RADIAL COMPARISON

OF

SOLAR SYSTEM BODIES

Haluk Akcam - Feb. 20, 2004

There are several thousands, perhaps millions of objects in our solar system. But, only a few of them are large scale bodies. We can find 30-35 bodies, which can maintain a diameter more than 1000 km. The shape of these large objects resemble a spheroid, due to accretion. A tabular list of them with recent figures can be found here, for a brief skim.

The Sun, a G2-V spectral type star, is the largest and central figure of the system, with a volumetric mean radius of 695990 km, resembling a huge furnace converting Hydrogen into Helium. There are some assumptions that the Sun may have a companion, but neither the observations nor the computations did verify the validity of this hypothesis, until now. In a descending order, next largest body is the giant planet Jupiter. It is about one thousandth of the Sun, and more massive than all other planets and satellites combined, suggesting a star candidate rather than a planet.

Next to Jupiter; Saturn, Uranus, and Neptune are the largest objects down to Earth, which is occupying the sixth place in regard to radial sizes. After Earth comes Venus and Mars. But the planetary sequence is broken here, because of two big satellites (Ganymede of Jupiter and Titan of Saturn), which are slightly larger than the planet Mercury. Down to the seventeenth place occupied by Pluto (a Trans-Neptunian Object in fact, but once regarded as a planet), there are five satellites (Callisto, Io, and Europa of Jupiter, Moon of Earth, and Triton of Neptune), which are quite larger than this tiny and remote quasi-planet.

The recent discoveries of some large scale TNOs (Trans-Neptunian Objects, orbiting beyond Neptune) have imported the possibility of the existence of more Pluto-like bodies at the edge of our solar system. But, since these bodies - also known as Kuiper Belt Objects - are too far and  too faint for observations, the radii of these objects cannot be measured precisely, for now. The largest TNO discovered until today is 2002LM60 (50000) Quaoar, and estimated to have a diameter of 1260 ±190 km after reprocessed images, which in turn placed this TNO in the 22nd rank of the list.

The Sun, and Jupiter next to it, are the first two largest bodies in the system, and the ratio between their radii is about 9.9554 ±0.0009. When we search for a similar ratio coupling with Jupiter, a simple comparison shows that the Earth becomes the most fitting object with a slight deviation. Namely; r_J² / r_S = 7022 ±1.21 km, and the closest radius to this value is the one of the sixth (6371.00 ±0.01) in the list, which is the Earth. The ratio between the radii of Jupiter and Earth is 10.9733 ±0.0010, ten percent more than the previous one. Thus, we get a comparison pattern of radii having the most similar ratio within.

After defining the first three succeeding components of a series, we need to find the fitting fourth one. Keeping the same ratio sequence linearity defined as 9.96 and 10.97, the value for the fourth radius becomes r_S × ( r_E / r_J )³ = 526.73 ±0.14, corresponding to the 28th or 29th body of the present list. Since the radii of corresponding bodies are not precise, we cannot assign a certain one as the fourth one, but estimate Tethys perhaps for it. The actual ratio then becomes 12.0254 ±0.0341 instead of hypothetical 12.0953 ±0.0032.

Now, we have a series with radii of the Sun > Jupiter > Earth > Tethys sequence. Between the Sun and Jupiter, there are no objects in the list. But, between Jupiter and the Earth, there are three bodies: Saturn, Uranus, and Neptune, which may constitute further similar series, with corresponding components found among the bodies between Earth and Tethys.

Following the ratio sequence of 9.96, 10.97, and 12.03 (12.10), the only and best fitting component for Saturn becomes Venus, with a ratio of 9.6222 ±0.0010. But surprisingly, Mars will be bypassed, and the component candidates for Uranus become Ganymede, Titan, Mercury, and Callisto. Also, for Neptune; Ganymede, Titan, Mercury, and Callisto will be found, but not Mars.

Yet, after the Saturn-Venus couple, there appears a vertical sequence with the Jupiter-Earth couple, as 10.97 and 9.62. The third term of this ratio sequence is ( r_S / r_V )² × ( r_E / r_J ) = 8.4374 ±0.0025, and thus the best fitting component for Mars is expected to have a radius of 28602.24 ±8.83. The only body with closest radius is then becomes Uranus. The actual ratio between the radii of Uranus and Mars is 7.4816 ±0.0022, 11 % less than the expected one.

Now, for Neptune remains Mercury, with a ratio of 10.0922 ±0.0119. If we would assign Mercury for Uranus, then the ratio would be 10.3955 ±0.0071. After disregarding Mars, and considering the radial similarities of Uranus and Neptune, the possibility of categorizing Uranus and Neptune under the same class, and assigning Mercury as the only couple for the both, seems more plausible. But then, Mars will remain alone, missing the larger component.

The origin of the asteroids, the minor planets orbiting on a main path between Mars and Jupiter, has been always a problem, in regard to the early formation of the solar system. For about two hundred years, most of the hypotheses have been focused on an exploded planet, but without sufficient material supporting the would be existence of such a body. Some of them also include the probability of Mars being a  satellite of the exploded planet. On the other hand, there has been controversial opinions claiming that the gravitational field of the massive body of Jupiter would not let any other planet orbiting in such a vicinity. Besides, neither the total mass nor the diverse composition of the asteroids are found supporting the possible existence of such single body, which is believed to be exploded.

Total mass of the asteroids is round 2.3 × 10²¹ kg, and Ceres, the largest one, comprises one third of it. Data for the three largest asteroids are: Ceres; mass 8.7 × 10²⁰ kg, density 1.98 g/cm³. Pallas; mass 3.18 × 10²⁰ kg, density 4.2 g/cm³. Vesta; mass 3.0 × 10²⁰ kg, density 3.9 g/cm³. Thus, total mass of the three becomes 1.488  × 10²¹ kg, about two third of all. Out of three largest one, we get a mean density of about 2.84 g/cm³, and then the radius of the would be planet - disregarding the chemical diversity of the asteroids - can be around only 580 km!

It is possible to suppose that Jupiter might have captured the large scale bodies out of the debris. Total mass of the Jovian satellites is 393.12 × 10²¹ kg, and the mean density is 2.404 g/cm³. When we add all Jovian satellites and all asteroids, with a 395.42 × 10²¹ kg mass, and 2.406 g/cm³ mean density, the radius of the would be planet can reach up only to 3400 km! Nothing more than a replica of Mars can be brought up, even with all of these asteroids and Jovian satellites.

Another possibility is that Jupiter itself might have absorbed the rest of the debris. Because, present volume of this giant planet seems unusually large. As an example, we may assume that Jupiter was once a relative small giant planet with a medium radius between the logarithmic scale of the Sun and the Earth: ( r_S × r_E )^½ = 66589 km. In that case, the space it occupies would be 86.4 % of the present Jupiter. If we assume that the mean density of the previous Jupiter was 1.314 g/cm³, then the corresponding mass would be 1.625 × 10²⁷ kg, or 85.6 % of the present one. The wanted mass is about 2.738 × 10²⁶ kg, which corresponds to a planet (density = 2.021) with a radius of 31858 km.

Then, we may further assume that Neptune was a larger planet, orbiting around the Sun with a = 2.8 AU, exactly in the fifth position of the Titius-Bode sequence, where the asteroid belt is placed today. If we assume that the radial ratio between Earth and previous Jupiter was the same of Mars and previous Neptune, then the previous Neptune would have a radius of 35431 km. It would be a planet three times larger than the present Neptune. With a mean density of 2.021 g/cm³, the mass of such a planet would be 3.766 × 10²⁶ kg. That is, 3.7 times more massive than the present Neptune.

Now, for a moment, our model presents a solar system fitting to the law of Titius-Bode: r_n = 0.4 + 0.15 × 2ⁿ, where n = -inf, 1, 2, 3, ..., 8. The table below shows the Bode's figures and actual possible distances of the bodies from the Sun, namely;  a × ( 1 - e )  and  a × ( 1 + e ) , in AU:

The stability of this model cannot endure, because of the interacting gravitational forces produced by these two massive planets having adjacent orbits. If we assume the Bode figures as semi-major axii of previous Neptune and previous Jupiter, their orbital periods become around 4.7 and 11.9 years, respectively. With these orbital parameters, heliocentric conjunctions occur each 7.7 years, with a mean distant of 2.4 AU. If a planet with a mass of 1.625 × 10²⁷ kg encounters another one with a mass of 3.766 × 10²⁶ kg, at a distance of 2.4 AU, the destiny of such a meeting ends up with a collision. The next planet, which is less than one fourth of the massive one, immediately explodes, and the debris strays away.

After the collision, we may further assume that the main part (8/11) of the exploded planet is absorbed by the giant counterpart, which in turn caused expansion and the birth of the present Jupiter. The scattered remnants (1/1000) might be partly captured by Jupiter forming the Jovian satellites, and the remained debris (6/1000000) constituted the present asteroid belt. Yet, about 3/11 part of the exploded planet might be ejected during the collision, and driven out to an orbit eleven times greater than the previous one, namely to the plane of present Neptune.

This scenario, although it lacks sufficient scientific evidence, is not unacceptable at all. But, the unfitting present orbit of Neptune in regard to Titius-Bode law cannot be counted as an evidence, of course. Since, Titius-Bode conjecture literally does not constitute a law, in scientific sense.

Before the formation of the asteroid belt, the model suggested here demonstrates an orbital planetary sequence with corresponding reciprocal radial ratios in this folded order: (Sun is regarded as the initial element, and Pluto being the representative of the outermost minor planets, or TNOs.)

With this model, the mean ratio of four sequential couples becomes 10.23. Further, we may assume that TNOs might be also remnants of a giant planet about one tenth of the Sun. Then, such a giant planet at the edge of the solar system with a = 38.8 AU would have a diameter similar to previous Jupiter. A mean density about 2 g/cm³, indicates a planet with 2.5 × 10²⁷ kg mass, which is about 1/800 of the Sun, or scattered some two hundred thousand replicas of Pluto, sharing a 1:3 orbital resonance with Uranus.

But, these are all assumptions based on the possible formation models of the solar system, which do not give an explicit clue to explain the radial classification of the present bodies, if there is any at all. Therefore, we should return to the present condition, and take a look at the main distribution of the bodies:

384-912

km

286-384

201-286

km

Ganymede

Titan

Mercury

Callisto

95-201

km

Io

Moon

Europa

Triton

Pluto

The table above is showing the possible classification of large scale objects of the solar system, based on volumetric mean radii. Out of the 39 largest bodies of the solar system (with H < 4.0), the first seventeen objects are shown here by name. The only Luminous one is the Sun. Four large bodies are constituting two adjacent subsets of the Large group. Moderate bodies are gathered in five subsets. Minor ones are not named, but given as radial limits. The next nineteen in the list and many others, which are not included there due to radial sizes, are to be placed in Minor-A1/A2 subset. Objects with a radius less than 95.5 km (down to 8 km) need to be placed in a fifth group, left to the minor one, which is not shown in the table. Minor group bodies are mostly irregularly shaped objects, due to accretion under low gravity with less mass. But, the Moderate, Large, and Luminous group bodies always maintain a spheroidal shape.

The comparison pattern of the table is based on radial proportions within the adjacent subsets, such as Large A1-A2 to Moderate A1-A2 is expected to be verified by Moderate C1-C2 to Minor C1-C2, and so on. A1-A2 subsets of Moderate and Large groups are constituting the two major couples: Earth-Jupiter, and Venus-Saturn. In a similar way, type A subsets and type C subsets are constituting other couples: Jupiter-Saturn, Earth-Venus, and Uranus-Neptune. The only luminous one, the Sun is shared by Jupiter and Saturn. Also, {Ganymede+ Titan+Mercury+Callisto} subset is shared by Uranus and Neptune. On the other hand, Mars does not constitute a couple with any members of the Large group, as well as the subset of {Io+Moon+Europa+Triton+Pluto}.

Estimated volumetric mean radius of a theoretically exploded planet between Mars and Jupiter, based on the calculated total mass and weighted mean density of the asteroid belt objects, is about 580 km. If existed, such a planet would be placed in A1-A2 subset of the Minor group. Again, if the total mass of the satellites of Jupiter would be added, the estimated radius of such a planet could place it only next to Mars, into Moderate-B. Therefore, we do not have any planet fit to subset Large-B. Same is valid for the vacant subset Large-D.

From an astrological point of view, it is then possible to assume that Mars and Pluto are the loose bodies, as well as the Moon. Sun, on the other hand, is in accordance only with Venus-Earth-Saturn-Jupiter quadruplet. Since Luminous C1-C2 subsets are not occupied, Uranus and Neptune seem also to be one of these loose bodies. If the bizarre companion hypothesis is not valid, the Sun can be freely added to this loose-body group.

To place a fictitious companion of the Sun in this table, we have only the group Luminous A to D subsets. Acceptable radius for A1-A2 are within 532,000 - 808,000 km. For the subset B, estimated radius is around 370,000 - 390,000 km. Subset C1-C2 is available for a radius around 230,000 - 267,000 km. The lowest possibility is for the subset D with a radius around 125,000 - 209,000 km. Yet, being a self-luminous body requires certain amount of mass, not less than round 1.5-1.6 × 10²⁹ kg. Therefore, with these figures, it is hardly possible to place a star into subset D, for the required minimum density of the body would be 4 to 20 g/cm³, in that case.

Radial comparison between the body and its largest satellite, gives also a clue about the structure of the solar system. The table below shows this relation in regard to major bodies. Some asteroids also do have satellites (like Ida-Dactyl), as well as some TNOs (like 1999TC36). But, in regard to Asteroid Belt or Kuiper Belt objects, the main body is not counted as a significant one, from the point of radial size, and not taken into consideration accordingly.

On the other hand, although some historical observations had reported a possible satellite of Venus, named Neith, we do not have any reliable information on this subject. The same procedure is valid also for Mercury. Therefore, we assume that the two inner planets do not possess any satellites. 

1.8777

±0.0992

3.6668

±0.0002

9.9554

±0.0009

18.1927

±0.0261

22.6144

±0.0199

26.5700

±0.0194

32.1487

±0.0822

305.4540

±4.1314

First three ratios, in which the inverse is more than 10 %, are unique in the solar system: Pluto-Charon, Earth-Moon, and Sun-Jupiter. Charon is the only known satellite of Pluto, so is the Moon for the Earth. Both planets are orbited by relative huge satellites, not seen anywhere else in the solar system. For instance, if the Sun would have a satellite same as the Earth has, the radius of such a planet would be 190,000 km, which is 20 times larger than Jupiter. When the distance of the Moon is also compared on radial basis, this planet would be closer than Mercury, with 0.28 AU. A case that is not possible. For Jupiter, same comparison gives a satellite with 19,000 km radius, which is 380 times larger than Ganymede, or 27 times larger than the Earth.

The numerical relation between radial ratios of Earth-Moon and Sun-Jupiter couples can be defined as:

[ ( r_S × r_M ) / ( r_J × r_E ) ]³ = 20.0135 ±0.0088

For the case Pluto-Charon and Sun Jupiter, the same relational value is 151.5581 ±23.8735, and for Pluto-Charon and Earth-Moon, it is 7.5727 ±1.1922. If we apply the same formula to the whole sequence of the table above, with the appropriate radii, the result will be as:

Relation values show that large planets, Neptune, Saturn, Jupiter, and Uranus seem to be in stable conformity with their satellites. The average value is 1.7714. But, Earth-Moon ratio seems small, and a satellite at least similar to Rhea or Titania would be expected in stead of the Moon, in order to see an equilibrium for the Earth, even if the Earth is the densest planet of the solar system. On the other hand, Pluto seems more unstable with Charon. Yet, Mars is an exception in this table, with its tiny little asteroid-like satellite Phobos.

With these figures, origin of the Moon seems quite disputable. It simply does not fit to the satellite formation history of large planets. Charon was probably not a satellite of Pluto, but perhaps a TNO, and even Pluto itself was plausibly not a planet either. The two miniature satellites of Mars are among the smallest objects of the solar system, and it is highly possible that these could be captured by Mars during a late period of the solar system formation. But, there are no suggestions for the case of the Moon.