(a) Cygnus X-1 is a source of X-rays that has been studied extensively by astronomers. The observations (X-ray, optical, and radio) show that it is a distance r ∼ 6,000 light-years from Earth. It consists of a very hot disk of X-ray-emitting gas that surrounds a black hole with mass ,and the hole in turn is in a binary orbit with a heavy companion star. Most of the X-ray photons have energies E ∼ 2 keV, their energy flux arriving at Earth is F ∼ 10⁻¹⁰ Wm⁻², and the portion of the disk that emits most of them has radius roughly 7 times that of the black hole (i.e., R ∼ 300 km).⁵Make a rough estimate of the mean occupation number of the X-rays’ photon states. Your answer should be in the region η ≪ 1, so the photons behave like classical, distinguishable particles. Will the occupation number change as the photons propagate from the source to Earth?

(b) A highly non spherical supernova in the Virgo cluster of galaxies (40 million light years from Earth) emits a burst of gravitational radiation with frequencies spread over the band 0.5–2.0 kHz, as measured at Earth. The burst comes out in a time of about 10ms, so it lasts only a few cycles, and it carries a total energy of roughly

is the mass of the Sun. The emitting region is about the size of the newly forming neutron-star core (10 km), which is small compared to the wavelength of the waves; so if one were to try to resolve the source spatially by imaging the gravitational waves with a gravitational lens, one would see only a blur of spatial size one wavelength rather than seeing the neutron star. What is the mean occupation number of the burst’s graviton states? Your answer should be in the region η ≫ 1, so the gravitons behave like a classical gravitational wave.

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