Question

1. In this question, you may assume that the core temperature in the Sun is T=15x106K. a Relatively low mass stars utstr eam Santo Toletin homology show that the central temperature in such stars scales as b) For hydrogen burning via the p-p chain to ignite, the temperature at the centre of the star must be larger than 4x106K. Calculate the minimum mass (in Solar masses) a star must have to ignite hydrogen. How does this compare to the mass of brown dwarfs?

just b

Answer 1

1) (b) core temperature of sun Te = ls 1 k. for Hydrogen burning via p-p chain to ignite, Core temperaturu or temperature of starus at the centru = 4000k - Tig let the mass of the stare: M & radius =R so, gravitational potential erergy = 4M² For the stane M = Np mp . No= No. of protons mp= mare of the protons so - GM INphp) (1) According to statistical mechanics Ek = NKT - (2) No No. of panticles (protons & Electrons) If we consider the start is consists of Hydrogen gas only then Number of rotons & Number of electrons are equal. so No = Nean so, from ear 18 (2) a) GM (p Mp) = KKT R = GMme (k= Boltzmann const.) KT If we assume T = Tig Cignition temperaturu) s. 'M= Main (minimum mass require for ignition) R= GM min mp - (3) * Tig

when Hydrogen ignition occure the of degeneracy pressure halts the gravitational collapse so, mass equation for this condition is M. 6 - 7 R - 4 Me mass of electron using the value of R from ea 18) in ed. 14 u) Momo - m- mmin : - - 3 Gmin - 3 - Main Here Tig= 4x100 = 1.05 x 1034 .3 4 - 6.6717" Nom? kg me = 9.1 x 10 kg -23 1:38 XIO 22 mp- 1.67 X10 2 -68 Mn = 1:05) X1 -23 x 638) X10 (667)x1022 x(4-1) xid" x (167)" x 10 -72 & 4x106 - 34 3/4 (0.000957 x 100 x 4x100 min 0.00544 x 3. 162x100 x x main = 0.0688 x 1031 kg

30 solar mass is approximately Mo = 2x10 kg so min = 0.344 Mo In Brown dwarf the fusion reaction doesn't start due to low mass. The minimum mass of a Brown dwaft to be stable under certain condition is MBO = 0.08 mo min 8 minimum mass of the start to ignite fusion 0.344m a mass of the Brown dwarf 0.08m. MRO MOD - min = 0.344 mo 0.08 mo -) Mmin = 4.3 MBD so, minimum mass require for a stare in p-p chain to ignite Hydrogen is 4.3. times the mass of a Brown Dwarf