Light from a massive star suffers 'gravitational red shift' i.e.,… - Vidyadip

Question

Light from a massive star suffers 'gravitational red shift' i.e., its wavelength changes towards the red end due to the gravitational attraction of the star. Obtain the formula for this gravitational red shift using the simple consideration that a photon of frequency v has energy $\text{h}\nu$ (h=planck's constant) and mass $\frac{\text{h}\nu}{\text{c}^2}.$ Estimate the magnitude of the red-shift for light of wavelength $5000\mathring{\text{A}}$ from a star of mass 10³²kg and radius 10⁶km. G = 6.67 × 10^-11Nm² kg^-2 and C = 3 × 108m s^-1.

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Answer

$\text{h}\nu^{'}=\text{h}\nu-\frac{\text{GMh}\nu}{\text{Rc}^2}$

i.e., $\nu'=\nu\Big(1-\frac{\text{GM}}{\text{c}^2\text{R}}\Big)$

where $\nu'$ is the shifted frequency.

Now, $\lambda'=\lambda\Big(1+\frac{\text{GM}}{\text{c}^2\text{R}}\Big)$ if $\frac{\text{GM}}{\text{Rc}^2}<1$

i.e., $\lambda'-\lambda=\frac{\lambda\text{GM}}{\text{c}^2\text{R}}=0.371\mathring{\text{A}}$

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