Does the sun really transmit our annual energy needs in 1 hour?

Abstract

This mini paper explores the mathematical claim that the amount of energy the sun delivers to the Earth in a single hour can meet global energy needs for a year. The underlying assumption is that the world's total annual energy consumption is around 170,000 TeraWatt-hours (TWh) [1], which is consistent with pre-2023 data.

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I. Introduction

The sun, an abundant natural energy source, continually showers the Earth with enormous quantities of energy. Some assert that the energy it delivers to Earth in just one hour could satiate the planet's annual energy needs. This claim's validity is often taken for granted, yet it is rarely scrutinized with mathematical rigor. This paper aims to create a simple model to investigate the veracity of this statement.

II. Model and Assumptions

We model the Earth as a sphere with a radius of approximately 6,371 kilometers (km), and the sun's energy is assumed to be uniformly distributed over the Earth's surface exposed to the sun. Solar radiation at the top of Earth's atmosphere, also known as solar constant, is approximately 1,361 Watts per square meter (W/m²) [2].

III. Calculations

1. First, we calculate the area of the Earth's surface that receives sunlight at any given moment. This is essentially a circle with the same radius as the Earth.

$\text{Area}=\pi r^2=\pi (6,371{)}^2\approx 127,516,117\text{ square km}$

2. Convert this to square meters:

$\text{Area}=127,516,117\text{ km²}\times (1,000,000\text{ m²/km²})=1.27516\times 10^{14}\text{ m²}$

3. Calculate the total power the Earth receives from the sun per second:

$\text{Power = Solar Constant}\times \text{Area}=1,361\text{ Watts/m²}\times 1.27516\times 10^{14}\text{ m²}\approx 1.7361\times 10^{17}\text{ Watts}$

4. To find the energy the Earth receives from the sun in one hour, we multiply the power by the number of seconds in an hour:

$\text{Energy = Power}\times \text{Time}=1.7361\times 10^{17}W\times 3,600\text{ seconds}\approx 6.25\times 10^{20}\text{Joules}$

5. Convert this energy to TeraWatt-hours $\text{(1 TWh}=3.6\times 10^{12}\text{ Joules)}$

$\text{Energy}=6.25\times 10^{20}\text{Joules}\times (1\text{ TWh}/3.6\times 10^{12}\text{ Joules})\approx 173,611\text{ TWh}$

IV. Conclusion

Our simple model suggests that the sun indeed delivers about $173,611\text{ TWh}$ to the Earth in just one hour, exceeding the global annual energy needs of $170,000\text{ TWh}$. However, this does not consider factors like the earth’s tilt, atmospheric absorption, seasonal variations, or the technology required to harness this energy, meaning that the actual obtainable amount of energy that could harness in one hour is likely to be a fraction of the figure we’ve obtained.

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References:
[1] https://ourworldindata.org/energy-production-consumption
[2] Kopp, Greg, and Judith L. Lean. "A new, lower value of total solar irradiance: Evidence and climate significance." Geophysical Research Letters 38.1 (2011).

Note: The calculations in this paper are based on a simplified model and should be treated as rough estimates.