Assume the angular momentum of a binary star system is quantised as J = nJ₀, where n is a positive integer. The system consists of two identical stars, each of mass m.
Physics · Gravitation · Satellites, Orbital Motion & Kepler’s Laws
JEE Advanced

Question

Assume the angular momentum of a binary star system is quantised as J = nJ₀, where n is a positive integer. The system consists of two identical stars, each of mass m.

G1 Physics · Accepted Answer

Step 1: For two identical stars of mass m separated by d, each orbits the centre of mass at radius d/2. Force balance givₑ Gm²/d²=mω²(d/2), hence ω²=2Gm/d³. Step 2: Total angular momentum is J=2[m(d/2)²ω]=(md²/2)ω, so J²=Gm³d/2 and d=2J²/(Gm³). With J=nJ₀ this givₑ the allowed separations; lowest n=1 givₑ 2J₀²/(Gm³). Step 3: The total energy is E=-Gm²/(2d)=-G²m⁵/(4J²), so for n=1 it is -G²m⁵/(4J₀²), which is not among the listed expressions. Detailed subanswers are given below.