Dawn distance to Ceres ≈ distance to moon

[TEFLing]

From the exoplanets.EU database, only two exoplanets have well defined masses and radii, and are of earthlike size

Kepler-36 & 55 Cnc e

The former closely fits the apparent relation, the other is under dense like Ceres

Perhaps has a Venusian atmosphere maybe with oceans? Some articles online seem to say so

More data than 6 points to plot would be immensely preferable

 

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Kepler-93 b: density 7.2 +- 1.3, surface gravity 18.6 +- 3.2 (seems reasonable)
Kepler-138 d: density 1.4 +- 0.7, surface gravity 4.0 +- 1.7 (does not fit at all)
Note that while the individual values show large uncertainties, they are highly correlated as the original measurements are (uncorrelated) radius and mass and we consider M/r^2 and M/r^3. Kepler-138 d is clearly not on the line.

I had a look at all exoplanets in the database smaller than 2 times the radius of earth, where both mass and radius have an uncertainty given and where this uncertainty is not too large (typically below 10%). It is not surprising that more massive planets are more dense due to pressure, but there are also very light planets. Adding our observation bias (light planets rarely have a reliable mass estimate), my conclusion is "we cannot draw a conclusion".

Also note that all the gas giants would fit in this image - they are very light (~1-2 g/cm^3) and have a high surface gravity. They are completely away from any function that could be drawn through rocky planets.

 

[TEFLing]

Manipulating the data from the exoplanets.eu database is very hard

MS Excel seems to treat error values of INF and NAN and zero as legitimate numbers

Never the less, about 350 exoplanets have well defined masses and radii ( fractional error < 0.12 )

If you scatter plot DENSITY on y vs. GRAVITY on x...

The exoplanets appear to fall on two separate lines...

The plot looks like /_

The flatter line seems to be gaseous planets like Jupiter, having DENSITY ~= 0.1 x GRAVITY utilizing Earth normalized units ( Earth = 1 )

Cp. Jupiter has a surface gravity of about
3, and a density of about 0.3

The more vertical steeper line seems to represent earthy planets, with bulk compositions more like Ceres, Earth, or Mercury

Perhaps basic general rules could help differentiate the two basic kinds of planets??

 

[TEFLing]

Trying to upload figure

 

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Lines through the origin are lines of constant radius. The steeper the line the smaller the planet.
The dominant line corresponds to the radius of Jupiter, or the (nearly) universal size for very massive gas planets.

 

[TEFLing]

http://www.universetoday.com/25348/what-is-the-smallest-star/

That is a great observation...

R ~ 1 Jupiter ~ 10 Earth's

True all the way out to the ballpark vicinity of

(x,y) = (p,g) ~ (30,300)

Corresponding to the smallest Red Dwarf class stars also ( having approximately 30,000 Earth masses = 30 x 10^3 )

http://solstation.com/images/sol-m-j2.jpg

assuming HSE, and using a single zone

dP/dr = -pg​

P_center /R ~ p_average g_surface​

P ~ p g R ~ g²​

for earth

P ~ 400 GPa​

for a hyper Jupiter, near the transition from brown dwarf to red dwarf, central pressures might be about ~10⁵x higher

the bulk modulus of rocky material is of order 100s GPa

so, for Earth and the other terrestrial planets in our solar system, even core material experiences pressures comparable to their bulk modulii

whereas core conditions in super and hyper Jupiters are exotic

if the least massive stars form the slowest...

then wouldn't they evolve towards the MS in quasi HSE?

so before their First Light of fusion ignition, they ought to resemble sub-stellar brown dwarf ultra Jupiters...

and so they ought to have a rocky metallic ultra Earth core, of say 10-15 Earth masses, or something like that

how would that affect fusion processes inside the protostars?? Would First Light occur in a liquid superfluid H / He shell around a massive rocky metallic core, or anything like that??

http://www.sciencedaily.com/releases/2015/01/150122145418.htm

 

[TEFLing]

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I want to try one more order of magnitude calculation, using the one zone approximation of the HSE equation

if we assume an adiabatic equation for pressure

P = K p^y​

then the scaling relation becomes

P/R ~ p g​

P ~ p ( p R ) R​

P ~ p²​

assuming roughly constant radius ( R ~ 1 R_Jupiter ). That is consistent with the adiabatic approximation, for an adiabatic index of y=2.

I think that would mean, gas in the interior would be hotter than if it was driven down from the surface and simply compressed by the ambient pressures...

i.e. Implies an internal source or reservoir of heat energy

such as fission in the super Earth core, or left over heat from planetary formation and differentiation