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Online Auxiliary Material For
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The Global Energy Balance of Titan
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Calculation of Emission Angle and Latitude At the Reference Altitude
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We introduce the geometry to compute the new emission angle and latitude at the
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reference altitude of 500 km from the original CIRS emission angle and altitude, which are
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referenced to the solid surface of Titan. Figure S1 shows that the original emission angle on the
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solid surface and the new emission angle on the sphere at the reference altitude.
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Figure S1. Conversion of the emission angle between the solid surface and the reference altitude
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for the CIRS nadir observations.
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In Fig. S1, the reference altitude (500 km) and the solid body radius of Titan (2575 km)
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are represented by H and R , respectively. In addition, we represent the original emission angle
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at the solid surface and the new emission angle at the reference altitude by 0 and  ,




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respectively. Based on the geometry shown in Fig. S1, we have the conversion between the
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original emission angle and the new emission angle as below.
R H
R

sin 0 sin 
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Rsin 0 
   arcsin

 R  H 
(1)

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The above conversion works for the CIRS nadir radiance. For nadir observations, we
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found that the range of the new emission angle  is from 0 to ~ 57 at the altitude of 500 km,
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which shrinks from the range of 0-90 for the original emission angle at the surface. Equation
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
(1) shows that the new emission angle  is less than the original latitude 0 due to the factor
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R R H  < 0, which results in the shrinking of the range of the new emission angle. In order to
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
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fill the gaps of 57-90 in the new emission angel, we include CIRS limb observations into this
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study.
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Figure S2. Conversion of the emission angle between the solid surface and the reference altitude
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for the CIRS limb observations.
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Figure S2 shows the original emission angle and the new emission angle for a limb
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observation. The limb radiance ray is tangent of the sphere at the altitude Z, so the original
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emission angle 0 at the altitude Z is a right angle. The new emission angle  , which is
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referenced to the sphere at the reference altitude H , can be expressed as

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sin  
R Z
R H
 R  Z 
   arcsin

R  H 


(2)

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The conversion of emission angle between the solid surface and the reference altitude is
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straightforward, so we have the simple relationship shown in the equations (1) and (2). The
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conversion of latitude between the solid surface and the reference altitude is relatively
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complicated. We do not have the simple geometry for this conversion. For the CIRS nadir
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observations, we use a method of parameterization to compute the new latitude at the sphere of
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the reference altitude, which is outlined as three steps: i) Define the observational location at the
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surface and the spacecraft’s location in sphere coordinates; ii) Parameterize a line to connect the
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two locations, which represents the ray path originating from the observational location at the
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surface; and iii) Solve the ray path for where it intersects the reference altitude (500 km) to get
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the new location (latitude and longitude) at the reference altitude. The parameterization of ray
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paths also works for the CIRS limb observations except for setting the sphere at the tangent
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heights as the new surface.
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