Back of the Envelope #1: Do we get enough energy from the Sun?

Solar energy is in reality the only truly sustainable form of energy we can extract, as long as the Sun is around. So how much of it there is to go around on our little planet? Is it really going to be enough? The numbers in this post will all be approximate and the details depend on how you approximate things. But the overall result is robust: There may seem a energy-crisis on Earth but the Solar System is a world of plenty, not scarcity.

The Sun’s massive output

The first thing we need to estimate is the total amount of energy that the Sun blows into all directions. And in doing so we will have to ignore certain forms of energy in order to keep the problem simple. But then, this is supposed to be a back of the envelope calculation and not a detailed analysis.

So the things that I will ignore are things such as the energy stored in the particles that the Sun blows into the solar system that form the so-called Solar Wind. What I will also ignore are the energy the Sun deposits into its immediate surroundings that heat up the solar ‘atmosphere’ or the so-called corona. The main flow of energy that will interest us is that which comes in the form of electromagnetic radiation, including light, and fortunately this is easy to estimate.

According to the Stefan-Boltzmann law an object that is in thermal equilibrium at a temperature T emits a total amount of radiation-energy j* per square-meter of its surface area, given by

Screenshot (90)The sigma here is the so-called Stefan-Boltzmann constant and it takes the value

Screenshot (91)Now the Sun’s surface temperature (at least of the ‘surface’ that we see when we look at the Sun) is 5777 degrees Kelvin, or about 5500 degrees Celsius. So every square-meter of the Sun’s visible surface deposits around 63,156,942 Watts of energy into the Solar System in the form of electromagnetic radiation. The bulk of this is in the form of visible light and infrared radiation. So let’s say 63 million Watts per square meter. Now the Sun has an awful lot of square meters to emit light from.

The radius of the Sun is 675,900 km, which is about 676 million meters. Now recall the well known formula for the surface of a sphere with a radius r

Screenshot (92)When we use this to calculate the Sun’s total output of energy per second (i.e. in Watt) we find 362,557,968,941,219,827,656,595,631 Watts … which about 363 million-trillion Megawatts. To put this into perspective a little, consider the following: the US’s average energy usage per second in 2017 amounted to roughly 32 MW.

The Sun could power more than twelve million-trillion countries like the United States of America. We may have an energy problem on Earth, but we definitely don’t have one in the Solar System.

The share we get

Now the Sun pumps this humongous amount of energy into all directions and by the time it reaches the distance that the Earth is away from the Sun, which is about 150 million kilometres or 150 billion meters, this energy has been spread across a huge sphere of which the Earth only covers a tiny bit. In fact, the Earth having a radius of 6375 km, or 6,375,000 meters, captures a fraction of it that we can compute by dividing its total cross section by the total surface area across which the Sun’s energy has been distributed. The cross section of a circle of radius r is given by

Screenshot (93)and we need to divide this by the surface area of the sphere of the Earth’s orbit! It turns out the Earth captures only 0.0000000005% of the Sun’s total output. But then the Sun’s output was absolutely extraordinary, so let’s see how much that actually is: it is 1,779,731,794 Megawatts, or 1.8 Billion Megawatts which is still enough to power millions of countries like the United States of America. So we’re good! In fact at the typical distance of the Earth from the Sun every square meter still receives around 1360 Watts of energy. Because the Earth’s surface isn’t flat, hence the Sun’s angle often oblique, and because half of the time it is on the nightside of the Earth, this means that every square meter of upper atmosphere of the Earth gets around 320 Watts … on average. You would need three square meters to run an average blow-dryer there. That sounds mediocre but there are plenty of square-meters up there.

Nevertheless this is not entirely what we get for our use down below on the Surface of the Earth. One thing that the Sun does for us is that it makes weather … and weather comes at a cost.

So what is left down here with us?

The average global temperature is around 13.5 degrees Celsius or 286.5 degrees Kelvin. But using our friend the Stefan-Boltzmann law this means that every square meter of the Earth’s surface radiates out about … wait for it … 336 Watts. How can that happen? How can the Earth’s surface radiate out more energy than it receives from the Sun? Well, the mystery is not the fact that the Earth has a hot core although that does contribute a little. The reason for this is the fact that the Earth’s surface is covered in a blanket that allows the Earth to keep some of the heat it radiates inside its atmosphere. This blanket is formed by the gasses we tend to call greenhouse gasses and CO2 is the most well known of them. But Methane and Water vapour are also important greenhouse gasses. If it weren’t for our greenhouse blanket the average surface temperature on the Earth would be roughly -3 degrees Celsius and we would be a frozen world.

So the Sun delivers us the energy to warm ourselves and our greenhouse gasses deliver us the blanket we need to have a surface that radiates a bit too much but has the energy to do so thanks to the recycling of that heat in our atmosphere. However the actual locally measured temperatures are very different across the surface of the Earth and this means there are huge differences in the amount of heat that is deposited in different parts of the Earth’s land and water surfaces. These differences drive the phenomenon we call weather and weather draws its energy from the 320 Watts that the upper atmosphere of the Earth receives, on average, per square meter. All this energy swirling around in the atmosphere eventually also expresses itself in its temperature. Maintaining that temperature requires energy as in a real sense the Earth surface is “naturally unnaturally warm”.

So what remains down here with us?

Once the sunlight has penetrated the 100 km down into our atmosphere it has ‘lost’ about two-thirds of its energy so that on average about 110 Watts ‘survive’ to heat up the soil or water, or to be photosynthesized into chemical energy by plants. These 110 Watts per square meter are thus also a reasonable estimate for what the optimal energy is solar power can extract. Wind-, Water- and Wave-power all extract energy that is essentially delivered to the Earth in that other two-thirds that goes into making weather. It is the weather system that feeds the winds that drive the wind turbines, that transports the water that rains and flows down to the dams where its energy is extracted.

That 110 Watts may not sound like much. But it still amounts to 56 billion MW for the Earth as a whole. There is a lot of surface area on the Earth! Yet the energy is not delivered equally across the globe, so there will be issues of collecting it and transporting it. And evidently we cannot use all this energy for our consumption as important Eco-systems need this energy, for example to produce the oxygen we breathe. Even when solar panels would become 20% efficient in converting solar energy into electric energy, storing that energy for later use and using it later each come with their own efficiency losses. Using 10 square meters of Earth surface to run a blow-dryer may indeed not be the best use of that surface area. With a future population of around 10 billion people a maximally available energy budget per person of 5.6 Megawatt is impressive but actually not astronomically so when you imagine which services it has to supply, all the way from oxygen & food supply to that blow-dryer, your car and your bitcoin miner.

So there is an energy-crisis after all?

No there isn’t. The total amount of energy the Earth receives from the Sun is more than enough to make the Earth a fantastic and cozy home that offers us warmth, food and oxygen to breathe. But if we intend to use significant amounts of that energy for productive means or entertainment then perhaps we should ask ourselves whether we should really burden the resources of our planet with that strain. When child play starts putting excessive demands on the in-house environment then, in the old days, parents wouldn’t ask kids to tone it down and use screen-based ‘play’ as an alternative. They would simply tell them to take the game outside. Old-fashioned as that may sound, it is exactly what we Earthlings need to do.

The Solar System is an Economy of plenty, not of scarcity. It has plenty of energy, plenty of space, plenty of every conceivable resource and … best of all … if we go out there to make use of it we aren’t taking it away from anybody else. One day we must muster the courage to do exactly that. That day will mark a watershed such as has never been seen before in the economics of Humankind.

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