Yes, you can find this in the W3 Accessibility Recommendations under the title "Relative Luminance". The formula is as follows, but I recommend you look it up in the original link, as the text here has a strange format, so I am just including it just for reference:
the relative brightness of any point in a colorspace, normalized to 0
for darkest black and 1 for lightest white
Note 1: For the sRGB colorspace, the relative luminance of a color is
defined as L = 0.2126 * R + 0.7152 * G + 0.0722 * B where R, G and B
are defined as:
if RsRGB <= 0.03928 then R = RsRGB/12.92 else R = ((RsRGB+0.055)/1.055) ^ 2.4
if GsRGB <= 0.03928 then G = GsRGB/12.92 else G = ((GsRGB+0.055)/1.055) ^ 2.4
if BsRGB <= 0.03928 then B = BsRGB/12.92 else B = ((BsRGB+0.055)/1.055) ^ 2.4
and RsRGB, GsRGB, and BsRGB are defined as:
RsRGB = R8bit/255
GsRGB = G8bit/255
BsRGB = B8bit/255
The "^" character is the exponentiation operator. (Formula taken from
[sRGB] and [IEC-4WD]).
Note 2: Almost all systems used today to view Web content assume sRGB
encoding. Unless it is known that another color space will be used to
process and display the content, authors should evaluate using sRGB
colorspace. If using other color spaces, see Understanding Success
Criterion 1.4.3.
Note 3: If dithering occurs after delivery, then the source color
value is used. For colors that are dithered at the source, the average
values of the colors that are dithered should be used (average R,
average G, and average B).
Note 4: Tools are available that automatically do the calculations
when testing contrast and flash.