<blockquote id="quote"><font size="2" face="Verdana, Arial, Helvetica" id="quote">quote:<hr height="1" noshade id="quote"><i>Originally posted by Jim</i>
<br />...complex to set the gravity constant at one...<hr height="1" noshade id="quote"></blockquote id="quote"></font id="quote">
Here is an equivalent way of saying what I said in my previous post:
The acceleration of gravity at the surface of the Planet (not counting centrifugal force due to rotation) times T squared (where T is the light time from the Sun to the Planet, at its semimajor axis, i.e. approx. its average distance from the Sun) times the "Golden Ratio" ( 1.618034 = (1+sqrt(5))/2 ) to some whole power, equals the radius of the Planet. Like the NASA "Fact Sheets", I use the "volumetric average" radius, i.e. the geometric mean of the three semi-axes of the Planet's ellipsoidal shape (two of which are equal). Using the current figures on NASA's Fact Sheets, I find that all the rocky planets, except Venus, and spherical minor planets, on which data are available, conform to this law.
I approximated Ceres as an ellipsoid, used the larger of the two diameters (customary for asteroids) given by NASA as the equatorial, then found the volumetric average radius myself. For Charon, only an equatorial radius was given, so I used it as the average, since for Pluto the equatorial and polar radii are given as equal. Other minor planets are not very round and/or lack moons and have unknown compositions, so their surface gravity is unknown or undefined. The gas giant planets have surface accelerations which NASA gives arbitrarily as that at a depth of one bar pressure in their atmosphere.
The formula is:
(surf. accel.)*(Sun light time)^2 * 1.618...^n = radius of Planet
Here is a list of the values of "n" found:
Mercury 5.968
Venus 3.441
Earth 1.994
Mars 0.951
Jupiter -1.812
Saturn -2.932
Uranus -7.209
Neptune -9.604
Pluto -10.89
Charon -11.04
Ceres 2.818
For the outer planets (Jupiter & beyond) "n" is negative.
For the minor planets Pluto & Ceres the last digits above are not significant, and for Charon the next to last digit is hardly significant. For these planets, NASA gives the masses or surface accelerations only to two, or not quite three, decimal places. Agreement is excellent for Earth; good for Mercury, Mars, and Saturn; and at least fair for Jupiter, Uranus, Pluto, Charon and Ceres.
