Photosynthesizing Felines – A Back of the Envelope Calculation

The other day I was out weeding in the garden, amazed at the ability of weeds to take over all the tasty plants.  Just think of all the energy that must go into growing these weeds, just to be heartless ripped out of the ground and tossed away.  It also irked me to no end that the rabbits (which magically can squeeze through the tiny openings in the fence) only eat the stuff I’ve planted and not the weeds.  In that moment, for some reason, it struck me, “why don’t animals use photosynthesis?”  Cut out the middle-man, so to speak.  One of the primary limitations on the population of a species is limited food resources.  Any animal that could use photosynthesis to live could thrive by absorbing a few nutrients, drinking some water, and sitting around in the sun all day.

I’m sure a biologist could explain to me all the reasons this wouldn’t be biochemically feasible, but I wanted to see if there is a physics rationale so on to my back-of-the-envelope calculation.

Model I – First Pass

Assume a cat-sized animal could absorb all of the solar radiation striking its body and convert 100% into usable energy.  I want to find out how much energy the animal can absorb in, say, an hour of sunning.  I’ll need to have something to compare this to so I decided to figure out how much energy it would take for the same animal to climb a 10 ft tree.

A cat (the one sitting next to me, anyway) is about a foot long by 4″ wide which is about 0.3 m by 0.03 m so the area of the cat is $A_{cat} = 10^{-2} m^2$ (note: $3^2\approx 10$).  The power density of solar radiation (amount of solar power striking the surface of the Earth per square meter) is $R = 1.4 \frac{kW}{m^2}$ and there are roughly $3 \times 10^3 s$ per hour.  This means our cat can absorb $E = (10^{-2} m^2) \times (1.4 \frac{kW}{m^2}) \times (3\times 10^3 s) \approx 4\times 10^5 J$.

For a 10 lb cat (40 N), the work needed to climb 10 ft (3 m) is $W = (40 N) \times (3 m) = 120 J \approx 10^2 J$

Wow!  Our little guy could run up and down 1000 trees after an hour of sunning.  Maybe there is something wrong with the model.

Model II – Some Considerations

I can see  two problems with the first pass.  1)  Plants only use a narrow part of the spectrum for photosynthesis and 2) photosynthesis is not 100% efficient.

Figure 1: Light absorbed by plants during photosynthesis - Image from http://en.wikipedia.org/wiki/File:Chlorophyll_ab_spectra.png on 7/17/2011

As you can see above, the band of light absorbed during photosynthesis is pretty narrow, especially when you compare it to the whole solar spectrum.

Figure 2: The Solar Radiation Spectrum - Image from http://en.wikipedia.org/wiki/File:Solar_Spectrum.png on 7/17/2011

I could use the two figures to determine what percentage of the total light is absorbed for photosynthesis, but that defeats the point of a back-of-the-envelope calculation so I’m going to guess that only 10% of the total solar radiation contributes to photosynthesis.  In addition, I could look up how efficient photosynthesis is, but I’m going to ball-park it.  I know internal combustion engines tend to be around 20%-ish efficient, so I’ll say photosynthesis is 10% efficient.  That is, only 10% of the absorbed energy is used to create sugar molecules.  Thus the usable energy our solar cat could gather from the sun is $0.10 \times 0.10 = 0.01$ times $10^5 J$ or $E = 10^3 J$.  That is still enough energy to climb ten 10 ft trees after an hours worth of sunning.  I have to admit that I am really surprised.  I totally expected the energy amounts to be too small to be a feasible alternative to eating.  Maybe when I have some time I’ll look up actual values and attempt to be more rigorous.