1. Shot noise
σ_shot = √(rate×t) = √(243408.201×100) = 4933.642 electrons.
| Criterion | Points |
|---|---|
| Correct Poisson-noise formula | 8 |
| Correct arithmetic | 7 |
Worked Solutions
2. Dominant term
Shot noise (4933.642 e⁻) is smaller than read noise (3.5 e⁻) at 100 s, and dark-current shot noise (0.3162 e⁻) is negligible; read noise dominates the total at this exposure time for this faint source.
| Criterion | Points |
|---|---|
| Correct comparison of all three terms | 8 |
| Correct dominant-term identification | 7 |
3. Crossover exposure time
Setting √(rate×t)=3.5 and solving for t gives t = (3.5)²/243408.201 = 0.0 s.
| Criterion | Points |
|---|---|
| Correct algebraic solution | 8 |
| Correct final value matching the given check | 7 |
4. Regime interpretation
Exposures shorter than 0.0 s are read-noise-limited (SNR grows roughly linearly with t); longer exposures are shot-noise-limited (SNR grows only as √t). For a faint, read-noise-sensitive measurement, it is more efficient to use exposures at or somewhat beyond this crossover, rather than many very short exposures that each pay the full read-noise penalty.
| Criterion | Points |
|---|---|
| Correct regime identification on both sides of the crossover | 12 |
| Correct efficiency argument for exposure strategy | 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.