天光视亮度的表述方法

Visual Brightness of the Sky

视荧光学中通常用亮度的对数来表达视亮度。这种表达在暗弱光线下是不成立的。本文根据光学经典中的视觉实验统计结果，推导出暗弱光线下视亮度对于亮度的表达式。

Visual brightness is a human sense to light. The concept of visual brishtnes in visualoptics is obviously different from the conceal of luminance in photometry, which is proportional to light energy. Generally speaking, loprithm of luminance is used to expressvisual brightness as formulas (1) and (2). Howerer, this logrithmic regulation is unsuitable in case of dim light.For different luminance (L). there are different limit (△L) for human eyes to tell theminimum change of brightness. Fig. 2 shows the relationship between △L and L based onthe data from measurements and statistics. which are listed in the optical classics. Fittingthe data in Fig. 2, formulas (3) and (4) can be obtained for the part of lgL<2. Here △L/L stands for a human capacity for telling the minimum change of brightness. Apparentlytea reciprocal of △L/L is the basic element of visual brightness (V) as shown in formula(5).Formulas (6) and (7) can be obtained by integrating formula(5), and C1 and C2 areintegral constants. Taking (L→0. lgL→ and V→0)as the boundary conditions thenC2=0 can be obtained. Formula (6) is equivalent to formula (7) when lgL=-0. 419, sothat C1=98.84k. Let k=1/56. 23 for the sake of simplification. the expressions (8) and(9) can be obtaihed for visual brightness. It can be seen from formula (8) that the visualbrightness fits the logarithmic regulation when 2>lgL>-0. 419.Fig. 1 is the measurement result for sky luminance in the logarithmic scale for the sunrise, where the horizontal coordinate is the altitude for the sun. If this process of sunriseis illustrated in visual brightness. Fig. 3 is obtained. It can be seen from Fig. 3 that the skygets bright slowly at first, then rapidly. and slowly again which tallies with human sensebetter.