Initial Draft	27 February 1993

The Apparent Quantization of Surface Gravity
Among Large Objects in the Solar System

by

Gerald David Nordley*
1238 Prescott Ave
Sunnyvale CA 94089

Abstract:

The surface gravities of the larger objects in the Solar System can be predicted to a surprising degree of accuracy by a the simple equation: g = a*Dn where g is the surface gravity of the object, a and D are fitting parameters and n is an integer.  There is no obvious physical reason for this, however, it is suggested that these values of g may be "strange attractors" in the chaotic system of planetary accretion.

1. Introduction:

The surface gravities of large objects are of interest to science fiction writers who take the effort to build realistic extraterrestrial worlds for their stories.  Since the solar system is the only model of a solar system we presently have as an example, we look at what we find here to imagine what we might find elsewhere.  This writer has always been struck by the number of planets that seem to have near Earth-normal gravity, but had not noticed until recently that both Mars and Mercury have essentially the same surface gravity.  That occasioned a look at Jupiter, which proved to differ from Earth by about the same factor as Earth differed from Mars.  This raised questions.  

2.  Data:

The Mars-Earth-Jupiter ratio caused the author to wonder whether other objects might fall into similar families.  A list of objects, their masses, and their radii was created from the values listed in the Cambridge Atlas of the Universe.  The approximate surface gravity of these objects was determined by the equation:

g = G*M/R2	(1)

where g is the surface gravity, M is the mass of the object and R is the radius of the object.  The objects appeared to fall into families.  The Mars-Earth-Jupiter relationship suggested a power law with surface gravity being proportional to some factor D to an 

*FBIS; Major, USAF (ret.)
integer.  Letting the constant of proportionality by a, we arrive at the following equation:

g' = a·Dn	(2)

where g' is the predicted surface gravity.
	Starting with the Mars-Earth-Jupiter factor of about 2.6, values of a and D were adjusted iteratively to give a good eyeball fit; the quality of the data on the outer satellites didn't suggest that a more rigorous mathematical treatment was warranted.  D = 2.55 and a = 0.6 gave a reasonable fit.  
	Figure 1 was generated by substituting the observed surface gravities, g,  for g' in equation 2 and solving for non-integer values of n, which were used as the horizontal coordinates for the points plotted. 
	Integer valuses were obtained by rounding this value the nearest integer, the predicted "quantized" surface gravities (g') were generated by equation (2).  The difference between the actual surface gravity and the value predicted by assuming an integer value for n was calculated and divided by g to obtain the normalized deviation in table 1.  For example, the predicted value of Callisto's surface gravity is 0.24, or 24% higher than the actual value.  A minus sign indicates the predicted value is less than observed.  A standard deviation, s, was calculated by:

s2 = ( ∑( 1 - g'/g)2 )/N

where N was the number of objects in the data base. The results are presented in Table 1.

3. Discussion:

	The results of this exercise were almost spooky. A cluster at    n = 1 (g ≈ 1.5 m/s2) representing moon-sized objects is now apparent, and there may be statistically significant clustering at n = -1 (g ≈ 0.24 m/s2).  Indeed, almost all of the surface gravities are within 25% of the predicted values quantized by a factor of  2.55!  This clustering of surface gravities is readily apparent in figure 1.  
	Things become less ordered in the lower mass range. Removal of the very low mass Uranian satellites from this data base would make it even more dramatic, and the data here was a little less certain (we can always hope refinements will improve the fit).  We can also see that the inner satellites of the giant planets tend to have surface gravities than predicted while the outer satellites tend have lower, something that might be expected from the "miniature solar system" model of giant planet moon systems, wherein the inner moons are made of denser stuff.
	Note however, that the model works well for both Earth and Saturn, at opposite ends of the density scale.  

	The question becomes whether this clustering of surface gravities is simply a numerical coincidence or whether it is the predictable result of some underlying physical process.  The example of Bode's law comes to mind.
	One possible explanation might come from noting that the accretional process has most of the attributes of a very complex, chaotic system, and that such systems often exhibit "strange attractors" i.e. preferred numerical values for various quantitative properties of the system.  But saying this, I'm afraid, has all the virtues of naming a disease without knowing its root cause.  Why would there be strange attractors in this system and why at those values?

4.  Conclusions:

	If this is simply a numerical coincidence, it should be admired as such without any further worry or effort being spent.
	However, if this apparent quantization is the result of a process that, in all likelihood, would occur elsewhere as well, it has some definite implications for science fiction world builders and astronomers interested in the possibility of life elsewhere.
	For science fiction writers, planets with surface gravities of, say, 50% to 70% of Earth's surface gravity become unlikely, and planets which have approximately Earth's surface gravity (which we like to use anyway, being easier to build and easier for the reader to grasp) become more likely.  Adventurous souls can still go for the 38% family of planets.  But the short of crushing Jovian style gravity, planets with significantly "heavy" gravity, such as 18 m/s become improbable. 
	For SETI workers, a systematic preference for Earth-normal surface gravity would raise the probability of Earth like planets existing somewhat, and increase the possibility of finding "life as we know it" out there.
	While the input data to this exercise can be refined and statistical tests improved, the coincidence seems clear enough as it is.  I suspect that massive computational simulations, well beyond the scope of this paper and its author, will be required to shed meaningful light on whatever processes might lay behind this apparent clustering of surface gravities.  

References:

Audouze, Jean and Guy Israel, ed. The Cambridge Atlas of the Universe The Cambridge University Press, New York, 1988.

"Why Pluto is Rocky," News Notes, Sky and Telescope vol. 76 no. 6; Dec.88

Tabel 1.	Apparent Quantization of Surface Gravities
		Among the Larger Objects in the Solar System
_____________________________________________________________________________
    Name       Mass       Radius   Surface  Exponents Predicted devi-  Vari-
	             (kg)       (m)      gravity   real Int  gravity  ation  ance
                                    (m/s2)              (m/s2)  (s =.155)    
1   Moon       7.35E22    1738000    1.623   1.1   1   1.53   -.05757 .00331
2   Io         8.916E22   1815000    1.806   1.2   1   1.53   -.15274 .02333
3   Europa     4.873E22   1569000    1.321   0.84  1   1.53    .15847 .02511
4   Ganymede   1.49E23    2631000    1.436   0.93  1   1.53    .06535 .00427
5   Callisto   1.064E23   2400000    1.232   0.77  1   1.53    .24141 .05828
6   Dione      1.05E21     560000     .2234 -1    -1  .2353    .05328 .00284
7   Rhea       2.28E21     765000     .2599 -0.9  -1  .2353   -.0948  .00899
8   Titan      1.359E23   2575000    1.367   0.88  1   1.53    .11885 .01412
9   Iapetus    1.93E21     730000     .2416 -1    -1  .2353   -.02626 .00069
10  Ariel      1.6E21      580000     .3173 -0.7  -1  .2353   -.25853 .06684
11  Umbriel    9.3E20      595000     .1753 -1    -1  .2353    .34247 .11729
12  Titania    2.7E21      805000     .278  -0.8  -1  .2353   -.15359 .02359
13  Oberon     2.9E21      775000     .3221 -0.7  -1  .2353   -.2696  .07268
14  Triton     5.7E22     1750000    1.242   0.78  1   1.53    .23207 .05386
15  Mercury    3.3E23     2439000    3.701   1.9   2   3.901   .05411 .00293
16  Venus      4.87E24    6050000    8.877   2.9   3   9.949   .12072 .01457
17  Earth      5.976E24   6378000    9.802   3     3   9.949   .01502 .00023
18  Mars       6.418E23   3397000    3.711   1.9   2   3.901   .0514  .00264
19  Jupiter    1.899E27  71398000   24.85    4     4   25.37   .02071 .00043
20  Saturn     5.686E26  60000000   10.54    3.1   3   9.949  -.05591 .00313
21  Uranus     8.66E25   26145000    8.453   2.8   3   9.949   .177   .03133
22  Neptune    1.03E26   24300000   11.64    3.2   3   9.949  -.14515 .02107
23  Pluto      1.26E22    1150000     .6357   .06  0    .6    -.05611 .00315
____________________________________________________________________________
*The Pluto Data is derived from density and radius given in "Why Pluto is Rocky,"Sky and Telescope News Notes vol. 76 no. 6; Dec.88

Figure 1.  Clustering of Surface Gravities Along the Curve: g = a·Dn