Question

In a little over 5 billion years, our Sun will collapse to a white dwarf approximately 16,000 km in diameter. (Ignore the fact that the Sun will lose mass as it ages.) (a) What will our Sun’s angular momentum and rotation rate be as a white dwarf? (Express your answers as multiples of its present-day values.) (b) Compared to its present value, will the Sun’s rotational kinetic energy increase, decrease, or stay the same when it becomes a white dwarf? If it does change, by what factor will it change? The radius of the Sun is presently 6.96 * 108 m.

Answer 1

a)

M_s = mass of sun

R_s = initial radius of sun = 6.96 x 10⁸ m

R_d = final radius of whitedwarf = 8000 km = 8 x 10⁶ m

w_i = present rotation rate

w_f = final rotation rate

Angular momentum of Sun = angular momentum of sun as white dwarf = (2/5) M_s R_s² W_i²

using conservation of angular momentum

Angular momentum of Sun = angular momentum of sun as white dwarf

(2/5) M_s R_s² W_i = (2/5) M_s R_d² W_f

W_f = R²_s W_i/R²_d = (6.96 x 10⁸ /(8 x 10⁶ ))² W_i = 7569 W_i

b)

as the sun becomes white dwarf , its moment of inertia decreases and rotation speed increases , hence the rotational kinetic energy will increase

change = (2/5) M_s R_s² W_i² - (2/5) M_s R_d² W_f² = (2/5) M_s R_s² W_i² - (2/5) M_s (87 R_s)² (7569 W_i)²

= (2/5) M_s R_s² W_i² (1 - 87^2 7569^2) = - 4.34 x 10¹¹

factor = 4.34 x 10¹¹