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

Stellar Radii & Redshift Basics

Hey future astrophysicists! 🌠 In this lesson we’ll explore how the colour shift of light from stars (redshift) tells us about their motion and even gives clues about their size. We’ll use the handy formulas:

• \$\displaystyle \frac{\Delta \lambda}{\lambda}\$ – change in wavelength over original wavelength
• \$\displaystyle \frac{\Delta f}{f}\$ – change in frequency over original frequency
• \$\displaystyle \frac{v}{c}\$ – velocity of the star divided by the speed of light

Let’s dive in!

When a star moves relative to us, its light waves stretch or squeeze, just like the sound of a passing ambulance.

Redshift (moving away): Wavelengths get longer → light looks redder.

Blueshift (moving toward): Wavelengths get shorter → light looks bluer.

Mathematically:

\$ \displaystyle \frac{\Delta \lambda}{\lambda} = \frac{v}{c} \$

or equivalently for frequency:

\$ \displaystyle \frac{\Delta f}{f} = -\frac{v}{c} \$

Notice the minus sign for frequency: when the star moves away, frequency drops.

1. Measure the shift in a known spectral line. For example, the hydrogen alpha line normally at \$656.3\,\text{nm}\$ is observed at \$660.0\,\text{nm}\$.
2. Compute \$\displaystyle \frac{\Delta \lambda}{\lambda} = \frac{660.0 - 656.3}{656.3} \approx 0.0056\$.
3. Use \$v = c \times \frac{\Delta \lambda}{\lambda}\$ with \$c = 3.0 \times 10^8\,\text{m/s}\$.
4. Result: \$v \approx 3.0 \times 10^8 \times 0.0056 \approx 1.7 \times 10^6\,\text{m/s}\$ (about 1,700 km/s).

That’s the star’s radial velocity away from us!

While redshift gives us speed, we can combine it with other data (like luminosity and temperature) to estimate a star’s radius.

One simple relation (for main‑sequence stars) is:

\$ \displaystyle R \approx \sqrt{\frac{L}{4\pi \sigma T^4}} \$

Where:

• \$L\$ = luminosity
• \$T\$ = surface temperature
• \$\sigma\$ = Stefan–Boltzmann constant

So, redshift tells us how fast the star is moving, and with other observations we can piece together its size.

Suppose the sodium D line (normally \$589.0\,\text{nm}\$) is seen at \$587.0\,\text{nm}\$.

\$\displaystyle \frac{\Delta \lambda}{\lambda} = \frac{587.0-589.0}{589.0} \approx -0.00339\$.

Velocity:

\$ v = 3.0 \times 10^8 \times (-0.00339) \approx -1.02 \times 10^6\,\text{m/s}\$.

Negative sign → star moving toward us at ~1,020 km/s.

Remember:

• Use \$v = c \times \frac{\Delta \lambda}{\lambda}\$ for redshift (positive \$\Delta \lambda\$).
• For blueshift, \$\Delta \lambda\$ is negative, giving a negative \$v\$ (toward).
• Always check units: convert nm to m if you want m/s.
• When asked for “speed of the star relative to Earth”, use the magnitude of \$v\$.
• In multiple‑choice, look for the answer that matches the sign convention.

Good luck, and keep looking up! 🚀