Is nuclear fusion possible on planets in our Solar system? Apart from thermo-nuclear reactions in the Sun, (and in hydrogen bombs and other methods developed by scientists) does nuclear fusion occur naturally in the planets?  If not, why?
Also, do the Sun and the Outer planets - Jupiter through Neptune - have rocky cores similar to the Earth (consisting of iron, nickel, silicon etc) in addition to the gases?  I have read that there is not much of hydrogen or helium in the inner planets because of their low escape velocities and the solar winds from the Sun.  Hence only the heavier elements remained.
 A: The only planet in the Solar system where nuclear fusion occurs is Earth. And that is only because we have the means to achieve the combination of high pressure and high temperature to overcome the Coulomb barrier. Even the heaviest of the planets, Jupiter, is about ten times too small to achieve the pressure required to sustain fusion.
A: Nuclear fusion requires two nuclei to interact through the strong force, fusing together to form a nucleus with a lower total rest mass than the sum of the rest masses of the fusing nuclei. The difference in mass multiplied by $c^2$ is the energy released in the fusion.
i.e. It can be an exothermic process, and usually is when fusing together lighter elements to form heavier ones, at least until iron-peak elements are produced.
However, to get fusion going you need to overcome the Coulomb repulsion between the protons of the two fusing nuclei. This Coulomb barrier is smallest for nuclei with a single proton, and it turns out that the easiest fusion reaction to get going is the fusion of deuterium (1 proton + 1 neutron) with tritium (1 proton + 2 neutrons) to form helium. However, whilst Deuterium is relatively abundant in stars and planets (initially), there isn't much tritium, and the next easiest fusion reaction is deuterium plus a proton to form Helium-3 (2 protons plus a neutron).
The Coulomb barrier is so high, even for this reaction, that is requires  quantum tunnelling in order to get the "reactants" close enough together for the strong nuclear force to cause the fusion to take place. Even with this assistance, it still requires the reactants to be fast-moving and densely packed, meaning that the reaction rates depend on density and are extremely temperature sensitive (deuterium burning inside stars has a rate proportional to $T^{12}$). This means that deuterium burning effectively switches on at a temperature just below $\sim 10^{6}$ K.
The fact is that unless a ball of gas is at least 13 times the mass of Jupiter, then its interior temperature never becomes high enough to fuse deuterium (see for example the canonical paper by Burrows et al. 1997). You might think that after a large gas giant is born, it should shrink under its own gravity and the core should become hotter and hotter until fusion takes place. This does not happen because the core becomes degenerate - the pressure due to degenerate electrons obeying the Pauli exclusion principle, which is independent of temperature, provides enough support to allows the object to cool without getting any smaller or hotter. Thus Jupiter is an order of magnitude too small and its core an order of magnitude too cool for any fusion reaction to take place.
As far as the "terrestrial planets" go, well the central temperatures are even cooler than those for gas giant planets and so the possibility of any fusion is even more remote.
Note that we cannot absolutely say that no fusion occurs, because there is always some tiny probability of the reaction proceeding through the tunnelling process. But it is never of any energetic consequence during the evolution of any of the planets. Note also that nuclear fission does take place and is energetically significant in the core of the Earth.
A: Also, in the early formation of the solar system, Jupiter was the closest to undergoing nuclear fission which could have caused another sun- explaining it's extreme size. 
