Answer
a. 0.32 nm.
b. 3.2 nm.
Work Step by Step
a. First find the velocity from Hubble’s law, equation 33–4.
$$\frac{v}{c}\frac{H_0d}{c}=\frac{(21000m/s/Mly)(7.0 Mly)}{3.00\times10^8m/s}=4.9\times10^{-4}$$
Find the observed wavelength from the Doppler shift equation, 33–3.
$$\lambda=\lambda_0\sqrt{\frac{1+v/c}{1-v/c}}$$
$$\lambda=(656.3nm) \sqrt{\frac{1+4.9\times10^{-4}}{1-4.9\times10^{-4}}}=656.62nm$$
The shift is 656.62nm-656.3nm=0.32nm.
b. Repeat, with the new distance. First find the velocity from Hubble’s law, equation 33–4.
$$\frac{v}{c}\frac{H_0d}{c}=\frac{(21000m/s/Mly)(70 Mly)}{3.00\times10^8m/s}=4.9\times10^{-3}$$
Find the observed wavelength from the Doppler shift equation, 33–3.
$$\lambda=\lambda_0\sqrt{\frac{1+v/c}{1-v/c}}$$
$$\lambda=(656.3nm) \sqrt{\frac{1+4.9\times10^{-3}}{1-4.9\times10^{-3}}}=659.52nm$$
The shift is 659.52nm-656.3nm=3.2nm.
