Gyroscopes and Satellites
(See page 17, January, 1929, Proceedings)
Lieutenant Commander L. Wain- wright, Ph.D., U. S. Navy—This article enunciates a “Law of Satellites,” together with a suppositious proof thereof in five parts. Based thereon is a discussion of the motion of the earth’s axis as to the ecliptic and a criticism of the present types of gyro compasses, with suggestions for a different type. The “Law of Satellites,” however, has not as yet been passed by the celestial legislature; the proof given is readily shown to be fallacious; consequently, the discussion of the earth’s motion and the strictures on gyro compasses are quite erroneous.
The “Law of Satellites” is stated: “Any body, given rotation about an axis through its center of gravity (making it a gyroscope) and revolution about some other axis (making it a satellite), will seek, as a position of stability, its axis of rotation perpendicular to its plane of revolution, with rotation and revolution in the same direction,” and it is said that it “applies to all satellites that have rotation, both natural and artificial.” The five parts of the proof are: (1) “Physical Analysis”; (2) “Visual Proof”; (3) “Presumptive Proof”; (4) “Measurements, in the Case of the Earth”; and, (5) “Legends.” These will be treated separately in turn.
1. Physical Analysis.—Consider the spinner (any satellite, top, compass, moon, planet, etc.), divided by an axial plane at right angles to the axial plane through the center of revolution. There is thus an inner and outer half. Each particle of one half has a velocity of revolution plus a component of the velocity of rotation. Each particle of the other half has a velocity of revolution minus a component of the velocity of rotation. The two halves, by reason of their different velocities, have different centrifugal pulls.
There is thus set up a couple, with generally an arm, making a moment, an outside force acting on the gyroscope. This moment inverts the axis of rotation when rotation is retrograde, or opposite the direction of revolution. Inversion changes retrograde to forward rotation. After which the same moment erects the axis to the orbital plane.
The mass of each half can be considered as concentrated at the center of gyration of each half. The arm becomes zero when the two centers of gyration and the center of revolution are in line. That occurs twice in each revolution and is permanent when the axis of rotation becomes parallel to the axis of revolution. At all other times, there is an arm, therefore a moment, which inverts and erects, according to the law of satellites.
There are many ways of demonstrating the fallacy of this argument. Perhaps the most satisfying is to set up the complete equations for the motion, but this procedure is too lengthy for the present purpose. Briefly, the proposition violates the principle of the conversation of angular momentum, a corollary of Newton’s Laws of Motion.
It may suffice to consider a simple case to which the “Law of Satellites” would apply were it valid, and show that the supposed relations do not exist. In the figure, A and B represent two particles of equal mass which are joined by an imponderable rod, thus forming a species of dumb-bell. By G is denoted the centroid of the dumbbell and its axis of rotation, the revolution being about C.
The short thin arrows, oppositely directed, represent the components of velocity of A and B due to rotation. The long thin arrows represent the components of velocity of A and B due to revolution. The two thick arrows represent the total velocities of A and B. An easy construction gives V, the virtual axis about which the dumb-bell may be regarded instantaneously to be simply rotating. Then the hollow arrows represent centrifugal reactions; those directed away from V being the totals, while the shorter ones are the components normal to the dumb-bell.
As the centrifugal reactions as to V are proportional to the radial distances from V, the geometrical relations obviously give equality to the normal reactions, and so it is evident that the revolution about C has no effect upon the rotation about G. Further, as the motion is initially all in the plane of the paper, there is no component of velocity or acceleration normal to that plane, and the motion will continue entirely in that plane.
Thus, we have an example of a satellite in retrograde rotation, with no force tending to invert the axis of rotation, and hence a contradiction of the “Law of Satellites.”
2. Visual Proof. This part relates to the performance of a toy gyroscope. The exposition of the action is correct, but the explanation is not, the true reason for the behavior described being found in the friction of the bearings so long as true gimbal action exists.
3. Presumptive Proof. All the planets, nearly all the moons, of the solar system, have attained forward rotation. Many axes are nearly perpendicular to their orbital planes. Such a common effect can hardly arrive except as the result of a common cause. The Law of Satellites is submitted as the common cause.
The “Law,” however, being non-existant, cannot be accepted. For a very satisfying theory designed to account for the facts, the interested reader is referred to the Chamberlin-Moulton Planetesimal Hypothesis which is ably and fascinatingly set forth in Professor Chamberlin’s small book, The Origin of the Earth. [The University of Chicago Press, 1916, ($1.50).]
4. Measurements, in the Case of the Earth. This part deals with the obliquity of the ecliptic, and consumes a relatively great space in deducing, tabulating, and discussing numerical values. A table purporting to give the inclination of the axis of the earth at various epochs covers two pages, and judging from the context and tabular values, was computed tediously by point-to- point methods. As it is based on the mistaken “law,” it is, of course, valueless. However, it may not be amiss to note that the assumptions made imply an elementary differential equation for which a simple explicit solution exists in the form k (t—T) = log tan (i/2). Either the table might have been easily prepared from this formula, or better, the formula might have replaced the table, as only a few values were wanted for discussion. The precession of the equinoxes, as every astronomer knows, is caused by differential gravitational attractions of the earth’s equatorial bulge by the moon (80 per cent and sun 20 per cent). The obliquity of the ecliptic is virtually constant, its slight changes being due mostly to perturbation of the earth’s orbit by the planets, etc.
5. Legends. The Giinese have legends that the equator was once perpendicular to the ecliptic and that, prior to that time, the sun rose in the west. (From recent encyclopedias.)
Herodotus records that, on one of his visits to Egypt, a priest of that country told him the sun formerly rose in the west. It was so improbable that its effect was to make Herodotus a liar, till the Law of Satellites was discovered.
This need not be enlarged upon save to remark that although the “Law of Satellites” does not support Herodotus, it would seem that the imputation of mendacity should have rested not upon him, but rather upon the Egyptian priest.
When unconstrained, the axis of the rotor will become parallel to the axis of the earth. In azimuth, it will point true north. In tilt, it will point the latitude. The axes are usually (the author knows of no exception) constrained to the horizontal plane. Any gyroscope, with any constraint, will develop precessional tendencies, which need attention, to avoid errors.
This arrangement, proposed as a gyro compass, would not work. Even theoretically, the rotor axis would not parallel the axis of the earth unless it were so placed, with no moment of momentum about any of its axes but that of spin. For, in accordance with the principle of the conservation of angular momentum, a gyroscope with three degrees of angular freedom, and with all its moment of momentum about its axis of figure, will maintain that axis parallel to its original direction. Practically, bearing friction cannot be wholly eliminated, so an attempt to mount a gyroscope without angular constraint must fail. The bearing friction would cause the spin-axis to wander from its initial direction.
It is precisely to overcome the deflections set up by bearing friction that a gravitational torque is so applied as to make the spin-axis seek a plane nearly that of the meridian. A great advantage of this construction is that the compass when started up will assume the north-indication of its own accord, whereas with the free-gyro- scope construction the north-indication must be known and be initially set by hand. Constraining the spin-axis to a near-horizontal position is not an inherent necessity, but a matter of practical convenience. When there arises any reason for departing from this practice, it can be done. As a matter of fact, a well-known inventor of mechanical devices has constructed a compass in which the spin-axis is inclined to the horizontal at a large angle, but it is still gravity-constrained.
“By avoiding damping, the latitude error will be obviated.” But the compass will never settle on the north-indication, and it will be necessary to determine it from the phase of the indication, which will vary with a period of about an hour and a half. This “latitude error” is a result of the eccentric-pin damping of the Sperry compass, and is a well-determined offset which is allowed for mechanically. It does not exist in some other compasses in which the damping is accomplished otherwise.
“The ballistic deflection, the quadrantal error and the centrifugal forces are due to pendulous construction, which will be cured by eliminating the cause.” The ballistic deflection, properly regulated, is desired, as it causes the compass to follow shifts in the virtual meridian. The quadrantal error is reduced to negligibility in the Arma and similar compasses by the cross-stabilization, and in the Sperry compass by the mercury ballistic. Just what is meant by “centrifugal forces” is not clear, unless it is tautology for the other effects.
Altogether it is much to be regretted that such an article, so thoroughly shot through with error, should have been published in the Proceedings. It may serve to mislead, and it certainly has a tendency to expose to ridicule the membership of the Institute. Had it been referred to any competent mathematician, the fundamental fallacies would have been speedily detected, so it is suggested that the editorial policy of the Proceedings be amended to provide for such review.
Editor’s Notes: Referring to the last paragraph of this discussion, another law has been violated—the law of not jumping at conclusions. The paper was referred to a competent mathematician, one internationally known. He saw no objection to publishing the paper, although he disagreed with its conclusions. He suggested the clause under the title “Speculations in Support of the Egyptian and Chinese Traditions that in Remote Ages the Sun Rose in the West.”
It was hoped the article would bring forth discussions. It did!
R. C. M.
Latitude and Longitude from a Simultaneous Observation of Altitude, Azimuth, and Time
(See page 993, November, 1928, and page 142, February, 1929, Proceedings)
S. A. Vincent.—Commander de Aquino’s discussion of my November, 1928, article on the above subject explains how readily his excellent tables shorten the solution of the problem. In presenting this method I purposely gave the basic formulas, the derivation of which will be at once apparent to those familiar with spherical trigonometry. My fourth paragraph states that “several solutions are possible,’’ and that “should the method come into general use condensed tables can be prepared to shorten the work.”
As the commander states, the line of position is now the popular method. Quite justly, it is rapidly supplementing the old time sights and latitude calculations. On the other hand, on account of the increased speed of ships, and the very high speeds in aerial navigation, the run between sights is becoming more and more difficult to chart with accuracy, and more instantaneous locations are greatly needed. Indeed, it is imperatively needed in aerial work where there is really not time for much work, either tabular or graphical. The entire practicability of always getting a prompt location from celestial bodies hinges upon our ability to make some other observation simultaneously, or practically so, along with altitude, and from these to obtain immediately our latitude and longitude. The only two observations which I can think of which may be made along with altitude are azimuth, or rate of change of altitude. Similarly two azimuths (preferably varying 90°, more or less) can be substituted for the altitude and azimuth of my method. At present, perhaps neither of these can be observed with the needed accuracy on the sea or in the air. In regard to azimuth a maximum error of two or three minutes when the altitude is low, of five minutes when the altitude is 60°, or of fifteen minutes at 80°, is about the tolerable limit at sea. I fully agree with the commander that magnetic, or ordinary commercial gyro compasses will not suffice under average sea conditions. However, I understand that the new naval gun fire-control gyro compasses are within our requirements. If I have not been misinformed the method I have proposed is therefore already practical, but in any event it would appear to be in the near future.
In regard to rate of change of altitude, about the same percentage of error would be tolerable, but at present no special instrument has been devised for taking this observation. Until this is done, the rate of change of altitude cannot always be used to obtain an accurate position, particularly in aerial work, but a skillful observer on water may make serviceable use of it, under certain conditions.
Conditions are too controlling for any one method of navigation ever to drive out all others. At dawn and dusk, and frequently at night, when if one star can be sighted, two usually may be, and we have therefore two simultaneous lines of position, which the orthodox methods work out very well. In the daytime, with as a rule only one object in the sky, and under adverse conditions at dawn, dusk, and night, other methods may prove preferable. Quite a number of special methods are now being developed, radio signal directions, mechanical solvers, special slide rules, and for aerial work very elaborate sheets of altitude curves from which a reasonably close position can be read off without any calculation whatever. The world is moving on and we are doing things now that we formerly could not do. In the future we will do things we cannot do now. Mathematically speaking, latitude and longitude are both determined by an accurate simultaneous observation of chronometer time, altitude, and either azimuth or rate of change of altitude. When we can add either one of these last two to the usual observation the problem will be solved, but not till then.
I wish to express my appreciation to Commander de Aquino for his kindly interest in writing up the application of his tables to my method.