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Widdekind
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By inspection of the following planetary data (Earth-normalized units), we find that average density (p) scales linearly with surface gravity (g):
p = p0 + k x g (Nilssen's Law)
wherep0 = 0.536 [ 2.96 g/cm3 ]
k = 0.464 [ 1 - p0 ]
Since surface gravity scales as the product of average density times radius, we may substitute and solve explicitly for average density, and mass (M), as functions of radius (R):k = 0.464 [ 1 - p0 ]
p = p0 / (1 - k x R)
g = p0 x R / (1 - k x R)
M = p0 x R3 / (1 - k x R)
This relation substantially agrees with previous models*. As w/ Nilssen's Law of Stars**, surface gravity acts as a "compressive force" which increases density. Since, however, rocky worlds are solids, p0 represents the "uncompressed density" of rocks at zero-gravity. Its value (2.96 g/cm3) substantially agrees with the estimated densities of asteroids (~2.9 g/cm3***). g = p0 x R / (1 - k x R)
M = p0 x R3 / (1 - k x R)
* Carroll & Ostlie. Introduction to Modern Astrophysics, pg. 825.
** https://www.physicsforums.com/showthread.php?p=1992585#post1992585
*** Carroll & Ostlie. ibid., pg. 878. Asteroid Ida has a density of 2.2 - 2.9 g/cm3. Our "uncompressed density" is slightly higher, as we are here dealing w/ solids, whereas asteroids are rubble piles.
Note that the Moon is a "well-formed world", which obeys Nilssen's Law. It's density is less than Earth's only b/c of its smaller surface gravity. There is no evidence that the Moon's over-all composition is different from the Earth's. Indeed, Mercury was specifically excluded from our sample b/c of its anomalous composition*.** https://www.physicsforums.com/showthread.php?p=1992585#post1992585
*** Carroll & Ostlie. ibid., pg. 878. Asteroid Ida has a density of 2.2 - 2.9 g/cm3. Our "uncompressed density" is slightly higher, as we are here dealing w/ solids, whereas asteroids are rubble piles.
* Mercury is iron-rich, and over-dense, compared to the other planetoids, b/c it's missing its Upper Mantle material, and is composed only of its iron core & Lower Mantle. It was once a "much bigger planet", see: National Geographic Channel Journey to the Edge of the Universe (TV)
Code:
World Mass Radius Density Surface-Gravity
Venus 0.8150 0.949 0.954 0.905
Earth 1.0 1.0 1.0 1.0
Moon 0.0123 0.272 0.611 0.166
Mars 0.1074 0.533 0.709 0.378