Does one square centimenter of the sun core really radiate this amount of energy? I have been thinking that since the core of the sun maintains its temperature at 15 million degrees Kelvin, then every cubic centimeter of this core is receiving a certain amount of energy to keep it at this temperature. So I was thinking what would happen if we could take a small sphere, say, 0.6 cm in radius (so the area is 1 square cm) and put it in the earth's atmosphere. Would it radiate the same energy that it consumed to get to 15 million degrees in the first place ?
I made some calculations using Stefan-Boltzmann law, here's what I got :
$$
P = Area \times  5.67 \times 10^{-8} \times e \times T^4
$$
So for easier calculations let's assume that it is a black body, and the surrounding temperature is negligible compared to 15 million degrees.
so I got : 
$$
P = .0001 \times 5.7*10^{-8} \times ({15*10^{6}})^4 = 2.9\times10^{17} \,\mathrm{joules/second}
$$ 
Now, isn't this a huge amount of energy ?
I know the body has to be maintained at 15 million degrees to keep radiating energy at this rate, about 0.8 cubic cm of the sun core is about 120 kilograms, so the more important question is for how long will this rate of radiation continue.
I honestly feel that there are mistakes in what I have calculated, so please any correction would be greatly appreciated. 
 A: The core is under a lot of pressure, so you couldn't remove it from the sun's gravity well without it expanding and cooling to the point fusion stops.  Only a small part of the core's energy makes it to the surface.  
Each second the sun's core converts 600 million tons of Hydrogen into 595 million tons of helium.  The missing 5 million tons is converted to energy.  In one second this is equivalent to 1 billion mega-ton hydrogen bombs.  This enough energy to fuel America's energy needs for 7 million years.
This energy though takes a long time to escape the core.  In fact by the time a photon from the core travels 320 km to a point where the sun is cool enough that electrons bind to atoms again and work it's way to the surface it can take a million years. (link)
The energy managing to escape from the entire surface of the sun is on the order of $3.9*10^{26}J/s$ (based on 5800K and 696,000km radius of sun).  Fusion does produce a lot of energy and is the reason you hear it discussed a lot.
A: Yes a small marble of material from the center the sun would radiate hugely, and cool off quickly, if it was not at the center of the sun. But since it is at the center, it receives an almost equal radiation from its surroundings and does not cool off. 
The marble does not contain all that much energy. It would be something like a nuclear bomb. A bomb creates a momentary nuclear reaction that heats a small amount of material to a temperature around 15 million degrees. It then cools off, transferring that energy to the surrounding country side. See this and this.
In the sun, the marble produces power continuously, but the power is tiny. Volume for volume, the sum produces heat at about the same rate as a compost heap. It produces a lot of heat because it is large. See this Wikipedia article on the Sun.
The reason center of the Sun is so hot it that it has very poor cooling. Energy produced in the center either leaves or raises the temperature of the center. It takes millions of years for energy to reach the surface. 
Likewise, the surface of the Sun has poor cooling. The power radiated by the surface of the sun is large because the surface is large. 
The power per square meter is that of a black body radiator at 6000 K. See this calculator. It is about $7 MW/m^2$, or $7 W/mm^2$, which is not small. Energy leaves the surface of the Sun only through radiation. This makes the surface heat up until there is enough radiation. 
Some computer chips give off more than 7 Watts of waste heat through each square millimeter of surface. With good cooling, they stay below 200C. If space was full of air and you had a giant fan, you could keep the surface of the sun at that temperature. 
A: Your $15\times10^6$ temperature is at the Sun's core, where H fusion takes place. The radiated energy occurs at the Sun's "surface", at a temperature of only about 5700 K.
