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Question: The 6563 Å \(H_{\alpha}\)line emitted by hydrogen in a star found to be red- shifted by 15 Å. The sp...

The 6563 Å HαH_{\alpha}line emitted by hydrogen in a star found to be red- shifted by 15 Å. The speed with which the star is be receding from the earth is :

A

3.2×105ms−13.2 \times 10^{5}ms^{- 1}

B

6.87×105ms−16.87 \times 10^{5}ms^{- 1}

C

2×105ms−12 \times 10^{5}ms^{- 1}

D

12.74×105ms−112.74 \times 10^{5}ms^{- 1}

Answer

6.87×105ms−16.87 \times 10^{5}ms^{- 1}

Explanation

Solution

: Wavelength of HαH_{\alpha}lineλ=6563A˚\lambda = 6563Å

=6563×10−10m6563 \times 10^{- 10}m

Red shift observed in star is,

(λ′−λ)=15A˚=15×10−10m(\lambda' - \lambda) = 15Å = 15 \times 10^{- 10}m

And speed of light, c=3×108ms−1c = 3 \times 10^{8}ms^{- 1}

Let the velocity of the star with which it is receding away from the earth be v

∴\thereforeRed shift relation

λ′−λ=vcλ\lambda' - \lambda = \frac{v}{c}\lambda

∴v=cλ×(λ′−λ)\therefore v = \frac{c}{\lambda} \times (\lambda' - \lambda)

=3×108×15×10−106563×10−10=6.86×105ms−1= \frac{3 \times 10^{8} \times 15 \times 10^{- 10}}{6563 \times 10^{- 10}} = 6.86 \times 10^{5}ms^{- 1}