Introduction There are three general types of models designed to explain the formation of the Earth-Moon couple. They are (a) fission of a larger protoEarth to form Earth and Moon; (b) collision of a (Mars-sized?) body with a smaller protoEarth to form Earth and Moon or material which orbits the Earth and aggregates to form the Moon; and (c ) capture of the Moon by Earth in its present orbit as the Moon passed closely by in capturing orbit.

The fission and collision models, the latter being presently the most favoured in scientific circles, must be rejected; if only for the following planetary relationships.

1. The mass of the terrestrial planets are related by the formula:

m_x = (M_J + 100) x 1.23ⁿ

where n = -28 (Earth), -29 (Venus), -39 (Mars), -42 (Mercury)

For the seven major satellites the masses are related by the formula:

m_y = (M_pJ +100) x 1.23ⁿ

where n = -46.5 (Ganymede), -47 (Titan), -48 (Callisto), -49 (Io), -50 (Moon), -52 (Europa), -56 (Triton);

M_J = mass of Jupiter;

M_pJ = mass of all the planets and satellites LESS Saturn.

2. The specific gravities of the silicate satellites and uncompressed terrestrial planets can be calculated using the formula:

S.G. = 2 x 0.981ⁿ

where the n’s of interest here are -22 (Europa), -27 (Moon), -30 (Io), -33 (Mars), and -36 (Earth).

(c ) the radii of the four planets Uranus (U), Neptune (N), Earth (E), and Venus (V) give the following ratios:

R_U/R_E = R_N/R_V = 4.1 (= 1.19⁸ nearly);

R_E/R_V = R_U/R_N = 1.055 (= 1.23^1/4 nearly).

4. Where tilt = obliquity of the rotation axis to the orbital plane normal, R = radius of the planet,

U, N, E, and V are as for (c ) above, then

tilt_U/tilt_E = R_U/R_E closely (4.18 cw 4.1)

tilt_N/tilt_V = R_N/R_V closely, where tilt_V = (180 + 2.67)^o NOT (180 - 2.67)^o and tilt_N = 720° + measured tilt.

5. As for (d) above but Ms = Mars, Mn = Moon and the obliquities are measured from the Jupiter orbit plane normal, then

tilt_Ms/tilt_Mn = R_Ms/R_Mn closely (2.03 cw 1.96),

i.e. the two bodies obey the same tilt ratio formula as do the four planets.

6. In a spherical body pressure varies approximately as a straight line from the centre (X) outwards. In the on line and touching order Earth, Venus, Mercury, point X, Neptune, Uranus; the ratio distance centre of Uranus from X to distance centre of Earth from X is the same as the ratio impulse pressure on Earth to impulse pressure on Uranus (see paper A) and the forces of Earth to Uranus on point X is the same as the ratio of the forces of Earth to Uranus on a point at 5.3 AU with the present orbit distances from the Sun and the two bodies lying on the same radius from the Sun.

7. The impulse pressures of Mars, Io, Moon, and Europa plot on a straight line against distance when the bodies are touching in line (see paper C).

(h) The rotation of Uranus can be calculated using the rotations of Earth and Jupiter and applying the Single Body Breakup hypothesis.]

The above points cannot be explained by the fission and collision models. For example, the probability that fission of a protoEarth would result in mathematically simple mass, volume, and density relationships with the other satellites of the same form A = B x Cⁿ is so remote as to be absurd. Also, the tilt relationships between Earth and Uranus, and between Moon and Mars make it virtually impossible that collision could have taken place, even assuming no addition/subtraction of mass and volume.

In fact points (a), (b), (c ), (d), and (e) also rule out expansion of the Earth (a popular "theory" at present); and together with the relationship shown between the tilts of Jupiter and Saturn (see paper B) show that none of the planets can have suffered major impacts of a size sufficient to tilt them.

There is therefore only Moon capture as a possible model for the Earth-Moon couple. But the Moon-capture model advocated in the literature, i.e. both bodies in orbit about the Sun, cannot be accepted. This is because it suffers from the requirements of an extremely restricted angle of capture and in it being unable to account for the geochemical dissimilarity of the two bodies using the Nebula Theory. The first objection may be overcome but the second, despite many attempts, has yet to be satisfactorily solved.

However, it is possible to successfully argue another capture model if the Nebula Theory of Solar System formation is abandoned and a single body breakup hypothesis used.

The New Earth-Moon Model and Supporting Evidence. It was pointed out in paper C that the formula d = 2 x 0.724ⁿ gives approximately the spacing of Jupiter’s four major satellites when d is in Jupiter radii and n = -3 for Io, -5 for Europa, -6 for Ganymede, and -8 for Callisto. The argument in paper C was that the Moon on ejection went into orbit about protoJupiter at an orbit distance of n = -4 or about 7.28 R_J, and it was shown that at this distance its orbit speed would have been about 16.0 km.sec^-1.

Note that the Earth ejection speed from protoJupiter (see paper A) was (29.8 - 12.9) = 16.9 km.^sec-1. Thus if the two bodies approached near enough to one another while Earth was passing outwards from Jupiter, and curving in towards the Sun, for the Earth to exert a greater force on it than that by Jupiter, the Moon would be readily captured by the Earth. Capture would have been a random event but the dynamic conditions of Moon and Earth, if the bodies were near enough to one another and Jupiter lay to the rear of Earth, would have been favourable for their joining to form a couple. In the case of the Moon-Earth couple, Earth would have had to have passed behind the Moon.

Conclusions and Predictions. There is enough mathematical evidence derived from a simple postulated breakup sequence of a differentiated compound body to argue with confidence that the Moon and Earth originated during a continual (not necessarily continuous) breakup and ejection of internal matter from a protoJupiter. The Moon was ejected first and went into orbit (or was approaching orbit distance) about the protoJupiter. Earth coming soon after moved slightly faster than the Moon , and passing outwards through the rings of satellites captured the Moon because of similarities in their dynamics and because of fortuitous timing.

If the sequence is as predicted from this hypothesis then the geochemical makeup of the Moon must lie between that of Io and Europa. Basically, it should consist of a more aluminous and less ferrous silicate than Earth, with a probable mathematical relationship of the elements. (There certainly is one for Fe between Earth and Moon. Aluminium has not been tested.)

The hypothesised structure and differentiation of protoJupiter should allow some prediction of the internal composition of the Moon - also of Mars, Io, and Europa - but with the possibility that some surface variation may have been effected during ejection through protoJupiter.