Common Doppler scaling of spectral-line separations ($\Delta {\lambda }_{\mathrm{obs}}=D\Delta {\lambda }_{\mathrm{emit}}$)

If two spectral lines come from the same uniformly moving source along the same line of sight, each wavelength is multiplied by the same longitudinal Doppler factor $D$. Subtracting ${\lambda }_{2,\mathrm{obs}}=D{\lambda }_{2,\mathrm{emit}}$ and ${\lambda }_{1,\mathrm{obs}}=D{\lambda }_{1,\mathrm{emit}}$ proves the displayed identity. Recession has $D>1$ and approach has $D<1$. Defining positive radial velocity for recession gives $\beta =(D^2-1)/(D^2+1)$. A measured increase in wavelength separation therefore cannot be reconciled with strictly approaching longitudinal motion merely by changing the speed's magnitude.