Question

Review I Constants Periodic Table Part A The spectral lines of two stars in a particular eclipsing binary system shift back and forth with a period of 10 months. The lines of both stars shift by equal amounts, and the amount of the Doppler shift indicates that each star has an orbital speed of 7.0x104 m/s relative to the other What are the masses of the two stars? Assume that each of the two stars traces a circular orbit around their center of mass. (Hint: See Mathematical Insight Measuring Stellar Masses.) Express your answer using two significant figures. 0図? MSun Submit Request Answer

Answer 1

Given,

The period, T = 10 months

= (10 * 30 * 24 * 60 * 60) sec = 2.6 * 10⁷ sec

Speed of each star, v = 7 * 10⁴ m/s

We know, v = 2$\pi$r / T

=> r = vT / 2$\pi$

r = (7 * 10⁴ * 2.6 * 10⁷) / 2$\pi$

r = 18.2 * 10¹¹ / 2$\pi$

r = 2.897 * 10¹¹ m

By using Kepler's law,

T² = r³ / (m₁ + m₂)

T = 10 months

T = (10 /12) years = 0.833 Years

m₁ + m₂ =  r³ / T²

= (2.897 * 10¹¹)³ / (0.833)²

= 35.04 * 10³³  kg

From Doppler shifts, the massive star of mass m₂ is twice as m₁.

m₁ + 2m₁ = 35.04 * 10³³ kg

3 m₁ = 35.04 * 10³³ kg

m₁ = 11.68 * 10³³ kg = 5.872 * 10⁶³ solar mass

So, m₂ = 2 * m₁

= 2 * 11.68 * 10³³

m₂ = 23.36 * 10³³ kg = 1.174 * 10⁶² solar mass