ner_tags: O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Software I-Software I-Software I-Software I-Software O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-CelestialObject O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-CelestialObject O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-CelestialObject O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O
tokens: in Eq. (12) , each locally reflected vector I , and hence each local reflection matrix R 1 , is rotated such that the reference plane is no longer the local meridian plane, but the planetary scattering plane. The local rotation angle β depends on the local viewing angle θ and the location of surface area d O on the planet. The geometric albedo A G of the planet with radius r at distance d is given by where π F 0 (0°, λ ) is the reflected total flux at wavelength λ and phase angle α equal to 0°. In P Y M IE DAP, the integration in Eq. (12) is replaced by a summation over locally reflected Stokes vectors. In order to do so, we divide the planetary disk on the sky in pixels, and compute the locally reflected Stokes vector at the center of each pixel. A pixel contributes to the disk signal when its center is within the disk radius. The integration is then given by with N the number of illuminated and visible pixels on the planetary disk, and with subscript n indicating that μ, μ 0 , φ − φ 0 , and β depend on the location of the pixel on the planet. In addition, μ 0 , φ − φ 0 , and β at a given location of the pixel on the planet depend on the planet phase angle α ( Appendix B provides relations that can be used to derive these angles). In the summation, d O n refers to the area of the pixel as measured on the surface of the planet. Although not explicitly indicated in Eqs. (14) and (12) , R 1 will usually also depend on the location (of a pixel) on the planet. Typical horizontally inhomogeneities would be the surface coverage and altitude, and the atmospheric composition and structure. The obvious horizontal variations on Earth are of course the oceans and the continents, and in the atmosphere, the clouds. Horizontal inhomogeneities can be taken into account by using different local reflection vectors R 1 across the planet (in that case, R 1 in Eq. (14) would include a subscript n ). Given a model planet and a planetary phase angle α , the steps to evaluate Eq. (14) are the following: Divide the planet into pixels small enough to be able to assume that the planet properties across each pixel are horizontally homogeneous, and to be able to follow the limb and the terminator of the planet. Calculate for (the center of) each pixel the angles μ (i.e., cos θ ), φ , and the rotational angle β . These angles are independent of the location of the sun or star with respect to the planet. Calculate for the given phase angle , and for (the center of) each pixel, μ 0 (i.e., cos θ 0 ) and φ 0 . These angles depend on the location of the sun or star with respect to the planet. Calculate for (the center of) each pixel the column vector R 1 of the locally reflected light using the appropriate database file (see Sect. 5 ). Perform the summation described by Eq. (14) . The pixels can be defined on the planet, for example, using a latitude and longitude grid, in which case μ d O , the projected area of the pixel (see Eq. (12) ), that is, the pixel area as “seen”