The Sun has a radius of 7:0 108 m, approximately, and a period of rotation
about its axis equal to 24.7 days. What is the Doppler shift of a spectral line with
laboratory wavelength 500 nm, in the light emitted (a) from the center of the Sun
and (b) from the edges of the Sun’s disk at its equator?
Ans.: (a) Dk ¼ 1:8 108 nm, (b) Dk ¼
0:00344 nm
λ=λ01+βcosθ1−β2≈λ0(1+βcosθ+12β2)\lambda= \lambda_0 \frac{1+ \beta cos \theta}{\sqrt{1- \beta^2}} \approx \lambda_0(1+ \beta cos \theta+ \frac{1}2 \beta^2)λ=λ01−β21+βcosθ≈λ0(1+βcosθ+21β2)
∇λ=λ−λ0≈λ0(Vc)cosθ+λ02(Vc)2\nabla \lambda=\lambda- \lambda _0 \approx \lambda_0 (\frac{V}c) cos \theta+ \frac {\lambda _0}2(\frac{V}c)^2∇λ=λ−λ0≈λ0(cV)cosθ+2λ0(cV)2
∇λ≈λ02(Vc)2\nabla \lambda \approx \frac{\lambda_0}{2}(\frac{V}c)^2∇λ≈2λ0(cV)2
∇λ0≈λ0(Vc)+λ02(Vc)2\nabla \lambda_0 \approx \lambda_0(\frac{V}c)+ \frac {\lambda _0}2 (\frac{V}{c})^2∇λ0≈λ0(cV)+2λ0(cV)2 and ∇λ180≈−λ0(Vc)+λ02(Vc)2\nabla \lambda_{180} \approx-\lambda_0(\frac{V}c)+\frac{\lambda_0}2(\frac{V}c)^2∇λ180≈−λ0(cV)+2λ0(cV)2
∇λ=∇λ0+∇λ1802≈λ02(Vc)2\nabla \lambda=\frac{\nabla \lambda_0+ \nabla \lambda{180}}{2} \approx\frac{\lambda_0}2(\frac{V}c)^2∇λ=2∇λ0+∇λ180≈2λ0(cV)2
