Studying the albedos of the planets and moons of the Solar System dates back at least a century^1,2,3,4. Of particular interest is the relationship between the albedo measured at superior conjunction, known as the ‘geometric albedo’, and the albedo considered over all orbital phase angles, known as the ‘spherical albedo’^2,5,6. Determining the relationship between the geometric and spherical albedos usually involves complex numerical calculations^7,8,9,10,11, and closed-form solutions are restricted to simple reflection laws^12,13. Here we report the discovery of closed-form solutions for the geometric albedo and integral phase function, which apply to any law of reflection that only depends on the scattering angle. The shape of a reflected light phase curve, quantified by the integral phase function, and the secondary eclipse depth, quantified by the geometric albedo, may now be self-consistently inverted to retrieve globally averaged physical parameters. Fully Bayesian phase-curve inversions for reflectance maps and simultaneous light-curve detrending may now be performed due to the efficiency of computation. Demonstrating these innovations for the hot Jupiter Kepler-7b, we infer a geometric albedo of \(0.2{5}_{-0.02}^{+0.01}\), a phase integral of 1.77 ± 0.07, a spherical albedo of \(0.4{4}_{-0.03}^{+0.02}\) and a scattering asymmetry factor of \(0.0{7}_{-0.11}^{+0.12}\).

研究太阳系行星和卫星的反照率至少可以追溯到一个世纪^1,2,3,4。特别令人感兴趣的是在上合相时测量的反照率（称为“几何反照率”）与考虑到所有轨道相位角的反照率（称为“球形反照率” ^{2,5,6）之间的关系}。确定几何反照率和球面反照率之间的关系通常涉及复杂的数值计算^7,8,9,10,11，而封闭形式的解决方案仅限于简单的反射定律^12,13. 在这里，我们报告了几何反照率和积分相位函数的封闭形式解的发现，它适用于仅取决于散射角的任何反射定律。由积分相位函数量化的反射光相位曲线的形状和由几何反照率量化的次食深度现在可以自洽地反转以检索全局平均的物理参数。由于计算的效率，现在可以执行反射图的完全贝叶斯相位曲线反转和同时光曲线去趋势。为了展示热木星 Kepler-7b 的这些创新，我们推断出几何反照率\(0.2{5}_{-0.02}^{+0.01}\)，相位积分为 1.77 ± 0.07，球面反照率\( 0.4{4}_{-0.03}^{+0.02}\)和\(0.0{7}_{-0.11}^{+0.12}\)的散射不对称因子。