Quick questions on Spectroscopy: emission, absorption and stellar spectra, HSC Physics Module 7

Electrons in atoms occupy discrete energy levels $E_1, E_2, E_3, \dots$ When an electron drops from a higher level $E_i$ to a lower level $E_f$, a photon of energy:

Continuous spectrum. A hot dense object (a glowing solid, liquid or high-pressure gas, or the interior of a star) emits a smooth distribution of wavelengths. The peak wavelength shifts with temperature (Wien's law); the total intensity follows the Stefan-Boltzmann law. The spectrum approximates a blackbody curve.

In practice, spectra are recorded by sending starlight through a slit, collimating it, dispersing it with a prism or diffraction grating, and imaging the result onto a CCD. A diffraction grating with line spacing $d$ produces principal maxima at:

A hot dense object (a glowing solid, liquid or high-pressure gas, or the interior of a star) emits a smooth distribution of wavelengths. The peak wavelength shifts with temperature (Wien's law); the total intensity follows the Stefan-Boltzmann law. The spectrum approximates a blackbody curve.

A hot, low-density gas (a discharge lamp, the corona of a star, a nebula) emits only at specific wavelengths corresponding to its atoms' allowed downward transitions. The spectrum looks like bright lines on a dark background.

Continuum light passing through a cool gas loses the photons whose energies match the gas atoms' allowed upward transitions. The result is a continuous spectrum crossed by dark lines (Fraunhofer lines). Stellar spectra are predominantly of this type: the photosphere produces near-continuum light that is absorbed by the cooler outer atmosphere.

Identify the absorption lines by wavelength and match to laboratory spectra. The most prominent lines in a Sun-like star are hydrogen Balmer lines (H-alpha at $656.3$ nm), neutral sodium, ionised calcium, magnesium and iron lines. Helium was discovered in 1868 from a solar absorption line that did not match any known terrestrial element.

The relative strengths of different lines depend on temperature, because each transition has an optimal temperature for being populated. The shape of the underlying continuum (Wien's law: $\lambda\_{\text{peak}} T =$ constant) gives an independent temperature estimate. Together these classify stars into the spectral sequence O, B, A, F, G, K, M, from hottest (blue-white) to coolest (red).

All lines are shifted from their laboratory wavelengths by the Doppler effect:

A rotating star has one limb moving toward us and the other away, so each spectral line is broadened symmetrically into a profile whose width measures the equatorial rotation speed.

Distinguish between emission and absorption line spectra and state one source of each. [2 marks]

A spectral line of rest wavelength $656.3 \text{ nm}$ (H-alpha) is observed at $\lambda = 658.5 \text{ nm}$ in a distant galaxy. Calculate the recession velocity. [3 marks]

A star's spectrum shows strong hydrogen Balmer lines and weak helium lines. (a) Use Wien's law $\lambda_{\max} T = 2.898 \times 10^{-3} \text{ m K}$ to estimate temperature if $\lambda_{\max} = 480 \text{ nm}$. (b) Classify the star's spectral type.