I think I've cracked the code to land requirements for food.

Instead of measuring outputs, which can vary, I'm going to measure inputs to determine the maximum possible output. Virtually all organic energy (otherwise known as food) is generated by phototrophs. Virtually all energy use is from materials that were once biological. Firewood, coal, natural gas and oil are all either biological or are the result of biological decay. So it makes sense that solar power would be the largest limiting factor on the amount of space it would take to support one person.

Accounting for solar energy from the sun, and subtracting for energy loss to the atmosphere, rotation of the planet, angle of the ground, heat, reflection and weather, an estimated 164 Watts per square meter per day.

Watts = Joules/Second
Joules = Watts x Second
Joules = 164 x 1 day
Joules = 164 x (24 x 60 x 60)
Joules = 164 x 86400
Joules = 14,169,600

So the sun provides 14, 169,600 joules of energy per square meter every day.

4184 Joules = 1 (food) Calorie
14,169,600 Joules = 3386.6 Calories

So the sun provides 3387 Calories per square meter per day. We need those calories, but obviously cant eat sunlight directly. It has to be converted to food by plants first. Now we must look into how efficiently the plants can convert the sunlight into energy. Professor Emeritus of Biophysics, Biochemistry, and Plant Biology Dr. Govindjee of Illinois University states that the average photosynthetic efficiency of plants is around 0.1%, while crop plants average in at 1-2% efficiency. Sugar can can reach up to 8% efficiency, which is close to Dr. Bolton's suggested theoretical maximum efficiency. Unfortunately, we can't eat most plants, only their fruits. The average sunlight to fruit efficiency is near 0.25%. So lets put that number into the equation.

3387 Calories x 0.0025 = 8.47 Calories

So that means we could theoretically generate 8.5 calories per square meter per day from the sun using plants, provided that we could capture all the light provided. How much land do we need to satisfy a 2,700 calorie worldwide average diet (according to National Geographic)?

2700 calories/(8.5 calories/m2/day) = 317.6 m2

There are 4047 square meters per acre, so using this measure we can safely state that 1/13th of an acre is the absolute minimum that can be used to provide food for a single human if there are no external inputs and perfect energy conversion.

If you want to eat an around a third of your calories through animal products as is average, then the land must be recalculated to factor in the ten percent rule. This rule states that no more than ten percent of the energy within an organism within a trophic level can be passed on to another organism in a higher trophic level by consumption.

2700 calories x 1/3 = 900 calories

1800 calories (plant food) / 8.5 calories/m2/day = 211.7 m2
(900 calories (animal product) / 8.5 calories/m2/day) / 10% (10 percent law)=  1058.8 m2

212 m2 + 1059 m2 = 1271 m2 or a third of an acre

An average American has a 3640 calorie diet and around 40% animal product, so the equation changes slightly.

((3/5 x 3640 calories)/8.5 calories/m2/day) + ((2/5 x 3640 calories)/8.5 calories/m2/day) / 0.10)
= 256.9 m2 + 1712.9 m2
= 1973 m2 or a half acre

In conclusion, it is theoretically impossible to use less than 1/2 acre for an average American diet. If anyone suggests otherwise, they must be getting energy from somewhere else, either by the introduction of animal feed, power or other external inputs.

Problems:

• Theoretical capacity is always greater than actual capacity.
• The sun is irregular by day, season, and weather, and the efficiency level stated is therefore only rarely accurate. It is almost always lower.
• It is also very difficult to get plants to capture every photon hitting a square meter and still be able to harvest them easily. A lot of sunlight is lost simply because it hits the ground and not the plant.
• This doesn't account for growing plants to maturity, only their production at maturity.
• This system doesn't account for plants grown to save seed.
• The average consumer eats 0.4 cows, 0.7 pigs etc. but you can only grow whole cows.
• Food isn't produced consistently year round, and storage methods lose calories.
• This calculation doesn't account for other biologically necessary elements like nutrients, water etc., all of which are recyclable but not necessarily available in sufficient quantity on this size of land.

Sources:
http://www.colorado.edu/geolsci/courses/GEOL3520/Solar_Power.pdf
http://zebu.uoregon.edu/disted/ph162/l4.html
http://www.ccfg.org.uk/conferences/downloads/P_Burgess.pdf
http://www.life.illinois.edu/govindjee/whatisit.htm
https://www.researchgate.net/journal/1751-1097_Photochemistry_and_Photobiology" Photochemistry and Photobiology (Impact Factor: 2.27).01/2008; 53(4):545 - 548.
https://en.wikipedia.org/wiki/Photosynthetic_efficiency
http://www.nationalgeographic.com/what-the-world-eats/
https://en.wikipedia.org/wiki/Ten_percent_law


This is just a series of links and summaries on the topic:

One Acre Feeds a Person - States that on a modern American diet about one acre is needed per person using conventional methods. This article relies on a study by Cornell University and does not discuss the sustainability or efficiency of this diet.

Per Square Mile - I love this blog. He talks about how the more people live together the less land they need per person, even on a hunter gatherer type lifestyle. Rough numbers are 1 person needs 1 square mile, but 100 people only need 32 square miles. That's 205-640 acres/person. Read it.

Can New York State Feed Itself? - The answer is no. Also noted is the fact that diet can change land requirements from a just under a half acre a person to over two acres a person. Uses the same Cornell University study.

129 Persons Fed from 1 Acre - Raederle seems to think that a revised aquaponic, sprouts, and permaculture system could allow everyone to live in the state of Texas. This is a very good study of most of the factors involved. The solution provided ignores a lot of remaining factors and feels like an attempted biosphere, which judging from our other biosphere experiments, won't work . Problems include the impossibility of growing anything between multiple eight story buildings and trees due to shade, and building eight story buildings out of mud. My purpose is not to argue about overpopulation. Instead, Ill note the interesting fact that Raederle thinks you can feed 12 people year round on 0.03 acres using aquaponics, or 100 ft2/person. Sounds fishy.
This article bugged me enough to write the light energy efficiency analysis.

Over Population is a Myth - I like this website. I think it would help if they produced a video that acknowledge that overpopulation is theoretically possible, and at what point they project that would be. Again, it seems like everyone wondering how many acres are needed to feed a person is also concerned about overpopulation. I would also like a definition of overpopulation which they could reference to point out we aren't there yet. I understand that this is difficult.