● TITLE: Origin of the moon: dynamical considerations
● AUTHOR: MacDonald, Gordon J.F.
● SOURCE: Annals N.Y. Acad. Sci., v. 118, Art. 20, pp. 739-782 (7.5.1965).
● ABSTRACT: No abstract is given in publication.

● COMMENT:
In the following no figures, equations, calculations, eccentricity considerations and theories of capture etc. from the text are given. (We remind that this publication was written before Apollo missions to the moon). Some sentences, which are a small part of the rest of the text, are given below.

● SOME SENTENCES FROM
INTRODUCTION:
P. 741
The moon is a unique body in the solar system. Other planets have their satellites, but only the Earth possesses a satellite whose orbital angular momentum about its primary exceeds the rotational momentum of the primary. For all other planets, the orbital angular momentum of the satellites is a small fraction of the rotational momentum of the planet. While other satellites are more massive than the moon, no other planet possesses a satellite whose mass is such a substantial fraction of the mass of the primary. These unique dynamical characteristics of the moon have led to a wide diversity of theories regarding its origin, and, in present as in past centuries, the moon's origin remains a matter of debate.

The mystery regarding the moon has deepened further as a result of a new interpretation of the data on the speed at which it is receding from the Earth. It has long been known that, as a result of tides raised by the moon on the Earth, angular momentum is transferred from the Earth's rotation into the moon's orbital motion. This long-term transfer moves the moon outward in its orbit. The rate at which the moon recedes from the Earth can be obtained from astronomical observations. ...
.
.
.

● SOME SENTENCES FROM THE NEXT CHAPTER
DISTRIBUTION OF ROTATIONAL MOMENTUM IN THE SOLAR SYSTEM:
.
.
.
p. 743
In the case of the Earth, we know that the Earth loses angular momentum due to gravitational tidal interaction with the Sun and the moon. The torque which gives rise to this angular momentum transfer is inversely proportional to the sixth power of the distance r between the disturbing bodies (moon and Sun) and the Earth. The height of the tides depends on r -3 while the tidal attraction on the bulge depends on r -3 leading to the r -6 dependence. ...
...The planet may also be slowed down by viscous interaction with neutral gas or dust. As far as the Earth is concerned, the angular momentum changes due to these processes are entirely negligible. ...
.
.
.
In examining the other members of the solar system, clearly there has been a tidal exchange of angular momentum between the rotational motion of the planets Mercury and Venus, and their orbital motion due to the tides raised on these planets by the Sun. MacDonald (1962) indicates that the angular momentum associated with rotation in both of these planets would be transferred into orbital motion in a time comparable to or less than the age of the solar system.
.
.
.
p. 744
FIGURE 1 illustrates the dependence of the angular momentum density on the mass of the planets. Venus and Mercury show a large deficiency of angular momentum compared to the major planets and Mars, consistent with the hypothesis that tidal interaction with the Sun has transferred the angular momentum of rotation into their orbital motion. The Earth also shows a deficiency in angular momentum for its mass, but the deficiency is much less than that shown by either Venus or Mercury. The rotational momenta of the major planets are consistent with the hypothesis that there is a simple relationship between the rotational momentum and mass of a planet.
A relationship between the angular momentum density and the mass follows from dimensional considerations. The rotational angular momentum density is proportional to R2Omega, where R is the radius of the planet and Omega is its angular velocity. ...
.
.
.
p. 747
Since the angular momentum of rotation depends on the difference between two large quantities, the angular momentum of the aggregating particles and the orbital momentum of the planets, it is not surprising that a detailed theory cannot be constructed. However, a simple relationship between angular momentum density and mass is expected on dimensional grounds...
.
.
.
The low and high values of angular momentum density correspond to rotational periods of 13.1 and 9.9 hours respectively. This author concludes that any theory of lunar origin requiring an initial rotational period for the earth much less than about 10 hours, must at the same time provide an explanation for the anomalously high initial density of rotational angular momentum of the earth.

● SOME SENTENCES FROM THE NEXT CHAPTER
ENERGY REQUIREMENT ON THE THEORIES OF LUNAR ORIGIN:
p. 748
A further requirement on the theories of lunar origin follows from considerations of the dissipation of rotational energy. ...
... if the dissipation process takes place at the surface of the Earth, such as would be the case if friction in the ocean tides drains the kinetic energy of rotation, then the thermal energy is released near the surface from which the friction takes into space.
.
.
.
FIGURE 2 illustrates the variation of the rotational kinetic energy of the earth, (C/2)Ω2 , with angular velocity Ω. The moment of inertia C about the axis of rotation is a function of the density distribution within the earth, but also depends weakly on the angular velocity. As the angular velocity Ω increases, the figure of equilibrium becomes flatter, producing a greater change in the moment of inertia.
.
.
.
● SOME SENTENCES FROM THE NEXT CHAPTER
DYNAMICAL HISTORY OF THE EARTH-MOON SYSTEM:
p. 750
A further requirement on lunar origin theories involves the time-scale for the development of the present Earth-moon system. The rate of tidal transfer of angular momentum from Earth to the moon determines the time in which the present configuration has developed.
In order to illustrate the effects of tidal friction, we first consider a rotating, perfectly elastic Earth. A moon moves in a circular orbit about the Earth; the orbit lies in the equatorial plane of the Earth. At a given point on the Earth's surface, the tide raised by the moon is high when the moon is directly overhead. The Earth's surface on the side toward the moon is distended, with the greatest distance from the center of the Earth to the outer surface along a line between the center of the moon and center of the Earth (Figure 3). The high tide occurs on the equator, with the tidal bulge passing around the Earth at a rate proportional to the difference in the angular velocity of the Earth Ω and the mean motion of the moon n. Let us suppose that the Earth is no longer perfectly elastic, but that there is friction within the body of the Earth; the Earth does not respond instantaneously to a changing external force, but deformation lags behind the changing external force. In this case, the rotation of the Earth carries the lagging tide forward, so that the tide is not high when the moon is directly overhead, but is high at
p. 751
some later time. Let δ denote the angle between the line of centers of the Earth and the moon and the line passing through the height of the tide and the center of the Earth. The angle is thus a measure of the degree of anelasticity within the Earth. The bottom half of Figure 3 illustrates the case of the lagging tide.
The moon's attraction on the tidal bulge is greater on the bulge facing the moon than it is on the bulge on the side of the Earth
p. 752
opposite to the moon, since the tidal force decreases with distance. As a result, the moon exerts a torque acting about the rotational axis, tending to retard the rotation of the Earth. Conversely, the tidal bulge exerts a torque on the moon, tending to move it along its orbit. The action of the moon on the tidal bulge tends to decelerate the Earth, removing angular momentum from the Earth. The angular momentum removed from the Earth appears in the moon's motion since the tidal bulge tends to pull the moon along its orbit. The net effect is that angular momentum is transferred from the rotational motion of the Earth to the orbital motion of the moon. At the same time, the kinetic energy associated with the rotation of the Earth decreases because of the frictional processes.
The moon does not revolve in a plane of the Earth's equator; its orbit is eccentric. We first consider the effects of inclination and obliquity. The Earth is inclined to the ecliptic by about 23o, and the orbital plane is inclined by about 5o. The rotation of the Earth does not carry the bulge forward in the orbital plane, but at an angle to it. As a result of the Earth's obliquity and the moon's inclination, there is a component of torque acting on the Earth tending to increase its obliquity and, similarly, a component on the moon tending to decrease its inclination.
...The gross effect of the tidal force is to increase the moon's distance from the Earth.
.
.
.
... it is sufficient to consider the orbit as circular.
.
.
.

p. 755
● SOME SENTENCES OF THE CHAPTER
MAGNITUDE OF THE DECELERATING TORQUE:
The torque acting on the moon provides a secular acceleration so that, over a long period, a discrepancy will arise between the position of the moon and that predicted from classical celestial mechanics in which frictional processes are neglected. A new analysis by Munk and MacDonald (1960) shows that modern observations (over the past 250 years) can be used to obtain a reliable estimate of the torque acting on the moon.
.
.
.
Munk and Macdonald obtain that, at present, the torque acting on the moon equals 3.9X1019 dyne cm.
.
.
.
By substitution of the numerical parameters into (18), we obtain δ=2.25O. The component of the torque in the orbital plane of the moon does work at a rate ... = 2.5X1019 erg/sec, ...

● FIRST PARAGRAPH OF THE NEXT CHAPTER
CIRCULAR ORBIT:
The general case of an inclined eccentric satellite contains many complications. In the case of a satellite in a circular orbit in an equatorial plane, the equations reduce into a single equation for the mean distance a. Gauss's equation for the mean distance (Brouwer and Clemence, 1961) is
da/dt=2 a S / n

Page 756
where S is the component of force acting on the satellite and directed perpendicular to the radius vector of the satellite in the orbit plane. S is positive in the direction of increasing longitude in the orbit. The mean motion, n, and a are related by Kepler's third law n2a3= ...
.
.
.

● SOME SENTENCES FROM THE NEXT CHAPTER
DETAILED CALCULATION OF THE SECULAR CHANGES IN THE MOON'S ORBIT:
.
.
.
p. 757
Thus, the rate at which the angular momentum of the Earth changes depends on the torque exerted by the tidal bulge on the moon. The angular momentum of the Earth can change through changes in its angular velocity and moment of inertia. ...
Since the angular velocity is a vector, the direction of the rotational axis in space changes corresponding to a change in the inclination of the Earth's axis to the plane of the ecliptic.
.
.
.
● SOME SENTENCES FROM THE CHAPTER
CRITIQUE ON THEORIES OF LUNAR ORIGIN:
.
.
.
p. 772
● DARWIN'S THEORY
The fission theory of the lunar origin is perhaps the best known of the lunar theories. It has failed to receive serious attention in the past 20 years because of a critical analysis given by Jeffreys. However, even though newer versions of the fission hypothesis have been propounded, Darwin's theory is still of more than just historical interest. Darwin (1962) noted that if the moon were combined with the Earth, the resulting Angular momentum would require that the combined mass would rotate with about a four-hour period. Darwin next suggested that the fundamental resonant frequency of the Earth would be about two hours, so that the period of solar tides would coincide with the free oscillation of the Earth. As a result of the ensuing resonance, the height of the tide would increase until a mass separated from the Earth. Later workers suggested that such an origin might explain the Pacific Ocean as a scar left by the
p. 773
departing moon. Further, it was noted that the density of the moon and the density of the upper mantle coincided; this has been used both in support of the Darwin theory of lunar origin and as a means of estimating the constitution of the upper mantle.
Jeffreys, after first accepting Darwin's theory, later criticized it (Jeffreys, 1930) on the basis that internal friction would damp Darwin's tidal bulge so that the disruption by resonance is highly unlikely. Recently, seismological studies have established that the free period of Earth, with its present structure, is less than an hour, about 54 minutes, for the mode of oscillation required for disruption (Benioff et al., 1961; Ness et al., 1961; Alsop et al., 1961). Even with a redistribution of mass, it is unlikely that the period could be lengthened to two hours, provided that the material of the Earth is solid. Alternatively, if the Earth were initially liquid, in which case the period might approximate two hours, then severe difficulties arise with respect to the Earth's thermal history (MacDonald, 1959). A further difficulty for the theory is that the mass thrown off by the Earth would initially have an angular velocity greater than the rotational velocity of the Earth. The tidal action of the thrown-off moon, traveling in such a retrograde orbit, would tend to bring the moon back toward the Earth, and the thrown-off mass would never escape.
.
.
.
● THE WISE-CAMERON THEORY
In the Wise-Cameron version of the Darwin hypothesis, it is supposed that the Earth becomes rotationally unstable as a result of a catastrophic change in the moment of inertia. ... .
.
.
p. 774
After fission, in the Wise model, the Earth would be rotating with a period of 2.65 hours. ...
.
.
.
● REFERENCES
Benioff, H., Press, F., Smith, H., "Excitation of the free oscillations of the Earth by earthquakes," Journal of Geophysical Research," v. 66, pp. 605-619 (1961)
Ness, N.F., Harrison, J.C., Slichter, L.B., "Observations of the free oscillations of the Earth," Journal of Geophysical Research, v. 66, pp. 621-629 (1961)
Alsop, L.L., Sutton, G.H., Ewing, M., "Free oscillations of the Earth observed on strain and pendulum seismographs," Journal of Geophysical Research, v. 66, pp. 631-641 (1961)

and more.


● RECOMMENDATION:
It is recommended to learn the full text of MacDonald's paper before any research on the origin of the moon.