1. Strehl ratios
S_K = exp[-(2π×100/2200)²] = 0.922; S_J = exp[-(2π×100/1250)²] = 0.777.
| Criterion | Points |
|---|---|
| Correct K-band arithmetic | 10 |
| Correct J-band arithmetic | 10 |
Worked Solutions
2. Comparison
Improvement factor = 0.922/0.812 = 1.14× higher Strehl ratio at K band with the upgraded system.
| Criterion | Points |
|---|---|
| Correct ratio arithmetic | 8 |
| Correct interpretation | 7 |
3. Wavefront-error budget
If three equal independent terms combine as √(3σ_i²) to go from 160 nm to 100 nm in quadrature difference, then σ_i = √[(160²-100²)/3] ≈ 72 nm per independent upgrade (an illustrative decomposition, not a claim about any specific real upgrade's exact contribution).
| Criterion | Points |
|---|---|
| Correct RSS-difference algebra | 10 |
| Correct final per-term value | 5 |
4. Science case justification
High-contrast J-band imaging benefits strongly from the proposed upgrade because Strehl ratio rises steeply with decreasing wavefront error at shorter wavelengths (S_J=0.777 versus a much lower current-system J-band Strehl); a higher Strehl ratio concentrates more light into the diffraction-limited core rather than a broad halo, directly improving the contrast needed to detect faint companions close to a bright host star.
| Criterion | Points |
|---|---|
| Correct quantitative grounding in the computed Strehl values | 12 |
| Correct physical connection to high-contrast imaging | 8 |
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 (arcsec/radian, nm/m, mm/m) and for comparative-magnitude or comparative-resolution statements given in the wrong direction (e.g., "finer" versus "coarser" resolution, or "more" versus "less" light-gathering power) -- these are the most common sources of order-of-magnitude mistakes in this course.