Monday, May 31, 2010

Human Plants

My favorite Far Side cartoon depicts a small child hiding in the back of a beehive. The beehive has a rock-sized hole through its wall. In the foreground, a bee detective tries to figure out how a rock could have been thrown from the inside. The cartoon itself is not that funny, but there’s a great caption.
Artist: G. Larson
Medium: Ink on paper.
Title: It was late, and I was tired.

After rereading the estimation below, I believe the same title applies…

How close to the Sun would you need to be to become a plant? I should probably elaborate. There have been many great advances in biotechnology. If scientists can grow ear-shaped cartilage on the back of a mouse, why not grow a food-making plant inside a human? If this worked, it would stop world hunger. We’d be mobile symbiotes walking around producing our own (very) local food using only the green energy of the Sun. But is there enough energy from the Sun to do this. From a conservation of energy standpoint, how close to the Sun would we need to be to absorb all the energy we need?

***********WARNING: Math Ahead***********

If the FDA is to be trusted, the average person should intake about 2000 Calories per day, which is a total power consumption of about 100 W. The solar flux reaching the Earth’s surface is 1370 W/m2. The maximum amount of body area that can absorb sunlight at any given time is about 1.0 m2, meaning that on Earth we’d be able to absorb 1370 W of power. If we could convert 100% of that power into useable food energy, we would have about 13.7 times more energy than we would need. But how far can we get from the Sun and still meet our energy needs?

To compute just how far away from the Sun we could go and still absorb enough energy to survive, we need to know how the density of solar energy decreases as you go further out. The energy density decreases as the inverse radius squared (~1/r2). This is easy to see if you remember that the total amount of solar energy available from the Sun is the same at all distances but the area over which that energy spreads out grows as the radius squared. From this we can construct an equation to compute the farthest distance we plant-a-noids could go away from the Sun,

power needed = (solar flux at Earth) · (body area) · (Earth-Sun distance)2 / (max distance)2

Solving for the maximum distance, we get,

max distance = [(solar flux at Earth) · (body area) · (Earth-Sun distance)2 / (power needed)]1/2
= [(1370 W/m2) · (1.0 m2) · (1.0 A.U.)2 / (100 W)]1/2
= 5.5×108 km.

**************************************************

The furthest a human-plant hybrid could go from the Sun would be about 500 million kilometers1. That’s somewhere between Mars and Jupiter.

“Wait a minute,” you say. “You assumed the plants were 100% efficient. Shouldn’t you assume a more realistic efficiency?” Really? You were fine with genetically engineering some sort of human-shaped lichen, but thermodynamic efficiency ruffles your feathers? Carnot would be so proud. It’s absolutely true the efficiency matters here. In fact, you might be able to tell from my phrasing of the question that I originally thought Plant Man would need to be a lot closer to the Sun. If you were only 20% efficient at converting energy to food, you would need to be closer to the Sun than the Earth is to produce enough energy. In real plants, only a small percentage of the energy absorbed goes into creating food. My number is only an upper bound. If you want to impress me, see if you can calculate how close to the Sun you’d actually after to be.

[1] In hindsight, Plant Man reminds me a lot of Nuclear Man from Super Man IV: The Quest for Peace. Spoiler alert: This is the worst Superman movie ever made.

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