Energy production in star For 3rd year physics honours Sukla Debnath

Early universe

Formation of star

Primordial nucleosynthesys

As the Universe expand and temperature fall the reaction rate decreased Below 10 9 k the nutron and proton combined to form deuterium nucleus a number of reaction then led to the formation of helium nucleus, the mostTightly bound nucleus involved in nucleosynthesys.

Energy production in star The three possible source of energy are –  Chemical Reaction  Gravitational contraction  Thermonuclear reaction

Chemical reaction : Coal burning Burning is conversion of chemical energy into electromagnetic reaction Coal burning yield 4 × 10 8 J/Kg which is not sufficient For our SUN, if the Sun is made of coal of 2×10 38 Kg it give energy 2× ×4×10 8 J = 8×10 38 J Sun has radiate 1.2×10 43 J energy in 10 9 year

Gravitational Contraction The Gravitational potential energy is given by, V=-3/5 (GM 2 /R) The energy radiated during collapse would be ΔE = -E f = 3/10 (GM 2 /R) o For Sun ΔE = 2× J

Thermonuclear reaction In the special theory of relativity, Einstein demonstrate that the total mass energy was conserved. The relaton between mass and energy is, E = m c 2 Where E=energy equivalent to mass m=mass c=speed of light The rest energy of proton, E= (1.67× )(3×10 8 ) 2 J =940 MeV

What is the source of enormous energy of stars Nuclear fission or fusion ? Binding energy curve :

What is the source of enormous energy of stars Nuclear fission or fusion ? From binding energy curve it is seen that the nuclei atomic number nearer to A=60 is most stable. If the heavy nuclei split apart into smaller nuclei having masses nearer to A=60,they produces energy and this is fission process. When the lightest element like hydrogen and helium combined to form a Comparably heavy nuclei up to A=60,the process is nuclear fusion and fusion Can release enormous energy. Both fission and fusion reaction have the potential to convert a small amount of mass into large amount of energy. However stars are made from light element. Thus nuclear fusion is quite possible source of energy of the star.

Nuclear time scale For Sun E= M s C 2 =1.8 ×10 47 J At current luminosity of sun this would be spent in T einstein = E/L =1.4 ×10 13 year The rest energy of proton, E= (1.67× )(3×10 8 ) 2 J =940 MeV

Classical approach The repulsive force between like Charged particles results a potential barrier that gets stronger as the Particle get closer

C oulomb potential barrier The coulombic potential is given by, u(r)=1/4πε 0 (z 1 z 2 e 2 /r) Z 1, z 2 are the no of proton in each nuclei and r is their distance. The K.E of the reduced mass µ, ½ µ v 2 =3/2 KT classical Fusion is possible if the average particle K.E is equals to or greater than the coulomb potential energy. For proton z=1, r= m T classical =1.1×10 10 k The central temperature of the sun 10 7 k much lower than k. So classical physics is unable to explain the phenomena.

If the gas is in thermal equillibrium with Temperature T, the atoms have a range of Velocities described by Maxwell-Boltzmann Distribution function. The central temperature of the sun is T≈2×10 7 K The K.E of a proton at this temp. is≈2 keV The electrostatic P.E of two protons Apart is 1MeV. The relative fraction of proton with thermal energy of 1MeV is only exp(- 1MeV/2keV) = exp(-500) Maxwell-Boltzmann statistics does not help

Quantum mechanics to rescue Heisenberg uncertainty principle state That the position and momentum of a particle are not Precisely defined, ΔxΔ The uncertainty in position means that if two proton can get close enough to each other There is some probability that they will be found within the coulomb barrier. This is known as tunneling

Quantum tunneling The tunneling is possible if the protons come within 1 de Broglie wavelenth for each other λ= The temperature require to sustain the nuclear reaction is, T quantum T q ≈10 7 k The temperature required for nuclear reaction is consistent with the estimated temperature of the sun. So without the quantum effect fusion would not be possible in the sun and such high luminosities could never be achieved.

Proton-proton cycle

The reactions are -

The CNO cycle For more massive stars the pp chain can still occur, but there is another sequence of Reactions that become more favorable for converting hydrogen to helium that is Called CNO cycle.

CNO cycle The reactions are-

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