “synergic curves”, as will be explained later.
Since any rocket escaping from the Earth’s neighbourhood must reach 11.2 km/sec, we can substitute this constant value in Equation III.2.a. and see how the mass-ratio R varies with the assumed exhaust velocity c. The equation – ignoring “gravitation loss” for the moment – then becomes
Neglecting gravitational loss assumes that the rocket’s acceleration is infinite (t=0) and this limiting case is shown by the curve n=∞ in Figure 7. It will be seen that with the best present-day fuels (c less than 2.5 km/sec) a mass-ratio of about 100 would be required – about ten times the limit that is practicable even with a very small payload.
When exhaust velocities of around 4.5 km/sec are available, which should be the case when high-energy fluorine-based fuels can be handled, the mass-ratio necessarily would be reduced to rather more than 10. This is still too high a value, thus confirming the conclusion already reached in the last chapter that it is impossible to build a single-stage, chemically-propelled rocket to escape from the Earth, even with no payload.
When one allows for the gravitational loss caused by the rocket’s finite acceleration, the picture is even blacker. Assuming that the rocket maintains a constant acceleration of ng (where n is not likely to exceed values of 5 to 10) it is easy to show (see Appendix) that the required mass-ratio for escape is given by the increased value
This function has been plotted in Figure 7 for various values of n, and the enormous losses incurred when n is low will be readily seen. The reductio ad absurdum case occurs when n=0, and the rocket has merely enough thrust to hang motionless in the air above its launching site until the fuel is exhausted!
Figure 7 will repay careful study, since it shows at one and the same time the paramount importance of high exhaust velocities and high accelerations. If a 10 km/sec fuel were available, the problem of building a single-stage rocket to reach escape velocity would be relatively easy. For a rocket accelerating at 5g, the mass-ratio required would be less than 4, whereas with present propellants the figure would be about 200.
The above discussion is quite valid as far as it goes and has been used by many to prove that space-flight must remain impossible. There are, however, few cases in scientific history of “negative predictions” surviving the passage of time. When, taking all factors into account, anything can be proved to be impossible, that usually means that it will be done in some different manner and employing a new and unforeseen technique. Demonstrations of the impossibility of heavier-than-air flight (a popular recreation among conservative scientists at the end of the last century) overlooked the petrol engine: those who believed that atomic power would never be released did not imagine the self-sustaining chain reaction and the ubiquitous neutron.
Much of technological progress consists of pincer movements around insoluble problems which eventually become left so far behind that their very existence is forgotten. In the case of astronautics, two solutions were put forward to overcome the difficulties discussed above. The first accepted the need for very high mass-ratios and proposed a method of construction – the step-rocket – which made them engineering possibilities. The second was much more daring: it proposed that the escape from Earth should not take place in one stage, but in two or more, the rocket actually being refuelled in space. This technique of orbital refuelling not only makes possible reductions in the overall masses required for interplanetary voyages, but, as we shall see later, opens up a whole range of important subsidiary projects.
The Earth-Moon Journey
The simplest of all journeys into space, and the first which will actually be accomplished, is the journey to or around the Moon, which will now be considered in detail. The conclusions reached in this chapter will apply, it should be noted, both to guided missiles, uncontrolled projectiles, or manned spaceships. They must all obey the same fundamental laws.
As far as energy requirements are concerned, Figure 2 shows that the Moon is, dynamically speaking, very nearly at “infinity”, despite its astronomical nearness. It needs a velocity of 11.2 km/sec to project a body to infinity – and 11.1 km/sec to project it so that it just reaches the Moon (385,000 kms or 240,000 miles at mean distance). This velocity difference is so small that it is frequently ignored and it is assumed that the full escape velocity is needed for the mission.
A body leaving the Earth in the direction of the Moon would be subject of the gravitational fields of both bodies, but for three-quarters of the way that of the Moon is completely negligible, as is shown in Figure 8. This diagram gives the accelerations produced by Earth and Moon in cm/sec2: in order to show the values over the region where both are significant, the scale here has been multiplied by 100.
Since the fields are opposing, they have been drawn on opposite sides of the horizontal axis, and it will be seen that there is a point – the so-called “neutral point” – at which both fields are equal and the resultant (represented by the dotted line) vanishes. Up to this point the body would have an acceleration towards the Earth: thereafter, the force acting upon it would be directed to the Moon. It might be mentioned here that, contrary to the vivid descriptions given by many writers, absolutely no physical phenomena of any kind would take place in a rocket passing this point. Since the machine would be in a “free fall”, with only gravitational forces acting upon it, its occupants would be weightless and so would be quite unaware of the fact that the actual direction of the fall had altered. Nor would it be possible for a body with insufficient speed to be stranded at the neutral point: the equilibrium would be quite unstable owing to the movement of the Moon and the (very small) perturbations produced by the Sun and the planets.
As it receded from the Earth the rocket’s velocity would decrease according to the escape-velocity curve in Figure 1, and it would thus pass the neutral point at a speed of about 1.6 km/sec in its fall towards the Moon. The Moon’s escape velocity is 2.34 km/sec, and this is the speed with which the rocket, if it started from rest at a great distance, would crash into the Moon’s surface. In the case we have taken the rocket has a certain additional energy since it left Earth with a speed slightly in excess of minimum requirements. Allowing for this, we find that in the fall towards the Moon it would reach a terminal speed of about 2.8 km/sec (6,300 m.p.h.). Clearly, if a safe landing is to be made, this speed must be neutralized by the further application of rocket power.
Just as the Earth has its characteristic circular velocity of about 8 km/sec, so has the Moon, the value for a point near its surface being 1.65 km/sec (equivalent to a period of 1.8 hours). It may seem a little odd to speak of satellites of satellites, but from the point of view of the Sun this is what the Moon already is! If, therefore, when a rocket was falling past the Moon its speed was reduced to its appropriate value by firing its motors in the direction of flight, then it might continue to circle our satellite, perhaps taking observations automatically and radioing them back to Earth. If the fuel reserves were sufficient, it might at a later time be accelerated again into an orbit which would return it to our planet. The velocities on the return journey would be identical with those on the outward one: the rocket would cross the neutral point at its minimum speed, and then accelerate more and more rapidly until it reached the Earth at 11.1 km/sec – the speed with which it originally started.
It will be seen, therefore, that as soon as it becomes possible to build rockets which can escape from the Earth at all, a considerable range of interesting possibilities will be opened up.
The above discussion leads us to the conception of the “characteristic velocity” which a rockets needs if it is to carry out any particular mission. For a rocket which is required to reach the Moon, but may be allowed to crash on it unchecked or shoot past into space, this velocity, as we have seen, is 11.1 km/sec, or a little less than the velocity of escape. If it is desired to make a landing to set down instruments or, later, human beings, then the machine’s fall into the Moon’s field must be counteracted. This means that in some way the rocket must be reorientated in space so that its motors point towards the Moon, and rocket braking must be employed. To put it picturesquely, the rocket must “sit on its exhaust” and so descend slowly on to our satellite’s surface.
If this manoeuvre was carried out in the most economical manner possible, it would require the combustion of exactly as much fuel as the escape from the Moon. Both missions are identical apart from the change in sign: it requires just as much energy to accelerate in space as to decelerate. The Moon’s escape velocity being 2.34 km/sec, the characteristic velocity for the whole trip is 11.1+2.34 or 13.44 km/sec. The rocket must therefore be designed as if it had to reach this speed, and this is the figure which must be substituted in Equation III.2.a. to obtain the mass-ratio required for the mission. The rocket, of course, never reaches this speed, since it divides its efforts between the two ends of the voyage: however, it would be capable of doing so if it burnt its fuel in one prolonged burst.
Mass Ratio Requirements
This figure of nearly 13.5 km/sec is a theoretical minimum value: it does not allow for gravitational loss at the take-off from Earth and an exactly corresponding, though much smaller, loss at the lunar landing. Taking these factors into account, the characteristic velocity for a voyage from rest on the Earth’s surface to rest on the Moon’s is about 16 km/sec (36,000 m.p.h.). With the most powerful chemical fuels ever likely to be available this would require an effective mass-ratio of about 35 and hence would involve the use of rockets of at least three stages, or else the orbital refuelling techniques mentioned before.
For a return journey the characteristic velocity must be doubled: it would therefore be about 32 km/sec. However, an interesting and important complication arises here. The descent on to the Moon could only be carried out by rocket braking, since there is practically no atmosphere. In the case of the Earth, the final landing could certainly be by parachute or some equivalent aerodynamic means. Indeed, it is possible that the greater part of the 11.1 km/sec which the rocket would acquire on its long fall back from the neutral point could be destroyed by air-resistance, by the technique of “braking ellipses.”
This procedure was worked out in great detail by the early German writers and is as follows. Suppose that in its fall towards the Earth the rocket is aimed so that it passes through the highest levels of the atmosphere – at an altitude of about 100 kilometres. It will suffer a certain amount of retardation due to air-resistance, which, if the altitude is chosen correctly, can be of any desired value. (There would be no great danger of the rocket becoming incandescent at these altitudes, for it would have only one-fifth of the speed of a meteor at this level and the air-resistance would therefore be only a twenty-fifth as great.) After “grazing” the atmosphere, the rocket would again emerge into space, where the frictional heating produced on its walls could be lost by radiation. It would now, however, be travelling at a speed substantially less than escape velocity, and so after receding from Earth to a considerable distance would return again along a very elongated ellipse. At “perigee” it would re-enter the atmosphere, cutting through it at a lower level but at less speed than on the first contact.
It this way, after a series of diminishing ellipses, the rocket could shed most of its excess speed without using any fuel. Indeed, it has been calculated that the entire landing on the Earth could be carried out in this manner, the final “touch-down” being by parachute. Before this can be settled definitely much more extensive knowledge of the upper atmosphere will be required, but undoubtedly substantial savings of fuel can be effected in this way.
Taking the most optimistic view we can calculate the “characteristic velocity” for the round trip as follows: –
The more pessimistic estimate, which assumes that the whole of the landing on Earth would have to be done by rocket braking, would be about 32 km/sec.
These performances would demand effective mass-ratios of about 70 and 1,000 respectively with the best conceivable chemical fuels, from which it will be seen what an important role air-braking can play if it proves practicable. But even the lower figure of 70 would require, for a ship large enough to carry men and their equipment, an initial mass of many thousands of tons at take-off. This demonstrates once again the virtual impossibility of a return voyage to the Moon, with landing, in a chemically-propelled rocket.
The economics of the Earth-Moon voyage would, however, alter drastically is orbital refuelling was employed.
So far, no mention has been made of the duration of the lunar journey. If the rocket maintained its initial speed of 11 km/sec (25,000 m.p.h.) it would reach the Moon after 10 hours, but since its velocity is steadily decreasing the figure is considerably greater. For a body leaving the Earth’s neighbourhood at the minimum speed which enables it to reach the Moon at all (11.1 km/sec), the journey to the Moon’s orbit takes about 116 hours (see Appendix, Equation V.1). This, however, ignores the acceleration of the Moon’s field towards the end of the journey, which would produce a small but appreciable reduction of transit time.
This figure of 116 hours is therefore the maximum length of time a free projectile could take on the direct journey to the Moon. A rocket which had to engage in retarding manoeuvres would, of course, be longer on the journey.
Five days is not a great deal of time in which to make a voyage to another world and it would decrease very sharply if the rocket left the Earth with any appreciable excess speed over the minimum 11.1 km/sec. (See Appendix, Equations V.2, 3.) Some typical values for these transit times are tabulated below.
The Sun’s Gravitational Field
In our discussions of lunar journeys in the last chapter, it was assumed that the Earth and Moon formed a more or less closed system and that the effects of other gravitational fields could be ignored. This is true, to a very high degree of accuracy, of the minute fields of the planets. The Sun’s field, however, is far more powerful since it holds the Earth firmly in its orbit at a distance of over 90,000,000 miles, and it may well be asked if we were justified in ignoring it in our calculations.
To a first approximation the answer is – luckily – “yes”. Although the Sun’s influence is relatively large, the variation over the whole width of the Moon’s orbit is very small – less than 1 per cent. of its absolute value. In other words, the Sun acts almost equally on Earth and Moon and on any object between them: a negligible error is therefore introduced if we ignore it completely. Its effect only appears in the third or later significant figures when more accurate calculations are required.
When we come to consider, not journeys from a planet to its nearby satellite, but from one planet to another, the situation is totally different. We must now alter our point-of-view from the Earth, holding its solitary Moon in its gravitational grip, to the Sun, keeping all the planets moving in its far more extensive field.
Everything that has been said about the Earth’s field in Chapter 2 applies, with a suitable alteration of scale, to the Sun’s. At the surface of the Sun, the acceleration of gravity is 28 times that at the surface of the Earth. If we use this value for “g” and the appropriate value of the Sun’s radius (695,500 kms or 432 000 miles) in the equations derived in the Appendix to Chapter 2, we can find the magnitudes of the solar escape and circular velocities and the dimensions of the orbits for the bodies moving in the Sun’s field, exactly as we have done in the case of the Earth. We can also calculate the work needed to lift a body from the Sun’s surface to infinity, and can express this in terms of a vertical distance – the “depth” of the Sun’s gravitational pit. A comparison of the two sets of figures is instructive:
We are not, of course, concerned with leaving the actual surface of the Sun, but if we move from one planetary orbit to another we are required to move up or down the slope of the Sun’s gravitational crater, which is precisely similar in shape to that of Earth (Figure 2), except that it is 3,000 times as deep. It is therefore important to consider the locations of the Earth and planets on the slopes of this imaginary crater.
The usual scale-drawing of the Solar System, as found in most school atlases and any astronomy book, shows the inner planets crowded round the Sun with the outer worlds at progressively increasing distances – up to 6,000,000,000 kms (3,750,000,000 miles) for Pluto, the most distant of the Sun’s children. The “energy diagram” of the system, however, presents a completely different picture. Far from being near the Sun in the gravitational sense, even the innermost planet, Mercury, is very remote from it. Whereas the full depth of the imaginary crater is nearly 20,000,000 kms, all the planets are crowded together on its very uppermost slopes, within 250,000 kms of the level plain into which it slowly flattens. This means that the work done in moving between the planetary orbits is only a small fraction of what it might well have been had the scale of the Solar System been different: indeed, we will presently see that crossing such an immense distance as that between Earth and Mars may require less energy than the journey between Earth and Moon.
In other words, the Sun’s gravitational field, though of enormous extent, is very “flat” in the region of the planets and the climb up its slopes requires relatively little energy. Superimposed on this field are the much smaller fields of the individual planets. These are effective only over very short distances, but their gradients are relatively steep and so we have the paradox arising that the first thousand miles of an interplanetary journey may require more energy than the next score of millions. This state of affairs is depicted in Figure 9, which will be explained in more detail presently.
The planets, at their varying distances, are travelling round the Sun in orbits which are, in most cases, very nearly circular, and all are moving in the same direction and lie approximately in the same plane. The velocities of motion can be calculated from Equation II.8 with suitable values for the constants: they range from 48 km/sec in the case of Mercury, the innermost planet, to 5 km/sec for Pluto, at the known limits of the Solar System. It will be seen, therefore, that a body on any of the planets already possesses, by virtue of its orbital motion, a very large part of the energy needed for interplanetary voyages.
If we calculate, from Equations II.5 and II.8, the velocity of escape and the orbital velocity in the Sun’s field at the position of the Earth, the values obtained are 42 and 30 km/sec respectively. The first figure is the velocity which a body, at rest in the Earth’s orbit, would have to be given to project it past all the outer planets and far away from the Solar System – indeed, to the stars themselves, after many millions of years. The second figure is, of course, the velocity which the Earth already possesses: the difference (12 km/sec) is therefore the additional speed which must be imparted to a body, moving with the Earth but free of its gravitational field, to send it completely out of the Solar System.
Similar calculations can be made for all the other planets, and some of the results are shown in Figure 9. As far as the writer knows, this form of representation of the Solar System is due to Dr. R. S. Richardson, of Mount Wilson Observatory.
It will be seen that this drawing bears a considerable similarity to Figure 2, but whereas in the earlier diagram the ordinates were in terms of distance (and hence energy), here they are in the more convenient form of velocity. The diagram must be imagined as extending downwards ten times further than shown, to the 618 km/sec escape velocity needed to leave the actual surface of the Sun. The left-hand branch of the curve shows the additional “transfer velocity” needed by a body, already moving in a circular orbit round the Sun, to permit it to leave the Solar System. Even for the orbit of Mercury this additional velocity is only 20 km/sec – a very small fraction of the enormous values needed in the neighbourhood of the Sun.
On the right-hand side of the figure, the subsidiary escape velocity curves for the individual planets have been superimposed, so that we can see at a glance the total velocity needed to leave the Solar System from the surface of the five inner planets – assuming that this feat was carried out by (1) accelerating to escape from the planet and then, when this has been achieved, accelerating to escape from the Sun’s field. As we shall see on page 66, it would be much more efficient to make the velocity change in one operation, as near to the planet as possible. Nevertheless, Figure 9 demonstrates the important fact that the energies needed to cross the great spaces of the Solar System are no more, and are often much less than, those needed to leave the planets themselves. It also shows that, from the energy viewpoint, the surface of Jupiter is much nearer the Sun than is that of Mercury!
We will now consider the velocities required to make the interplanetary journey which is perhaps of the greatest interest – that from the Earth to Mars. The case examined will that in which the maximum possible use is made of the planets’ existing velocities. Obviously, if one had quite unlimited supplies of energy one could travel from one planet to another by any route one fancied, but for a long time to come only the orbits of minimum energy – the astronautical equivalents of great circle routes in terrestrial navigation – will be of practical interest.
Figure 10 (a) shows the orbits of the two planets drawn to scale, although as the orbit of Mars is actually somewhat eccentric (e=0,093) the average values of its radius and orbital speed have been taken for simplicity. This approximation will give results which are slightly too pessimistic for journeys when Mars is at its closest to the Sun, and vice versa, but the variations are very small.
The path of a spaceship in the Sun’s controlling field must follow one of the curves – ellipse, parabola or hyperbola – discussed in Chapter 2, and any of these could in principle be employed for interplanetary travel. But the path which, as will be almost intuitively obvious, is the easiest one to use is the ellipse which is tangent to both planetary orbits, and with the Sun at one focus.
It is easy to calculate, from Equation II.10, the velocity which a body needs to travel in such a path. When it was nearest the Sun, i.e. in the neighbourhood of the Earth, its speed would be 32.7 km/sec. As it grazed the orbit of Mars this would drop to 21.5 km/sec. These speeds do not differ very greatly from those of the planets themselves – 29.8 and 24.2 km/sec respectively.
To project a body, which is already moving in the Earth’s orbit, out to Mars we need thus only give it an additional speed of 32.7 – 29.8 or less than 3 km/sec in the direction of the Earth’s motion. It would then drift outwards away from the Sun along the ellipse of Figure 10 (a) until it reached the orbit of Mars. Its velocity would then be 24.2 – 21.5 or 2.7 km/sec too low for it to remain here and it would start to drop back to the Earth’s orbit again. If, however, it was now given this “transfer” velocity of 2.7 km/sec it would remain in the Martian orbit. It could then land on Mars, using rocket braking against the planet’s gravitational field, or could become a third satellite of the little world, taking observations until it was time to start on the homeward voyage. A rather subtle point arises here. If we wish a spaceship to reach Mars from the surface of the Earth, it must still have an excess speed of 2.9 km/sec when it has escaped from the Earth. If it started at 11.2 km/sec, it would have no residual speed left when it had done this. Therefore, since the problem is one of kinetic energies, the required starting speed is given by squaring these velocities, adding them, and taking the square root. The result is 11.6 km/sec: the arithmetical sum of 14.1 km/sec gives the correct answer only if the ship waited until it had completely escaped from the Earth before accelerating into the voyage orbit – obviously an uneconomical procedure.
Similarly for the landing on Mars, when we have to neutralise the 5 km/sec produced in falling through Mars’ gravitational field, and change orbital velocities by 2.7 km/sec, the required total change would 5.7 and not 7.7 km/sec.
For the complete journey, therefore, the ship must be designed to make speed alterations of 11.6 and 5.7 km/sec at the two ends of its voyage – a total (since this time, of course, we have to add arithmetically!) 17.3 km/sec.
In practice this minimum value would have to be increased to about 20 km/sec to allow for gravitational losses at the landings and take-offs, course corrections, etc. Nevertheless, this is not such a large increase over the 16 km/sec needed for the Earth-Moon journey – yet the total distance covered is more than a thousand times greater!
The return journey, apart from the possible use of air-resistance braking in the Earth’s atmosphere, would be carried out in an identical manner and would require the same total velocity. The characteristic velocity for the round trip would therefore be about 40 km/sec, or just under 30 km/sec if 100 per cent. air-braking could be used at the Earth landing.
It will be realised that such journeys could only be carried out at the times when the planets were in the correct relative positions, so that the rocket would arrive at the Martian orbit at the point also occupied by the planet.
On the above lines it is possible to calculate the “characteristic velocity” needed for any interplanetary journey, and a table of such values for the more important cases is given below.
These values must be regarded as no more than approximations based on rather conservative assumptions, so that the actual values would certainly be somewhat less. No allowance has been made, for example, for the fact that a spaceship taking off from the Equator would possess an additional half a kilometre a second velocity owing to the Earth’s rotation. And if, as some believe, the whole of the landing on Earth can be effected by air-braking alone, the figures for the return journeys would be reduced by 10 or 11 km/sec.
The characteristic velocities for interplanetary journeys are considerably higher than for the lunar voyage, and the necessary mass-ratios are very much higher still since they increase as the power of the characteristic velocity. (Equation III.2.a.)
Assuming the use of a fuel giving an exhaust velocity of 4.5 km/sec, which we have seen is probably the maximum that can ever be obtained from chemical propellants, the return Martian journey with landing on Mars would demand an effective mass-ratio of about 7,300, which is utterly beyond realization. (It would mean in practice that for every ton taken on the round trip several score thousand tons of fuel would be required at the take-off!) Even assuming the use of atmospheric braking for the whole of the final Earth landing, we still obtain mass-ratios for the round trip of 790 or more. Using step construction, this would require a rocket with an initial mass comparable to that of a battleship.
This does not mean that interplanetary travel is impossible with chemical fuels, but it does mean that it is impossible to build spaceships capable of reaching the planets from the Earth’s surface, landing on them, and returning to the Earth in a single operation, carrying all the fuel for the complete mission. If the task could be broken down into its components, it would become easier by several orders of magnitude, and would enter the realm of engineering possibility. In other words, we are again compelled to consider orbital refuelling.
As an example of the sort of thing that might be done on these lines, consider a journey to Mars starting from an orbit just outside the limits of the Earth’s atmosphere, the spaceship having been refuelled as suggested in Chapter 4 and described in more detail in Chapter 8. It would then escape from this orbit and enter the cotangential ellipse taking it to Mars, the total velocity for this manoeuvre being about 3.6 km/sec. On approaching Mars and accelerating into its orbit, the ship would not land but would become a satellite of Mars at a distance of a few hundred kilometres from the surface. At this height it would be possible to learn an immense amount about the planet by telescopic observation.
The spaceship would continue to circle Mars in a free orbit until the planet was in the correct position for the return journey. This would involve a waiting period of 455 days, which, though long, means that it would be possible to observe a complete cycle of seasons over the two hemispheres. The total characteristic velocity for the mission would be as follows: –
Assuming a rocket exhaust velocity of 4.5 km/sec, this could be accomplished with a mass-ratio of 3.6 or, for the return trip back into the orbit round Earth, 13. Such figures could be achieved by a spaceship of relatively few steps, the construction of which would be further simplified by the fact that it would never have to withstand high accelerations since it would always be operating in low gravitational fields. […] Although the complete project, including the fuelling of the ship in its orbit and the eventual retrieving of the crew by an auxiliary rocket when they had returned to the Earth’s neighbourhood, would be exceedingly expensive and would require the combustion of several thousand tons of fuel, there would never be any question of handling such quantities in a single operation or in a single machine. The largest amount to be dealt with at any one time would be a few hundred tons.
As a more remote prospect, if the materials for refuelling spaceships could be found on any of the planets and extracted without undue difficulty, the economics of the entire project would change radically. (There would be little transatlantic flying even today if aircraft had to carry their fuel for the round trip.) But even when such co-operation makes it possible to budget for one-way trips only, and even when orbital refuelling techniques are exploited to the utmost, flight to the other planets will remain a fabulously expensive enterprise, which can be carried out only at infrequent intervals. It would still be worth doing on purely scientific grounds and for the profounder reasons discussed in Chapter 10: but even a flourishing world-state could not afford it very often.
All this is assuming that rocket exhaust velocities appreciably greater than 4.5 km/sec can never be attained. If this limitation can be circumvented in any way the whole picture will be altered. To take a specific case, consider the round trip to Mars which, as we have seen, requires a characteristic velocity of 40 km/sec if the mission is to be carried out as a single operation. The effective mass-ratios needed for this journey if high exhaust velocities can be obtained are listed below.
The rate at which the figures decrease with relatively modest increases in exhaust velocity is astonishing. The value of exhaust velocity at which interplanetary travel begins to look a practical proposition rather than a prodigious scientific feat is about 10 km/sec – four times the value attainable today and twice that which seems the ultimate limit for chemically-propelled rockets. It is, therefore, natural to ask if such performances can be obtained by any application of atomic power.
It can be said at once that the energies released by nuclear reactions are of such a magnitude as to make the requirements of interplanetary travel look very modest indeed. At a very conservative estimate, the fifty or so pounds of fissile material in the first atomic bombs liberated 10,000,000 mile-tons of energy. This is more than sufficient to take a mass of 1,000 tons to the Moon and to bring it back to Earth – a feat which would require the combustion of millions of tons of chemical fuel. This fantastic disproportion – 50 pounds of plutonium doing the work of millions of tons of chemicals – becomes even more astonishing when one considers that less than 0.1 per cent. of the total energy is actually liberated in present atomic explosions.
If even this 0.1 per cent. could be used to produce a propulsive jet, the “exhaust velocities” obtained would be about a thousand times those possible with chemical reactions. Instead of trying to design spaceships consisting of 90 per cent. fuel – and then having to discard section after section to get a sufficiently high final velocity – it would be quite literally true to say that the fuel was a completely negligible fraction of the machine’s mass – much less than 1 per cent. of the total.
This is certainly an attractive prospect after the rather depressing figures given earlier in this chapter. As we have now succeeded in liberating atomic energy both at controlled, low-energy and at uncontrolled, super-high energy levels, it may well be asked why so much time has been spent discussing the almost crippling limitations of chemical propellants, when atomic energy can open up not merely the nearer planets but the entire Solar System with equal ease.
The answer can be given at once. The controlled use of atomic energy is not going to be simple even for fixed generating stations with virtually no limitations of mass. And of all the possible uses of atomic energy, the application to aircraft and rocket propulsion appears the most difficult, and raises the most stubborn technical problems.
On the other hand, it is the one which offers the greatest dividends if it can be achieved. In its “terrestrial” applications atomic energy offers nothing essentially new. It can perform, perhaps more economically, what can also be done in other ways. But as a means of propulsion in space it offers – in theory at least – a solution to difficulties which would otherwise be totally insuperable.
When the time comes to write the history of atomic power and its impact on human affairs, it may well be found that all its other applications – countless though they may be – will be overshadowed by the fact that through its use Mankind obtained the freedom of space, with all that that implies.
First by land, then by sea, man grew to know this planet; but its final conquest was to lie in a third element, and by means beyond the imagination of almost all men who had ever lived before the twentieth century. The swiftness with which mankind has lifted its commerce and its wars into the air has surpassed the wildest fantasy. Now indeed we have fulfilled the poet’s dream and can “ride secure the cruel sky”. Through this mastery the last unknown lands have been opened up: over the road along which Alexander burnt out his life, the businessmen and civil servants now pass in comfort in a matter of hours.
The victory has been complete, yet in the winning it has turned to ashes. Every age but ours has had its
, its Happy Isles, its North-West
Passage to lure the adventurous into the unknown. A lifetime ago men could
still dream of what might lie on the poles – but soon the North Pole will be
the cross-roads of the world. We may try to console ourselves with the thought
that even if Earth has no new horizons, there are no bounds to the endless
frontier of space. Yet it may be doubted if this is enough, for only very
sophisticated minds are satisfied with purely intellectual adventures. El Dorado
The importance of exploration does not lie merely in the opportunities it gives to the adolescent (but not to be despised) desires for excitement and variety. It is no mere accident that the age of
was also the age
of Leonardo, or that Sir Walter Raleigh was a contemporary of Shakespeare and
Galileo. “In human records”, wrote the anthropologist J. D. Unwin, “there is no
trace of any display of productive energy which has not been preceded by a
display of expansive energy.” And today, all possibility of expansion on Earth
itself has practically ceased. Columbus
The thought is a sombre one. Even if it survives the hazards of war, our culture is proceeding under a momentum which must be exhausted in the foreseeable future. Fabre once described how he linked the two ends of a chain of marching caterpillars so that they circled endlessly in a closed loop. Even if we avoid all other disasters, this would appear a fitting symbol of humanity’s eventual fate when the impetus of the last few centuries has reached its peak and died away. For a closed culture, though it may endure for centuries, is inherently unstable. It may decay quietly and crumble into ruin, or it may be disrupted violently by internal conflicts. Space travel is a necessary, though not in itself a sufficient, way of escape from this predicament.
Those new frontiers are urgently needed. The crossing of space – even the mere belief in its possibility – may do much to reduce the tensions of our age by turning men’s minds outwards and away from their tribal conflicts. It may well be that only by acquiring this new sense of boundless frontiers will the world break free from the ancient cycle of war and peace. One wonders how even the most stubborn of nationalisms will survive when men have seen the Earth as a pale crescent dwindling against the stars, until at last they look for it in vain.
No doubt there are many who, while agreeing that these things are possible, will shrink from them in horror, hoping that they will never come to pass. They remember Pascal’s terror of the silent spaces between the stars, and are overwhelmed by the nightmare immensities which Victorian astronomers were so fond of evoking. Such an outlook is somewhat naive, for the meaningless millions of miles between the Sun and its outermost planets are no more, and no less, impressive than the vertiginous gulf lying between the electron and the atomic nucleus. Mere distance is nothing: only time that is needed to span it has any meaning. A spaceship which can reach the Moon at all would require less time for the journey than a stage-coach once took to travel the length of
When the atomic drive is reasonably efficient, the nearer planets would be only
a few weeks from Earth, and so will seem scarcely more remote than are the
antipodes today. England
It is fascinating, however premature, to try and imagine the pattern of events when the Solar System is opened up to mankind. In the footsteps of the first explorers will follow the scientists and engineers, shaping strange environments with technologies as yet unborn. Later will come the colonists, laying the foundations of cultures which in time may surpass those of the mother world. The torch of civilisation has dropped from falling fingers too often before for us to imagine that it will never be handed on again.
We must not let our pride in our achievements blind us to the lessons of history. Over the first cities of mankind, the desert sands now lie centuries deep. Could the builders of
and Babylon – once the wonders of the world –
have pictured London or ? Nor can we imagine the citadels
that our descendants may build beneath the blinding sun of Mercury, or under
the stars of the cold Plutonian wastes. And beyond the planets, though ages
still ahead of us in time, lies the unknown and infinite promise of the stars. New York
There will, it is true, be danger in space, as there has always been on the oceans or in the air. Some of these dangers we may guess: others we shall not know until we meet them. Nature is no friend of man’s, and the most he can hope for is her neutrality. But if he meets destruction, it will be at his own hands and according to a familiar pattern.
The dream of flight was one of the noblest, and one of the most disinterested, of all man’s aspirations. Yet it led in the end to that silver Superfortress driving in passionless beauty through August skies toward the city whose name it was to sear into the conscience of the world. Already there has been half-serious talk in the
concerning the use of
the Moon for military bases and launching sites. The crossing of space may thus
bring, not a new Renaissance, but the final catastrophe that haunts our
generation. United States
That is the danger, the dark thundercloud that threatens the promise of the dawn. The rocket has already been the instrument of evil, and may be so again. But there is no way back into the past: the choice, as Wells once said, is the Universe – or nothing. Though men and civilizations may yearn for rest, for Elysian dream of the Lotus Eaters, that is a desire that merges imperceptibly into death. The challenge of the great spaces between the worlds is a stupendous one; but if we fail to meet it, the story of our race will be drawing to its close. Humanity will have turned its back upon the still untrodden heights and will be descending again the long slope that stretches, across a thousand million years of time, down to the shores of the primeval sea.
 Anyone unduly disconcerted by the occasional appearance of the mathematical fiction “infinity” can substitute “a few million miles”.
 See, for example, Cleaver, “Interplanetary Flight: Is the Rocket the only Answer?”: Journal of the B.I.S., 6, 127-48 (June 1947); of Seifert, Mills and Summerfield, “The Physics of Rockets”: American Journal of Physics, 15, 121-40 (March-April 1947).
 It compares quite favourably with the ten weeks of
first voyage! Columbus
 If we desire to leave the Earth and escape from the Solar System, this calculation shows that the starting speed should be 16.4 km/sec.
 There is an apparent discrepancy here as it has been stated that the first atomic bomb was equivalent to 20,000 tons of T.N.T. But the greater part of the millions of tons mentioned above would be used merely to transport a smaller quantity of fuel out of the Earth’s field: it is this factor which reduces still further the efficiency of the chemical fuel against the virtually weightless atomic fuel in this (highly theoretical) calculation.
 James Elroy Flecker (1884–1915), “To a Poet aThousand Years Hence”. Ed.