A binary star consists of two stars A and B that orbit one another, as illustrated in Fig. 1.1.

Question 8

A binary star consists of two stars A and B that orbit one another, as illustrated in Fig. 1.1.

Fig. 1.1

The stars are in circular orbits with the centres of both orbits at point P, a distance d from the centre of star A.

(a) (i) Explain why the centripetal force acting on both stars has the same magnitude. [2]

(ii) The period of the orbit of the stars about point P is 4.0 years.

Calculate the angular speed ω of the stars. [2]

(b) The separation of the centres of the stars is 2.8 × 10⁸ km.

The mass of star A is MA. The mass of star B is MB.

The ratio MA/MB is 3.0.

(i) Determine the distance d. [3]

(ii) Use your answers in (a)(ii) and (b)(i) to determine the mass MB of star B.

Explain your working. [3]

 [Total: 10]

Reference: Past Exam Paper – June 2016 Paper 42 Q1

Solution:

(a) (i) The gravitational force provides the centripetal force. From Newton’s 3^rd law, the forces have the same magnitude.

(ii)

{Period T = 4 years = (4.0 × 365 × 24 × 3600) seconds}

ω = 2π / T

ω = 2π / (4.0 × 365 × 24 × 3600)

ω = 5.0 (4.98) × 10^–8 rad s^–1

(b)

(i)

{Centripetal force = mrω²

The centripetal forces acting on the stars are equal in magnitude.

Centripetal force on star A = MA d ω²

Centripetal force on star B = MB (2.8×10⁸ – d) ω² }

(centripetal force =) MA d ω² = MB (2.8×10⁸ – d) ω²

{ MA d ω² = MB (2.8×10⁸ – d) ω²

MA d = MB (2.8×10⁸ – d)

M_A / M_B = (2.8×10⁸ – d) / d

3.0 = (2.8×10⁸ – d) / d }

MA / MB = 3.0 = (2.8×10⁸ – d) / d

{3d = 2.8×10⁸ – d

3d + d = 2.8×10⁸}

d = 7.0 × 10⁷ km

(ii)

{The gravitational force provides the centripetal force.

Note that the separation should be in metres, not kilometres.

We want to find M_B. So, we equate the gravitational force to the centripetal force on star A so that M_A gets eliminated.

GM_AM_B / r² = M_Adω²

For the gravitational force, r is the separation of the stars.

For the centripetal force, d is the distance of star A from point P.}

GM_AM_B / (2.8×10¹¹)² = M_Adω²

M_B = (2.8 × 10¹¹)² × dω² / G

M_B= (2.8×10¹¹)² × (7.0×10¹⁰) × (4.98×10^–8)² / (6.67×10^–11)            

M_B= 2.0 × 10²⁹ kg