φ_{Y} = [−1.853, −1.235, −1.853, −0.412, −1.853, 0.412, −1.853, 1.235,

−1.853, 1.853, −1.647, 1.853, −0.823, 1.853, 0.000, 1.853, 0.823, 1.853, 1.647]

φ_{Z} = [−2.029, −2.254, −1.127, −2.254, −0.225, −2.254, 0.676, −2.254,

1.578, −2.029, 2.254, −1.127, 2.254, −0.225, 2.254, 0.676, 2.254, 1.578, 2.254]

\sqrt{2}

\sqrt{2}

\sqrt{2}

\sqrt{2}

FWHM(km s^{−1}) = 1.20671cΔσλ = 36.18 (\frac{Δσ}{cm^{−1}} ) (\frac{λ}{μm}) . r_{p−a} = r_{eq} [1 − e^{2} cos^{2} (ϕ)]^{1/2} ,

e = [\frac{r_{eq}^{2} − r_{p}^{2}}{r_{eq}^{2}} ]^{1/2} .

r_{gm} = (r_{eq}.r_{p−a})^{1/2} .

θ_{p} = \frac{r_{gm}}{D_{H}} and Ω = πθ_{p}^{2}.

S̄_{S} = \frac{∫_{P assband} S_{S}(ν)R(ν)dν}{∫_{P assband} R(ν)dν} ,

S̄_{Calib} = K_{Beam}\frac{∫_{P assband} S_{C}(ν)R_{P} (ν)dν}{∫_{P assband} R_{P} (ν)dν} ,

K_{Beam}(θ_{p}, θ_{Beam}) = \frac{1 − exp(−x)}{x}; x = 4 ln (2) (\frac{θ_{p}}{θ_{Beam}} )^{2} ,

\frac{dV}{dQ} ∝ V => \frac{dV}{dS̄} ∝ V

\frac{dS̄}{dV} = f(V).

f(V) = K_{1} + \frac{K_{2}}{V − K_{3}} ,

S̄ = ∫_{V_{0}}^{V_{m}}f(V)dV = ∫_{V_{0}}^{V_{m}}(K_{1} + \frac{K_{2}}{V − K_{3}} ) dV,

S̄ = K_{1}(V_{m} − V_{0}) + K_{2} ln (\frac{V_{m} − K_{3}}{V_{0} − K_{3}} ) .

S_{S}(ν) = S_{S}(ν_{0}) (\frac{ν}{ν_{0}} )^{α_{S}} .

S̄_{S} = \frac{S_{S}(ν_{0})}{ν_{0}^{α_{S}}} [\frac{∫_{P assband} ν^{α_{S}}R(ν)dν}{∫_{P assband} R(ν)dν} ] ,

S_{S}(ν_{0}) = S̄_{S} [\frac{ν_{0}^{α_{S}} ∫_{P assband} R(ν)dν}{∫_{P assband} ν^{α_{S}}R(ν)dν} ] = K_{4}S̄_{S}.

K_{4} = \frac{1}{ν_{0}} \frac{∫_{P assband} R(ν)dν}{∫_{P assband} \frac{R(ν)}{ν}dν} .

K_{4P} = \frac{1}{ν_{0}} \frac{∫_{P assband} R_{P} (ν)dν}{∫_{P assband} \frac{R_{P} (ν)}{ν}dν} .

K_{4E} = \frac{1}{ν_{0}} \frac{∫_{P assband} R_{E}(ν)dν}{∫_{P assband} \frac{R_{E}(ν)}{ν}dν} .

S′_{S}(ν_{0}) = ν^{(α_{Snew}−α_{S_{0}})} [\frac{∫_{P assband} R(ν)ν^{α_{S_{0}}} dν}{∫_{P assband} R(ν)ν^{α_{Snew}}dν} ] S_{S}(ν_{0}) = K_{C}(α_{Snew})S_{S}(ν_{0}).

K_{C}(α_{Snew}) = ν_{0}^{(α_{Snew}+1)} [\frac{∫_{P assband} R(ν)ν^{−1}dν}{∫_{P assband} R(ν)ν^{α_{Snew}}dν} ] .

S(ν) = ∮_{4π} B(θ, ϕ)I_{ν}(θ, ϕ)dΩ,

A_{Beam} = ∮_{4π} B(θ, ϕ)dΩ.

A_{Beam} = \frac{πθ_{Beam}^{2}}{4 ln (2)} .

K_{4P} K_{C}(point, α_{s} = 3) = (1.0113, 1.0087, 1.0065) × (0.9070, 0.9180, 0.8952) = (0.9173, 0.9260, 0.9011)
K_{4E}K_{C}(extended, α_{s} = 3) = (0.9939, 0.9920, 0.9773) × (0.9727, 0.9821, 1.0037) = (0.9668, 0.9742, 0.9809).

P = \frac{π}{4 ln(2)} (\frac{θ_{Beam}}{θ_{Pix}} )^{2} erf^{2} [(\frac{θ_{Pix}}{θ_{Beam}} ) [ln(2)]^{1/2}] ,

2 / π

P = 1 − 0.0018θ_{Pix} − 0.00037θ_{Pix}^{2}, U_{P} = 0.0023 − 0.00113θ_{Pix}.

Measurement = R_{Source}B_{Source} + R_{Tel}B_{Tel} + R_{Inst}B_{Inst}, ε_{T el} = 0.0336λ^{−0.5} + 0.273λ^{−1}, B_{Tel} = (1 − ε_{Tel})ε_{Tel}B(T_{M1}, ν) + ε_{Tel}B(T_{M2}, ν),

B(T, ν) = \frac{2hν^{3}}{c^{2}} \frac{1}{e^{hν/kT} − 1}.

B_{Inst} = B(T_{Inst}, ν).

B_{Source} = (V_{Obs} − R_{Inst}B_{Inst} − R_{Tel}B_{Tel})/R_{Source}, A_{eff}Ω = (γλ)^{2}, R_{P} = \frac{V_{Uranus}}{M_{Uranus}} and R_{E} = \frac{1}{(1 − ε)^{2}} \frac{V_{Telescope}}{M_{Darksky}} ,

\frac{R_{E}}{R_{P}} = Ωη_{P} .

A_{eff} = \frac{R_{P}}{R_{E}} η_{P} (γλ)^{2}.