When a star like our Sun exhausts its fuel, thermonuclear reactions in its core

Question

When a star like our Sun exhausts its fuel, thermonuclear reactions in its core cease, and it collapses to become a white dwarf. Often the star will blow off its outer layers and lose some mass before it collapses. Suppose a star with the Sun’s mass and radius (the radius of the Sun is 6.96×108m ) is rotating with period 25 days and then it collapses to a white dwarf with 51 % of the Sun’s mass and a rotation period of 131 s. a) What is the radius of the white dwarf? b) compare your answer to the radius of the earth. c) compare your answer to the radius of the sun.

Explanation / Answer

Let Ms be the mass of Star

and m be the mass of Dwarf

According to question

m = 0.51 Ms

Let the radius of Star be Rs = Radius of Sun = 696 x 10^(6) m

and the radius of dwarf br Rd

Let the Moment of Inertia of Star and Dwarf be Is and Id

Is = (2/5) Ms x Rs^(2)

Id = (2/5)Md x Rd^(2)

and Angular velocity of Star and Dwarf be Ws and Wd

Time period of Star be Ts = 25 days = 25 x 24 x 3600 = 2160000 seconds

Time period of Dwarf be Td = 131 seconds

We know that

Ws = ( 2pi / Ts ) = ( 2 x 3.14) / (2160000) = 2.91 x 10^(-6) rad/s

Wd = ( 2pi / Td ) = (2 x 3.14) / (131) = 0.048 rad/s

BY CONSERVATION OF ANGULAR MOMENTUM

Is x Ws = Id x Wd

(2/5) Ms x Rs^(2) x 2.91 x 10^(-6) = (2/5)Md x Rd^(2) x 0.048

Ms x Rs^(2) x 2.91 x 10^(-6) = 0.51 x Ms x Rd^(2) x 0.048

Rd^(2) = 1.19 x 10^(-4) Rs

Rd = 0.011 Rs

Rs is the Radius of Star = Radius of Sun

Radius of Dwarf is 0.11 times the radius of Sun.....(Answer c)

Rd = 0.11 x 696 x 10^(6) = 7588396.715 m = 7588.4 km .......(Answer a)

Radius of Earth be Re = 6371 km = 6371 x 10^(3) m

Rd / Re = ( 7588396.715 / 6371000 ) = 1.191

Rd = 1.191 Re

Radius of Dwarf is 1.191 times the radius of Earth......(Answer b)

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