The local cherry trees are currently covered with pink and white popcorn. Nature is beginning to unfurl her solar collectors . I found myself wondering how many watts of power an average oak around here produces. A detailed calculation would be very difficult, but fortunately a reasonable approximation is easy - a good example of play to create an estimate of something rather involved

To get the average you need the average amount solar power that falls where you live. You can start with the number for the Earth in orbit, correct for the elliptical orbit, take into account the geometry of the Earth, the fact it spins on its axis, then correct for the seasons and the weather ... ugh .. or you can look up the number that was obtained experimentally close to where you live. Now approximate an average tree and calculate the amount of energy that falls on the tree during the time when it is in leaf. Now consider what is meant by power produced. Some of it goes into keeping the tree alive, but a reasonable definition might be the energy in the biomass it produces from photosynthesis. That's about all you need. The details of the quick mental calculation are in the footnote, but the bottom line is a fifty foot oak averages about 80 watts - a tenth of a horsepower - during the full leaf season in New Jersey.^{1} This is the average through day and night, but during the day in bright direct sunlight the tree is probably up around 200 watts. This is an unexpected coincidence with the power requirement of a human- someone who eats 2,000 calories of food a day averages about 100 watts of input power from the food.

The low efficiency of the tree's photosynthesis is impressive - 0.5% in this case. It turns out deciduous trees average 0.2 to 0.5% efficiency, so the approximation isn't terribly wrong as back of the envelopes go

Photosynthesis keeps life running on Earth. Plants convert carbon dioxide, water and energy from sunlight into a sugar and oxygen. Everything you eat ultimately came from the process. It isn't terribly efficient - some crops manage to top one percent efficiency, but not by much. By contrast photovoltaic cells with 20 percent efficiencies are common. Theoretical efficiencies for photosynthesis are higher - something like seven percent, but Nature didn't find a need to work out how to get there.

It wouldn't make much sense for animals or people to be able to do photosynthesis. We may be able to receive a half square meter of sunlight under ideal conditions if we ran around in warm sunny climates wearing little or nothing. If we sat outside all day that might bring in a half watt or so. About what your smartphone uses and forty times less than your brain needs. We need plant life - a lot of it.

This sort of number might seem frivolous, but you can set food, or any other plant product, limits by working with a related number - how much biomass of a crop can be produced per hectare. Back of the envelopes tell us that some activities like eating meat at American levels don't scale to the world's population - there simply isn't enough available land area to raise the required crops. This is a big problem as societies gain wealth and begin to eat more meat as making meat is very inefficient often requiring at least ten kilograms of vegetation per kilogram of animal produced.

Thinking in terms of power per unit area is a very useful approach for thinking about diffuse energy. You can easily set limits to how much of certain types of energy can be produced in an area. Conversely you can consider how much power a society consumes per unit area to see if a county can be self sufficient with a mix of alternate energy types. These are great tools for dealing with claims that seem a bit suspect and by setting conservative limits even crude back of the envelope calculations can be compelling. This is also a good approach for estimating how many years of vegetation are required to produce a year's worth of fossil fuel consumption - another very difficult problem where the trick is to set limits.

But back to the trees. Oxygen, the 'waste' product of photosynthesis, is sort of useful to us. How many trees are needed to support you? There are a number of ways to come up with a number. Another back of the envelope suggests about eight fifty foot oaks in New Jersey would keep me breathing. I'll stay away from boring you, but the next time you take a walk in a park or forest remember to thank the trees - our partners in life.

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^{1} Here's the mental calculation I did - partly to show the approximations you can make when you know you're just looking for a ballpark number. If you play with numbers these things long enough you remember numbers and conversion constants.

° the average solar energy per day for this part of the country for April through September is about 4 kWh/m^{2} per day and trees are in leaf about 220 days a year.

° an average mature oak is about 50 feet tall. I approximate the leaf area by a disk with a radius of 6 meters or an area of about 100 m^{2}

this → 88,000 kWh for the oak per leaf season

° a 50 foot oak has a dry mass of about a ton - I'll call it a metric tonne. It produces about 100 kg of new dry wood a year

° dry wood has an energy content of about 16 MJ/kg or about 4.4 kWh/kg

this → 440 kWh of energy from the dry wood and an efficiency of 440/88,000 or 0.5%

440 kWh/leaf season → 440kWh/(220 d * 24h/d) or a bit over 80 watts average

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