| Topic: |
Science > Physics |
| User: |
"Fred" |
| Date: |
03 Feb 2008 03:43:25 AM |
| Object: |
Density of hydrogen on Jupiter |
I am looking for a way to solve this problem. The answer would be
interesting but the formula is more important.
Assume that Jupiter consists entirely of hydrogen. (I realize that it doesn't.)
The density of hydrogen along the radius will increase as one approaches the
center.
Is there a formula that can identify the density at any point on the radius?
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| User: "Androcles" |
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| Title: Re: Density of hydrogen on Jupiter |
03 Feb 2008 04:17:13 AM |
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"Fred" <fredjacobson@roadrunner.com> wrote in message
news:47a58c71$0$17350$4c368faf@roadrunner.com...
|I am looking for a way to solve this problem. The answer would be
| interesting but the formula is more important.
|
|
|
| Assume that Jupiter consists entirely of hydrogen. (I realize that it
doesn't.)
|
| The density of hydrogen along the radius will increase as one approaches
the
| center.
|
| Is there a formula that can identify the density at any point on the
radius?
|
Water when solidified has a lower density (ice floats) but I have
no experience with hydrogen density when liquefied, which it
certainly would be at some point given the pressure at Jupiter's
core. It is after all the second most massive body in the solar
system. Butane easily liquefies under pressure as you are probably
aware.
For a planet that is all gas and smaller than Jupiter so that it
doesn't liquefy you could work out a density gradient based
on a square law but at the surface where gravity is maximised
and pressure is zero the problem of temperature is an issue.
Such a planet would boil away rather than be held together
by gravity - as you know, smaller bodies such as our own
Moon are airless.
So the planet has to be large enough to have sufficient gravity
to hold together and small enough to not liquefy... any real
ball of gas will have a liquid boundary, making the density
problem an issue.
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| User: "Thomas" |
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| Title: Re: Density of hydrogen on Jupiter |
03 Feb 2008 04:58:05 AM |
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Hi Fred,
The density of a gaseous mass in a state of 'hydrostatic
equilibrium' (i.e. the pressure gradient force exactly cancelling the
gravitational force) is proportional to 1/r^2 (see section b) of my
page http://www.plasmaphysics.org.uk/research/starformation.htm ,where
I have shown this in connection with the star formation problem).
Thomas
Fred wrote:
I am looking for a way to solve this problem. The answer would be
interesting but the formula is more important.
Assume that Jupiter consists entirely of hydrogen. (I realize that it doesn't.)
The density of hydrogen along the radius will increase as one approaches the
center.
Is there a formula that can identify the density at any point on the radius?
.
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| User: "tadchem" |
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| Title: Re: Density of hydrogen on Jupiter |
03 Feb 2008 12:42:19 PM |
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On Feb 3, 5:58 am, Thomas <thomas.s...@gmail.com> wrote:
Hi Fred,
The density of a gaseous mass in a state of 'hydrostatic
equilibrium' (i.e. the pressure gradient force exactly cancelling the
gravitational force) is proportional to 1/r^2 (see section b) of my
pagehttp://www.plasmaphysics.org.uk/research/starformation.htm,where
I have shown this in connection with the star formation problem).
Thomas
<snip repost>
Thomas,
Your Equation (8) assumes an isothermal, ideal gas. Such is almost
NEVER the case. In gas clouds there are spatial variations in
temperature. Planets and stars are condensed phases where the
compressibility never follows the gas laws. Start with a more
realistic equation of state:
http://en.wikipedia.org/wiki/Equation_of_state
Tom Davidson
Richmond, VA
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| User: "Thomas" |
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| Title: Re: Density of hydrogen on Jupiter |
04 Feb 2008 04:02:04 AM |
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On 3 Feb, 18:42, tadchem <tadc...@comcast.net> wrote:
On Feb 3, 5:58 am, Thomas <thomas.s...@gmail.com> wrote:
Hi Fred,
The density of a gaseous mass in a state of 'hydrostatic
equilibrium' (i.e. the pressure gradient force exactly cancelling the
gravitational force) is proportional to 1/r^2 (see section b) of my
pagehttp://www.plasmaphysics.org.uk/research/starformation.htm,where
I have shown this in connection with the star formation problem).
Thomas
<snip repost>
Thomas,
Your Equation (8) assumes an isothermal, ideal gas. Such is almost
NEVER the case.
In the absence of any localized heat sources or sinks, a gas in
hydrostatic equilibrium must necessarily be isothermal (any
temperature differences will level out with time).
In gas clouds there are spatial variations in
temperature. Planets and stars are condensed phases where the
compressibility never follows the gas laws.
As you can easily show over the virial theorem (which says that the
average kinetic energy is -1/2 times the average gravitational
potential energy), the average kinetic temperature of a gas mass
corresponding to the sun is about 10^7 K, and for Jupiter still 10^5
K. The latter corresponds to an energy of about 10 eV, so it will
hardly allow any molecular bonds to form (i.e. a condensation is
impossible).
Thomas
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| User: "tadchem" |
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| Title: Re: Density of hydrogen on Jupiter |
04 Feb 2008 03:36:05 PM |
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On Feb 4, 5:02 am, Thomas <thomas.s...@gmail.com> wrote:
On 3 Feb, 18:42, tadchem <tadc...@comcast.net> wrote:
On Feb 3, 5:58 am, Thomas <thomas.s...@gmail.com> wrote:
Hi Fred,
The density of a gaseous mass in a state of 'hydrostatic
equilibrium' (i.e. the pressure gradient force exactly cancelling the
gravitational force) is proportional to 1/r^2 (see section b) of my
pagehttp://www.plasmaphysics.org.uk/research/starformation.htm,where
I have shown this in connection with the star formation problem).
Thomas
<snip repost>
Thomas,
Your Equation (8) assumes an isothermal, ideal gas. Such is almost
NEVER the case.
In the absence of any localized heat sources or sinks, a gas in
hydrostatic equilibrium
Read what you just wrote: "a gas in hydrostatic equilibrium"!
Do you want to correct that or to continue sounding ignorant?
must necessarily be isothermal (any
temperature differences will level out with time).
Does the word "adiabatic" mean anything to you? There is a
gravitational gradient present. Kinetic energy will vary with
gravitational potential, regardless of temperature, because the fluid
is free to move.
In gas clouds there are spatial variations in
temperature. Planets and stars are condensed phases where the
compressibility never follows the gas laws.
As you can easily show over the virial theorem (which says that the
average kinetic energy is -1/2 times the average gravitational
potential energy), the average kinetic temperature of a gas mass
corresponding to the sun is about 10^7 K, and for Jupiter still 10^5
K.
I'm glad you recognize that fact. Can you recognize that the
gravitational potential energy is not the same throughout the mass?
Do you realize that as individual molecules of the fluid fall or climb
in the gravity well, their kinetic energy (and therefore temperature)
will change?
The latter corresponds to an energy of about 10 eV, so it will
hardly allow any molecular bonds to form (i.e. a condensation is
impossible).
Your high school physics doesn't begin to tell the real story.
There are at least FOUR major states of matter: solid, liquid, gas,
and plasma.
When thermal energies exceed ionization energies, a plasma is present,
regardless of the density.
In the *real* world, sufficient pressure will compress even a plasma
into a metallic(!) state, describable as hydrogen ions embedded in a
Fermi "sea" of free electrons.
http://burro.astr.cwru.edu/stu/advanced/jupiter.html
Thomas
Tom Davidson
Richmond, VA
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| User: "Thomas" |
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| Title: Re: Density of hydrogen on Jupiter |
05 Feb 2008 12:24:34 PM |
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On 4 Feb, 21:36, tadchem <tadc...@comcast.net> wrote:
Does the word "adiabatic" mean anything to you? There is a
gravitational gradient present. Kinetic energy will vary with
gravitational potential, regardless of temperature, because the fluid
is free to move.
I repeat: we are assuming a state of *equilibrium* , i.e. a (quasi)
steady state. Nothing is macroscopically moving or otherwise changing.
For a collisional gas and in the absence of any heat sources or sinks,
this also logically implies that the temperature is constant
throughout the volume, which in turn implies a 1/r^2 density
dependence as pointed out.
In gas clouds there are spatial variations in
temperature. Planets and stars are condensed phases where the
compressibility never follows the gas laws.
As you can easily show over the virial theorem (which says that the
average kinetic energy is -1/2 times the average gravitational
potential energy), the average kinetic temperature of a gas mass
corresponding to the sun is about 10^7 K, and for Jupiter still 10^5
K.
I'm glad you recognize that fact. Can you recognize that the
gravitational potential energy is not the same throughout the mass?
Do you realize that as individual molecules of the fluid fall or climb
in the gravity well, their kinetic energy (and therefore temperature)
will change?
As indicated above, in a state of a quasi-static equilibrium there are
no net energy changes anywhere in the volume, and hence the latter is
isothermal in the absence of any heat sources or sinks (particle
collisions will level out any initial temperature differences). And
the only heat sink here is the 'surface' of the mass as defined by the
point where the density gets small enough so that individual atoms can
exist, which enables inelastic collisions and thus a 'radiative
cooling'. This is why the 'surface' is cooler than the interior of the
gas ball (I have explained this in more detail on my page
http://www.plasmaphysics.org.uk/research/sun.htm for the case of the
sun).
When thermal energies exceed ionization energies, a plasma is
present, regardless of the density.
Yes, I agree, and this is exactly why an increase of pressure can only
turn a gas/plasma into a fluid below a certain temperature.
In the *real* world, sufficient pressure will compress even a plasma
into a metallic(!) state, describable as hydrogen ions embedded in a
Fermi "sea" of free electrons.
I would grant you that point if you could show me some experimental
data proving that hydrogen can be turned into a fluid at 10^5 K or
above. Otherwise, I would consider this claim just part of an
*imaginary* world.
The only circumstance that could change something about the 1/r^2
behaviour in my view is when the density becomes so high that the
atomic nuclei would be pushed into each other. But this would require
densities about 10^15 times higher than near the surface, i.e.
assuming a 1/r^2 density behaviour, this should occur at a fraction
3*10^-8 of the radius (about 20 m for the sun and 2 m for Jupiter),
and I would not want to speculate what the physical state of matter so
close to the center is.
Thomas
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| User: "tadchem" |
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| Title: Re: Density of hydrogen on Jupiter |
05 Feb 2008 03:58:14 PM |
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On Feb 5, 1:24 pm, Thomas <thomas.s...@gmail.com> wrote:
On 4 Feb, 21:36, tadchem <tadc...@comcast.net> wrote:
Does the word "adiabatic" mean anything to you? There is a
gravitational gradient present. Kinetic energy will vary with
gravitational potential, regardless of temperature, because the fluid
is free to move.
I repeat: we are assuming a state of *equilibrium* , i.e. a (quasi)
steady state. Nothing is macroscopically moving or otherwise changing.
For a collisional gas and in the absence of any heat sources or sinks,
this also logically implies that the temperature is constant
throughout the volume, which in turn implies a 1/r^2 density
dependence as pointed out.
YOU are "assuming" a state of equilibrium; a condition never mentioned
in the OP.
YOU are "assuming" an ideal gas; a composition never mentioned in the
OP.
The OP *did* specify a Jupiter-sized mass of pure hydrogen.
One must use an equation of state for a real gas. The amount of
hydrogen involved IS sufficient to generate pressures that will
compress hydrogen to a liquid and then to a metal, so it is an error
to assume it is all gas.
As indicated above, in a state of a quasi-static equilibrium there are
no net energy changes anywhere in the volume, and hence the latter is
isothermal in the absence of any heat sources or sinks (particle
collisions will level out any initial temperature differences).
"No net energy changes within the volume" is a specification for an
adiabatic system. As pressure P varies (with depth) the available
energy (called enthalpy) H of a volume V of the hydrogen will vary
dH =3D V*dP
Requiring the energy to be in equilibrium is NOT the same as requiring
a uniform temperature.
When thermal energies exceed ionization energies, a plasma is
present, regardless of the density.
Yes, I agree, and this is exactly why an increase of pressure can only
turn a gas/plasma into a fluid below a certain temperature.
You are forgetting supercritical fluids.
In the *real* world, sufficient pressure will compress even a plasma
into a metallic(!) state, describable as hydrogen ions embedded in a
Fermi "sea" of free electrons.
I would grant you that point if you could show me some experimental
data proving that hydrogen can be turned into a fluid at 10^5 K or
above. Otherwise, I would consider this claim just part of an
*imaginary* world.
Done:
"Electrical resistivities were measured for liquid H2 and D2 shock
compressed to pressures of 93-180 GPa (0.93-1.8 Mbar). Calculated
densities and temperatures were in the range 0.28-0.36 mol/cm3 and
2200-4400 K. Resistivity decreases almost 4 orders of magnitude from
93 to 140 GPa and is essentially constant at a value typical of a
liquid metal from 140 to 180 GPa. The data are interpreted in terms of
a continuous transition from a semiconducting to metallic diatomic
fluid at 140 GPa and 3000 K."
http://prola.aps.org/abstract/PRL/v76/i11/p1860_1
The only circumstance that could change something about the 1/r^2
behaviour in my view is when the density becomes so high that the
atomic nuclei would be pushed into each other.
The experimental densities of metallic hydrogen are several times
higher than those of the frigid liquid:
# Specific Gravity, Liquid @ B.P., 1 atm: 0.0710
# Boiling Point @ 1 atm: -423.0=B0F (-252.8=B0C, 20=B0K)
http://www-safety.deas.harvard.edu/services/hydrogen.html
Note that the experiments reported above achieved densities of 0.28-
0.36 mol/cm3 (about the same on an atomic basis - 0.28-0.36 g/cc -
since hydrogen atoms weigh 1.008 g/mol), but the liquid only has a
density of 0.0716 g/cc. The frigid liquid can be taken as the limit
if density at which atoms are touching each other - any closer and
penetration occurs.
But this would require
densities about 10^15 times higher than near the surface,
Check your math. What density are you assigning for the "surface"?
i.e.
assuming a 1/r^2 density behaviour,
Only a valid assumptions where the gas is tenuous - i.e. several times
lower density than the density at the critical point - 0.0312 g/cm3 -
depending on how accurate you want to be.
this should occur at a fraction
3*10^-8 of the radius (about 20 m for the sun and 2 m for Jupiter),
and I would not want to speculate what the physical state of matter so
close to the center is.
It is not much of a speculation. The experimental data shows that
under lesser pressures hydrogen becomes a metallic liquid. This also
accounts for the giant (earth's x 20,000) magnetic field of Jupiter.
Thomas
Tom Davidson
Richmond, VA
"Don't tell fish stories where the people know you; but particularly,
don't tell them where they know the fish." --Mark Twain
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| User: "Eric Gisse" |
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| Title: Re: Density of hydrogen on Jupiter |
06 Feb 2008 12:26:30 AM |
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On Feb 5, 12:58 pm, tadchem <tadc...@comcast.net> wrote:
[...]
This is why I asked for the equation of state.
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| User: "tadchem" |
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| Title: Re: Density of hydrogen on Jupiter |
06 Feb 2008 03:38:33 AM |
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On Feb 6, 1:26 am, Eric Gisse <jowr...@gmail.com> wrote:
On Feb 5, 12:58 pm, tadchem <tadc...@comcast.net> wrote:
[...]
This is why I asked for the equation of state.
For all his qualifications (M.Sc. Physics, Ph.D. Astronomy) Thomas
should realize the importance of an equation of state. If he had taken
a course in Physical Chemistry somewhere in his career we would
realize that.
In his haste to get an an answer at his website, has has
oversimplified his treatment of hydrogen.
Even his premise there, that the photosphere is "Cooled by Inelastic
Collisions", is flawed.
Inelastic collisions do not remove energy from a system; they
randomize thermal energy. Cooling occurs when energy is removed from
the system, usually by EM radiation, but a state change of a component
can also do the trick.
Tom Davidson
Richmond, VA
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| User: "Eric Gisse" |
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| Title: Re: Density of hydrogen on Jupiter |
03 Feb 2008 03:57:14 AM |
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On Feb 3, 12:43 am, "Fred" <fredjacob...@roadrunner.com> wrote:
I am looking for a way to solve this problem. The answer would be
interesting but the formula is more important.
Assume that Jupiter consists entirely of hydrogen. (I realize that it doesn't.)
The density of hydrogen along the radius will increase as one approaches the
center.
Is there a formula that can identify the density at any point on the radius?
What equation of state would you like to assume?
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| User: "tadchem" |
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| Title: Re: Density of hydrogen on Jupiter |
04 Feb 2008 03:51:24 PM |
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On Feb 3, 4:43 am, "Fred" <fredjacob...@roadrunner.com> wrote:
I am looking for a way to solve this problem. The answer would be
interesting but the formula is more important.
Assume that Jupiter consists entirely of hydrogen. (I realize that it doesn't.)
Actually, it is about 90% hydrogen and the rest is almost all helium,
except for within the atmosphere where there is known to be about
0.35% other components:
http://burro.astr.cwru.edu/stu/advanced/jupiter.html
The density of hydrogen along the radius will increase as one approaches the
center.
Is there a formula that can identify the density at any point on the radius?
Skalafuris has a model which predicts pressure-dependent densities for
metallic hydrogen of 0.4 to 4.0 g/cc with a theoretical upper limit of
4.8 g/cc:
http://www.springerlink.com/content/h226824477441582/
HTH
Tom Davidson
Richmond, VA
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| User: "tadchem" |
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| Title: Re: Density of hydrogen on Jupiter |
03 Feb 2008 12:54:43 PM |
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On Feb 3, 4:43 am, "Fred" <fredjacob...@roadrunner.com> wrote:
I am looking for a way to solve this problem. The answer would be
interesting but the formula is more important.
Assume that Jupiter consists entirely of hydrogen. (I realize that it doesn't.)
OK. We've got a Jupiter-sized planet comprised solely of hydrogen.
The density of hydrogen along the radius will increase as one approaches the
center.
Is there a formula that can identify the density at any point on the radius?
The equation of state can be used to describe the density of a
substance given its temperature and pressure:
http://en.wikipedia.org/wiki/Equation_of_state
The pressure at any given radius is effectively the weight of all the
material *above* that position supported by a unit area. For example,
on earth the air above one square inch of ocean weighs about 14.7
pounds.
The pressure is thus the integral of the density through the distance
from the point of interest to the top of the atmosphere. When you form
the integral remember to take into consideration that the cross-
sectional area of your column of fluid is proportional to the square
of the distance from the planet's center.
Since this pressure feed back into the integral, you way want to
simplify the equation be expressing it in differential form and
applying a boundary condition such as lim(r -> infinity) of P(r) = 0
and a second specifying that the total mass of hydrogen is a specified
amount.
Tom Davidson
Richmond, VA
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