Constraints on the thermal and non-thermal emission in GRB 090902B and GRB 100724B. Top: for a given thermal fraction ϵ_th = 10^-2, 10^-1.5, 10^-1, 10^-0.5, and 1, the radius R₀ = ℓ/θ at the base of the flow is plotted as a function of the non-thermal efficiency f_Nth. The corresponding thermal efficiency f_th is also shown (top x-axis). Bottom: for a given magnetisation σ = 10^-1, 10^-0.5, 1, 10^0.5, 10¹, 10^1.5, and 10² at the end of acceleration phase, the Lorentz factor of the flow is plotted as a function of f_Nth (the unmagnetised case σ = 0 cannot be distinguished from the case σ = 0.1). Sets of parameters representative of the different classes of scenarios discussed in the paper are indicated: F (ϵ_th = 1, σ = 0) (standard fireball), M,is₁ (log ϵ_th = −0.5, σ = 0,) and M,is₂ (log ϵ_th = −1.5, σ = 0) (efficient magnetic acceleration: magnetisation is low above the photosphere and the dominant non-thermal mechanism is internal shocks), M,rec₁ (log ϵ_th = −0.5, σ = 10), and M,rec₂ (log ϵ_th = −1.5, σ = 10) (magnetised flow at large distance, the dominant non-thermal mechanism is magnetic reconnection). The initial radius is fixed to R₀ = 300   km, a typical value for long GRBs. The observational data (thermal flux, temperature, ratio of the thermal over the total flux) used for the calculation (see text) are taken from Abdo et al. (2009); Pe’er et al. (2012) for GRB 090902B (left column), and from Guiriec et al. (2011) for GRB 100724B (right column).