S2: Astrophysics: From Planets to The Cosmos Trinity term 2012

Figure 1: The primary minimum in the light curve of the eclipsing binary system AR Lacertae,
plotted as a function of phase (i.e. fraction of period). This system, whose period is 1.98 days
is also a spectroscopic binary: the radial velocity curves of the two components are sinusoidal
and have amplitudes of 116km s −1
9. AR Lac is an eclipsing spectroscopic binary with period 1.98 days. The radial velocity
curves of the two stars are sinusoidal with equal amplitudes of 116km.s −1 . The diagram
(Fig. 1) shows the primary minimum in the light curve as a function of orbital phase.
Determine the masses and radii of the two components. Assume the orbit is edge on.
10. By equating the gravitational acceleration at the surface of a neutron star to the centrifugal
acceleration at the equator, calculate the size of a 3M ⊙ neutron star that can spin with
a period of 1ms without disintegrating. What is the density implied? Compare this with
nuclear densities.
11. A mass m falls towards a black hole and ends up in an orbit at three Schwarzschild radii
from the balck hole (this the closest stable orbit to the black hole). Use the Newtonian
expression for the decrease in potential energy, and the assumption that half this decrease
will be radiated away (from the Virial theorem), to estimate the fraction of the rest mass
that will be radiated away in this process. How does that compare with the fraction of
the rest mass released in the p-p chain that you calculated in Ques 6?
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