granpa Posted February 9, 2016 Share Posted February 9, 2016 (edited) 20.66 earth masses of ice inside Saturn if density of metallic hydrogen = 2.2 g/cm^3The 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 kmjupiter Scale height: 27 kmhttps://en.wikipedia.org/wiki/Frenkel_lineBelow 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 Jupiter1.326 = density of Jupiter2.2 g/cm^3 = Metallic hydrogen density(1.326/2.2)^0.333) = radius of metallic hydrogen core = rmhc = 0.845*radius of Jupiter76,850km-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 km0.08 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 Jupiter1/((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/rmhcMass of Jupiter = 317.8 earth massesmass 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 Link to comment Share on other sites More sharing options...
pavelcherepan Posted February 9, 2016 Share Posted February 9, 2016 (edited) 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 Link to comment Share on other sites More sharing options...
granpa Posted February 9, 2016 Author Share Posted February 9, 2016 (edited) 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 Link to comment Share on other sites More sharing options...
Mordred Posted February 9, 2016 Share Posted February 9, 2016 (edited) 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 Link to comment Share on other sites More sharing options...
pavelcherepan Posted February 9, 2016 Share Posted February 9, 2016 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 Link to comment Share on other sites More sharing options...
granpa Posted February 9, 2016 Author Share Posted February 9, 2016 (edited) 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 Link to comment Share on other sites More sharing options...
pavelcherepan Posted February 9, 2016 Share Posted February 9, 2016 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 Link to comment Share on other sites More sharing options...
granpa Posted February 9, 2016 Author Share Posted February 9, 2016 (edited) 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 Link to comment Share on other sites More sharing options...
pavelcherepan Posted February 9, 2016 Share Posted February 9, 2016 Yeah, that does seem strange. I'll re-check my data. Link to comment Share on other sites More sharing options...
granpa Posted February 9, 2016 Author Share Posted February 9, 2016 (edited) sun radiates 3.9 x 10^33 ergs per secJupiter 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 * 10^9Jupiter 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 angstroms3.8angstroms*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 Link to comment Share on other sites More sharing options...
imatfaal Posted February 9, 2016 Share Posted February 9, 2016 ... (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 3 Link to comment Share on other sites More sharing options...
EdEarl Posted February 9, 2016 Share Posted February 9, 2016 (edited) 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 Link to comment Share on other sites More sharing options...
granpa Posted February 11, 2016 Author Share Posted February 11, 2016 (edited) 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 mass1 = density of metallic hydrogen1 = surface gravity of jupiter1 = radius of jupiter0.5 = pressure at exact center of jupitergravity G(x) = x + R^3*D/x^2integral of G(x) from 1 to R gives pressure at top of superdense coreintegral G(x) = x^2/2 - D*R^3/xintegral 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 hydrogenR is the bottom axisInside the superdense core would then be another slightly denser core and another inside that.For D = 1 a minimum of 7 levels are neededIf this process continues then eventually the hydrogen will be compressed into neutrons.That would be about 36 levelsIf 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^3The 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 kmjupiter Scale height: 27 kmhttps://en.wikipedia.org/wiki/Frenkel_lineBelow 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 Jupiter1.326 = density of Jupiter2.2 g/cm^3 = Metallic hydrogen density(1.326/2.2)^0.333) = radius of metallic hydrogen core = rmhc = 0.845*radius of Jupiter76,850km-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 km0.08 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 Jupiter1/((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/rmhcMass of Jupiter = 317.8 earth massesmass 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 bar287 696 bar54,300 km - 357 km = Saturn polar radius - 6 scale heights1 * (10 m/s^2) = surface gravity of SaturnDensity of Saturn = 0.687 g/cm^3Mass of Saturn = 95.16 earth massesmass of Saturns liquid hydrogen atmosphere = (0.08/0.687)*(1-0.6^3)*95.16 earth masses = 8.7 earth massesmass of Saturns metallic hydrogen core = (2.2/0.687)*(0.6^3)*95.16 = 65.8 earth massesmass of Saturns inner ice core = 95.16 - 65.8 - 8.7 = 20.66 earth massesThis 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 Link to comment Share on other sites More sharing options...
granpa Posted February 11, 2016 Author Share Posted February 11, 2016 Earth reaches 300000 bar at 1000 km depth (1000km) * (3g/cm^3) * (10m/s^2) in bar Link to comment Share on other sites More sharing options...
imatfaal Posted February 11, 2016 Share Posted February 11, 2016 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. Link to comment Share on other sites More sharing options...
granpa Posted February 11, 2016 Author Share Posted February 11, 2016 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 Link to comment Share on other sites More sharing options...
imatfaal Posted February 11, 2016 Share Posted February 11, 2016 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 Link to comment Share on other sites More sharing options...
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