A SPECIFIC GRAVITY RELATIONSHIP BETWEEN THE SATELLITES OF THE PLANETS AND THE PLANETS.

By T. Frank Lee, geologist, MAppSc, BSc, etc., January, 2009. 102 Mill Street, Ballarat, 3350, Victoria Australia.

It was fairly early discovered during my study of the Solar System that there was a mathematical relationship between the densities of the satellites. Later, it was found that this relationship - formula – extended to the planets. This is reasonable, for if the brown body became differentiated as described by the Single Body Breakup Hypothesis and the minor planets and satellites peeled off in succession and were ejected then some sort of density relationship should appear. This paper gives a summary of the early work as outlined in my 1994 book, with some comments on work done since that time.

In the book, beginning on p.216, is Section 6. Only the early part of this section will be given here.

6. A SPECIFIC GRAVITY RELATIONSHIP BETWEEN THE SATELLITES OF THE PLANETS LEADING TO A SUGGESTED ORIGIN OF THE SATELLITES (AND OF THE TERRESTRIAL PLANETS).

Table 1 of Yoder and Peale (1981) was seen while looking up details during the writing of sub-section 3(f). The table lists the physical dimensions of the four major satellites of Jupiter: Io, Europa, Ganymede, and Callisto. The specific gravities (S.G.s) in particular attracted attention. They are (with radius in km of the satellite in brackets): Io 3.53 (1816), Europa 3.03 (1569), Ganymede 1.93 (2631), and Callisto 1.79 (2400). Note the ratios of 1000x S.G. over radius: Io = 3530/1816 = 1.94383; Europa = 3030/1569 = 1.93117 i.e. 0.66% different; Ganymede = 1930/2631 = 0.73356; Callisto = 1790/2400 = 0.74583 i.e. 1.67% different.

There appears to be a close relationship between the two dense satellites and a close relationship between the two less dense satellites, while the two groups are distinct. (Actually not all that distinct, for the reciprocals of the Ganymede and Callisto values are not so vastly different to the above Io and Europa values.) (A much later written paper will be found in this website showing the relationship between these four satellites plus Titan and the Moon. F.L., 2009)

But the above, calculated in retrospect, merely suggest a relationship. It was the S.G. values which initially attracted attention. Assuming the hypothesis and the suggested origin of the satellites presented in this paper are correct it is to be expected that certain relationships appear. And this relationship is confirmed by the following ratios:

S.G._I/S.G._E = 3.53/3.03 = 0.858357 = 0.981089⁸; S.G._G/S.G._C = 1.93/1.79 = 0.927461 = 0.981350⁴.

(Note: Where M = Moon, S.G._M/S.G._I = 3.34/3.53 = 0.946176; S.G._E/S.G._M = 0.907186.

Av. 0.926681 or 0.981144⁴)

That is, the ratios appear to be related to the ratio 0.98 determined in Section 1, 0.981⁸ being less than 1% different from 0.98⁸, the worst case.

Io and Europa belong to the “silicate” satellites group and Ganymede and Callisto belong to the “icy”. The above result suggests a close relationship between the two groups. Yet initial testing suggested that the four values are not part of a single series, but are of two series; viz. for “silicate” satellites x = 3.03x0.981^-n; for “icy” satellites y = 2.00x0.981⁺ⁿ.

However, the two equations are linked, for 3.03/2.00 = 1.515 = 1.23085²; again the appearance of an important ratio discovered in Section 1.

The equation y = 2.00x0.981⁺ⁿ was examined first; with the values determined from it compared with S.G.s of known “icy” satellites. Tabling successive solutions of even powers of +n gives, for y = 2 x 0.981⁺ⁿ:

n y Satellite and value Satellite of

0 2.000 Triton 2.0 Neptune

2 1.925 Ganymede 1.93 Jupiter

4 1.852 Titan 1.85 Saturn

6 1.783 Callisto 1.79 (Cole gives 1.81) Jupiter

8 1.715 None

10 1.651 Ariel 1.65 Uranus

12 1.589 Titania 1.59 Uranus

14 1.529 ----

16 1.471 ----

18 1.416 Mimas 1.40 Saturn

20 1.363 ----

22 1.311 ----

24 1.262 Miranda 1.26 Uranus

26 1.215 Enceladus 1.20 Saturn

28 1.169 Iapetus 1.16 Saturn

30 1.125 None

32 1.083 None

34 1.042 None

36 1.003 Tethys 1.00 Saturn

The list has eleven coincident S.G. values out of nineteen. Can the missing values be explained? Yes they can. First consider the values for n = 13 to 23. The odd powers give:

13 1.559 None

15 1.500 Oberon 1.50 Uranus

17 1.443 Umbriel 1.44; Dione 1.43 Uranus; Saturn

19 1.389 Amalthea 1.382 calculated from dimensions Jupiter

21 1.337 Rhea 1.33 Saturn

23 1.287 none

(Except for Ganymede and Callisto the satellite S.G. values are taken from Cole, 1990; but note that I calculate Rhea S.G. as 1.216 (n = 26). I have no details of the Neptune satellite Nereid and so this has been omitted from consideration.)

So four of the missing satellites have been accounted for. But why the change to odd powers between n = 14 and n = 24? I cannot say; but I note that there is a symmetry of the even powers about n = 18, i.e. blank, blank, Mimas, blank blank. Taking the S.G.s and radii of Oberon, Umbriel, and Dione (from Cole, 1990: S.G. 1.50, 775 km; 1.44, 595 km; S.G. 1.43, 560 km, respectively) and determining the S.G. of the sum gives 1.471, i.e. equal to 2x0981¹⁶. A similar calculation cannot be done for Amalthea and Rhea as Cole does not give a S.G. for Amalthea and calculating the S.G. of Rhea gives 1.216 (i.e. 2x0.981²⁶) compared with the value of 1.33 given by Cole.

None-the-less, the exact determinations of such a large number of densities of “icy” satellites using such a simple power formula strongly suggests a relationship between the satellites even though they are satellites of four planets. The suggested origin given in this paper readily accounts for this as it requires the satellites came into being as Saturn separated from the parent brown body prior to the ejection of Uranus and Neptune. (A few years after this was written I brought the formula to the attention of an astrophysicist, who promptly swept it aside on the grounds that a formula can be determined for any group of points. Yes, but not a formula of a simple form, in this case A = BxCⁿ, that appears again and again in different groups of physical properties. About a month after the criticism a further five satellite values were published that fitted the formula perfectly.)

There are still four omissions from the list: noted as “None”. There is an explanation even for these. In sub-section 3(e) it was argued that the Earth’s water derived from collision of the planet with an “icy” satellite circling protoJupiter in the “inner” ring as Earth was ejected; and in sub-section 7 it will be shown that the probability of such a collision was quite high. The calculation of the above list, if all even powers of n should coincide with a satellite (and omitting between n = 13 and n = 23) requires that four satellites have been removed. It is natural to suspect that one of these missing satellites collided with Earth. If so, it was probably that of n = 8 (S.G. = 1.715 as the densities of 30, 32, and 34 are so low (1.125, 1.083, and 1.042) as to suggest appreciable NH₃, CO₂ or some such compound, which cannot be accounted for in the calculations of sub-section 3(e). It can be simply shown that if this was the satellite that collided with Earth then its radius was about 511 km.

Also, it will be argued in sub-section 7 that probability of capture of inner ring satellites by Mars would account for its two moons Phobos and Deimos. These two moons are “silicate” with densities near 2.0. The moons would be the “heavy” cores of the former “light icy” satellites (the water having been drawn to Mars over time. FL.2009) and so the satellites must have had much lower S.G.s, suggesting they were low on the above table. So these can account for two of the three n = 30, 32 or 34 missing satellites.

The next paragraph suggesting the last missing satellite is probably Charon is incorrect. This body has since been shown to have formed a string with Pluto and Triton, with a body “Asteroid” in opposition, forming immediately after the ejection of the “silicate”, (including Mars) and “medium icy” satellites, and being ejected immediately before Uranus and Earth. See 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” in the website.

An abstract of M. Bursa’s 1990 paper concerning the densities calculated for Saturnian satellites states, “the author estimates that the densities of Prometheus, Pandora, Epimetheus, Mimas, and Enceladus are (0.8-1.3)10³ kgm^-3 (7 refs.).” That is, some of the smallest satellites are at or below the density of ice. The outer layers of the “hydrous” shells would not only be expected to be light but also relatively incompetent and so produce small bodies. However, I have stopped the density investigation arbitrarily at S.G. 1 and made no attempt to see if the formula still applies for these very light satellites. (In passing I note here that the hypothesis can also explain quite simply why the small satellites of Saturn have lighter densities than those of Uranus.)

Now we look at the formula x = 3.03x0.981^-4.

The “silicate” satellites are few in number (This should be expected being much more competent bodies than the “icy”.) and it seemed at first that there was probably no point in looking for a fit for a sequence in n values. However, it is possible that the “silicate” satellites were the beginning of the breakup of the silicate core of protoJupiter (since proved conclusively to be so) and in Section 1 it was considered possible that Mars was transitional between satellite and true terrestrial planet. The suggestion, then, was that n values be inserted in the formula to cover the full arrange of “silicate” satellites and the terrestrial planets. The following table was constructed for the formula x = 3.03x0.981^-n.

n (n_a-n_b) x Body and values Satellite of

0 - 3.030 Europa 3.03 Jupiter

5 5 3.335 Moon 3.34 Earth

8 3 3.530 Io 3.530 Jupiter

11 3 3.742 !_↘ Mars uncompressed 3.7-3.8*

14 3 3.963 ! !↘

17 3 4.198 ! !---------- Earth uncompressed 4.0 – 4.5*

20 3 4.447 ! !↗

23 3 4.710 !↗----------Venus uncompressed 3.9 – 4.7*

30 7 5.387 Mercury 5.4 (uncompressed?)*

*Values from Table 4-1, Wood (1979), p.67

The values obtained using the formula for the three satellites, Mars and Mercury are quite close to the known values and the formula appears to be valid. The uncompressed values of Earth and Venus are too imprecise to be used for testing. At the same time, there appears to be a sequence relationship.

My enthusiasm for the formula x = 3.03x0.981^-n cooled a little when I noted, after testing it, the S.G. values for the “silicate” satellites given in Cole (1990). These differ slightly from those of Yoder and Peale and tend to make the fit look worse. After a brief consideration of the problem it was decided that despite initial testing, the first formula was valid for the “silicate” satellites for if y = 2.00x0.981ⁿ is used with negative values of n then the following table can be made.

n (n_a-n_b) y Body and value Satellite of

-22 22 3.050 Europa 3.04 Jupiter

-27 5 3.357 Moon 3.34 Earth

-30 3 3.556 Io 3.55 Jupiter

-33 3 3.767 Mars uncompressed 3.7-3.8

-36 3 3.990 !↘!↘

-39 3 4.226 ! !------ Earth uncompressed 4.0-4.5**

-42 3 4.476 !---!↗--- Venus uncompressed 3.9-4.7

-45 3 4.742 !↗

-48 3 5.022 None

-52 4 5.423 Mercury uncompressed 5.4 (Mercury 5.42)

** Cameron (1985) states in his introduction that Strom (1979) calculated uncompressed values for Earth and Mercury as 4.04 and 5.3 gmcm^-3reapectively. This places Earth’s value within 1.25% of the n = -36 value in the formula. For Mercury, if n is taken as -51, i.e. continuing the increases of 3 in the table, then y = 2x0.981^-51 = 5.32 and this agrees with Strom’s value of 5.3. (The uncompressed values for Earth and Venus are based on the planets’ internal composition and structure according to the Nebula Theory. The Hypothesis suggests a different central composition – not solid nickel/iron – and so the uncompressed values for Earth and Venus in the above list must be considered wrong. F.L., 2009)

The y values can be considered to be just as good as the x values and so it is possible to determine all the 20 satellite S.G. values less two given by Cole (1990) in his Tables V and VI, together with those of Mars and Mercury. Of the two not calculated one has been given no density figure (and calculating it by mass over volume gives an impossibly high value) and the other, Dione (1.43), is almost coincident with Umbriel (1.44), which does fit the formula, and when used with Umbriel and Oberon gives the correct n = 16 value.

I find it impossible to consider the formula y = 2.00x0.981ⁿ as accidental. There must be some physical basis for it to fit so many actual S.G. values so closely over such a broad range; and in addition to show a sequence pattern. To use the power of a number from +36 to -52 and obtain correct figures over this interval cannot be accidental. Even the changing of the number 0.981 to 0.98 (i.e. by 0.1%) gives end values of 0.96642 (c.w. 1.00256) and 5.71840 (c.w. 5.42302), clearly resulting in quite different S.G.s to the actual. (I note that in “Physics and Chemistry of the Solar System” by J. S. Lewis [Academic Press, 1997], Table 111.3 on p.54 the uncompressed pressures [=P] are for Earth 4.05, Venus 4.00, Mercury 5.30 giving n values in the equation for Earth -36 [difference 1.51%], Venus -36 [0.26%], and Mercury [0.38%]. That is, the “3” sequence of –n is throughout the silicate bodies. I shall not discuss the Earth/Venus similarity.)

I cannot see how the S.G. sequence can be explained by the Planetesimal (read Nebula) Theory. The theory argues that most, if not all, the satellites were either formed at approximately the heliocentric distance of their parent planets (within the zone of influence as the forming planet swept up planetesimals) or formed about the planets. Yet the tables show the satellites, based on density, as being sequentially related to one another even though they are irregularly distributed amongst the planets. But what cannot be explained here by the Planetesimal theory can be simply explained by the hypothesis presented in this paper. The “icy” group of satellites formed by successive “peeling” of the outer layers of the “hydrous” shell of protoJupiter, while the “silicate” satellites formed when the “hydrous” shell was largely removed and the mechanism operated on the outer layers of the silicate core. It is perhaps unlikely that the “hydrous” shell material would have formed competent small bodies without some more competent material present about which the “ice” could have coagulated, and I suggest this was “spray” of hydrous silicate being flung outwards from the surface of the silicate core, thus the water-ice/carbonaceous chondrite material mixture of the icy satellites can be explained.

What follows is surmise based on sub-sections 3(e) and 7 and the formula ------

January, 2009 comments. As has been mentioned above, it has since been established that internal breakup of the brown body was such that “peeling” of the “hydrous” and “silicate” satellites occurred simultaneously. See, for example, the poster presentation of breakup that is given in this website. Also, the “spraying” of matter during ejection of the planets must be taken into account. It is my opinion that much of the satellite swarm about Saturn, the “light icy” satellites (and also the comets), was due to spraying of the incompetent outer “icy” material immediately after Saturn split from protoJupiter. A meteorite study showed such spraying occurred with the “silicate” bodies.

I should also like to direct the readers’ attention to the application of the formula y = 2x0.981ⁿ to the densities, not uncompressed densities, of the gaseous planets. It will be found that for Jupiter n = +20 (0.20% different), Saturn +54 (0.024%), Uranus +22 (0.88%), and Neptune +6 (0.98%). Notice that the n numbers are all even. Calculating the average density for Jupiter + Saturn gives 1.11875 (closely 1.23^1/2), Neptune + Uranus gives 1.53342 (closely 1.23²). The ratio of the two groups is 1.37913 (0.06% from 1.38 or 0.15% from o.724^-1). The analysis can be carried further, for example silicates to gaseous, including the satellites, and so on but I have not been inclined to do so. Enough that the values show very close origin relationships between the planets and must reflect the bodies’ origin. Can the Nebula Theory give an explanation for this? The Hypothesis does.