Solar Cycle Data and Sunspot Pressure Balance
0.60 x 116.4 = 69.8. This hypothetical smoothed maximum would still be above the ~50 threshold used here for "weakest historical cycles," so it would not be exceptionally weak by historical standards, though it would be substantially weaker than both Cycle 24 and Cycle 25.
| Criterion | Points |
|---|---|
| Correct arithmetic | 8 |
| Correct comparative statement | 7 |
P_mag(3500 G) = 48,741 Pa, giving a pressure-balance fraction of 3.90 relative to P_phot=12,500 Pa -- more than a factor of 10 (specifically 3.9x), meaning the simple pressure-balance approximation would require reducing internal gas pressure by more than a factor of 10 relative to the ambient photosphere at this strong field.
| Criterion | Points |
|---|---|
| Correct P_mag arithmetic at 3500 G | 8 |
| Correct greater-than-10x conclusion | 7 |
Revised ratio = (1.10 x 160.8)/116.4 = 1.52, compared to the lab's original ratio of 1.38 -- a modest but non-negligible increase, illustrating why sunspot-number ratios computed soon after a cycle's maximum should be treated as provisional until SILSO's official smoothing is finalized.
| Criterion | Points |
|---|---|
| Correct revised-ratio arithmetic | 5 |
| Correct comparison to the original ratio | 5 |
Solving B^2/(2 mu0)=P_phot for B gives B=sqrt(2 mu0 P_phot) = 0.1772 T = 1772 G. This lies between the lab's 2000 G and 2500 G table entries, consistent with the lab's numerically identified threshold falling in that same range.
| Criterion | Points |
|---|---|
| Correct algebraic rearrangement | 7 |
| Correct numeric threshold and unit conversion to Gauss | 8 |
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 (Gauss vs Tesla, nT vs T, km/s vs m/s, hours vs seconds), for comparative-magnitude statements given in the wrong direction, and for level-1/level-2/level-3 data-provenance conflation (e.g., presenting the order-of-magnitude flare free-energy estimate as a precise measurement, or presenting a Carrington-Dst estimate as a directly measured historical value) -- these are the most common sources of mistakes in this course.