Question

For the following, quote your numerical answers to two significant figures: a) A binary system consists of two stars orbiting a common center of mass. Star A is a white star of mass MA = 2.02Mo, where Mo = 1.989 x 1030 kg is the mass of the Sun. Its companion, Star B, is a white dwarf with mass Mg = 0.978M. The orbital period of the two objects is observed to be t = 50.09 years. What is the length of the semi-major axis, a, of their orbit? b) The distance of closest approach between the two stars is known to be rmin = 7.989 AU, where AU is an "astronomical unit," defined from the Earth-Sun distance: 1 AU is equal to 1.496 x 1011 meters. Find the orbital parameters c (scale factor) and € (eccentricity). Because the stars are similarly massive, the center of mass cannot be approximated as being within one star or the other. What is the radial distance rA from the center of mass to star A? What is the similar distance rg to star B? Express your answer as a function of the radial separation r and the masses of the stars; don't plug in any numbers yet.

Answer 1

a) The orbital period and length of semi major axis of the binary system is defined by the equation

P2=4π2a3G(m1+m2)

41-a3 PG(m + m2

Here, P = 50.09 years = 1.579*10⁹ s

m₁ = 2.02 m_sun

m₂ = 0.987 m_sun

So,

a = ((1.579*10⁹)² *6.67*10^-11 *(2.02+.0987)*1.989*10³⁰/(4*3.14²))^1/3

a = 2.6*10¹² m

b) The distance of closest approach = 7.989*1.496*10¹¹ = 1.195*10¹² m

S, eccentricity is given by

r_p = a(1-e)

e = 1-r/a = 1-1.195/2.6 = 0.54

the scale factor of the orbit is the distance from the mid point to any of the focal point.

c = a*e = 2.6*10¹² *0.54 = 1.4*10¹² m

c)

Assuming that the star is at the origin of the co-ordinate system, and the distance between the star and planet is r,

the center of mass is given by

r_cm =r* M_b / (M_a + M_b)

The distance from A to center of mass is

R_A = r_cm

The distance to B from center of mass is

R_B = r-r_cm

Since it is said not to plug in numbers, I am writing just the expression.