Energy production in star

For 3rd year physics honours

Sukla Debnath

Early universe

Formation of star

Primordial nucleosynthesys

Primordial nucleosynthesys

As the Universe expand and temperature fall the reaction rate

decreased Below 109 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 × 108 J/Kg which is not sufficient

•For our SUN , if the Sun is made of coal of 2×1038 Kg it give energy 2× 1030×4×108 J = 8×1038 J Sun has radiate 1.2×1043 J energy in 109 year

Gravitational Contraction

The Gravitational potential energy is given by , V=-3/5 (GM2 /R)

The energy radiated during collapse would be ΔE = -Ef = 3/10 (GM2/R)

o For Sun ΔE = 2× 1041 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 c2

Where E=energy equivalent to mass m=mass c=speed of light

The rest energy of proton,E= (1.67× 10-27 )(3×108)2 J =940 MeV

What is the source of enormous energy of starsNuclear fission or fusion ?

Binding energy curve :

What is the source of enormous energy of starsNuclear 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 fusionCan 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= Ms C2 =1.8 ×1047 J At current luminosity of sun this would be spent in

Teinstein = E/L =1.4 ×1013 year

The rest energy of proton,E= (1.67× 10-27 )(3×108)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

Coulomb potential barrier

The coulombic potential is given by, u(r)=1/4πε0 (z1z2e2/r)

Z1 , z2 are the no of proton in each nuclei and r is their distance.The K.E of the reduced mass µ , ½ µ v2 =3/2 KTclassical Fusion is possible if the average particle K.E is equals to or greater than the coulomb potential energy.

For proton z=1, r=10-15mTclassical =1.1×1010 k

The central temperature of the sun 107 k much lower than 1010 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≈2×107 K

The K.E of a proton at this temp. is≈2 keV

The electrostatic P.E of two protons 10-15

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 stateThat the position and momentum of a particle are notPrecisely 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 tunnelingThe tunneling is possible if the protons come within 1 de Brogliewavelenth for each other λ=

The temperature require to sustain the nuclear reaction is, Tquantum

Tq ≈107 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

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-

Thank you