Interior temperature of the Sun: Given that the Sun has mass M = 2 × 1030 kg and

Question

Interior temperature of the Sun: Given that the Sun has mass M = 2 × 1030 kg and radius R = 6.96 × 108 m. (a) Estimate the pressure at the center of the Sun using P GM2/R4 with G the universal gravitational constant. Express your answer in Pascals. (b) Assuming the Sun is made entirely of hydrogen and is a solid sphere of uniform density, estimate the number density of hydrogen. Express your answer in unit of “hydrogen atom per cubic centermeter”. (c) Using your answers to (a) and (b) and the ideal gas law P = nkT where n is the number density and P is the pressure, estimate the temperature at the interior of the Sun.

Explanation / Answer

Since m = 2 x 1030 kg and radius is 6.96 x108 m

(a) Since P = GM2 / R4 = 6.67 x 10-11x (2 x 1030)2/ (6.96 x 108)4 = 0.0113 x 1017 kg2/m4, to convert it into Pascal which is 1 Pa = 1N/m2 = 1 kg/ms2. Also 1kg/m2 x 9.80665 = 1Pa, so for kg2/m4 , it will be kg2/ m4 x 9.80665 x 9.80665 = 1Pa. So, P will be 0.0113 x 1017 x 9.80665 x 9.80665 = 1.0867 x 1017 Pa.

(b) To calculate the number of hydrogens, which is total number of hydrogens per cubic centimeter, the total number of hydrogens is estimated to be equal to weight of the sun, so to find the number density of hydrogen atoms, n will be n = M/V, where is mass of the sun and V is volume since it is a solid sphere,

n = 2 x 1030/ (4/3 x pi x r3)

n = 2 x 1030 x 10-10/ 1408.020 = 0.001420 x 1020 hydrogens/cm3.

(c) Using the ideal gas law P = nkT, where P is the pressure in pascals, n is the number density of hydrogen per cubic centimeter, k is boltzmann's constant and T is the temperature in Kelvin.

P = nkT

1.0867 x 1017 = 1.420 x 1017 x 1.38 x 10-23 x T

T = 1.0867 x 1017/1.420 x 1017 x 1.38 x 10-23

T = 0.55 x 1023 K.

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