Question

(please type not write down the answer) astronomy

3. A main sequence star of mass 15 M⊙ has a luminosity of approximately 10,000 L⊙ . a. At what rate does mass vanish as H is fused to He in the star’s core? Note: When we say “mass vanish” what we really mean is “gets converted into energy and leaves the star as light”. b. At what rate is H converted into He? To do this you need to take into account that for every kg of hydrogen burned, only 0.7% gets converted into energy while the rest turns into helium. c. Assuming that only the 10% of the star’s mass in the central regions will get hot enough for fusion, calculate the main sequence lifetime of the star. Put your answer in years, and compare to the lifetime of the Sun. It should be much, much shorter.

Answer 1

Given that :

mass of star, M_S = 15 M

mass of sun, M = 1.989 x 10³⁰ kg

luminosity of star, L_S = 10000 L

luminosity of sun, L = 3.9 x 10²⁶ W

(a) The rate at which mass vanishes is given by,

using a formula, we have

L_S = E / t = m_E c² / t

m_E / t = L_S / c²

where, c = speed of light = 3 x 10⁸ m/s

m_E = mass that is converted to energy

m_E / t = [(10000) (3.9 x 10²⁶ W)] / (3 x 10⁸ m/s)²

m_E / t =4.33 x 10¹³ kg/sec

m_E / t = 4.33 x 10¹³ kg/sec

(b) The rate at which H is converted into He is given by,

m_E = (0.007) m_H

using a formula, we have

m_H / t = L_S / (0.007) c²

where, m_H = mass of hydrogen that is converted to helium in a fusion reaction

m_H / t = [(10000)(3.9 x 10²⁶ W)] / (0.007) (3 x 10⁸ m/s)²

m_H / t = 1.58*10¹⁶​ kg/sec

m_H / t = 1.58*10¹⁶​ kg/sec

(c) Mass of helium available for fusion is 0.1 M_S.

Total mass that will be converted into energy during the entire main sequence lifetime is given as :

m_MS = (0.007) x (0.1) x (15) x (1.989 x 10³⁰ kg)

m_MS =2.088 x 10²⁸ kg

Total energy produced during the entire main sequence lifetime is given by,

E_MS = m_MS c² = (6.96 x 10²⁶ kg) (3 x 10⁸ m/s)²

E_MS = 1.87 x 10⁴⁵ J

Now, the main sequence lifetime of the star will be given as :

t_MS = E_MS / L_S = (1.87 x 10⁴⁵J) / (10000) (3.9 x 10²⁶ J/s)

t_MS = 1.605 x 10¹³ sec