use ∆λ / λ . ∆f / f . v / c for the redshift of electromagnetic radiation from a source moving relative to an observer

Stellar Radii – Using the Doppler Red‑shift

In this note we explore how the Doppler shift of spectral lines can be used to infer the motion of a star relative to the observer and, in combination with binary‑star dynamics, to determine stellar radii. The key relationships are

• Fractional change in wavelength: \$\displaystyle \frac{\Delta\lambda}{\lambda}\$
• Fractional change in frequency: \$\displaystyle \frac{\Delta f}{f}\$
• Ratio of source speed to the speed of light: \$\displaystyle \frac{v}{c}\$

1. The relativistic Doppler formula (non‑relativistic limit)

For speeds much smaller than the speed of light (\$v \ll c\$) the observed wavelength \$\lambda{\text{obs}}\$ of a spectral line emitted at rest wavelength \$\lambda0\$ is shifted by

\$\frac{\Delta\lambda}{\lambda0}= \frac{\lambda{\text{obs}}-\lambda0}{\lambda0}= \frac{v}{c}\$

Similarly for frequency \$f\$ we have

\$\frac{\Delta f}{f0}= \frac{f{\text{obs}}-f0}{f0}= -\,\frac{v}{c}\$

The sign convention is:

• \$v>0\$ (receding) → red‑shift (\$\Delta\lambda>0\$, \$\Delta f<0\$)
• \$v<0\$ (approaching) → blue‑shift (\$\Delta\lambda<0\$, \$\Delta f>0\$)

2. Measuring radial velocity from spectral lines

From an observed spectrum we identify a known laboratory line (e.g., H‑α at \$\lambda0 = 656.28\,\$nm). The measured wavelength \$\lambda{\text{obs}}\$ gives the radial velocity:

\$v = c\,\frac{\lambda{\text{obs}}-\lambda0}{\lambda_0}\$

Typical uncertainties are of order \$10^{-4}\,c\$ for high‑resolution spectrographs, corresponding to a few km s\$^{-1}\$.

3. Binary stars – linking radial velocity to orbital parameters

In a spectroscopic binary the observed radial velocity varies sinusoidally with orbital phase. The semi‑amplitude \$K\$ of the velocity curve is related to the orbital elements by

\$K = \frac{2\pi a_1 \sin i}{P\sqrt{1-e^2}}\$

where

• \$a_1\$ – semi‑major axis of the primary’s orbit about the centre of mass
• \$i\$ – inclination of the orbital plane
• \$P\$ – orbital period
• \$e\$ – eccentricity

From Kepler’s third law

\$\frac{(a1+a2)^3}{P^2}= \frac{G(M1+M2)}{4\pi^2}\$

we can solve for the masses \$M1\$ and \$M2\$. Once the masses are known, the radii can be estimated using the mass–radius relation for main‑sequence stars or, for eclipsing binaries, directly from the eclipse geometry.

4. Example calculation – Determining the radius of a main‑sequence star

1. Measure the H‑α line shift at several orbital phases and obtain \$K = 45\,\$km s\$^{-1}\$.
2. From the light curve of the eclipsing binary determine \$P = 3.2\,\$days and \$i = 87^\circ\$.
3. Assume a circular orbit (\$e=0\$) and solve for \$a_1\$:

   \$a_1 = \frac{K P}{2\pi \sin i} \approx 6.2\times10^{6}\,\text{km}\$

4. Use the mass function and an estimate of \$M2\$ (from spectral type) to find \$M1 \approx 1.2\,M_\odot\$.
5. Apply the empirical main‑sequence relation \$R/R\odot \approx (M/M\odot)^{0.8}\$:

   \$R1 \approx 1.2^{0.8}\,R\odot \approx 1.15\,R_\odot\$

5. Summary of key equations

6. Practical tips for A‑level examinations

• Always state whether the shift is a red‑shift or blue‑shift before substituting numbers.
• Check that the speed you calculate is much less than \$c\$; otherwise use the full relativistic formula

  \$\frac{\lambda{\text{obs}}}{\lambda0}= \sqrt{\frac{1+v/c}{1-v/c}}\$

• When given the period \$P\$ in days, convert to seconds before using it in Kepler’s law.
• Remember that the inclination \$i\$ appears as \$\sin i\$; for eclipsing binaries \$i\approx90^\circ\$ so \$\sin i\approx1\$.

By mastering the use of \$\Delta\lambda/\lambda\$, \$\Delta f/f\$ and \$v/c\$ you can confidently analyse Doppler shifts, determine stellar radial velocities, and, through binary‑star dynamics, estimate stellar radii – a powerful set of tools for the Cambridge A‑Level Physics syllabus.