ASTR 310 Problem Set 02: Optical Depth Attenuation and the Boltzmann Equation for Vega

Problem set for Unit 2.

Problems

1. Transmitted fraction 15 points

A stellar atmosphere layer has optical depth τ = 4.5 at a particular wavelength. Compute the transmitted intensity fraction I(τ)/I(0), and state what fraction of the incident intensity is absorbed or scattered away.

2. Matching optical depth to transmitted fraction 15 points

A different wavelength shows a transmitted fraction of exactly 10% through the same physical layer. Solve for the optical depth τ at that wavelength (invert the exponential relationship), and state whether this wavelength has higher or lower opacity than the τ = 4.5 wavelength in Problem 1.

3. Boltzmann ratio for Vega 20 points

Vega has T_eff = 9602 K (a standard published value). Using the hydrogen n=1→n=2 Boltzmann ratio (ΔE = 10.2 eV = 1.634 × 10⁻¹⁎ J, g2/g1 = 4), compute Vega's n2/n1 ratio and compare it (as a ratio, not just qualitatively) to the Sun's value (5778 K) from Lecture 04/Lab 02.

4. Physical interpretation 15 points

Vega is spectral type A0V and shows much stronger Balmer absorption lines than the Sun (G2V). Using your Problem 3 ratio, explain quantitatively why this is consistent with the Boltzmann-equation prediction, and state one piece of additional physics (not computed in this problem set) that would be needed to predict Balmer-line strength correctly for a star hotter than about 10,000 K.

Due and Scope

Submit a complete derivation with equations, units, labeled quantities, and a short narrative interpretation for each problem. Show every step; a correct final number without a visible derivation receives partial credit at most, and quoting a lecture number without adapting it to this problem's specific inputs receives no credit for that step.

OpenStax Companion Reading

OpenStax Astronomy 2e (local reference copy: references/openstax-astronomy-2e-extracted.txt), Chapter 5.5 and Chapter 17.3.

References and Data Sources