Extend this unit's real sunspot-number comparison and pressure-balance tool to new hypothetical and threshold cases.
Suppose a future solar cycle reaches only 60% of Solar Cycle 24's smoothed maximum SSN. Compute this hypothetical cycle's smoothed maximum SSN, and state whether it would be weaker or stronger than the weakest well-documented historical cycles (which have reached smoothed maxima below 50).
Compute the sunspot magnetic pressure for an unusually strong umbral field of 3500 G (a real, though rare, published maximum sunspot field strength), and compute the pressure-balance fraction relative to the photospheric gas pressure used in the lab. State whether the simple pressure-balance approximation would require the internal gas pressure to be reduced by more than, or less than, a factor of 10 relative to the ambient photosphere at this field strength.
Recompute the Cycle 25/Cycle 24 smoothed-maximum ratio if Cycle 25's true smoothed maximum is later revised upward by 10% (a plausible outcome, since SILSO smoothed values are only finalized after a cycle's maximum has clearly passed). Compute the revised ratio and compare it to the value computed in the lab.
Algebraically solve P_mag=B^2/(2 mu0)=P_phot for B, and evaluate the resulting threshold field strength in both Tesla and Gauss using the photospheric pressure from this unit's data. Compare your algebraic threshold to the numerical threshold you identified in the lab's four-field-strength table.
Submit a complete derivation with equations, units, labeled quantities, and a short narrative interpretation for each problem. Show every step; a correct final number without a visible derivation receives partial credit at most, and quoting a lecture number without adapting it to this problem's specific inputs receives no credit for that step.
OpenStax Astronomy 2e (local extracted reference copy: references/openstax-astronomy-2e-extracted.txt) covers the Sun in Chapter 15 ("The Sun: A Garden-Variety Star"): 15.1 The Structure and Composition of the Sun, 15.2 The Solar Cycle, and 15.3 Solar Activity above the Photosphere. Chapter 15’s section numbering was cross-checked directly against its own embedded figure numbering this session (Figures 15.1 through 15.24 all appear under their matching section headers with no offset), so this citation is verified against the local extracted text rather than extrapolated by analogy from a prior course. OpenStax’s treatment throughout Chapter 15 is descriptive and qualitative; nearly every quantitative derivation in this course (the hydrostatic solar-atmosphere/scale-height argument, the sunspot magnetic-pressure balance, the Parker solar-wind critical-point argument and its numerical transonic solution, flare free-energy budgets, the Chapman-Ferraro magnetopause standoff distance, and the Dessler-Parker-Sckopke ring-current/Dst relation) goes well beyond OpenStax Astronomy 2e’s introductory level; each lecture states explicitly where this course extends beyond the assigned reading.
materials/ASTR370/data/ where referenced above.