Friday, February 16, 2018

Ask a Physicist: How Many Calories are in that Sunbeam?

Last week, Joe from Massachusetts wrote in to ask:

Life is possible through the transfer of the sun's energy, through photosynthesis, and animals eating and us eating them. Is it possible to measure how much energy a person receives from the sun in order to live an average life, say 85 years being the average? Tall order, yes?
Great question—it's awesome that you recognize the flow of energy involved in biology and physics: it means that ultimately, almost all life on Earth is fusion-powered in a sense. Perhaps surprisingly, your question isn't that tall of an order! Although energy takes many forms, one of the amazing things about physics is that, at the end of the day, it's all interconvertible—going from calories to a more standard unit of energy like joules is a simple scaling conversion. That means that if you multiply a person's daily caloric requirements by the number of days in 85 years, you can find a rough answer to your question!

However, it's interesting to note that the amount of solar energy that goes into a person's daily food is wildly variable depending on their diet. The general rule in biology is that, for every level you go up the food chain (trophic level is the scientific term), you lose 90% of the energy involved. If a blade of grass absorbs 1000 calories of energy as sunlight over the course of its lifetime, it'll provide about 100 calories to the cow that eats it—the rest go into the processes involved with keeping the grass alive, helping it reproduce, etc. Those 100 calories of grass turn into about 10 calories of edible beef, and the rest goes to the cow. The upshot of this is that a vegetarian can get by on about 1/10th the amount of total energy from the sun that a carnivore can, which is part of the reason why vegetarianism is better for the environment—and may play an increasingly important role in helping us feed everybody as the global population increases.

Delicious, maybe, but definitely not the most efficient way to get your day's calories.
Image Credit: Martin Abegglen, via Wikimedia (CC BY-SA 2.0)
But let's get to the core of your question. In physics, a calorie is the amount of energy that it takes to heat a milliliter (one cubic centimeter) of water by 1°C. Confusingly enough, though, a physics calorie isn't the same thing as a food Calorie—calorie with a capital C. A food Calorie is equal to 1000 physics calories, meaning that it's the amount of energy that it'd take to heat a full liter of water by 1°C.

One of the more common measures of energy is the joule, a unit that's closely related to the familiar watt; a 60-watt lightbulb uses 60 joules of energy per second. A physics calorie is 4.184 joules, so a food Calorie is 4,184—enough to keep a 60-watt bulb burning for (4184/60=) ~70 seconds!

Assuming a person takes in 2,000 Calories a day, and lives 85 years as you suggested, they'd consume:
62 million Calories! Multiplying again by 4184 to find the number of joules, we get an answer in the hundreds of billions—at which point it becomes handy to start using SI notation: over the course of a lifetime, a person consumes about 260 Gigajoules—billion joules—of energy as food. For scale, gasoline has an energy density of about 130 Megajoules per gallon, so if a person could run on gasoline like a car, they'd need 2000 gallons to keep them going from birth 'til death.

So what is that, in terms of pure sunlight? At noon on a sunny day in the summer, when the sun is at its highest in the sky, there's about 1120 joules of energy striking a square meter of Earth's surface each second, but that number changes over the course of the day, and over the course of the seasons. According to this image, the average spot in the US receives around 1700 kilowatt-hours per square meter of land in a year. Since a watt is just one joule per second, we know that a kilowatt is 1000 joules per second, and a kilowatt-hour is 1000 times the number of seconds in an hour (3600), for a total of 3.6 million joules. If each square meter of sunlight gets 1700 of those in a year, more multiplication tells us that 6.12 Gigajoules land on a square meter of soil as sunlight annually!

Since that's just about a hundredth of a person's total lifetime caloric requirement (620 GJ), that tells us that powering a person for 85 years requires all the energy that lands on a square that's ten meters to a side, for a full year.

When you think about the number of people on Earth, that already seems like a lot—and that number assumes you're catching the rays with perfectly efficient solar panels and plugging those panels directly into people...but we don't run on that kind of juice!

That's more like it! ...although calling it "juice" is more than a little generous.
Image Credit: Sunny Delight Beverage Co.
That means we've got to take into account something called photosynthetic efficiency. When a photon of sunlight hits a leaf (more specifically, the magnesium atom at the center of a chlorophyll molecule)—its electromagnetic energy gets converted to chemical energy, as the excitation of the magnesium is used to tear the carbon atoms off of CO2 and stick them together into carbohydrates. This reaction can only be caused by photons of certain frequencies, so a leaf only ends up turning about 5% of the energy that hits it into chemical energy. If that weren't bad enough, that 5% figure doesn't even take into account the "trophic level" losses mentioned above—5% is the plant's share, meaning we get a tenth of that—another factor of ten in the amount of sunlight it takes.

So what's all this total up to? If you need 260 GJ of energy per person, and only 0.5% of the energy that hits a plant as sunlight lands on a person's plate, we need to multiply our earlier figure—100 square meters—by a factor of 200.

While 20,000 square meters sounds like a lot, it's only 2% of a square kilometer. If we wanted to get a lifetime's worth of calories from sunlight in a single day, how much area would we need to plant and harvest? To find that, we just have to multiply that 20,000 square meters by the number of days in a year—which gives us a number in the ballpark of 7.3 square kilometers. For a sense of scale, imagine a circular field two miles across at its center; that's about 7.3 km2. A field like that, planted somewhere in the midwest, harnesses enough energy each day to feed a person for a full lifetime!

Thanks for writing in!

—Stephen Skolnick


  1. Something seems amiss in your analysis. If 90% of a day's sunlight is needed to support a person for 85 years then that area could only support about 31,000 people per year! (365 days x 85 years) Am I missing something?

  2. That figure you calculated of 1,900 Gigajoules of energy hitting Arizona each day is about 7 orders of magnitude too low. AZ gets minimum 5kwh/m2/day, which is 18 Megajoules/m2/day. Arizona is about 295 billion m2, so it should be receiving over 5300 Petajoules of energy each day.

    Perhaps you didn't convert square miles into square meters? 1,900 gigajoules/day is about right for 114000 m2, but AZ is 114000 Miles2.

  3. Whew! Yeah, thanks for pointing that out, folks—should have figured something was off there. Wolfram insists that 1900 GJ is the correct figure for total sunlight incident per day in AZ, but obviously that must be per square kilometer or something. We'll get this fixed, pronto.

  4. This article is why you need to think about your results. If an area the size of Arizona could only feed 31025 people (as the poster above and I calculate) the math is way off.