Spectra & Composition

Section 8 of 8

Assumptions and Synthesis

Part 7: Assumptions and Limitations

Spectral classification assumes a single star. Unresolved binaries blend two spectra, complicating classification. Spectroscopic binaries are identified by two sets of absorption lines (double-lined) or periodic velocity oscillations (single-lined) — Lecture 4’s territory.

The Bohr model is approximate. It works beautifully for hydrogen but breaks down for multi-electron atoms, which need full quantum mechanics. The concept — discrete levels producing discrete lines — remains exactly right.

Line identification requires lab data. We match stellar lines to laboratory-measured wavelengths. If an element’s spectrum hasn’t been measured in the lab, we can’t identify it in a star. Historically, helium was found in the Sun’s spectrum before it was found on Earth — its lines matched no known element, so it was named for the Greek sun god (Helios).

Interstellar absorption adds extra lines. Gas and dust between us and a star imprint additional features (interstellar sodium D lines, diffuse interstellar bands). These are separated from the star’s own lines by checking whether they share the star’s Doppler shift (interstellar lines have their own, different velocity).

Part 7 takeaway: spectroscopy is robust but not foolproof. Binaries blend spectra, interstellar gas adds false lines, and precise composition requires careful modeling. These caveats don’t undermine the method — they refine it.

Summary: Observable → Model → Inference

We observe	We use	We infer
Dark lines at specific wavelengths	Atomic physics: each element has unique energy levels (Bohr model)	Composition — which atoms are present
Relative strengths of different species’ lines	Boltzmann distribution: temperature controls level populations	Temperature and spectral type (OBAFGKM)
Lines shifted from lab wavelengths	Doppler formula $\Delta\lambda/\lambda_0 = v_r/c$	Radial velocity — motion toward/away
Width and shape of lines	Pressure broadening, rotation, turbulence	Atmospheric density, rotation, magnetic fields
Molecular bands in planetary spectra	Quantum mechanics of molecular vibrations	Atmospheric composition and greenhouse effect

Self-Assessment Checklist

After working through this reading, you should be able to:

☐ Sketch the three types of spectra and explain when each appears (Kirchhoff’s laws)
☐ Explain why stars show absorption spectra using the two-layer model (photosphere + atmosphere)
☐ Use the Bohr model to explain why spectral lines occur at specific wavelengths
☐ Calculate the wavelength of a hydrogen transition from energy levels
☐ Recite the OBAFGKM sequence and explain it as a temperature sequence
☐ Explain why H Balmer lines peak in strength at A-type temperatures
☐ Apply the Doppler formula to compute radial velocity from a line shift
☐ Distinguish spectral type, luminosity class, and metallicity
☐ Explain the greenhouse effect using Kirchhoff’s laws and molecular absorption bands
☐ Calculate a planet’s equilibrium temperature using Stefan-Boltzmann

Key Equations Reference

Equation	Use	Notes
$E_n = -13.6\ \text{eV}/n^2$	Hydrogen energy levels (Bohr model)	$n = 1, 2, 3, \ldots$ ; negative = bound
$E = h\nu = hc\,\lambda^{-1}$	Photon energy from wavelength	Connects quantum levels to spectral lines
$\lambda = hc/\Delta E$	Wavelength of a spectral line	$\Delta E$ = energy gap between levels
$\Delta\lambda/\lambda_0 = v_r/c$	Doppler shift (non-relativistic)	$v_r \gt 0$ receding (redshift); $v_r \lt 0$ approaching (blueshift)
fraction in level $n \propto e^{-\chi_n/k_BT}$	Boltzmann distribution (qualitative)	$\chi_n$ = excitation energy above ground state; explains spectral type = temperature
$T_{\text{eq}} = (L_\star(1-A)/16\pi\sigma d^2)^{1/4}$	Planetary equilibrium temperature	No greenhouse; bare energy balance
$T_{\text{eq}} \approx 279\ \text{K}\,(1-A)^{1/4}(d/\text{AU})^{-1/2}$	Solar-system shortcut	For planets orbiting the Sun

Key Constants

Constant	Value	Units
Planck’s constant	$h = 6.626 \times 10^{-27}$	erg·s
Speed of light	$c = 2.998 \times 10^{10}$	cm/s ( $= 3.0 \times 10^5$ km/s)
Boltzmann’s constant	$k_B = 8.617 \times 10^{-5}$	eV/K
Stefan-Boltzmann constant	$\sigma = 5.67 \times 10^{-5}$	erg cm⁻² s⁻¹ K⁻⁴
Wien’s constant	$b = 2.898 \times 10^6$	nm·K
Hydrogen ionization energy	13.6	eV
$1\ \text{eV}$	$1.602 \times 10^{-12}$	erg
Solar luminosity	$L_\odot = 3.828 \times 10^{33}$	erg/s
$1\ \text{AU}$	$1.496 \times 10^{13}$	cm

Glossary

Absorption line: A dark feature in a spectrum where a cool gas absorbs photons from a hotter background source at a specific wavelength set by an atomic energy-level gap. It appears at the same wavelength the gas would emit if heated.
Albedo: The fraction $A$ of incident light a surface or planet reflects rather than absorbs. $A = 0$ is perfect absorption; $A = 1$ is a perfect mirror. The Moon has $A \approx 0.12$ , Earth $\approx 0.30$ , Venus $\approx 0.77$ . Only the absorbed fraction $(1-A)$ heats the planet.
Balmer series: Hydrogen spectral lines from transitions to or from the $n = 2$ level. The visible members are Hα (656 nm), Hβ (486 nm), Hγ (434 nm), and Hδ (410 nm) — the most prominent features in many stellar spectra.
Blueshift: A shift of spectral lines to shorter wavelengths ( $\Delta\lambda \lt 0$ ), indicating the source is approaching the observer.
Bohr model: A quantum model of hydrogen in which the electron occupies discrete energy levels $E_n = -13.6\ \text{eV}/n^2$ and absorbs or emits a photon only when it transitions between levels. Approximate for multi-electron atoms, but the core idea — discrete levels yield discrete lines — is exactly right.
Boltzmann distribution: The rule that the fraction of atoms in an excited level $n$ scales as $e^{-\chi_n / k_B T}$ , where $\chi_n$ is the excitation energy above the ground state. Its exponential sensitivity to temperature is why spectral type is a temperature sequence: temperature, not abundance, sets which levels are populated and therefore which lines appear.
Doppler shift: The change in observed wavelength caused by a source’s motion along the line of sight: $\Delta\lambda/\lambda_0 = v_r/c$ . Receding sources shift to longer wavelengths (redshift); approaching sources shift to shorter wavelengths (blueshift).
Equilibrium temperature: The temperature at which a planet radiates exactly as much energy as it absorbs from its star, assuming no atmosphere: $T_{\text{eq}} = (L_\star(1-A)/16\pi\sigma d^2)^{1/4}$ . The greenhouse effect raises the real surface temperature above this baseline.
Greenhouse effect: Warming of a planet’s surface caused by atmospheric gases absorbing outgoing infrared radiation and re-emitting part of it back downward. It is Kirchhoff’s third law applied to a planet: a cool atmosphere absorbing from the warm surface continuum.
Kirchhoff's laws: Three rules linking the appearance of a spectrum to the physical conditions of its source: (1) a hot dense object yields a continuous spectrum; (2) a hot low-density gas yields bright emission lines; (3) a cool gas in front of a hotter continuum yields dark absorption lines at the same wavelengths it would emit. Worked backward, a spectrum’s appearance reveals the source’s configuration.
Metallicity: The mass fraction $Z$ of elements heavier than helium in a star or gas cloud ( $Z_\odot \approx 0.014$ ). It modulates the strength of metal absorption lines but, at fixed temperature, does not change a star’s spectral type — a second-order effect.
Photosphere: The visible “surface” of a star — the depth at which it becomes opaque (optical depth $\tau \approx 1$ ) and from which the continuous spectrum escapes. For the Sun it sits near 5,800 K; the slightly cooler atmosphere just above it carves the absorption lines.
Radial velocity: The component of a star’s velocity along the observer’s line of sight, measured from the Doppler shift of its spectral lines. The perpendicular (transverse) motion produces no shift and must be measured astrometrically.
Redshift: A shift of spectral lines to longer wavelengths ( $\Delta\lambda \gt 0$ ), indicating the source is receding from the observer.
Spectrum: The intensity of light as a function of wavelength, obtained by dispersing light through a prism or diffraction grating. A spectrum is quantitative data: its continuum slope, emission peaks, and absorption lines each encode physical conditions of the source.