Spectra & Composition

Useful constants: $h = 6.626 \times 10^{-27}\ \mathrm{erg\,s}$; $c = 2.998 \times 10^{10}\ \mathrm{cm/s}$ ($= 3.0 \times 10^5\ \mathrm{km/s}$); $k_B = 8.617 \times 10^{-5}\ \mathrm{eV/K}$; $1\ \mathrm{eV} = 1.602 \times 10^{-12}\ \mathrm{erg}$; $b = 2.898 \times 10^6\ \mathrm{nm\,K}$; $\sigma = 5.67 \times 10^{-5}\ \mathrm{erg\,cm^{-2}\,s^{-1}\,K^{-4}}$; $L_\odot = 3.828 \times 10^{33}\ \mathrm{erg/s}$; $1\ \mathrm{AU} = 1.496 \times 10^{13}\ \mathrm{cm}$. Recommended shortcut: $hc \approx 1240\ \mathrm{eV\,nm}$.

Conceptual

1. ⭐ Kirchhoff’s Laws Applied. You observe three astronomical sources:

• Source A: smooth rainbow of colors, brightest in the infrared
• Source B: bright colored lines on a dark background
• Source C: rainbow with dark lines, brightest in the blue-green

For each, (a) identify which Kirchhoff law applies, (b) describe the physical setup, and (c) give an astronomical example.

2. ⭐ The OBAFGKM Misconception. A student says: “O stars show strong helium lines because they have more helium than cooler stars.” Explain why this is wrong. What actually controls the strength of helium lines?

3. ⭐⭐ Doppler Direction. Two stars are observed with Hα lines at:

• Star X: $\lambda_{\text{obs}} = 656.8\ \text{nm}$
• Star Y: $\lambda_{\text{obs}} = 655.8\ \text{nm}$

(a) Which star is approaching and which is receding? (b) Without calculating, which star has the larger radial velocity magnitude? (c) Verify your answer to (b) by computing both velocities.

Calculation

4. ⭐ Hβ from the Bohr Model. Calculate the wavelength of the Hβ line ($n = 4 \to 2$). Show all steps: energy levels, energy difference, unit conversion to ergs, and application of $\lambda = hc/\Delta E$. Compare to the observed value of 486.1 nm.

5. ⭐⭐ Doppler Velocity. A star’s sodium D line (rest wavelength $\lambda_0 = 589.0\ \text{nm}$) is observed at $\lambda_{\text{obs}} = 589.4\ \text{nm}$. (a) Is the star approaching or receding? (b) Calculate the radial velocity in km/s. (c) Express the fractional shift $\Delta\lambda/\lambda_0$ as a percentage — how does it compare to $v_r/c$?

6. ⭐⭐ Equilibrium Temperature. Calculate the equilibrium temperature of a planet at $d = 1.5\ \text{AU}$ from a Sun-like star, with albedo $A = 0.25$. (a) Use the ratio method $T_{\text{eq}} = 279\ \text{K} \times (1-A)^{1/4}(d/1\,\text{AU})^{-1/2}$. (b) Compare to Mars’s actual surface temperature ($\sim 210\ \text{K}$). (c) What does the comparison tell you about Mars’s greenhouse effect?

Synthesis

7. ⭐⭐ Multi-Clue Stellar Diagnosis. You observe a star with:

• Strong H Balmer absorption lines (Hα, Hβ, Hγ all prominent)
• Weak metal lines
• Hα observed at 656.6 nm
• Continuum peak near 350 nm

(a) Estimate the effective temperature from the continuum peak using Wien’s law. (b) What spectral type is this star? Justify using the Balmer line strength. (c) Calculate the radial velocity from the Hα shift. (d) Is this a dwarf or a giant? Can you tell from the information given?

8. ⭐⭐ Stellar vs. Planetary Spectroscopy. Explain how Kirchhoff’s third law applies to both: (a) a G-type star’s spectrum showing Ca II absorption lines; (b) Earth’s atmosphere absorbing outgoing infrared radiation at CO₂’s 15 μm band. In each case, identify the “hot continuum source” and the “cooler absorbing gas.”

9. ⭐⭐⭐ Venus’s Runaway Greenhouse. Venus has $d = 0.72\ \text{AU}$, albedo $A = 0.77$ (highly reflective clouds), and an atmosphere that is 96.5% CO₂ at 92 bar surface pressure. (a) Calculate Venus’s equilibrium temperature using the ratio method. (b) Compare to the actual surface temperature of 735 K. (c) Calculate the greenhouse warming $\Delta T = T_{\text{surface}} - T_{\text{eq}}$. (d) The greenhouse warming on Venus is about 15× larger than on Earth. Using the molecular-absorption concept from Part 6, explain qualitatively why a thicker CO₂ atmosphere produces a much larger greenhouse effect. (e) Could Earth experience a runaway greenhouse? What would need to happen?