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By Robert H. Goddard
Dover Publications, Inc.Copyright © 2002 Dover Publications, Inc.
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A METHOD OF REACHING EXTREME ALTITUDES
By ROBERT H. GODDARD (WITH 10 PLATES)
A search for methods of raising recording apparatus beyond the range for sounding balloons (about 20 miles) led the writer to develop a theory of rocket action, in general (pp. 6 to 11), taking into account air resistance and gravity. The problem was to determine the minimum initial mass of an ideal rocket necessary, in order that on continuous loss of mass, a final mass of one pound would remain, at any desired altitude.
An approximate method was found necessary, in solving this problem (pp. 10 to 11), in order to avoid an unsolved problem in the Calculus of Variations. The solution that was obtained revealed the fact that surprisingly small initial masses would be necessary (table VII, p. 46) provided the gases were ejected from the rocket at a high velocity, and also provided that most of the rocket consisted of propellant material. The reason for this is, very briefly, that the velocity enters exponentially in the expression for the initial mass. Thus if the velocity of the ejected gases be increased five fold, for example, the initial mass necessary to reach a given height will be reduced to the fifth root of that required for the lesser velocity. (A simple calculation, p. 50, shows at once the effectiveness of a rocket apparatus of high efficiency.)
It was obviously desirable to perform certain experiments: First, with the object of finding just how inefficient an ordinary rocket is, and secondly, to determine to what extent the efficiency could be increased in a rocket of new design. The term "efficiency" here means the ratio of the kinetic energy of the expelled gases to the heat energy of the powder, the kinetic energy being calculated from the average velocity of ejection, which was obtained indirectly by observations on the recoil of the rocket.
It was found that not only does the powder in an ordinary rocket constitute but a small fraction of the total mass (¼ or 1/5), but that, furthermore, the efficiency is only 2 per cent, the average velocity of ejection being about 1,000 ft./sec. (table I, p. 12). This was true even in the case of the Coston ship rocket, which was found to have a range of a quarter of a mile.
Experiments were next performed with the object of increasing the average velocity of ejection of the gases. Charges of dense smokeless powder were fired in strong steel chambers (fig. 2, p. 13), these chambers being provided with smooth tapered nozzles, the object of which was to obtain the work of expansion of the gases, much as is done in the De Laval steam turbine. The efficiencies and velocities obtained in this way were remarkably high (table II, p. 15), the highest efficiency, or rather "duty," being over 64 per cent, and the highest average velocity of ejection being slightly under 8,000 ft./sec., which exceeds any velocity hitherto attained by matter in appreciable amounts.
These velocities were proved to be real velocities, and not merely effects due to reaction against the air, by firing the same steel chambers in vacuo, and observing the recoil. The velocities obtained in this way were not much different from those obtained in air (table III, p. 30).
It will be evident that a heavy steel chamber, such as was used in the above-mentioned experiments, could not compete with the ordinary rocket, even with the high velocities which were obtained. If, however, successive charges were fired in the same chamber, much as in a rapid fire gun, most of the mass of the rocket could consist of propellant, and the superiority over the ordinary rocket could be increased enormously. Such reloading mechanisms, together with what is termed a "primary and secondary" rocket principle, are the subject of certain United States Letters Patent (p. 6). Inasmuch as these two features are self-evidently operative, it was not considered necessary to perform experiments concerning them, in order to be certain of the practicability of the general method.
Regarding the heights that could be reached by the above method; an application of the theory to cases which the experiments show must be realizable in practice indicates that a mass of one pound could be elevated to altitudes of 35, 72, and 232 miles; by employing initial masses of from 3.6 to 12.6, from 5.1 to 24.3, and from 9.8 to 89.6 pounds, respectively (table VII, p. 46). If a device of the Coston ship rocket type were used instead, the initial masses would be of the order of magnitude of those above, raised to the 27th power. It should be understood that if the mass of the recording instruments alone were one pound, the entire final mass would be 3 or 4 pounds.
Regarding the possibility of recovering apparatus upon its return, calculations (pp. 51 and 53) show that the times of ascent and descent will be short, and that a small parachute should be sufficient to ensure safe landing.
Calculations indicate, further (pp. 54 to 57), that with a rocket of high efficiency, consisting chiefly of propellant material, it should be possible to send small masses even to such great distances as to escape the earth's attraction.
In conclusion, it is believed that not only has a new and valuable method of reaching high altitudes been shown to be operative in theory, but that the experiments herein described settle all the points upon which there could be reasonable doubt.
The following discussion is divided into three parts:
Part I. Theory.
Part II. Experiments.
Part III. Calculations, based upon the theory and the experimental results.
IMPORTANCE OF THE SUBJECT
The greatest altitude at which soundings of the atmosphere have been made by balloons, namely, about 20 miles, is but a small fraction of the height to which the atmosphere is supposed to extend. In fact, the most interesting, and in some ways the most important, part of the atmosphere lies in this unexplored region; a means of exploring which has, up to the present, not seriously been suggested.
A few of the more important matters to be investigated in this region are the following: the density, chemical constitution, and temperature of the atmosphere, as well as the height to which it extends. Other problems are the nature of the aurora, and (with apparatus held by gyroscopes in a fixed direction in space) the nature of the α, β, and γ radioactive rays from matter in the sun as well as the ultra-violet spectrum of this body.
Speculations have been made as to the nature of the upper atmosphere—those by Wegener being, perhaps, the most plausible. By estimating the temperature and percentage composition of the gases present in the atmosphere, Wegener calculates the partial pressures of the constituent gases, and concludes that there are four rather distinct regions or spheres of the atmosphere in which certain gases predominate: the troposphere, in which are the clouds; the stratosphere, predominatingly nitrogen; the hydrogen sphere; and the geocoronium sphere. This highest sphere appears to consist essentially of an element, " geocoronium," a gas undiscovered at the surface of the earth, having a spectrum which is the single aurora line, 557µµ, and being 0.4 as heavy as hydrogen. The existence of such a gas is in agreement with Nicholson's theory of the atom, and its investigation would, of course, be a matter of considerable importance to astronomy and physics as well as to meteorology. It is of interest to note that the greatest altitude attained by sounding balloons extends but one-third through the second region, or stratosphere.
No instruments for obtaining data at these high altitudes are herein discussed, but it will be at once evident that their construction is a problem of small difficulty compared with the attainment of the desired altitudes.
A METHOD OF REACHING EXTREME ALTITUDES By ROBERT H. GODDARD
PART I. THEORY
METHOD TO BE EMPLOYED
It is possible to obtain a suggestion as to the method that must be employed from the fundamental principles of mechanics, together with a consideration of the conditions of the problem. We are at once limited to an apparatus which reacts against matter, this matter being carried by the apparatus in question. For the entire system we must have: First, action in accordance with Newton's Third Law of Motion; and, secondly, energy supplied from some source or sources must be used to give kinetic and potential energy to the apparatus that is being raised; kinetic energy to the matter which, by reaction, produces the desired motion of the apparatus; and also sufficient energy to overcome air resistance.
We are at once limited, since sub-atomic energy is not available, to a means of propulsion in which jets of gas are employed. This will be evident from the following consideration: First; the matter which, by its being ejected furnishes the necessary reaction, must be taken with the apparatus in reasonably small amounts. Secondly, energy must be taken with the apparatus in as large amounts as possible. Now, inasmuch as the maximum amount of energy associated with the minimum amount of matter occurs with chemical energy, both the matter and the energy for reaction must be supplied by a substance which, on burning or exploding, liberates a large amount of energy, and permits the ejection of the products that are formed. An ideal substance is evidently smokeless powder, which furnishes a large amount of energy, but does not explode with such violence as to be uncontrollable.
The apparatus must obviously be constructed on the principle of the rocket. An ordinary rocket, however, of reasonably small bulk, can rise to but a very limited altitude. This is due to the fact that the part of the rocket that furnishes the energy is but a rather small fraction of the total mass of the rocket; and also to the fact that only a part of this energy is converted into kinetic energy of the mass which is expelled. It will be expected, then, that the ordinary rocket is an inefficient heat engine. Experiments will be described below which show that this is true to a surprising degree.
By the application of several new principles, an efficiency manyfold greater than that of the ordinary rocket is possible; experimental demonstrations of which will also be described below. Inasmuch as these principles are of some value for military purposes, the writer has protected himself, as well as aerological science in America, by certain United States Letters Patent; of which the following have already been issued:
The principles concerning efficiency are essentially three in number. The first concerns thermodynamic efficiency, and is the use of a smooth nozzle, of proper length and taper, through which the gaseous products of combustion are discharged. By this means the work of expansion of the gases is obtained as kinetic energy, and also complete combustion is ensured.
The second principle is embodied in a reloading device, whereby a large mass of explosive material is used, a little at a time, in a small, strong, combustion chamber. This enables high chamber pressures to be employed, impossible in an ordinary paper rocket, and also permits most of the mass of the rocket to consist of propellant material.
The third principle consists in the employment of a primary and secondary rocket apparatus, the secondary (a copy in minature of the primary) being fired when the primary has reached the upper limit of its flight. This is most clearly shown, in principle, in United States Patent No. 1, 102,653.
By this means the large ratio of propellant material to total mass is kept sensibly the same during the entire flight. This last principle is obviously serviceable only when the most extreme altitudes are to be reached. In order to avoid damage when the discarded casings reach the ground, each should be fitted with a parachute device, as explained in United States Patent No. 1,191,299.
Experiments will be described below which show that, by application of the above principles, it is possible to convert the rocket from a very inefficient heat engine into the most efficient heat engine that ever has been devised.
STATEMENT OF THE PROBLEM
Before describing the experiments that have been performed, it will be well to deduce the theory of rocket action in general, in order to show the tremendous importance of efficiency in the attainment of very high altitudes. A statement of the problem will therefore be made, which will lead to the differential equation of the motion. An approximate solution of this equation will be made for the initial mass required to raise a mass of one pound to any desired altitude, when said initial mass is a minimum.
A particular form of ideal rocket is chosen for the discussion as being very amenable to theoretical treatment, and at the same time embodying all of the essential points of the practical apparatus. Referring to figure 1, a mass H, weighing one pound is to be raised as high as possible in a vertical direction by a rocket formed of a cone, P, of propellant material, surrounded by a casing K. The material P is expelled downward with a constant velocity, c. It is further supposed that the casing, K, drops away continuously as the propellant material P burns, so that the base of the rocket always remains plane. It will be seen that this approximates to the case of a rocket in which the casing and firing chamber of a primary rocket are discarded after the magazine has been exhausted of cartridges, as well as to the case in which cartridge shells are ejected as fast as the cartridges are fired.
Let us call
M=the initial mass of the rocket,
m=the mass that has been ejected up to the time, t,
v=the velocity of the rocket, at time t,
c=the velocity of ejection of the mass expelled,
R=the force, in absolute units, due to air resistance,
g=the acceleration of gravity, dm=the mass expelled during the time dt,
k=the constant fraction of the mass dm that consists of casing K, expelled with zero velocity relative to the remainder of the rocket, and
dv=the increment of velocity given the remaining mass of the rocket.
The differential equation for this ideal rocket will be the analytical statement of Newton's Third Law, obtained by equating the momentum at a time t to that at the time t+dt, plus the impulse of the forces of air resistance and gravity,
(M - m)v=dm(I - k)(v - c)+vkdm + (M-m-dm)(v+dv)+[R+g(M-m)]dt.
If we neglect terms of the second order, this equation reduces to
c(I - k)dm=(M - m)dv+[R + g(M - m)]dt.
A check upon the correctness of this equation may be had from the analytical expression for the Conservation of Energy, obtained by equating the heat energy evolved by the burning of the mass of propellant, dm(1—k), to the additional kinetic energy of the system produced by this mass plus the work done against gravity and air resistance during the time dt. The equation thus derived is found to be identical with equation (1).
REDUCTION OF EQUATION TO THE SIMPLEST FORM
In the most general case, it will be found that R and g are most simply expressed when in terms of v and s. In particular, the quantity R, the air resistance of the rocket at time t, depends not only upon the density of the air and the velocity of the rocket, but also upon the cross section, S, at the time t. The cross section, S, should obviously be as small as possible; and this condition will be satisfied at all times, provided it is the following function of the mass of the rocket (M—m),
S=A(M - m)
where A is a constant of proportionality. This condition is evidently satisfied by the ideal rocket, figure 1. Equation (2) expresses the fact that the shape of the rocket apparatus is at all times similar to the shape at the start; or, expressed differently, S must vary as the square of the linear dimensions, whereas the mass (M—m) varies as the cube. Provision that this condition may approximately be fulfilled is contained in the principle of primary and secondary rockets.
The resistance, R, may be taken as independent of the length of the rocket by neglecting "skin friction." For velocities exceeding that of sound this is entirely permissible, provided the cross section is greatest at the head of the apparatus, as shown in United States Patent No. 1,102,653.
Excerpted from Rockets by Robert H. Goddard. Copyright © 2002 Dover Publications, Inc.. Excerpted by permission of Dover Publications, Inc..
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