Air is treated as an ideal gas, ρ = P / (R·T), with the specific gas constant for air R = 287 J/(kg·K). Local pressure thins with altitude, P(h) = P₀·e−h/8000. The envelope traps a volume V of hot air at temperature Tenv, while the same volume of ambient air outside sits at Tamb. Because both are evaluated at the same local pressure, the buoyant lift is exactly:
F_net = g·V·P(h)/R·(1/T_amb − 1/T_env) − m_load·g − k·v|v|
Holding the burner raises Tenv toward a ceiling (~120°C above ambient) following Newton's law of approach; releasing it lets Tenv relax back toward Tamb by Newton's law of cooling. Venting dumps heat instantly (the real vent flap), and dropping ballast removes mass instantly — both classic ways pilots trade one variable for altitude control. The quadratic term k·v|v| is aerodynamic drag from the envelope's frontal area.