Figure 1
Like many of those interested in maintaining planted aquaria, I have been impressed and inspired by those of Takashi Amano, as pictured in his three volumes of "Nature Aquarium World."1 And like others, I have noted that the lighting used seemed to be more intense per unit volume for small tanks than for large tanks. This relationship seemed to differ from the standard 2 to 4 watts per gallon rule, often recommended on aquarium-related web sites. Figure 1 is a "scatterplot" of the watts per gallon used in the 161 Amano tanks, pictured in the three books.
1. Neptune City, NJ: T.F.H. Publications, Inc., 3 Vols., 1996, et.seq..
Figure 1
Using the statistics program, WinSTAT(R),2 we can employ regression analysis to draw a best fit curve through these data points, as in Figure 2. This plot confirms the observation that the number of watts used per gallon of tank size is not a linear function of volume but instead decreases with increasing volume.
2. WinSTAT(R) is an extension of Microsoft Excel, and was developed by my son, Robert Fitch, who also kindly applied it to the data I extracted from the Amano books.
Figure 2
We now repeat the same plot, but transform both axes to a logarithmic scale, as shown in Figure 3. This has two advantages: 1) The "bunched up" data at the lower end of the scale get "spread out" and become more readable. 2) Any power function appears as a straight line when logarithmically scaled.
Figure 3
Figure 4 shows the same relationship for watts as a function of liters, instead of gallons.
Figure 4
This Amano difference was also noted and examined in 1998 by Erik Olson of "The Krib" web site in an article entitled "Lighting Level: What's Ideal."3 He wrote that "For the smallest sizes, 8 watts per gallon is too little! For the larger tanks above 100 gallons, 2 watts per gallon is too much. Perhaps a better 'formula'...is obtained by calculating watts per 'estimated surface area' (calculated by taking volume to the 2/3 power). This might seem a strange calculation at first, but when you stop to think of it, it does make sense, as the light falls on an area, not a volume."
Figure 5 shows the three proposed relationships. The red line is a best fit to the data, when the light intensity is assumed to be linearly proportional to volume, i.e. volume to the 1st power. The green line is a best fit, if the intensity is proportional to surface area, i.e. volume to the 2/3 power. Neither of these lines fits nearly as well as the magenta line obtained by regression analysis. The exponent in this case is 0.46 --- very nearly one-half. In other words the intensity appears to be nearly proportional to the square root of the volume. In still other words, the intensity increases much more slowly than volume and somewhat more slowly than surface area.
3. Http://www.thekrib.com/Plants/Tech/Lighting/
Figure 5
The best-fit power function for volume measured in gallons is:
Watts = 22.65 x Gallons0.46
The equation for volume in liters is:
Watts = 12.25 x Liters0.46
To compare the Amano lighting to your own tank size, you can use the following calculator:
Finally, I don't suggest for a moment that Takashi Amano follows any such calculation. in determining the lighting intensity he uses for his tanks. One look at the scatter plot of Figure 1 shows considerable variance in data points for identically sized tanks. More likely, intuition and experience have been his guides. Nevertheless, I hope you agree that the exercise was worth the effort, if only to dispel earlier notions of lighting requirements.