Hi, I was in bed a few nights ago and for some reason (don't ask me
why) I started thinking:

If the Sun was compressed down to the size of, say, a tennis ball
(small enough to handle basically) and it was placed in a box (cube)
of the best (most effective) insulating material we have produced, how
thick would the walls of the box have to be to insulate all the heat
ie. the outside of the box would be the same temperature as the room
temperature to the touch?

I am a science student so realise there might be fewer molecules in
the small Sun so less bonds breaking etc. and I know it isn't possible
but try and think of it as the Sun at its regular size and a giant box
round it to make it easier to think of without any 'realistic'
problems there are by shrinking the Sun!!

I've set the amount as the minimum because I've asked this more for
comments than an answer as I don't think one would be available but
with maths and physics etc I may be wrong.
Thanks for taking the time to read all this, I look forward to any
comments. Please feel free to ask for clarification of the idea in my
head!!
Ed

Well first off, if you compressed our sun down to the diameter of a
tennis ball, it'd be a black hole.

   But whatever size, if you envelope it in insulation, and the inside
& outside are at different temperatures, there's no way to permanently
stop heat transfer through the envelope. In time, the two regions will
approach thermal equilibrium, and the insulation itself will radiate
equally with its environment.

What if it was 100% insulation? Surely if that was physically possible
you could stop heat transfer? ALso, surely after so many layers of
atoms of insulatory material the vibrational energy would be lost with
each layer to nothing?
I know it isn't possible, that's why I said imagine it as regular sized

You could do it if you had perfect insulation, which would store
energy without getting hotter. If you had perfect insulation, of
course, you'd only need one "layer." Perfect insulation violates
conservation of energy and isn't possible.

Let's try something with real insulation: If the box+insulation start
out at room temperature and then you place the sun in the box, the
outside of the box will start to heat up. In the long run, the box
will be putting 4e26 Watts into the room just as the sun is putting
4e26 Watts into the box. However, it takes time for the sun to heat up
all that insulation. How much time depends on the properties of the
insulation and how much of it there is. No matter how much real
insulation you use, the surface temperature of the box will have to
increase from room temperature. All you can do is slow it down.

It's possible to calculate how much insulation you'd need to keep the
box surface temperature increasing slowly enough so that it didn't
rise more than (say) 1 degree in 100 years. But I can tell you the
answer already: a lot.

Well it's not as straightforward as just making the envelope "100%
insulation". What is insulation? Is it meaningful to speak of
something being 100% insulation? If you've got a source of heat, for
example the Sun, then what does it mean to insulate the surrounding
space from the Sun's heat? What exactly does insulation do?

   I'll use the Sun itself as a good example of insulation.
Energy-carrying light leaving the Sun's surface races away at 300,000
kilometers/second. Earth is 150,000,000 km from the Sun, and it only
takes about eight minutes for this radiation to travel to Earth. But
the Sun's radiant energy is generated deep within, at the core, where
conditions induce nuclear fusion. How long does it take for the
fusion-borne light to get from the core to the surface, a distance of
only about 700,000 km?

   It actually takes *millions* of years for the released light
radiation to arrive at the surface. But light travels at the speed of
light. How can it take so gawd-aweful long? It's because the Sun's own
dense layers act as insulation. The fusion generated photons can only
travel very short distances before interacting with matter. It gets
absorbed, then re-emitted. Absorbed & re-emitted again. And again. And
again... Each such interaction takes time. Not only that, but the
direction of each re-emission is determined randomly, so it's not
always traveling outwards, away from the core. The radiation spends a
lot of time wandering in zig-zaggy paths, which add up to a very, VERY
large total distance for it to travel before at long last reaching the
surface. It's like the difference between water flowing through a
straight length of pipe, with endpoints at A & B, and one which
meanders all over the place but still has endpoints A & B. It'll take
a lot longer for a unit of water to make it from A to B in the
meandering pipe, even though both paths are carrying water at the same
number of liters per second.

   So we see here that what thermal insulation does is to make heat
take a longer *time* to travel between two regions than it would in
the absence of the barrier.

   But even so, like water in the pipes, the heat is being carried
through the barrier at the same wattage. The Sun's light radiation
takes a tremendous time to get to the surface, but once there it still
pours out at the same total rate it was produced at the core. It has
to. Otherwise the energy wouldn't be conserved. It has to show up
eventually.

   So you can build any sort of elaborate envelope around the Sun. It
can retard the flow of heat, but cannot lessen the wattage of that
flow. There is no insulation of finite mass & volume which can conceal
the Sun's radiant heat indefinitely. (In fact, if we want to get
thoroughly technical, there's the issue of solar neutrinos, which are
a fusion product and which carry some portion of the released energy.
Neutrinos have extremely low rates of interaction with matter, and
almost all solar neutrinos zip right out of the Sun, millions of years
ahead of the photon radiation. Even a hypothetical barrier to capture
the energy of solar neutrinos would boggle the mind.)

The answer is no matter how thick the insulation is heat will flow out.
Thus the exterior will never be the same temperature as the room.

Your insilation can be infinately thin as long as it's a perfect
reflector on the inside.

You could also just make your reflector perfect at reflecting stuff
that can travel through a vacuum and then create a vacuum between the
reflector and the outter peice of the insulated container.

Once you have your reflector it's cool to think about what happens
inside the bottle.

Either use an ideal thermos bottle or a box with mirrored walls. That
will take care of some of the heat transfer methods. Since vacuum in
the thermos does not transfer heat and ideally reflective surfaces do
not emit any, a thermos with a levitating inside part will preserve it
just like it would preserve coffee.

"Either use an ideal thermos bottle or a box with mirrored walls. That
will take care of some of the heat transfer methods. Since vacuum in
the thermos does not transfer heat and ideally reflective surfaces do
not emit any, a thermos with a levitating inside part will preserve it
just like it would preserve coffee."

   But a vacuum does carry heat, in the form of radiation. (The three
modes of heat transfer are conduction, convection, radiation.) And
ideal thermos would eliminate conduction & convection, but not
radiation.

A black hole might do it.  No light escape and that likely means that
nothing else is escaping either.  Another way would be to put the sun
by itself in an infinately far away part of the universe the space
would eventually spread the energy so thin that it wouldnt hit the
earth at all.  This means spreading out the light so much that between
the individual photons you could fit a planet the size of earth. 
Thats pretty far away. Another trick would be to make a solar panel
box and use all the energy created to power massive air
conditioners..that ones really a stretch though.

matt

The Sun, at the size you have mentioned, will turn into a black
hole.Since nothing can escape from a black hole,there will be no
radiation.Thus, if the space inside the insulating box is more than
the boundary of the event horizon of the  black hole,any box will be
able to hold the "thing"(its no longer the sun) safely.

but of course a black hole does radiate. Due to the curvature of
space, the area around the event horizon emits Hawking radiation. This
is a simple calculation of quantum field theory in a curved space
time. Effectively it comes down to particle-antiparticle pairs being
created in the vacuum around the horizon and one of the pairs escaping
while the other falls into the black hole (this is a somewhat
simplified view). Eventually the black hole will radiate completely
and you'll be left with perfectly thermalised radiation. So I'm afraid
that putting it in a black hole will not work.

Insulation capable of not meling under 1,000,000 degrees
Sun spots that press miles out from around it, nutrinos, gamma rays, protons...
Pressing it down or supernova won't help, no material could wistand
from imploding within the star.  If you could take the heat from the
sun and transfer it into energy somehow then disperse the energy
somewhere safe.  Some supersized solar panels? Boron and Phospate,
silicon would turn to plasma under those conditions.  Absorbing the
heat rather than insulating from it would be your best bet.