1. RSS error budget
σ_total = √(8.0²+4.0²+6.0²) = 10.77 mmag; the Poisson term (8.0 mmag) is the largest single term and dominates the total.
| Criterion | Points |
|---|---|
| Correct RSS arithmetic | 12 |
| Correct dominant-term identification | 8 |
Worked Solutions
2. Aperture trade-space
Collecting-area ratio (D_Keck/D_Gemini)² = (10/8.1)² = 1.524; in the background-limited (√t) regime, reaching the same SNR with Gemini's smaller aperture requires roughly 1.52× the exposure time Keck would need.
| Criterion | Points |
|---|---|
| Correct area-ratio arithmetic | 8 |
| Correct application to the √t scaling | 7 |
3. Resolution requirement check
Keck's AO-corrected diffraction-limited K-band FWHM (46.7 mas) is finer than the required 30 mas separation, so the requirement is achievable in principle at the diffraction limit; whether it is achievable in practice depends on the AO system actually reaching a high enough Strehl ratio (Lecture 11) to place most of the companion's light in a resolvable core rather than a smeared halo.
| Criterion | Points |
|---|---|
| Correct numeric comparison | 10 |
| Correct caveat about Strehl ratio, not just the nominal FWHM | 5 |
4. Proposal synthesis
This proposed observation is technically justifiable: the 10.77 mmag total error budget is dominated by Poisson noise (Problem 1), meaning a longer exposure, not better calibration, is the most effective lever if higher photometric precision is needed; and the required 30 mas resolution lies within Keck's AO-corrected diffraction limit (Problem 3), provided the AO system reaches a Strehl ratio sufficient to concentrate the companion's light into a resolvable core rather than a seeing-like halo. A revised proposal should therefore request exposure time informed by the Poisson term and explicitly state the assumed AO Strehl ratio as a stated risk.
| Criterion | Points |
|---|---|
| Coherent synthesis connecting both prior problems | 12 |
| Explicit, quantitatively grounded recommendation with stated risk | 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.