19.4 earth masses of ice inside Saturn

[granpa]

20.66 earth masses of ice inside Saturn if density of metallic hydrogen = 2.2 g/cm^3

The surface of the gas giant is defined as the point where the pressure of the atmosphere is 1 bar,
Scale height = the vertical distance over which the density and pressure fall by a factor of 1/e.
saturn Scale height: 59.5 km
jupiter Scale height: 27 km
https://en.wikipedia.org/wiki/Frenkel_line
Below the Frenkel line the fluids are "rigid" and "solid-like", whereas above it fluids are "soft" and "gas-like".

(1-((1.326/2.2)^0.333)) * (76850km-160km) * (0.08g/cm^3) * 2.53 * (10m/s^2) * (1/((1.326/2.2)^0.333)) in bar
= 285 047 bar (according to google) = pressure at which hydrogen becomes metallic inside Jupiter

1.326 = density of Jupiter
2.2 g/cm^3 = Metallic hydrogen density
(1.326/2.2)^0.333) = radius of metallic hydrogen core = rmhc = 0.845*radius of Jupiter
76,850km-160 km = polar radius of Jupiter - 6 scale heights = Frenkel line
(1-((1.326/2.2)^0.333)) * (76,850km - 160km) = depth of liquid hydrogen = 11 898 km
0.08 g/cm^2 = estimated density of 0.75 liquid hydrogen (0.071) + 0.25 liquid helium (0.125)
2.53 * (10 m/s^2) = surface gravity of Jupiter
1/((1.326/2.2)^0.333) accounts for increase of gravity with depth = (integral of 1/x^2 from surface to rmhc)/(integral of 1 from surface to rmhc) = 1/rmhc

Mass of Jupiter = 317.8 earth masses
mass of Jupiters liquid hydrogen atmosphere = (0.08/1.326)*(1-0.845^3)*317.8 earth masses = 7.6 earth masses

Edited February 9, 2016 by granpa

[pavelcherepan]

granpa, so what is your discussion point here?

 

EDIT: Looked up the estimated composition of Saturn and it doesn't have anywhere near enough oxygen to form 19.4 Earth masses of water ice. Unless your "ice core" includes ammonia and other potential ices, although Saturn is only estimated to have about 0.01% of ammonia by volume, so that won't change things drastically.

 

http://www.sciencedirect.com/science/article/pii/S0032063399000471

Edited February 9, 2016 by pavelcherepan

[granpa]

https://en.wikipedia.org/wiki/Volatiles

 

In planetary science, volatiles are the group of chemical elements and chemical compounds with low boiling points that are associated with a planet's or moon's crust and/or atmosphere. Examples include nitrogen, water, carbon dioxide, ammonia, hydrogen, methane and sulfur dioxide. In astrogeology, these compounds, in their solid state, often comprise large proportions of the crusts of moons and dwarf planets.

In contrast with volatiles, elements and compounds with high boiling points are known as refractory substances.^[1]

Planetary scientists often classify volatiles with exceptionally low melting points, such as hydrogen and helium, as gases (as in gas giant), while those volatiles with melting points above about 100 K are referred to as ices. The terms "gas" and "ice" in this context can apply to compounds that may be solids, liquids or gases. Thus, Jupiter and Saturn are referred to as "gas giants", and Uranus and Neptune are referred to as "ice giants", even though the vast majority of the "gas" and "ice" in their interiors is a hot, highly dense fluid that gets denser as the center of the planet is approached

 

 

granpa, so what is your discussion point here?

 

EDIT: Looked up the estimated composition of Saturn and it doesn't have anywhere near enough oxygen to form 19.4 Earth masses of water ice. Unless your "ice core" includes ammonia and other potential ices, although Saturn is only estimated to have about 0.01% of ammonia by volume, so that won't change things drastically.

 

http://www.sciencedirect.com/science/article/pii/S0032063399000471

your link is describing the atmosphere not the core

the core is similar to Neptune

my point was "20.66 earth masses of ice inside Saturn if density of metallic hydrogen = 2.2 g/cm^3"

Edited February 9, 2016 by granpa

[Mordred]

Please include the link of the pages (obviously wiki in this case), when your doing a copy/paste.

 

However the question still stands what did you want to discuss? This far you've only stated what other articles state.

 

Do you have a particular question on those articles? Or are you asking us to check the math in your opening post?

Edited February 9, 2016 by Mordred

[pavelcherepan]

I'm checking a few papers at the moment, but they all seem to give different results from what you've come up with:

 

 

 

using five-layer models, and a different, fixed equation of state, found rock/ice core masses of about 5 M_⊕ for Jupiter and 7 M_⊕ for Saturn, these planets containing about 50 M_⊕ and 25 M_⊕ of heavy elements, respectively.

 

http://www.sciencedirect.com/science/article/pii/S0032063399000434

 

So it does assume/model that the total amount of heavy elements in Saturn is about 25 Earth masses, but the ice/rock core is only about 7 Earth masses.

 

Another paper gives a range that covers your result as well, but says that the range can be reduced by 7 Earth masses, depending on the amount of helium sedimentation:

 

 

In the case of Saturn, ranges from nearly 0 to 10 MZ, enve M, the mass of the core being between 8 and 25 M. Note, however, that the mass of the solid core might be reduced by up to ∼7 M depending upon the extend of sedimented helium, a process that is required to explain the present-day luminosity of the planet (Fortney & Hubbard 2003; Guillot 2005).

 

http://iopscience.iop.org/article/10.1086/431325/pdf

[granpa]

Saturn has a hot interior, reaching 11,700 °C at its core, and it radiates 2.5 times more energy into space than it receives from the Sun.

 

I think hydrogen is dissolving in the metallic hydrogen and getting subducted down so deep it becomes metallic hydrogen.

I think that explains the present-day luminosity of the planet

Edited February 9, 2016 by granpa

[pavelcherepan]

Your estimate for metallic hydrogen density contradicts with current estimates of ~0.6-0.8 g/cm³, which invalidates all your calculations.

 

http://www.ptep-online.com/index_files/2011/PP-26-07.PDF

[granpa]

Your estimate for metallic hydrogen density contradicts with current estimates of ~0.6-0.8 g/cm³, which invalidates all your calculations.

 

http://www.ptep-online.com/index_files/2011/PP-26-07.PDF

0.8 g/cm^3? Lets see how that works.

 

https://en.wikipedia.org/wiki/Jupiter#Internal_structure

 

The core region is surrounded by dense metallic hydrogen, which extends outward to about 78% of the radius of the planet.^%5B32%5D

 

mass of Jupiters metallic hydrogen core = (0.8/1.326)*(1-0.78^3)*317.8 earth masses = 100.75 earth masses

 

mass of Jupiters liquid hydrogen atmosphere = (0.08/1.326)*(1-0.78^3)*317.8 earth masses = 10.075 earth masses

 

wheres the other 207 earth masses?

Edited February 9, 2016 by granpa

[pavelcherepan]

Yeah, that does seem strange. I'll re-check my data.

[granpa]

sun radiates 3.9 x 10^33 ergs per sec
Jupiter receives (3.9 * 10^33 ergs)/(0.5 * 10^9) = 7.8 × 10^24 ergs from the sun per sec

(4*(5.202 au)^2 * 3.14 in miles^2)/( 3.14*(71492km*69911km) in miles^2)= 0.5 * 10^9
Jupiter is radiating 0.9 times the energy it receives = 7 x 10^24 ergs per second of its own energy.

285000 bar *2*10^13cm^3 in ergs = 5.7 × 10^24 ergs

growth of metallic hydrogen core = 2*10^13 cm^3 per sec
(2*10^13 cm^3)/(4*pi*(0.845*(76,850 km-160 km))^2) = 3.8 angstroms
3.8angstroms*32000000*10^9 in km = 12160 km per billion years

 

unfortunately 12000 km per billion years is too large

One explanation would be that hydrogen releases energy when it becomes metallic and therefore the core wouldnt need to grow as fast to explain the excess energy

Edited February 9, 2016 by granpa

[imatfaal]

...

(4*(5.202 au)^2 * 3.14 in miles^2)/( 3.14*(71492km*69911km) in miles^2)= 0.5 * 10^9

...

 

au, miles and km - all in the same equation. You gotta show some love to SI and stop abusing units. How you got the right answer God only knows - but I got 5.085*10^8 as well

[EdEarl]

sun radiates 3.9 x 10^33 ergs per sec

Jupiter receives (3.9 * 10^33 ergs)/(0.5 * 10^9) = 7.8 × 10^24 ergs from the sun per sec

(4*(5.202 au)^2 * 3.14 in miles^2)/( 3.14*(71492km*69911km) in miles^2)= 0.5 * 10^9

Jupiter is radiating 0.9 times the energy it receives = 7 x 10^24 ergs per second of its own energy.

 

285000 bar *2*10^13cm^3 in ergs = 5.7 × 10^24 ergs

growth of metallic hydrogen core = 2*10^13 cm^3 per sec

(2*10^13 cm^3)/(4*pi*(0.845*(76,850 km-160 km))^2) = 3.8 angstroms

3.8angstroms*32000000*10^9 in km = 12160 km per billion years

 

unfortunately 12000 km per billion years is too large

One explanation would be that hydrogen releases energy when it becomes metallic and therefore the core wouldnt need to grow as fast to explain the excess energy

285000 bar *2*10^13cm^3 in ergs = 5.7 × 10^24 ergs

X bar * Y cm^3 in ergs = XY ergs

1 bar = 1.019 716 213 kilogram-force/square centimeter

1 erg = 1.019716213e-10 kilogram-force centimeter

 

bar/erg = 1e-10/cm^3

I don't understand, it seems units are not correct.

 

PS

NVM imatfaal explained

Edited February 9, 2016 by EdEarl

[granpa]

Adding mass to a jupiter size planet has no effect on its radius so all additional mass (up to 60 jupiter masses) goes into a superdense central core of density D+1 and radius R:
1 = jupiter mass
1 = density of metallic hydrogen
1 = surface gravity of jupiter
1 = radius of jupiter
0.5 = pressure at exact center of jupiter

gravity G(x) = x + R^3*D/x^2
integral of G(x) from 1 to R gives pressure at top of superdense core
integral G(x) = x^2/2 - D*R^3/x
integral from 1 to R = -0.5*(R-1)*(2*D*R^2+R+1)

this pressure should not change as R changes.
Unfortunately it changes greatly for large values of D.
Only solution is to make D small.
D can even equal one.
D = 1 corresponds to a superdense core twice as dense as metallic hydrogen



R is the bottom axis

Inside the superdense core would then be another slightly denser core and another inside that.
For D = 1 a minimum of 7 levels are needed

If this process continues then eventually the hydrogen will be compressed into neutrons.
That would be about 36 levels

If D = 1 and R = 0.5 the pressure at the exact center = about 2

20.66 earth masses of ice inside Saturn if density of metallic hydrogen = 2.2 g/cm^3

The surface of the gas giant is defined as the point where the pressure of the atmosphere is 1 bar,
Scale height = the vertical distance over which the density and pressure fall by a factor of 1/e.
saturn Scale height: 59.5 km
jupiter Scale height: 27 km
https://en.wikipedia.org/wiki/Frenkel_line
Below the Frenkel line the fluids are "rigid" and "solid-like", whereas above it fluids are "soft" and "gas-like".

(1-((1.326/2.2)^0.333)) * (76850km-160km) * (0.08g/cm^3) * 2.53 * (10m/s^2) * (1/((1.326/2.2)^0.333)) in bar
= 285 047 bar (according to google) = pressure at which hydrogen becomes metallic inside Jupiter

1.326 = density of Jupiter
2.2 g/cm^3 = Metallic hydrogen density
(1.326/2.2)^0.333) = radius of metallic hydrogen core = rmhc = 0.845*radius of Jupiter
76,850km-160 km = polar radius of Jupiter - 6 scale heights = Frenkel line
(1-((1.326/2.2)^0.333)) * (76,850km - 160km) = depth of liquid hydrogen = 11 898 km
0.08 g/cm^2 = estimated density of 0.75 liquid hydrogen (0.071) + 0.25 liquid helium (0.125)
2.53 * (10 m/s^2) = surface gravity of Jupiter
1/((1.326/2.2)^0.333) accounts for increase of gravity with depth = (integral of 1/x^2 from surface to rmhc)/(integral of 1 from surface to rmhc) = 1/rmhc

Mass of Jupiter = 317.8 earth masses
mass of Jupiters liquid hydrogen atmosphere = (0.08/1.326)*(1-0.845^3)*317.8 earth masses = 7.6 earth masses

Where the hell did the second half go?

Saturn reaches 287 696 bar at 0.6 from center
(1-0.6) * (54300km-357km) * (0.08g/cm^3) * (10m/s^2) * (1/0.6) in bar
287 696 bar

54,300 km - 357 km = Saturn polar radius - 6 scale heights
1 * (10 m/s^2) = surface gravity of Saturn

Density of Saturn = 0.687 g/cm^3
Mass of Saturn = 95.16 earth masses
mass of Saturns liquid hydrogen atmosphere = (0.08/0.687)*(1-0.6^3)*95.16 earth masses = 8.7 earth masses
mass of Saturns metallic hydrogen core = (2.2/0.687)*(0.6^3)*95.16 = 65.8 earth masses
mass of Saturns inner ice core = 95.16 - 65.8 - 8.7 = 20.66 earth masses

This makes sense because Neptune is 17 earth masses and is mostly ice 
And Uranus is 14.5 earth masses and is mostly ice

 

 

 

if the shells are evenly spaced then 13 shells gives 95 jupiter masses. (each shell being twice as dense as the previous)

sum (r/s)^3*2^(s-r), r=1 to s, s=13

for D=1 pressure=0.5 at R=0.75

for D=1.83 pressure=0.5 at R=0.726

D=1.83 means 2.83 times denser hence size=0.7071 Edited February 11, 2016 by granpa

[granpa]

Earth reaches 300000 bar at 1000 km depth

(1000km) * (3g/cm^3) * (10m/s^2) in bar

[imatfaal]

Earth reaches 300000 bar at 1000 km depth

(1000km) * (3g/cm^3) * (10m/s^2) in bar

 

Surely the lithostatic pressure should be the integral and not the simple multiplication - g varies with depth and over roughly 1/6 of the earth's radius will vary enough that it must be taken into account.

[granpa]

Doesnt change much

plot 3.3*x+((5.5-3.3)*(5/6)^3)/x^2, x=5/6 to 1

https://www.wolframalpha.com/input/?i=plot+3.3*x%2B%28%285.5-3.3%29*%285%2F6%29^3%29%2Fx^2,+x%3D5%2F6+to+1

[imatfaal]

Doesnt change much

plot 3.3*x+((5.5-3.3)*(5/6)^3)/x^2, x=5/6 to 1

https://www.wolframalpha.com/input/?i=plot+3.3*x%2B%28%285.5-3.3%29*%285%2F6%29^3%29%2Fx^2,+x%3D5%2F6+to+1

 

Not sure I follow that - but in checking my approach to reply I noticed that the received model for density of the earth has g constant for about the first 2000km and then rising to max at r=3500km with g =10.68m/s2. So any objection would be theoretically correct for the wrong sort of planet - but completely moot for earth