Our planet has a very hot core of molten stuff. How long would it stay
hot if the Sun were to go out?

Hi there!

What an interesting question!

I think the main thing to understand is whether the sun (or lack of
sun) has any appreciable effect on the eventual dissipation of the
earth's internal heat. By appreciable I mean, if we are talking 3-4
billion years 'til final cooling (see below), will the lack of the sun
change that by a billion or so years?

The answer appears to be NO.

First, consider where the "liquid" core is located:

"Earth has two cores--the inner core and the outer core.  The inner
core is solid and is about 5100 kilometers down (approximately 3150
miles).  Surrounding the inner core is the outer core, which is
believed to be liquid, and starts about 2900 kilometers below the
surface of the earth (1800 miles)."
http://newton.dep.anl.gov/askasci/gen99/gen99175.htm

The Earth's crust serves as insulation against heat loss. Heat can
only escape through breaks in the crust. There are several ways for
this to occur:
 
"With the moon the heat escaped by two mechanisms. One was volcanic
activity induced by meteorite impacts. The meteorites penetrated
through the crust and into the core... The second way heat escaped was
through conduction. Rock is a poor conductor of heat, however, so in
larger planetary bodies most of the heat must escape by other means."

"Mars is an example of another way internal heat can be lost. Olympus
mons, the large Martian volcano is a hot spot, and it probably tapped
directly into the core of the planet, and piped the heat in the form
of magma directly to the surface."

"The third way a planet can lose its heat is to establish convection
cells in the mantle. That is, hot, low density material from the lower
mantle, heated by the core, flows upward towards the surface where the
heat escapes through volcanic activity. The cooled, now denser
material then sinks back toward the core to be heated again. This
creates a cycle of movement, but it is very slow, only centimeters per
year, taking on the order of a billion years of so for a complete
cycle."

This third way is the way Earth's core loses temperature. 

In conclusion

"... the earth is cooling off, and as time goes by there will be less
and less heat to escape until there is none left at all. At that point
the earth will die a heat death ... will probably take another 4-5
billion years to loose all its heat -and become a dead planet."

But the good news is: "... by that time the sun will expand into its
red giant phase and burn the earth to a crisp before that final heat
death comes."

Note: All of the quotes above are from: The Heat History of the Earth.
I urge you to read it in full. I found it fascinating!

 http://geollab.jmu.edu/Fichter/PlateTect/heathistory.html  

I hope this helped you understand a little more about our Earth's
geology.

--K

Search term used: "source of heat" earth core

Very comprehensive answer. Thank you! (The comments and discussion
that followed was quite elucidating as well.

An additional factor to consider is the very real possibility that the
earth is creating heat in its core in a kind of natural nucear
reactor.  This link is from a recent news story dealing with just this
issue.

http://www.evworld.com/databases/shownews.cfm?pageid=news110602-07

The article begins:
"Source: UPI [Jun 11, 2002]  Thousands of miles beneath our feet, a
giant nuclear reactor seems to be at work deep within Earth's core,
and preliminary research suggests it may be the mysterious power
source behind the planet's magnetic field and thermal energy, upon
which all life on the planet depends for its survival, scientists told
United Press International."

Furthermore, Dr Tom J. Chalko, MSc, PhD, Head of Geophysics Division,
Scientific E Research P/L, Melbourne, Australia says on his site
http://sci-e-research.com/geophysics.html
that "all heat generated inside Earth is of radionic origin. In other
words, Earth in its entirety can be considered a nuclear reactor
fuelled by spontaneous fission of various isotopes in the super-heavy
inner core, as well as their daughter products of decay in the mantle
and in the crust."

This would certainly require us to adjust our notion of how much heat
the earth has to lose but one of the most incredible proposed outcomes
has the earth not going out with a wimper but with a bang.  Dr. Chalko
proposes that global warming may cause the earth to fail to cool the
nuclear reactor within causing a separation of materials that will
"enrich the nuclear fuel in the core to the point of creating
conditions for a chain reaction and a gigantic atomic explosion."

At first glance then, it seems that the earth would likely take even
longer to lose all of its heat than the figures given above because it
is not merely losing heat but creating it.

The answer presented above neglects (I think) a fundamental issue. The
earth's core is hot because of radioactive heating. Basically the core
is
fully of radioactive elements which when they decay produce heat. If
this
was not the case the earth would have become a solid mass a long time
ago.

In the early 1900 Kelvin estimated the age of the earth as being less
than 10
million years since this was how long it would have taken the earth to
cool
down from a molten state to the current temperatue. This in fact posed
a lot of problems for biology since there was no way that life could
have evoled so
fast. However he was wrong since he did not know about radioactivity.

The simple answer is that the temperature of the earth's core has
nothing to
do with the presence of the sun. Hence even if the sun was to go out
the core would stay hot for several billion years. We would all perish
but that is another story.

There is a lot of good information above, but I hope to add a
clarification.
The surface of the earth would lose its heat very rapidly without any
solar radiation, just as it does at night. Atmospheric water vapor
(clouds) would become scarce (for comparison with Mars see
http://humbabe.arc.nasa.gov/mgcm/faq/liquid.html), which is
responsible for much of the insulating properties of our atmosphere.
As stated previously, the core temperatures do not contribute much to
surface temperatures currently, but in this situation all surface heat
would be from the core.
The formula for heat transfer can be found at:

http://hyperphysics.phy-astr.gsu.edu/hbase/thermo/heatra.html#c1

It is directly proportional to the difference between the hot and cold
regions. Assuming the worst case scenario: if the surface approached
absolute zero (-273 C) from its current temperature of 15 C, and a
core temperature of about 3,000 C, the increase in heat loss would be
about 10%.