MadSci Network: Earth Sciences

Re: How is Jupiter's core 30,000 degrees Celsius.How do I calculate that myself

Date: Fri Dec 21 09:53:08 2007
Posted By: David and John Free, Post-doc/Fellow, MFA, MFA
Area of science: Earth Sciences
ID: 1192205229.Es

Hello Gabriel

The calculations published, see refs (1) to (7), give estimated 
temperatures in the range 10,000 to 30,000 K. So you are well in line 
with this!

Two perhaps useful approaches are these:

1.	When we compress a gas it gets hot
2.	Jupiter was made from “solar material” 4 billion years ago and 
its current rate of cooling is known, ref (16). A similar estimate 
for “age of the Earth” can be found in ref (13). They got it all wrong as 
they did not allow for radioactive heating within the Earth.
A third approach is, as in ref (8), to think that SOME thermonuclear 
fusion processes are taking place within Jupiter’s core, despite them 
being rare “spontaneous” events unable to sustain continuous fusion 

So maybe, if you want 30,000 degrees you should start with the compressed 
gas idea 1. above. You will find no need to think of ways to maker it 

If we compress a perfect gas the temperature rise can be calculated from 
                 T1/T2 = (Density1/Density2) raised to power alpha

This comes from the gas laws PV=RT and, for adiabatic compression 
PV^gamma = constant. Henhce T2/T1 = Density ratio power (gamma - 1)

We know Jupiter’s gas is 95% H2 at the radius where its pressure is one 
Earth’s atmosphere.   Here (see NASA data below) the temperature T1 is  
165 K, (although ref (10) says 1360).
So you would be happy if  T1/T2 equalled 1/200, for 200 times 165 is 

For diatomic H2 (alpha = 3/2) this would require compression to a 
density  of only 0.45 grams/ml and we know that densities higher than 
that exist within Jupiter. For dividing Jupiter’s mass by its volume 
(taken at one atmosphere pressure radius) we get a MEAN density of 1.3 
grams/millilitre. According to ref (14) this density, 1.3 is that of 
metallic hydrogen. That the mean density of Jupiter is that of metallic 
hydrogen is no coincidence. It simply means that a major part of Jupiter 
is metallic hydrogen. Its properties (as far as known) are well described 
in David Ceperley’s page ref (15).

As the density increases (atoms closer together) the electron-states of 
the atoms begin to overlap (coupled oscillators) and the allowed states 
of hydrogen become continuous conduction bands. It becomes a metal. 
Attempts to further compress it gives rise to “exclusion principle 
repulsion forces” as the electrons (not allowed to share states) are 
forced into higher energies.
The electrons are free to move as in a metal (metallic hydrogen) and this 
is a very hot metal that readily conducts any heat generated deep within 
it! The transition (gradual-with-depth change from 1.08 to 1.3 g/ml)  
takes place at around 2.5 million atmospheres pressure, ref (14). Once 
all is in the metallic state the material is extremely stiff but the 
pressures very high! So the work done PdV is a huge number times a tiny 
one, neither of which are known but can only make it hotter!
If you’d like the temperature profile with depth within Jupiter you could 
build a model that solves the equation

                          -dP/dR = D(R)G(R)

This says the increase of pressure as we descend 1 meter into Jupiter is 
Jupiter’s gravity at radius R times its gas density at radius R
G(R) is proportional to the mass within R and decreases as the square of 
R, so we can calculate it from known conditions – see below -  at 
Jupiter’s surface (taken as where the gas pressure is one Earth 
atmosphere). All we need to do is guess several “reasonable” density 
profiles D(R). (They MUST contain ALL Jupiter’s mass within a radius of  
71,492  Km, and you will find all these reasonable profiles can (by means 
of the compressed gas gets hot or other ideas) can be made to give 30,000 

Good luck with your continued researches


 NASA Data on Jupiter from Dr. David R. Williams,
NSSDC, Mail Code 690.1 NASA Goddard Space Flight Center Greenbelt, MD 

Bulk parameters
                                   Jupiter      Earth   Ratio 
Mass (1024 kg)                      1,898.6     5.9736      317.83 
Volume (1010 km3)                 143,128     108.321      1321.33
Radius (1 bar level) (km)
    Equatorial                     71,492       6,378.1      11.209    
    Polar                          66,854       6,356.8      10.517
Volumetric mean radius (km)        69,911       6,371.0      10.973
Ellipticity                         0.06487     0.00335      19.36 
Mean density (kg/m3)                1,326       5,515         0.240 
Gravity (eq., 1 bar) (m/s2)        24.79        9.80          2.530 
Acceleration (eq., 1 bar) (m/s2)   23.12        9.78          2.364 
Escape velocity (km/s)             59.5        11.19          5.32
GM (x 106 km3/s2)                 126.686       0.3986      317.8 
Solar irradiance (W/m2)            50.50     1367.6           0.037
Black-body temperature (K)        110.0       254.3           0.433
Jovian Atmosphere
Surface Pressure: >>1000 bars  
Temperature at 1 bar: 165 K (-108 C)
Temperature at 0.1 bar: 112 K (-161 C)
Density at 1 bar: 0.16 kg/m3
Wind speeds
   Up to 150 m/s (<30 degrees latitude)
   Up to  40 m/s (>30 degrees latitude)
Scale height: 27 km
Mean molecular weight: 2.22 g/mole
Atmospheric composition (by volume, uncertainty in parentheses)
    Major:       Molecular hydrogen (H2) - 89.8% (2.0%); Helium (He) - 
10.2% (2.0%)
    Minor (ppm): Methane (CH4) - 3000 (1000); Ammonia (NH3) - 260 (40);
    Hydrogen Deuteride (HD) - 28 (10); Ethane (C2H6) - 5.8 (1.5);


(1)     The Interior of Jupiter
Most of the interior of Jupiter is liquid (primarily hydrogen and about 
10% helium). The central temperatures are thought to lie in the 13000-
35000 degree ... - 4k - Cached - 
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(2)      Jupiter
The Interior of Jupiter This picture illustrates the internal structure 
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(3)     Jupiter
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Models of Jupiter's interior that are described in the chapter can be 
downloaded from or viewed by 
going HERE. - 2k - Cached - Similar pages

(6)     Jupiter :: The interior -- Britannica Online Encyclopedia
Britannica online encyclopedia article on Jupiter, The interior: The 
atmosphere of Jupiter constitutes only a very small fraction of the 
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(7)     Nov 13: Jupiter II
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(8)      D-D Fusion in the Interior of Jupiter?, Title Page
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File Format: Microsoft Word - View as HTML
Smaller giants (0.15MJupiter) begin separation in less than a billion 
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(13)     Carslaw and Jaeger        The Conduction of Heat      Oxford 
University Press

(14)    Handbook Of Chemistry and Physics      CRC Press  59th edition 
page B-28

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