The escape velocity is
v = 2 G M R = 2 ⋅ 6.67 ⋅ 1 0 − 11 N m 2 / k g 2 ⋅ 2 ⋅ 1 0 30 k g 0.9 ⋅ 6.4 ⋅ 1 0 6 m = 6.8 ⋅ 1 0 6 m / s . 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}. v = R 2 GM = 0.9 ⋅ 6.4 ⋅ 1 0 6 m 2 ⋅ 6.67 ⋅ 1 0 − 11 N m 2 /k g 2 ⋅ 2 ⋅ 1 0 30 kg = 6.8 ⋅ 1 0 6 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 = G M c 2 R = 6.67 ⋅ 1 0 − 11 N m 2 / k g 2 ⋅ 2 ⋅ 1 0 30 k g ( 3 ⋅ 1 0 8 m / s ) 2 ⋅ 0.9 ⋅ 6.4 ⋅ 1 0 6 m = 2.6 ⋅ 1 0 − 4 . 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}. z G = c 2 R GM = ( 3 ⋅ 1 0 8 m/s ) 2 ⋅ 0.9 ⋅ 6.4 ⋅ 1 0 6 m 6.67 ⋅ 1 0 − 11 N m 2 /k g 2 ⋅ 2 ⋅ 1 0 30 kg = 2.6 ⋅ 1 0 − 4 .
Therefore, the change of wavelength of the carbon line is
Δ λ = z G ⋅ λ 0 = 2.6 ⋅ 1 0 − 4 ⋅ 700 n m = 0.18 n m . \Delta\lambda = z_G\cdot\lambda_0 = 2.6\cdot10^{-4}\cdot700\,\mathrm{nm} = 0.18\,\mathrm{nm}. Δ λ = z G ⋅ λ 0 = 2.6 ⋅ 1 0 − 4 ⋅ 700 nm = 0.18 nm .
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