Two evolving giants are each about to fill their Roche lobes

PHYSICS · Multi-part · Fermi University Physics · Astrophysics & Cosmology · Celestial mechanics and planetary systems · Stellar death and compact objects

Stars from the Inside, Ch Accretion power: mass transfer, discs and the weighing of black holes

Problem

Two evolving giants are each about to fill their Roche lobes. System A: a

5\,M_\odot

giant with a deep convective envelope transfers mass to a

2\,M_\odot

companion, so

q \equiv m_{\rm donor}/m_{\rm accretor} = 2.5

. System B: a metal-poor

0.8\,M_\odot

giant with a deep convective envelope, in a globular cluster, transfers mass to a

1.4\,M_\odot

neutron star, so

q = 0.57

. A donor with a deep convective envelope behaves like an

n = 3/2

polytrope, for which

\zeta_{\rm ad} \equiv \mathrm{d}\ln R/\mathrm{d}\ln m_1 \approx -1/3

. Under conservative mass transfer the Roche lobe responds as

\zeta_L \equiv \mathrm{d}\ln r_L/\mathrm{d}\ln m_1 \approx 2.13q - 1.67

. Transfer is dynamically stable only if

\zeta_{\rm ad} \ge \zeta_L

. ((a)) Evaluate

\zeta_L

for System A and System B and decide which transfer is dynamically stable. ((b)) System B's transfer is conservative: no mass or orbital angular momentum leaves the binary. It runs until the giant's envelope is exhausted, leaving a helium white-dwarf core of

0.2\,M_\odot

; every gram the donor loses is gained by the neutron star. Using conservation of the orbital angular momentum of a circular orbit,

J = m_1m_2\sqrt{Ga/M}

(with

M = m_1 + m_2

fixed), find the ratio of the final to the initial orbital separation. ((c)) Name the class of binary this system belongs to while the transfer in part (b) is under way, and name the two objects the binary consists of once the transfer has finished.

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