1 2 Online Auxiliary Material For 3 The Global Energy Balance of Titan 4 5 Calculation of Emission Angle and Latitude At the Reference Altitude 6 We introduce the geometry to compute the new emission angle and latitude at the 7 reference altitude of 500 km from the original CIRS emission angle and altitude, which are 8 referenced to the solid surface of Titan. Figure S1 shows that the original emission angle on the 9 solid surface and the new emission angle on the sphere at the reference altitude. 10 11 Figure S1. Conversion of the emission angle between the solid surface and the reference altitude 12 for the CIRS nadir observations. 13 14 In Fig. S1, the reference altitude (500 km) and the solid body radius of Titan (2575 km) 15 are represented by H and R , respectively. In addition, we represent the original emission angle 16 at the solid surface and the new emission angle at the reference altitude by 0 and , 1 1 respectively. Based on the geometry shown in Fig. S1, we have the conversion between the 2 original emission angle and the new emission angle as below. R H R sin 0 sin 3 4 Rsin 0 arcsin R H (1) 5 The above conversion works for the CIRS nadir radiance. For nadir observations, we 6 found that the range of the new emission angle is from 0 to ~ 57 at the altitude of 500 km, 7 which shrinks from the range of 0-90 for the original emission angle at the surface. Equation 8 (1) shows that the new emission angle is less than the original latitude 0 due to the factor 9 R R H < 0, which results in the shrinking of the range of the new emission angle. In order to 10 fill the gaps of 57-90 in the new emission angel, we include CIRS limb observations into this 11 study. 12 13 Figure S2. Conversion of the emission angle between the solid surface and the reference altitude 14 for the CIRS limb observations. 15 16 Figure S2 shows the original emission angle and the new emission angle for a limb 17 observation. The limb radiance ray is tangent of the sphere at the altitude Z, so the original 2 1 emission angle 0 at the altitude Z is a right angle. The new emission angle , which is 2 referenced to the sphere at the reference altitude H , can be expressed as 3 4 sin R Z R H R Z arcsin R H (2) 5 The conversion of emission angle between the solid surface and the reference altitude is 6 straightforward, so we have the simple relationship shown in the equations (1) and (2). The 7 conversion of latitude between the solid surface and the reference altitude is relatively 8 complicated. We do not have the simple geometry for this conversion. For the CIRS nadir 9 observations, we use a method of parameterization to compute the new latitude at the sphere of 10 the reference altitude, which is outlined as three steps: i) Define the observational location at the 11 surface and the spacecraft’s location in sphere coordinates; ii) Parameterize a line to connect the 12 two locations, which represents the ray path originating from the observational location at the 13 surface; and iii) Solve the ray path for where it intersects the reference altitude (500 km) to get 14 the new location (latitude and longitude) at the reference altitude. The parameterization of ray 15 paths also works for the CIRS limb observations except for setting the sphere at the tangent 16 heights as the new surface. 17 3