understand that the lines in the emission and absorption spectra from distant objects show an increase in wavelength from their known values

Stellar Radii

In this lesson we will explore how the spectra of distant stars can be used to determine their radii. The key idea is that the spectral lines we observe are shifted to longer wavelengths (red‑shifted) compared with laboratory values. By measuring this shift we can infer the star’s radial velocity, and together with its luminosity we can calculate its radius.

Learning Objective

Understand that the lines in the emission and absorption spectra from distant objects show an increase in wavelength from their known values, and use this information to estimate stellar radii.

1. Why Spectral Lines Shift

The increase in wavelength is a consequence of the Doppler effect for light. If a star is moving away from the observer, each photon is stretched, giving a longer observed wavelength \$ \lambda{\text{obs}} \$ than the rest wavelength \$ \lambda{0} \$.

The fractional shift is defined as

\$\frac{\Delta \lambda}{\lambda{0}} = \frac{\lambda{\text{obs}}-\lambda{0}}{\lambda{0}} = \frac{v_{\text{r}}}{c}\$

where \$v_{\text{r}}\$ is the radial velocity (positive for recession) and \$c\$ is the speed of light.

2. Measuring the Red‑Shift

Typical strong absorption lines used in stellar spectroscopy are listed below.

For each line the radial velocity is obtained from the formula above, converting \$c = 3.00\times10^{5}\,\text{km s}^{-1}\$.

3. From Radial \cdot elocity to Distance (Optional)

If the star belongs to a galaxy whose recession follows Hubble’s law, the distance \$d\$ can be estimated by

\$d = \frac{v{\text{r}}}{H{0}}\$

with \$H_{0}\approx70\ \text{km s}^{-1}\,\text{Mpc}^{-1}\$.

4. Determining Stellar Radius

Once the distance is known, the absolute luminosity \$L\$ can be derived from the apparent magnitude. The radius \$R\$ follows from the Stefan‑Boltzmann law:

\$L = 4\pi R^{2}\sigma T_{\text{eff}}^{4}\$

Re‑arranged for \$R\$:

\$R = \sqrt{\frac{L}{4\pi\sigma T_{\text{eff}}^{4}}}\$

where \$\sigma = 5.67\times10^{-8}\ \text{W m}^{-2}\,\text{K}^{-4}\$ and \$T_{\text{eff}}\$ is the effective temperature obtained from the star’s spectral type.

5. Worked Example

1. Measure \$ \lambda_{\text{obs}} \$ for the Hα line: \$658.00\ \text{nm}\$.
2. Calculate \$ \Delta\lambda = 658.00 - 656.28 = 1.72\ \text{nm}\$.
3. Find \$ v{\text{r}} = c\,\frac{\Delta\lambda}{\lambda{0}} = 3.00\times10^{5}\,\frac{1.72}{656.28} \approx 785\ \text{km s}^{-1}\$.
4. Assuming the star is in a galaxy obeying Hubble’s law, \$ d = 785/70 \approx 11.2\ \text{Mpc}\$.
5. From photometry, the absolute luminosity is \$L = 2.5\times10^{28}\ \text{W}\$.
6. Spectral type indicates \$T_{\text{eff}} = 8\,000\ \text{K}\$.
7. Compute radius:

   \$\$R = \sqrt{\frac{2.5\times10^{28}}{4\pi(5.67\times10^{-8})(8\,000)^{4}}}

   \approx 1.2\times10^{9}\ \text{m} \approx 1.7\,R_{\odot}\$\$

6. Key Points to Remember

• Red‑shift means \$ \lambda{\text{obs}} > \lambda{0} \$.
• The fractional shift directly gives the radial velocity via \$v{\text{r}} = c\,\Delta\lambda/\lambda{0}\$.
• Knowing \$v_{\text{r}}\$ can lead to a distance estimate (Hubble’s law) for extragalactic objects.
• Stellar radius follows from the Stefan‑Boltzmann law once \$L\$ and \$T_{\text{eff}}\$ are known.
• Accurate spectroscopy and careful calibration of laboratory wavelengths are essential for reliable results.