We’ve seen that it takes about 1031 J to melt the entire mantle. Let’s calculate how long this magma ocean remains liqui

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We’ve seen that it takes about 1031 J to melt the entire mantle.
Let’s calculate how long this magma ocean remains liquid. We’ll
make some radically simplifying assumptions: Energy radiates from
the surface as a black body and there is no atmosphere.
Furthermore, the magma ocean solidifies from the base towards the
surface, and heat is carried to the surface rapidly. Step 1: Draw a
schematic of the problem. Step 2: Black body radiation radiates
energy away from a surface at the rate of sT4, where s is the
Stephan-Boltzmann constant, 5.67x10-8 W/m2/K4 (a watt is J/s)
Approximately how long will it take the magma ocean to emit 1031 J
of energy? Assume the temperature of a magma at the surface is
~2000 K. Express you answer in years (1 year is 3.15 x 107 s). You
should get an answer more than a year, but less than the age of the
Earth. To do this problem, think carefully about units to address
missing information. Draw a picture if you are struggling to start.
Step 3: Add to your picture in a different color the consequences
of one or two of the simplifying assumptions (e.g. add an
atmosphere, the solidification occurs at the surface or mid-mantle,
or the radiation occurs as a grey body) and discuss in words how
you think it will change the cooling time of the magma
ocean.