CONAN.utils.cosine_atm_extended

CONAN.utils.cosine_atm_extended#

CONAN.utils.cosine_atm_extended(phi, Fd=0, Fn=0, delta_deg=0, A2=0.0, delta2_deg=0, CA08_convert=False)#

Extended atmospheric phase curve with first and second harmonics:

F(phi) = Fmin1 + A1*(1 - cos(phi + delta))

  • A2*(1 - cos(2*phi + delta2))

F(ϕ)=Fn + (Fd-Fn/2cosδ)[cosδ - cos(ϕ+δ)] + A2[cosδ2-cos(2ϕ+δ2)]

Fd and Fn are the TOTAL observed fluxes at occultation (phi=pi) and transit (phi=0) respectively — exactly as in the original single-cosine model. A1 is derived from them after subtracting the second harmonic’s contribution at those two events.

Since cos(2*0 + delta2) = cos(2*pi + delta2) = cos(delta2), the second harmonic contributes the SAME offset at both phi=0 and phi=pi:

A2*(1 - np.cos(delta2))

so the first harmonic must account for:

Fd_1st = Fd - A2*(1 - np.cos(delta2)) Fn_1st = Fn - A2*(1 - np.cos(delta2))

and A1 is then computed exactly as in the original model:

A1 = (Fd_1st - Fn_1st) / (2 * np.cos(delta))

= (Fd - Fn) / (2 * np.cos(delta)) <- unchanged by A2!

This means the day-night contrast Fd - Fn and the offset delta are completely decoupled from A2 and delta2. A2 only affects the shape of the curve between transit and occultation.

Parameters:

  • phi (array-like) – Phase angle in radians (phi = 2*pi*phase circular, or f + omega - pi/2 for eccentric orbit).

  • Fd (float) – Total observed flux at occultation (phi = pi).

  • Fn (float) – Total observed flux at transit (phi = 0).

  • delta_deg (float) – Hotspot offset in degrees. Positive = eastward (peak before eclipse).

  • A2 (float) – Amplitude of the second harmonic (default 0 → original model). Physically, A2 >= 0. Conversion from harmonic coefficients: A2 = sqrt(C2^2 + D2^2).

  • delta2_deg (float) – Phase offset of the second harmonic in degrees. Conversion from harmonic coefficients: delta2 = -arctan(D2/C2).

  • CA08_convert (bool, optional) – If True, also return the equivalent Cowan & Agol 2008 model parameters (C1, D1, C2, D2) that would produce the same phase curve. Default is False.

Returns:

res – delta : first harmonic offset (radians) delta2 : second harmonic offset (radians) delta_phase : first harmonic offset in phase units A1 : first harmonic amplitude = (Fd-Fn)/(2*cos^n1(delta)) A2 : second harmonic amplitude Aatm : = A1 (semi-amplitude of the day-night variation) Fmin : global minimum of the phase curve Fmax : global maximum Focc : total flux at occultation [= Fd, by construction] Ftr : total flux at transit [= Fn, by construction] pc : phase curve array

Return type:

SimpleNamespace

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