What is the average density of a rocky world according to Nilssen's Law?

[Widdekind]

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)​

where

p₀ = 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):

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***).

* 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*.

* 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)​

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

 

[Vanadium 50]

I don't think surface gravity is a sensible independent variable. When coalescing, the protoplanets didn't say to themselves, "Hey, let's form a configuration with such-and-such surface gravity!" Surface gravity depends on radius and composition, not the other way around.

For a sphere of uniform density, surface gravity increases linearly with radius. So it's not surprising that you get something like this. However, even you must admit that you are being selective with your data: tossing Mercury out and the Moon in, because one agrees and the other doesn't . There are 9 bodies in the solar system as large as or as heavy as the moon. There are 126 ways to pick 4 of them, so it's not that hard to find a pattern when one has the ability to cherry pick the data.

 

[Widdekind]

Peer-reviewed Journal articles exclude Mercury from consideration b/c of its anomalous composition*. It is well known that Mercury "was once a much bigger planet"**, whose outer layers were ablated by Solar processes. Nilssen's Law only applies to "well-formed (rocky) worlds". For example, Nilssen's Law does not apply to Iron meteorites, whose average densities might be ~1.5 (in Earth units, where 1.0 = 5.52 g/cm³), even for micro-meteorites. Moreover, Nilssen's Law does not apply to icy worlds either. Again, ices have very different bulk properties from "standard composition" rock.

* S.N. Raymond, et al. Exotic Earths: Forming Habitable Worlds with Giant
Planet Migration, pg. 13. (See: http://arxiv.org/PS_cache/astro-ph/p.../0609253v1.pdf )
** National Geographic Channel Journey to the Edge of the Universe (TV)​

Nilssen's Law is apt & accurate for "well-formed rocky worlds", but does not cover other cases like:

• disturbed planets (Mercury)
• Iron asteroids
• Icy worlds

where the bulk composition is distinctly different from terrestrial, Earth-like rock.

 

[Widdekind]

Nilssen's Law is limited in scope, to "well-formed" rocky worlds, of Earth-like Bulk Composition:

• Venus
• Earth
• Moon
• Mars
• (rocky) Asteroids & Meteorites

However, it is true that this is a limited sample. Nilssen's Law does not hold for Mercury (closer to Sun), nor for icy worlds (further from the Sun). This suggests that there is an implicit temperature dependence. For, the aforesaid Earth-like rocky worlds all formed within the Sun's "water zone" (roughly 273 K < T < 373 K, neglecting Green House Effects).

Conversely, Mercury formed inside the Sun's "steam zone" (T > 373 K), while icy worlds all formed out in the Sun's "ice zone" (T < 273 K).

So, you raised an important caveat:

Nilssen's Law only applies to "well-formed" (non-disrupted) Water Zone worlds​

Evidently, other relations govern Steam Zone & Ice Zone (as it were) worlds.

 

[Vanadium 50]

It is well known that Mercury "was once a much bigger planet", whose outer layers were ablated by Solar processes.

No, it is not. And "I saw it in a TV show" is hardly a peer-reviewed publication. There are no fewer than three competing theories.

 

[Widdekind]

No, it is not. And "I saw it in a TV show" is hardly a peer-reviewed publication. There are no fewer than three competing theories.

What are those theories?

Mercury formed in the Sun's "Steam Zone", as I pointed out above. I already acknowledged that planetoids' Bulk Composition, & hence Bulk Properties, will depend on their Formation Zone. Indeed, Carroll & Ostlie (pg. 893) show that there are 4 such zones:

1. Steam Zone
2. Water Zone
3. Ice Zone
4. Methane-Ice Zone

Indeed, I analyzed the Surface Gravities & Densities of other major Solar System planetoids, and found that their properties varied by Formation Zone:

1. Io, Europa (Water Zone)
2. Ganymede, Callisto, Titan (Ice Zone)
3. Triton, Pluto (Methane-Ice Zone)

Indeed, Io & Europa both basically obey Nilssen's Law. Io's actual density is 4% too high, while Europa's is 8% too low. Thus, both are "well-formed rocky worlds", but Io is somewhat more "Mercury-like" (over-dense), while Europa is somewhat "ice-like" (under-dense). This is b/c "Jupiter must have been hotter in the past than it is today, [so] Io would have been close enough to have had most of its volatiles evaporate away" (much like Mercury, from the Sun). And, "the characteristics of these worlds [= Jovian moons] are consistent with a decreasing average density with increasing distance from Jupiter, implying that the relative amount of water-ice crust increases w.r.t. the rock core" (pg. 837).

Meanwhile, Ganymede, Callisto, & Titan, all w/in the Sun's Ice Zone, all have extremely similar bulk properties. And, Triton & Pluto, w/in the Sun's Meth-Ice Zone, also have extremely similar properties.

Thus, the central star does not alone determine the "formation zone" (Steam, Water, Ice, Meth-Ice). Rather, major planets, like Jupiter, which coalesced out of its own sub-nebula (pg. 838), can impact planetoid formation in their vicinity*.

* Io (~422,000 km) is somewhat "steam-like"; Europa (~671,000 km) is somewhat ice-like; Gannymede (~1,070,000 km) is fully ice-like. This indicates information about the massive Jovian sub-nebula's temperature structure.​

But, the fundamental underlying principal, of planetoid formation, is that their Bulk Properties depend on their Bulk Composition, which is determined by the Temperature Zone they coalesce in.

SUMMARY: There are four (4) Nilssen's Laws, one for each Formation Zone (Steam, Water, Ice, Meth-Ice). That cited above is but the second such Law.

 

[Widdekind]

I analyzed all the spheroidal Icy Moons, of all the Gas Giants, plus Pluto & Charon. I found that Icy Worlds all (basically) obey a similar simple scaling law (Earth units):

p = p_0,ice + k_ice x g​

where:

p_0,ice = 0.29 (~1.6 g/cm^3)
k_ice = 0.44​

However, seven (7) of the smallest Icy Worlds are under-dense, perhaps b/c they are "rubble piles", have undergone extensive off-gassing [sp?], etc.

Attached are figures visualizing the data, and these fit-lines. Steam Zone worlds (ie., Mercury) are drawn in red; Water Zone worlds are in blue (also including Io & Europa); Ice Zone worlds are in green.

Please excuse the lack of graphical annotations. If anyone knows how to annotate graphs in SciLab, please speak up.


ADDENDUM: Note that k_ice (0.44) is almost the same as k_rock (0.46). Only the "uncompressed density" increases. So, both Rocky Worlds & Icy Worlds have essentially the same "bulk compressibility". This suggests that Steam Zone worlds (eg., Mercury) might obey something like:

p = 0.8 + 0.5 x g​

 

[LURCH]

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.

Most planetary models depict the Earth with a substantial metal core, and the Moon without one. Are you saying these models are innacurate? If so, are you claiming that the Earth does not have a metal core, or that the Moon does?

 

[Widdekind]

Most planetary models depict the Earth with a substantial metal core, and the Moon without one. Are you saying these models are innacurate? If so, are you claiming that the Earth does not have a metal core, or that the Moon does?

Thank you for your reply.

ALLEGATION: The Moon's Bulk Composition is (essentially) the same as Earth's. The Moon is ~30% Iron, ~2% Nickel, etc., (essentially) the same as Earth.

CLAIM: Yes, the Moon has a metallic iron core.

WAGER: $10

CAVEAT: The Moon is probably cold & geologically "dead". For, a planetoid's Cooling Time (t_cool) scales with its Surface Gravity (g):

$$t_{cool} \approx \frac{Heat Energy}{Surface Area}$$
$$\approx \frac{M}{R^{2}}$$
$$= g$$​

Thus, at ~1/6^th g_earth, the Moon cooled ~6x more quickly.

 

[Vanadium 50]

You owe someone $10. I'd suggest a PF contribution would be in order.

The moon contains much less iron than the Earth - the Earth's core is about a third of the total earth, and the lunar core is probably around 2% of the moon: certainly less than 4%. The iron was detected by seeing how the moon interacted with the Earth's magnetic field.

The whole process that lead to this "law" is deeply flawed. Earth and Venus have nearly the same mass and radius, so wherever one data point is, the other one is right there as well. Add Mars, and now you have two data points. Two points determine a line. So the only independent test is whether anything else ends up on this line.

Out of all the other objects, only one, the moon, ends up on this line. You dismiss the others for a variety of ad hoc reasons. For example, you reject Mercury (which doesn't end up on the line) because it's composition is different than the earth's, but accept the moon (which does end up on the line) even though it's composition is also different. This isn't how science operates.

 

[Widdekind]

Please look at the graph again. Venus, the Earth, the Moon, and Mars all lie precisely upon the line, to w/in one pixel on your screen. The only 2 of the 6 data points slightly off the line, are Io & Europa, which "should" be Ice Worlds (having formed out beyond the Sun's Snow Line), but coalesced near the hot proto-Jupiter, which boiled out most of their volatiles.

Do you understand, that Mercury formed in an entirely different Formation Zone from the four above worlds??

If I have not been clear, I appologize:

1. Mercury = STEAM Zone (T > 373 K)
2. Venus, Earth, Moon, Mars = WATER Zone (373 K > T > 273 K)

Nilssen's Second Law (p₀ = .536, k = .464) only applies to Water Zone worlds.

It is well known that Mercury's composition is different from the other Inner Planets. I cited an academic article that also threw out Mercury from its consideration of terrestrial planets (last page of the .pdf).

The Moon is 30% Iron. Since it only masses 0.0123 M_earth, it should have ~0.003 M_earth of iron. It's iron core has a radius of ~900-950 km*.

* From scaling the Earth's interior dimensions, see: http://chianti.geol.ucl.ac.uk/~dario/earthfg.gif. This yields 950 km. However, the Moon is cold & dead. It's iron core is much colder than the Earth's, and so will be relatively smaller, depending on the Thermal Expansion properties of iron.​

I would be interested to read your articles, please cite them. Thank you.

 

[Vanadium 50]

Do you understand, that Mercury formed in an entirely different Formation Zone from the four above worlds??

Yes, and I understand that "formation zone" is your own invention, implemented to explain why your so-called law fails on the vast majority of test cases.

The Moon is 30% Iron.

Saying that over and over again doesn't make it true. Data from magnetic measurements of the moon, such as by the Lunar Prospector magnetometer indicates that the lunar core is ten times smaller than you propose. Your claim that it is somehow ten times smaller because of thermal expansion is ludicrous: iron doesn't shrink that much when cooled. Nothing does.

I would be interested to read your articles, please cite them.

Too bad. I am not going to give you a list of articles that I have published, because that would obviously reveal my identity. I chose to post under a pseudonym for a reason.