1. Transmitted fraction
I(τ)/I(0) = e⁻⁵·⁵ = 0.0111, so about 1.1% of the incident intensity is transmitted and about 98.9% is absorbed or scattered away.
| Criterion | Points |
|---|---|
| Correct exponential arithmetic | 8 |
| Correct percent-absorbed statement | 7 |
Worked Solutions
2. Inverted optical depth
0.10 = e⁻τ ⇒ τ = −ln(0.10) = 2.303. Since 2.30 > 4.5, this wavelength has higher opacity (more absorbing/scattering material along the line of sight) than the τ = 4.5 wavelength in Problem 1.
| Criterion | Points |
|---|---|
| Correct inversion of the exponential relationship | 8 |
| Correct higher/lower opacity comparison | 7 |
3. Boltzmann ratio for Vega
n2/n1 (Vega, 9602 K) = 1.780e-05; n2/n1 (Sun, 5778 K) = 5.113e-09. Ratio = 3481.6, i.e., Vega's n=2 population fraction is about 3482 times the Sun's.
| Criterion | Points |
|---|---|
| Correct Boltzmann-ratio arithmetic for Vega | 12 |
| Correct computed ratio relative to the Sun | 8 |
4. Physical interpretation
The much larger n2/n1 ratio at Vega's temperature directly predicts much stronger Balmer absorption, consistent with Vega's observed strong Balmer lines. Correctly predicting the further decline in Balmer strength above roughly 10,000 K requires the Saha equation (Lecture 04), which accounts for hydrogen ionization removing the bound electron needed for a Balmer transition.
| Criterion | Points |
|---|---|
| Correct quantitative connection to the computed ratio | 8 |
| Correct identification of the Saha equation as the missing physics | 7 |
Grading Notes
Award full marks for correct method, units, and a numeric result consistent with the arithmetic shown (allow reasonable rounding). Verify that the student re-derives the number for this problem's specific inputs rather than quoting a lecture example without adaptation. Watch specifically for unit-conversion errors (AU/m, years/seconds, km/Mpc, Msun/kg) -- these are the most common source of order-of-magnitude mistakes in this course.