This is a 2D-native counterpart to the 3D orbit-camera version. Instead of flying a camera around a rendered planet, two flat diagrams carry the exact same physics:
Sky-plane occultation. The star and planet are drawn as circles in the plane of the sky (the view a distant telescope actually gets), and the light-curve dip is the exact analytic overlap area of two circles — not a linear "distance from center" taper. That overlap fraction directly sets how deep the transit reads in each wavelength band below.
d = separation of disc centers
overlap(R,r,d) = r²·acos((d²+r²−R²)/(2dr)) + R²·acos((d²+R²−r²)/(2dR))
− ½·√((−d+r+R)(d+r−R)(d−r+R)(d+r+R)) [d between |R−r| and R+r]
transit fraction = overlap / (π·r²)
Disequilibrium phase map. The right-hand heatmap sweeps the same steady-state photochemistry model — production balanced by loss, dP/dt = 0 — over the full biosphere × volcanism grid at your current UV setting, so you can see the whole biosignature/false-positive landscape at once, not just the one point your sliders currently sit on:
O₂ production = 0.50·bio + 0.015·UV (photosynthesis + weak CO₂/H₂O photolysis)
O₂ loss = 0.05 + 0.04·UV (surface oxidation + UV-driven recombination)
CH₄ production = 0.35·bio + 0.15·volcanism (biogenic methanogenesis + geologic outgassing)
CH₄ loss = 0.08 + 0.20·UV (UV photolysis dominates)
steady state: [X]_ss = production_X / loss_X
The extra transit depth from one atmospheric scale height is H = k_B·T/(μ·m_H·g); Δdepth ≈ 2·R_p·H/R_s². For an Earth-radius planet around a Sun-like star, H ≈ 8 km gives only ~0.2 ppm per scale height — real surveys target smaller, cooler M-dwarf hosts where the same physics yields a far larger signal.
Why simultaneous O₂ + CH₄ matters: the two gases react (CH₄ + 2O₂ → CO₂ + 2H₂O) faster than either persists alone, so finding both at once means something keeps replenishing them. Following Lovelock's 1965 disequilibrium argument (formalized by Krissansen-Totton et al. 2018), co-existing O₂ and CH₄ is treated as a stronger biosignature than either gas alone; O₂ with no CH₄ is flagged as a possible abiotic false positive (Meadows 2017). This toy model is illustrative, not a literature retrieval code.