Question

A binary star system consists of two stars of masses m1 and m2. The stars, which gravitationally attract each other, revolve around the center of mass of the system. The star with mass mi has a centripetal acceleration of magnitude a1.

Find a2, the magnitude of the centripetal acceleration of the star with mass m2 Express the acceleration in terms of quantities given in the problem introduction.

Answer 1

a₂ = G(m₁ + m₂)/(R - √[G(m₁ + m₂)/a₁])²

Explanation:

Since the gravitation force of attraction of the center of mass on m₁ equals the centripetal force, then

G(m₁ + m₂)m₁/R₁ = m₁a₁ where (m₁ + m₂) = total mass of the system and R₁ = distance of m₁ from the center of mass

So, G(m₁ + m₂)m₁/R₁² = m₁a₁

G(m₁ + m₂)m₁/m₁a₁ = R₁²

R₁² = G(m₁ + m₂)/a₁

R₁ = √[G(m₁ + m₂)/a₁]

The distance of m₂ from the center of mass is thus R₂ = R - R₁ = R - √[G(m₁ + m₂)/a₁] where R is the distance between the two stars.

Also, the gravitation force of attraction of the center of mass on m₂ equals the centripetal force, then

G(m₁ + m₂)m₂/R₂ = m₂a₂ where (m₁ + m₂) = total mass of the system and R₂ = distance of m₂ from the center of mass

So, G(m₁ + m₂)m₂/R₂² = m₂a₂

G(m₁ + m₂)m₂/m₂R₂² = a₂

a₂ = G(m₁ + m₂)/R₂²

Substituting R₂ into the equation, we have

a₂ = G(m₁ + m₂)/(R - √[G(m₁ + m₂)/a₁])²