THE SINGLE BODY BREAKUP HYPOTHESIS FOR THE SOLAR SYSTEM

NOTE: The subject matter and conclusions of the paper “Further Study of the Major Satellites of the Solar System and of the Lesser Bodies Triton, Pluto, and Charon – A Revision of Ideas of Origin”, dated May to September, 2004 supersedes any mathematics and conclusions reached in this earlier paper when the mathematics and conclusions are on properties treated in both papers.

PAPER C

The Major Satellites and Mars (but Triton excluded)

It is the intention in this paper to look at the major satellites and Mars to compare some of their major physical properties and to suggest a possible (probable?) mode of formation. Why is Mars included with the satellites? Because it has characteristics of both planets and satellites. But its origin can be shown to be linked more closely to the satellites, as will be illustrated below.

It was suggested in paper B that, excluding Saturn, the planets and satellites once formed a "protoJupiter", the structure of which was a Mercury core surrounded by a series of shells composed successively outwards of the mass and volume of Venus, Earth, Mars, "silicate" satellites, Pluto group, "medium icy" satellites, "light icy" satellites, Neptune, Uranus, and Jupiter. It was further suggested that this protoJupiter began to break up internally and the matter to be ejected from the body by a series of impulse forces. Paper A gives the breakup stage immediately prior to the ejection of Earth and Uranus.

Initial internal breakup was the separation of the gaseous component of the core (Neptune/Uranus) and the silicate/"icy" component. There is little difficulty in arguing or comprehending that before the terrestrial planets could be formed the surrounding less competent satellite shells had first to be removed. As this is an outline paper and not a rigid treatise, satellite removal prior to planet removal will be assumed; and it will be assumed that "onion skin" stripping of the core was the mode of separation followed by impulse force ejection. Should such not have been so then some (perhaps all) of the following relationships would not appear and the assumptions would have been wrong.

1. Mass Relationship. It is an empirical fact that the masses of the major satellites are equated by the formula,

m_x = (M_pJ 100) x 1.23ⁿ

where m_x = mass of the satellite

M_pJ = sum of the masses of all the planets and satellites LESS Saturn;

n = -46.5 for Ganymede, -47 Titan, -48 Callisto, -49 Io, -50 Moon, -52 Europa, and Triton –56.

Mars does not fit this sequence very well -n = -39 gives a +7.13% difference from the measured mass - and more nearly fits the formula for the terrestrial planets of,

m_x = (M_J 100) x 1.23ⁿ

where M_J = mass of Jupiter

and n = -39 (-2.65% different).

2. Specific Gravity (Density) Relationship It is an empirical fact that the specific gravities of the satellites and the uncompressed terrestrial planets obey the formula,

S.G. = 2 x 0.981ⁿ

where n is an integer.

For the "icy" satellites, n = -2 for Triton, +2 Ganymede, +4 Titan, and +6 Callisto. For the "silicate" satellites and uncompressed terrestrial planets, n = -22 for Europa, -27 Moon, -30 Io, -33 Mars, ---.

The above formulae suggest a close relationship between the satellites and the terrestrial planets groups, and there are other suggestive relationships not touched on here.

Consider, now, the separation of the "medium icy" and "silicate" satellites from the silicate core. Assume that the silicate/"icy" sphere lay beside the Neptune/Uranus sphere and that the "icy" shell peeled off in layers to form the three major "icy" satellites in line and touching one another and the silicate/"silicate" satellite core as in the following figure – from left to right Callisto, Titan, and Ganymede, "silicate"/"silicate satellite" sphere.

This was then almost certainly followed by a similar stripping of four satellites – Charon, "Asteroid", Pluto, and Triton - followed by the "silicate" satellites Europa, Moon, Io, and Mars as below.

The central four satellites are outside this discussion and can be ignored.

(c ) Radii Comparison of the Satellites There are various ways of comparing the two strings of satellites, e.g. within each string, between each string, and each string to the enclosing body, but only two will be given here as suggestive support for the breakup as outlined.

(c1) The radii in km of the satellites are:

Mars 3398, Io 1829, Moon 1738, Europa 1560,

Ganymede 2640, Titan 2560, Callisto 2400.

(c1a) Taking ratios of successive couples between the two groups gives:

Mars/Ganymede = 1.28712, within 4.64% of 1.23;

Io/Titan = 0.71436, within 1.32% of 0.724;

Moon/Callisto = 0.72417, within 0.02% of 0.724;

which suggests a relationship.

(c1b) More interestingly, taking ratios of successive couples from Triton to Ganymede to Mars and on into the terrestrial planets gives:

Titan/Callisto = 1.06667, within 1.29% of 1.23^1/4 (= 1.06667¹);

Ganymede/Titan = 1.03125, within 0.49% of 1.23^1/8 (= 1.06348^1/2);

Europa/Ganymede = 0.59091, no fit;

Moon/Europa = 1.11410, within 0.46% of 1.23^1/2;

Io/Moon = 1.05236, within 0.07% of 1.23^1/4;

Mars/Io = 1.85785, within 2.62% of 0.724^-2;

Earth/Mars = 1.87699, within 1.61% of 0.724^-2;

Venus/Earth = 0.94857, within 0.10% of 1.23^-1/4;

Mercury/Venus = 0.40314, no fit.

The Mercury/Venus value can be ignored as Mercury is the remainder of the sequence breakup. Then the bodies within the three groups "icy", "silicate", planet show 1.23 relationships, Europa/Ganymede shows no fit (another group – the Pluto group - lies here), and Mars shows an inverse relationship between Io (1.85785¹) and Earth (1.87699^-1). Mars thus shows a relationship to both satellites and planets; it is not clearly one or the other.

The pattern of the ratios clearly points to an intimate relationship between the bodies and the groups of bodies. Development was not unconnected or random. The single body breakup hypothesis gives one to expect such an intimate relationship.

4. Impulse Pressure Test Consider first the "silicate" satellites. These are postulated as having successively peeled off the silicate core to lie on a diameter and touching in the order Europa, Moon, Io, Mars, silicate core. Assume that at this stage an impulsive force originated at the centre of protoJupiter, a force of sufficient size to drive outwards the four smaller, in-line satellites but not the silicate core (or the gaseous core adjacent to it and on the other side of the centre of protoJupiter). If this breakup and ejection took place then two dynamic conditions can be predicted, conditions which must still exist to-day. They are:

(d1) On the basis that the impulsive force reduced as a straight line variation from the centre of the system outwards along the internal bodies’ alignment, the impulse pressures per unit cross sectional area on the four bodies must plot as a straight line variation when the impulse pressure is plotted on the y-axis against distances equal to the centre points of the touching spheres on the x-axis.

Assuming the velocities of ejection of the four bodies were translated directly into linear orbit speeds, then for Io and Europa:

impulse pressure = impulse force divided by cross-sectional area = m_x.v_x/pr_x²,

where m, v, and r have their usual meanings in dynamics.

The ejection velocity of Mars would have been its present velocity about the Sun less the velocity of the composite body (= 12.9 km.sec^-1 at 5.3 AU), i.e.(24.1 - 12.9) = 11.2 km.sec^-1, and so its impulse pressure, too, can be calculated. Plotting the three impulse pressure values gives the following graph.

A near straight line curve results. This suggests that the development as given above could have taken place. In turn it also suggests that the Moon, in the sequence as given, had had induced on it at commencement of its outward movement an impulse pressure of 1.24x10¹⁷ kg.km.sec^-1 (measured off the graph). By inserting this value in the impulse formula we obtain for the Moon a velocity of ejection of 16.01 km.sec^-1.

It is an empirical fact that the formula d = 2 x 0.724ⁿ gives approximately the distance of each major satellite of Jupiter from Jupiter measured in Jupiter radii (R_J), where n = -3 for Io, n = -5 for Europa, n = -6 for Ganymede, and n = -8 for Callisto. But note that when n = -4 is inserted in the formula a value of 7.28R_J is obtained and a body at this distance from Jupiter would have an orbit velocity about Jupiter of approximately 16.0 km.sec^-1. That is to say, the velocity of ejection and consequently orbit speed about Jupiter of the Moon (if it went into orbit) as determined by impulse pressure closely approximates the orbit velocity determined by the distance formula for n = -4. This suggests that the Moon was part of the series of satellites given, originally lay in the order given within protoJupiter, and was ejected into an orbit of about 7.3R_J about protoJupiter.

Turning to the "medium icy" satellites, assume a similar formation and ejection sequence took place as argued for the "silicate" satellites and prior to them. The bodies developed would have lain on a common diameter and touched in the order Callisto, Titan, Ganymede, silicate core. They would then have suffered an impulse force which drove them out of protoJupiter.

Knowing the velocities of orbit of Ganymede and Callisto we can calculate their bodies’ impulse pressures. For Titan, since captured by Saturn, we can make a calculated guess of its ejection velocity in the same way as was done for the Moon, above. In the formula d = 2 x 0.724ⁿ the Moon was placed between Io and Europa at the vacant n = -4 position. There is a vacant position at n = -7, between Ganymede and Callisto where Titan must be placed to give the correct satellite sequence. Assuming this position to be correctly filled it is possible to determine the satellite’s impulse pressure because its orbit velocity can be calculated. Plotting the impulse pressures against spacing of the three satellites as for the "silicate" satellites gives the following graph,

and it is seen that an almost straight line curve is obtained. This leads to the conclusion - in conjunction with the "silicate" satellite curve - that the satellites were formed as two groups and were ejected by two separate impulses, one immediately after each group formed.

(d2) Further evidence of such an ejection will appear if the impulse force acted at a slight angle to the diameter on which the satellites lay immediately after formation. The rotation axes of the bodies (Yes! They would have been rotating.) would have been tilted by the component of the impulse force at right angles to the join line of the body centres. It is a mathematical fact that the ratio of any two obliquities of the rotation axes (here called for brevity, tilts) must ideally equal the ratio of the radii of the same two bodies. The ideal is unlikely to be realised but the two ratios should be approximately equal.

Only the tilts of Mars and the Moon have been measured. Tilts for the other satellites have been "theoretically" calculated using assumptions but because of rotation lock by Jupiter none have been actually measured. Thus this tilt test can only be applied to Mars and the Moon in the "silicate" satellite group. Applying the test, the ratio for the radii of Mars and Moon = 1.96.

The tilts of the two bodies must be taken at (very closely) the location of Jupiter. Not considering refinements and possible variations (see Lee; Jan., 1998) the ratio of the tilts is about:

= (24.98 + 1.85 - 1.31)^o/(11.233 + 1.309)^o = 2.03,

i.e. the two ratios differ by 3.7% and the dynamic requirement obtains. The origin of the two bodies as suggested above is thus supported.

Conclusions The tests based on the assumption of the origin of the satellites deriving from the internal breakup of a protoJupiter accompanied by intermittent impulsive ejection of the bodies as described have been:

1. the mass relationships;
2. the density relationships;
3. the ratios of their radii;
4. the impulse pressure relationships - straightline variations as required;
5. the ratio relationship of the tilts of the rotation axes of Mars and Moon to the ratio of their radii.

All have supported the contention that development of the satellites came about in the early stages of the internal breakup of a protoJupiter body. None of the five relationships can be explained by the Nebula Theory, which requires random formation of the bodies and unrelated points of origin for the non-Galilean bodies.