2. Main-sequence stars get their energy from the fusion of hydrogen to helium in

Question

2. Main-sequence stars get their energy from the fusion of hydrogen to helium in their cores. Since most of the hydrogen in stars is outside the core, stars only get to use about 15% of their hydrogen while on the main sequence. In lecture 20 we learned that fusing 1 gram of hydrogen to helium produces 6.4 × 1018 ergs of energy. (a) A star with a mass of 100 M (the mass of the sun is 2 × 1033 gm) has a luminosity of 106 L (the luminosity of the sun is 4 × 1033 ergs/sec). How many years will this star remain on the main sequence? (First calculate the amount of hydrogen the star will convert to helium. Do not forget that only 75% of its mass is hydrogen. Then calculate how much energy this produces. Finally calculate how long the energy can last at the luminosity of the star.) (b) A star with a mass of 0.25 M has a luminosity of 0.002 L. How long does it remain on the main sequence?

Explanation / Answer

a)

mass of sun, M=2x1033gm

mass of star = 100M =200x1033gm

luminosity of sun,L = 4 × 1033 ergs/sec

luminosity of star = 106L = 106 x 4 × 1033 ergs/sec

75% of star's mass is hydrogen, so total hydrogen = 75/100*200x1033gm =150x1033gm

stars only get to use about 15% of their hydrogen while on the main sequence

So 15/100*150x1033gm =225x1032gm of hydrogen is used while on the main sequence

We know that 1gm of hydrogen to helium produces 6.4 × 1018 ergs of energy

so 225x1032gm of hydrogen produces 225x1032*6.4 × 1018 ergs of energy = 1440x1050erg of energy

luminosity of star = 106 x 4 × 1033 ergs/sec

So star's life time = 1440x1050erg/106 x 4 × 1033 ergs/sec = 3.4x1017sec

we know 1 year =3.15x107sec

so 3.4x1017sec =3.14x1017/3.15x107 = 1.08x1010years

b)

mass of sun, M=2x1033gm

mass of star = 0.25M =0.25x2x1033gm

luminosity of sun,L = 4 × 1033 ergs/sec

luminosity of star = 0.002L = 0.002x 4 × 1033 ergs/sec

75% of star's mass is hydrogen, so total hydrogen = 75/100*0.25x2x1033gmgm =0.375x1033gm

stars only get to use about 15% of their hydrogen while on the main sequence

So 15/100*0.375x1033gm =5.625x1031gm of hydrogen is used while on the main sequence

We know that 1gm of hydrogen to helium produces 6.4 × 1018 ergs of energy

so 5.625x1031gm of hydrogen produces 5.625x1031*6.4 × 1018 ergs of energy = 36x1049 erg of energy

luminosity of star = 0.002 x 4 × 1033 ergs/sec

So star's life time = 36x1049 erg/0.002 x 4 × 1033ergs/sec = 4500x1016 =4.5x1018 sec

we know 1 year =3.15x107sec

so 4.5x1018 sec =4.5x1018/3.15x107 years= 1.43x1011years

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