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

Stars are cosmic storytellers, revealing their secrets through light. By analyzing stellar spectra, astronomers decode information about a star's temperature, composition, and motion. This detective work allows us to understand the properties and behavior of stars across the universe.

Stellar motion adds another layer to the cosmic narrative. By measuring proper motion and radial velocity, we can track how stars move through space. This information helps us understand stellar dynamics and the structure of our galaxy.

Stellar Spectra and Properties

Analysis of stellar spectra

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• Stellar spectra contain absorption lines caused by elements in the star's atmosphere absorbing specific wavelengths of light
  • Each element produces a unique pattern of absorption lines (hydrogen, helium, calcium)
  • The strength of the lines depends on the abundance of the element more abundant elements create stronger lines
• The overall shape of the spectrum depends on the star's temperature
  • Hotter stars have peak intensity at shorter wavelengths appear bluer (Sirius, Vega)
  • Cooler stars have peak intensity at longer wavelengths appear redder (Betelgeuse, Antares)
  • This relationship between temperature and peak wavelength is described by Wien's displacement law
• The star's radius can be determined using the Stefan-Boltzmann law: $L = 4\pi R^2 \sigma T^4$
  • $L$ is the star's luminosity, determined from its apparent brightness and distance measured using parallax or spectroscopic parallax methods
  • $T$ is the star's surface temperature, determined from its spectral type by comparing its spectrum to standard stars (OBAFGKM)
  • $\sigma$ is the Stefan-Boltzmann constant $5.67 \times 10^{-8}$ W m$^{-2}$ K$^{-4}$
  • Solving for $R$ gives the star's radius larger radius means more luminous at same temperature

Electromagnetic Spectrum and Blackbody Radiation

• Stars emit radiation across the electromagnetic spectrum, from radio waves to gamma rays
• The intensity distribution of this radiation closely follows that of a blackbody
• Blackbody radiation is the thermal radiation emitted by an ideal absorber and emitter of electromagnetic radiation
• The peak wavelength of a star's blackbody radiation curve is inversely proportional to its temperature, as described by Wien's displacement law
• Spectral line broadening occurs due to various factors, including the star's rotation and temperature, affecting the width of absorption lines in the spectrum

Doppler effect in stellar measurements

• The Doppler effect causes the wavelength of light to change if the source is moving relative to the observer
  • Approaching motion causes a blueshift shorter wavelengths (moving car horn sounds higher pitched)
  • Receding motion causes a redshift longer wavelengths (moving car horn sounds lower pitched)
• A star's radial velocity (motion towards or away from Earth) can be measured from the Doppler shift of its spectral lines
  • The shift is proportional to the star's radial velocity larger shift means faster velocity
  • 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 star's rotation can also be measured from the Doppler shift of its spectral lines
  • The side of the star rotating towards us is blueshifted, while the side rotating away is redshifted
  • This causes the spectral lines to broaden, with the amount of broadening depending on the star's rotation speed faster rotation means broader lines

Stellar Motion

Proper motion and space velocity

• Proper motion is a star's angular motion across the sky, perpendicular to our line of sight
  • It is measured in arcseconds per year (Barnard's star 10.3 arcsec/year, Proxima Centauri 3.85 arcsec/year)
  • Proper motion is caused by the star's transverse velocity motion perpendicular to our line of sight
• A star's space velocity is its total velocity through space, including both radial and transverse components
  • The transverse velocity can be calculated from the proper motion and distance: $v_t = 4.74 \mu d$, where $v_t$ is in km/s, $\mu$ is the proper motion in arcseconds per year, and $d$ is the distance in parsecs
  • The radial velocity is measured from the Doppler shift of the star's spectral lines positive for receding, negative for approaching
  • The space velocity is the vector sum of the transverse and radial velocities calculated using Pythagorean theorem $v = \sqrt{v_r^2 + v_t^2}$