Re: ILE - The Ideal Lane-Emden Equation

On 20 Jul, 13:32, af...@xxxxxxxxxxxxxxxxxxx (John Park) wrote:
Thomas Smid (thomas.s...@xxxxxxxxx) writes:

[...]



the radii for the hydrogen masses in the solar system (Sun, Jupiter,
Saturn, Uranus, Neptune) follow the sequence 1, 0.1, 0.085, 0.037,
0.035, which quite accurately corresponds to an exponent 1/3 in the
mass-radius relationship, but not to any of those claimed above.

I replied:



Something to do with the fact that they're not doing nuclear fusion?

More to the point, you treat these five bodies as though they were
analogous despite the facts that: one has a surface temperature a good
5500 K greater than the others; in four the hydrogen is present as
molecules or a metallic phase, while in the other it is monatomic or in a
plasma; and in two cases hydrogen probably makes up only about 15% of
the total mass anyway.

It is in the first place a hard observational fact that the radius for
these bodies follows a R~M^1/3 law. Whatever you make out of this
with regard to their physics has to be consistent with it. It
definitely suggests that the radial density function is pretty much
identical in all cases (if you assume an ideal gas in hydrostatic

^^^^^^^^^^^^^^^^^^^^^^^^^^

equilibrium, then (as shown above) the radius is given by R=[M(R)*(3-
k)/4pi/n(R)/m]^1/3 , where k is the power index for the radial
density function n(r)~r^-k ; the figures show that k differs not by
more than about 10% between the sun and the hydrogen planets (Saturn
excluded for some reason)); so even if there is any fusion occurring
in the sun, it has an insignificant effect on its structure).

But it's pretty clearly not the _same_ ideal gas (plasma/H vs H2/CH4 &c); and
for the outer planets it's questionable whether it makes sense to regard
them as gases at all.

I think it makes sense to describe all these planets as consisting
largely of an ideal gas, because even for Neptune, the internal
temperature (as given by the gravitational potential over the virial
theorem) is still about 15,000 K, which should make it impossible for
organized structures of the matter in the form of a fluid or a solid
to develop. So essentially, we are still dealing here (like in the
case of the sun) with a 'plasma soup' of nuclei and electrons. Because
no atoms (and thus no excitation of atomic transitions) can exist at
this temperature and density, one can consider it as an ideal gas (in
fact more ideal than gases in the usual sense).

You've enough data to find a coincidence;
not enough to show a connection. (Note that Uranus and Neptune are so
similar they're effectively the same point.)

Well, the point is that these 'coincidences' confirm a mass-radius
relationship R~M^1/3, but invalidate other exponents.


Put it another way: if Uranus and Neptune had the same masses but were 75%
hydrogen instead of 15% and had orbits inside Venus', would you still expect
them to fit your curve?

The fact that they fit the curve suggests that they *do* consist
largely of hydrogen.

Thomas


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Relevant Pages

  • Re: ILE - The Ideal Lane-Emden Equation
    ... mass-radius relationship, but not to any of those claimed above. ... identical in all cases (if you assume an ideal gas in hydrostatic ... in the sun, it has an insignificant effect on its structure). ... for the outer planets it's questionable whether it makes sense to regard ...
    (sci.astro)
  • Re: ILE - The Ideal Lane-Emden Equation
    ... largely of an ideal gas, because even for Neptune, the internal ... this temperature and density, one can consider it as an ideal gas (in ... to the sun and giant planets). ...
    (sci.astro)