17.4 Using Spectra to Measure Stellar Radius, Composition, and Motion

Stellar Spectra and Properties

A star's spectrum is essentially its fingerprint. By spreading starlight into its component wavelengths, astronomers can figure out what a star is made of, how hot it is, how big it is, and even how fast it's moving. All of that from light alone.

Analysis of Stellar Spectra

Composition from absorption lines: When light passes through a star's outer atmosphere, elements there absorb specific wavelengths. This creates dark lines in the spectrum called absorption lines. Each element leaves its own unique pattern. Hydrogen produces one set of lines, helium another, calcium another, and so on. The stronger an absorption line appears, the more abundant that element is in the star's atmosphere.

Temperature from the spectrum's shape: The overall shape of a star's continuous spectrum depends on its surface temperature.

• Hotter stars peak at shorter (bluer) wavelengths. Sirius and Vega, for example, appear blue-white.
• Cooler stars peak at longer (redder) wavelengths. Betelgeuse and Antares appear distinctly reddish.

This inverse relationship between temperature and peak wavelength is Wien's displacement law: the hotter the star, the shorter the peak wavelength.

Radius from the Stefan-Boltzmann law: Once you know a star's luminosity and temperature, you can solve for its radius using:

$L = 4\pi R^2 \sigma T^4$

• $L$ = luminosity (total energy output), found from the star's apparent brightness and distance
• $T$ = surface temperature, determined from its spectral type (the OBAFGKM classification)
• $R$ = radius (what you're solving for)
• $\sigma$ = the Stefan-Boltzmann constant, $5.67 \times 10^{-8}$ W m$^{-2}$ K$^{-4}$

The key idea: at the same temperature, a larger star has a greater surface area and therefore higher luminosity. So if two stars have the same temperature but one is far more luminous, that star must be physically larger.

Electromagnetic Spectrum and Blackbody Radiation

Stars emit radiation across the entire electromagnetic spectrum, from radio waves to gamma rays. The intensity distribution across wavelengths closely matches that of a blackbody, which is an idealized object that perfectly absorbs and emits all radiation. Real stars aren't perfect blackbodies (those absorption lines are proof), but the match is close enough to be extremely useful.

Wien's displacement law applies directly to this blackbody curve: the peak wavelength shifts to shorter wavelengths as temperature increases.

One additional detail worth knowing: absorption lines aren't perfectly sharp. They get broadened by factors like the star's rotation speed and the temperature of its atmosphere. Faster rotation and higher temperatures both produce wider lines. This broadening carries useful information, as you'll see below.

Doppler Effect in Stellar Measurements

The Doppler effect is the change in wavelength that occurs when a light source moves relative to an observer. You've heard the analogy with sound: a car horn sounds higher-pitched as it approaches and lower-pitched as it moves away. Light works the same way.

• A star moving toward Earth has its light shifted to shorter wavelengths: a blueshift.
• A star moving away from Earth has its light shifted to longer wavelengths: a redshift.

Measuring radial velocity: By comparing the observed position of a star's spectral lines to their known rest wavelengths, astronomers calculate the star's radial velocity (its speed toward or away from us). The formula is:

$\Delta \lambda / \lambda = v_r / c$

where $\Delta \lambda$ is the wavelength shift, $\lambda$ is the rest wavelength, $v_r$ is the radial velocity, and $c$ is the speed of light. A larger shift means a faster radial velocity.

Measuring rotation: The Doppler effect also reveals how fast a star spins. One limb of a rotating star moves toward us (blueshifted) while the opposite limb moves away (redshifted). The net result is that spectral lines appear broadened rather than shifted. Faster rotation produces broader lines.

Stellar Motion

Stars aren't fixed in space. They move, and astronomers break that motion into components to measure it.

Proper Motion and Space Velocity

Proper motion is a star's apparent angular movement across the sky, perpendicular to our line of sight. It's measured in arcseconds per year. Most stars have tiny proper motions, but nearby fast-moving stars stand out. Barnard's Star has the largest known proper motion at 10.3 arcsec/year, while Proxima Centauri moves at 3.85 arcsec/year.

Proper motion reflects a star's transverse velocity, its actual speed perpendicular to our line of sight. To convert proper motion into a real velocity, you need the star's distance:

$v_t = 4.74 \mu d$

where $v_t$ is the transverse velocity in km/s, $\mu$ is the proper motion in arcseconds per year, and $d$ is the distance in parsecs.

Space velocity is the star's true total velocity through space. It combines two perpendicular components:

• Radial velocity ($v_r$): toward or away from us, measured via the Doppler shift
• Transverse velocity ($v_t$): across the sky, calculated from proper motion and distance

Since these components are perpendicular, you combine them with the Pythagorean theorem:

$v = \sqrt{v_r^2 + v_t^2}$

By convention, radial velocity is positive when a star is receding and negative when approaching.