Question

 

A type-Ia supernova may be the thermonuclear explosion of an accreting white dwarf that approaches the Chandresekhar limit. An alternative scenario, however, is that supernova Ia progenitors are white-dwarf binaries that lose orbital energy to gravitational waves until they merge, and thus exceed the Chandrasekhar mass and explode. a) Show that the oribtal kinetic energy of an equal-mass binary with seperation a and individual masses M is E_{k}=\frac{GM^{2}}{2a} and the total orbital energy (kinetic plus gravitational) is minus this amount. b) The power lost to gravitational waves by such a system is \dot{E}_{gw}=-\frac{2c^{5}}{5G}(\frac{2GM}{c^{2}a})^{5} By equating to the time derivative of the total energy found in (a), obtain a differential equation for a(t) , and solve it. c) What is the maximum initial seperation that a white-dwarf binary can have, if the compononets are to merge within 10 Gyr> Assume the white dwarfs have 1M_{sun} each, and the merger occurs when a=0 (Hint: Answer is 0.016 AU)

Answer 1

A type-Ia supernova may be the thermonuclear explosion of an accreting white dwarf that approaches the Chandresekhar limit. An alternative scenario, however, is that supernova Ia progenitors are white-dwarf binaries that lose orbital energy to gravitational waves until they merge, and thus exceed the Chandrasekhar mass and explode.

a) Show that the oribtal kinetic energy of an equal-mass binary with seperation a and individual masses M is E_{k}=\frac{GM^{2}}{2a} and the total orbital energy (kinetic plus gravitational) is minus this amount.

b) The power lost to gravitational waves by such a system is \dot{E}_{gw}=-\frac{2c^{5}}{5G}(\frac{2GM}{c^{2}a})^{5} By equating to the time derivative of the total energy found in (a), obtain a differential equation for a(t) , and solve it.

c) What is the maximum initial seperation that a white-dwarf binary can have, if the compononets are to merge within 10 Gyr> Assume the white dwarfs have 1M_{sun} each, and the merger occurs when a=0

6(A)