Black holes in orbit around a normal star are detected from the earth due to the frictional heating of infalling gas into the black hole, which can reach temperatures greater than 10⁶K. Assuming that the infalling gas can be modelled as a black body radiator, then the wavelength of maximum power lies in which region of the electromagnetic spectrum?

The energy of the gas at temperature T is approximately 3/2kT, where k is the Boltzmann constant. In a black body radiation model, this energy corresponds to a wavelength λ with energy hc/λ. Therefore:

hc/λ = 3/2kT

Solving for λ:

λ ≈ 2hc/(3kT)

Given T = 10⁶ K, we can estimate λ. Using h ≈ 6.63 x 10⁻³⁴ Js, c ≈ 3 x 10⁸ m/s, and k ≈ 1.38 x 10⁻²³ J/K:

λ ≈ 2 * (6.63 x 10⁻³⁴ Js) * (3 x 10⁸ m/s) / (3 * (1.38 x 10⁻²³ J/K) * (10⁶ K))

λ ≈ 10⁻¹⁰ m = 0.1 nm

This wavelength corresponds to the X-ray region of the electromagnetic spectrum.