Light as Information

Useful constants: $c = 2.998 \times 10^{10}~\mathrm{cm/s}$ ; $h = 6.626 \times 10^{-27}~\mathrm{erg\,s}$ ; $k_B = 1.38 \times 10^{-16}~\mathrm{erg/K}$ ; $b = 2.898 \times 10^{6}~\mathrm{nm\,K} = 0.2898~\mathrm{cm\,K}$ ; $1~\mathrm{nm} = 10^{-7}~\mathrm{cm}$ .

Conceptual

1. ⭐⭐ Kirchhoff’s laws in action. You see three spectra: (a) a smooth rainbow with dark lines; (b) a smooth rainbow with no lines; (c) a dark background with bright lines. For each, identify the physical source (hot dense object, hot thin gas, or cool gas in front of a hot source), state which of Kirchhoff’s three laws applies, and what geometry is required.

2. ⭐⭐ Blackbody or not? Source A has a smooth spectrum with a single broad peak; Source B has sharp spikes at specific wavelengths. Which is better described as a blackbody? What emission mechanism does the non-blackbody spectrum suggest (thermal continuum vs. atomic transitions)?

3. ⭐⭐ Why skies are red at sunset. Use Rayleigh scattering to explain why the sky is blue in daytime and the Sun looks red near the horizon. Distinguish what you see looking away from the Sun (scattered skylight) versus at the Sun near the horizon (transmitted beam). How does the atmospheric path length change, and how does $\lambda^{-4}$ make the effect strong?

4. ⭐ Two telescope goals. To (i) detect the faintest objects and (ii) resolve the smallest angular details, which telescope property matters most for each? Justify using the $D^2$ and $\lambda/D$ scalings.

5. ⭐⭐ Peak is not only. A star peaks at 250 nm. In 2-4 sentences, explain why it can still appear blue-white to human eyes.

Calculation

6. ⭐⭐ Wave relation. A radio wave has $\lambda = 3$ cm.

(a) What is its frequency $\nu$ ?
(b) Visible light has $\lambda = 500$ nm. How many times larger is $\nu_{\text{vis}}$ than $\nu_{\text{radio}}$ ?

7. ⭐⭐ Photon energy from wavelength. A green photon has $\lambda = 600$ nm.

(a) Convert this wavelength to cm.
(b) Compute its frequency.
(c) Compute its energy in erg using $E = hc/\lambda$ . Give answers to 2 significant figures.

8. ⭐⭐ Wien’s law temperature. A star’s spectrum peaks at $\lambda_{\text{peak}} = 700$ nm. Estimate the temperature, then compare to the Sun ( $\sim 5800$ K): hotter or cooler, and by roughly what factor?

9. ⭐⭐ Rayleigh scattering ratio. Compute $\sigma_{450}/\sigma_{900}$ using $\sigma \propto \lambda^{-4}$ . Which wavelength is scattered more, and by what factor?

10. ⭐⭐ Telescope scaling. Two telescopes observe at the same wavelength — Telescope A: $D = 2$ m; Telescope B: $D = 8$ m.

(a) Compute $A_B/A_A$ .
(b) Compute $\theta_B/\theta_A$ (diffraction-limited).
(c) Which is better for faint objects, and which gives sharper images?

11. ⭐⭐ Doppler sign and speed. The H $\alpha$ line ( $\lambda_0 = 656.3$ nm) is observed at 656.6 nm. Is the source approaching or receding? Estimate $v_r$ with the non-relativistic Doppler relation.

12. ⭐⭐ JWST versus Hubble.

(a) Compute the collecting-area ratio using $A \propto D^2$ with $D_{\text{JWST}} = 6.5$ m and $D_{\text{HST}} = 2.4$ m.
(b) Compute the angular-resolution ratio at the same wavelength using $\theta \propto \lambda D^{-1}$ .
(c) Interpret each ratio in one sentence.

Synthesis

13. ⭐⭐ Temperature plus absorption. A star’s spectrum peaks near 1.0 μm and shows deep absorption lines.

(a) Use Wien’s law to estimate the temperature.
(b) Explain what deep absorption lines imply about the atmosphere and viewing geometry (Kirchhoff’s laws).
(c) What additional information do line positions provide?

14. ⭐⭐⭐ Short-wavelength suppression. Use the Planck function $B_\lambda(T) = \frac{2hc^2}{\lambda^5}\frac{1}{e^{hc/\lambda k_B T}-1}$ to explain why intensity is suppressed at very short wavelengths.

(a) In the short-wavelength limit ( $\lambda \to 0$ ), what happens to the prefactor $\lambda^{-5}$ ?
(b) In that same limit, what happens to the exponent $x = hc/(\lambda k_B T)$ and therefore to $e^x - 1$ ?
(c) Which effect dominates the product, and why does that prevent the ultraviolet catastrophe?
(d) Connect your conclusion to the statement “photons are expensive when $h\nu \gg k_B T$ .”