Ricco's law

thumb|upright=1.2|Threshold data from Table 8 of Blackwell (1946) plotted as Figure 4 of [[Andrew Crumey|Crumey (2014). Curves are for background luminances ranging from 3.426 × 10⁻⁵ cd m⁻² (top) to 3.426 × 10³ cd m⁻² (bottom) at intervals of one log unit. The straight dotted sections correspond to Ricco's law.]]

Riccò's law, discovered by astronomer Annibale Riccò, is one of several laws that describe a human's ability to visually detect targets on a uniform background. It says that for visual targets below a certain size, threshold visibility depends on the area of the target, and hence on the total light received. The "certain size" (called the "critical visual angle"), is small in daylight conditions, larger in low light levels. The law is of special significance in visual astronomy, since it concerns the ability to distinguish between faint point sources (e.g. stars) and small, faint extended objects ("DSOs").

Derivation

Suppose that an achromatic target of angular area <math>A</math> is viewed against a uniform background luminance <math>B</math> (e.g. a disc of white light is projected on a white screen, or a nebula is seen through a telescope). For the target to be visible at all, there must be sufficient luminance contrast; i.e. the target must be brighter (or darker) than the background by some amount <math>\Delta B</math>. If the target is at threshold (i.e. only just visible) then the threshold contrast is defined as <math>C = \Delta B / B </math>. Riccò's law states that for targets below a certain size, threshold contrast is inversely proportional to target area, i.e.

<math display="block">CA = R</math>

for some constant <math>R</math>. Different values of background luminance <math>B</math> will yield different values of <math>R</math>.

This can be seen in contrast threshold data for different levels of background luminance, plotted on a single graph as <math>\log C</math> versus <math>\log A</math>. In each case (i.e. for each background <math>B</math>), the threshold curve for small targets is a straight line of gradient −1, i.e.

<math display="block">\log C = -\log A + \mathrm{constant}</math>

<math display="block">\log (CA) = \mathrm{constant}</math>

Targets for which the law holds are indistinguishable from point sources. Reading towards the right of each threshold curve, there is a target size at which the law begins to break down, i.e. the slope deviates from -1. This is called the "critical visual angle". Therefore, the contrast threshold required for detection is proportional to the signal-to-noise ratio multiplied by the noise divided by the area. This leads to the above equation.

Background dependency

thumb|Figure 5 from [[Andrew Crumey|Crumey (2014), showing <math>\sqrt R</math> as a function of <math>B^{-1/4}</math>]]

The "constant" R is actually a function of the background luminance B to which the eye is assumed to be adapted. It has been shown by Andrew Crumey for threshold contrast C

<math display="block">C\equiv\frac{\Delta B}B \propto\frac 1{A\sqrt{B.</math>

However, at very low background luminance (less than 10⁻⁵ candela per square metre), where the only perception is of 'dark light' (neural noise), the threshold value for the illuminance

<math display="block">\Delta I=A\Delta B</math>

is a constant (around 10⁻⁹ lux) and does not depend on B.