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Doppler shifted hydrogen absorption lines are seen in the spectrum of a star The hydrogen line at 656.28nm is seen to be shifted to656.50nm?

How fast is the star moving (Note: The speed of light is approximately 300,000 km/s, or 3 × 105 km/s)?

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  • Argent
    Lv 7
    2 years ago
    Favorite Answer

    The Doppler shift formula is λ₂/λ₁ = √((1+v/c) / (1-v/c)), where λ₁ is the unshifted wavelength, λ₂ is the shifted wavelength, v is the speed of the star, and c is the speed of light.

    Solve for v:

    (λ₂/λ₁)² = (1+v/c) / (1-v/c), so

    (λ₂/λ₁)² = (1+v/c) / (1-v/c) · c/c = (c+v) / (c-v), so

    (λ₂/λ₁)²(c-v) = c+v, so

    c(λ₂/λ₁)² - v(λ₂/λ₁)² = c + v, so

    c(λ₂/λ₁)² - c = v(λ₂/λ₁)² + v, so

    v((λ₂/λ₁)² + 1) = c(λ₂/λ₁)² - c, so

    v = (c(λ₂/λ₁)² - c) / ((λ₂/λ₁)² + 1), or

    v = c((λ₂/λ₁)² - 1) / ((λ₂/λ₁)² + 1.

    Here, λ₂ = 656.50 nm; λ₁ = 656.28 nm; c = 2.9979 × 10⁸ m/s.

    λ₂/λ₁ = 1.000335; (λ₂/λ₁)² = 1.00067, so

    v = (2.9979 × 10⁸ m/s)(1.00067 - 1) / (1.00067 + 1) = 100.4 km/s = 1.004 × 10⁵ m/s.

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