This is a 2D companion to the 3D "Hot Jupiter Atmospheric Escape" simulation. Instead of a spatial orbit view, it draws the actual restricted three-body effective potential along the star–planet line in the rotating (co-orbiting) frame — the same physics that produces the Roche lobe used by the energy-limited mass-loss formula, just plotted directly as a curve instead of inferred from a 3D scene.
Φ_eff(d) = −GM★/d − GM_p/|d−a| − ½Ω²(d−d_cm)²
Ω² = G(M★+M_p) / a³ — orbital angular speed
d = distance from the star along the line
L1 = local maximum of Φ_eff between the two bodies
Ṁ = ε·π·R_p³·F_XUV / (G·M_p·K_tide) — same formula as the 3D sim
K_tide = 1 − 3/(2ξ) + 1/(2ξ³), ξ = R_Roche/R_p
R_Roche = a·(M_p/3M★)^(1/3) — Eggleton-style formula
The curve's saddle point (L1) is found by numerically maximizing Φ_eff between the star and planet — its distance from the planet is compared live against the closed-form R_Roche used in K_tide, confirming the formula is the correct leading-order approximation to the real Roche geometry.
Orange particles are launched from the planet's day side (facing the star) with the same day-side kick used in the 3D sim, then integrated as d²s/dt² = −dΦ_eff/ds along this single axis. Particles that climb over L1 fall toward the star and are marked as escaped — visually the 1D analogue of the comet-like tail in the 3D version, but derived from the potential shape rather than a ballistic 3D trajectory.
The right-hand grid is a parameter-space map: every cell is an independent (orbital distance, planet mass) pair, colored by log₁₀ Ṁ from the same formula. The crosshair marks the sliders' current position, so you can see at a glance how sensitive escape is to being closer-in or lighter, without needing to sweep the sliders one at a time.