Question

A low mass star evolved as white dwarf with one solar mass and radius equal to 0.9 times the Earth radius. Estimate the escape velocity of the white dwarf and the Gravitational red shit of the carbon spectral line of wave length 700nm.

Answer 1

The escape velocity is

$v = \sqrt{\dfrac{2GM}{R}} = \sqrt{\dfrac{2\cdot6.67\cdot10^{-11}\,\mathrm{N\,m^2/kg^2}\cdot2\cdot10^{30}\,\mathrm{kg}}{0.9\cdot6.4\cdot10^6\,\mathrm{m}}} = 6.8\cdot10^6\,\mathrm{m/s}.$

Let us assume the observer to be very far from the white dwarf. Therefore, the gravitational redshift can be calculated as

$z_G = \dfrac{GM}{c^2R} = \dfrac{6.67\cdot10^{-11}\,\mathrm{N\,m^2/kg^2}\cdot2\cdot10^{30}\,\mathrm{kg}}{(3\cdot10^8\,\mathrm{m/s})^2\cdot 0.9\cdot6.4\cdot10^6\,\mathrm{m}} = 2.6\cdot10^{-4}.$

Therefore, the change of wavelength of the carbon line is

$\Delta\lambda = z_G\cdot\lambda_0 = 2.6\cdot10^{-4}\cdot700\,\mathrm{nm} = 0.18\,\mathrm{nm}.$