The average temperature of the Earth is determined by the thermal balance, that is, how much energy the planet receives and how much it radiates back into space. When drawing up a balance sheet, energy power values are usually used. Of ~340 W/m² of solar radiation received by the Earth, an average of ~100 W/m² is reflected into space and the remaining ~240 W/m² of solar energy goes to the Earth’s energy budget. The energy budget of the planet is considered in detail in the special literature [9] [10] [11] .

The total radiation power per unit of the surface of a completely black solid, P (W/m²), is related to its temperature T by the fundamental Stefan–Boltzmann law [12] [13] :

P = σ T 4 , (1)

where σ is the Stefan-Boltzmann constant:

σ = 2 π 5 k 4 15 c 2 h 3 = 5.6704 × 10 − 8   J ⋅ s − 1 ⋅ m − 2 ⋅ K − 4 (2)

Here k is Boltzmann’s constant, h is Planck’s constant, and c is the speed of light.

The surface of the Earth is usually approximately regarded as a completely black body. According to this assumption, in order to maintain heat balance, each square meter of the planet’s surface emits an average of 239 W/m² of incoming radiation at a temperature, T_r:

T r = 239 / σ 4 = 255   K (3)

However, the average temperature of the Earth’s surface, T_e, is higher and equal to 288 K. The temperature difference, T_e − T_r = 33 K, is explained by the contribution of the greenhouse effect [6] [7] . Thus, the average temperature of the earth’s surface is equal to the sum of temperatures

T e = T r + 33   K (4)

This increase in temperature due to the greenhouse effect, 33 K, determines the suitability for life and the modern climate of the earth. It should be noted that we distinguish between two stages of the greenhouse effect: the main stage, which lasted for billions of years and determined the Earth’s climate, and the current short (several decades) warming period, which is presumably associated with an increase in carbon dioxide due to the burning of organic fuels.

It is worth emphasizing that the thermal balance is compiled for an uninhabited planet [9] . To quantify the impact of humanity on climate change, one can use the idea of Kardashev, who proposed to evaluate the development of civilization by the amount of energy, E, that it is able to use [14] [15] .

Let’s call the total energy produced by humanity the energy of civilization. The second law of thermodynamics states that the efficiency of any real process is less than 100%: part of the energy is dissipated in the environment in the form of heat, the rest, free energy, can be used for the needs of humanity. According to the first law of thermodynamics, free energy cannot disappear, and in the process of producing useful work, it is also dissipated as heat in the atmosphere of the planet. Thus, on the scale of the Earth, all the energy of civilization is inevitably converted into the heat of civilization, which is dissipated in the atmosphere.

Since single- and diatomic air molecules do not absorb or emit electromagnetic radiation in the real temperature range, the heat of civilization is dissipated by increasing the kinetic energy of air molecules (mainly N₂, O₂ and Ar) when colliding with more heated objects. Air molecules are held by gravity and remain on Earth; thus, the energy of civilization accumulates on the planet in the form of kinetic energy of molecules, and its total value, E_t, is summed up over a time interval:

E t = ∑ i 1 i 2 E i (5)

where, E_i is the annual energy output; i₁ and i₂ are the beginning and end of the interval, respectively. Therefore, the influence of civilization’s energy on temperature is constantly increasing; this is the additional energy (compared to the pre-industrial era) that the planet receives from a developing civilization. Although only free energy is used for the development of civilization, thermal pollution is caused by the total energy generated by humanity.

The sources of this energy, fossil fuels, radioactive substances, winds, rivers, and tides, exist in nature regardless of whether the earth is inhabited or not. Questions arise: “Why does the energy from these resources affect the thermal balance of a planet with a developed civilization? And why are these resources actually neglected in the case of an uninhabited planet?” The main reason is the speed of energy release. In nature, these sources slowly dissipate their energy, while humanity forces them to release free energy for a short time. The ratio of speeds can reach the order of a million and a billion. Below are two illustrations. In nature, the energy of rivers is slowly dispersed over thousands of kilometers of riverbeds, whereas on hydroelectric dams it is released within seconds. Fossil fuels and radioactive substances have existed for millions and billions of years, but their internal energy is quickly released in the corresponding chain reactions. Obviously, if the speed of natural processes is millions of times slower, then the contribution of these sources to the heat balance of an uninhabited planet per second, that is, the power of these sources, is negligible. The same reasoning can be carried out with respect to the energy density per m²: in technology, it can be millions of times higher.

In order to assess the impact of the energy of civilization on global warming, let’s assume that the warming is caused only by the energy of civilization, and the influence of all other factors does not change during the period under consideration. Then it follows from Equation (4) that the difference in earth temperatures ΔT_e for the studied period is equal to the difference in radiation temperatures ΔT_r for the same period:

Δ T e = Δ T r (6)

Equation (6) makes it possible to estimate ΔT_e proceeding from the Stefan-Boltzmann equation, since the increment of the temperature of the surface layer ΔT_e turns out to be equal to the change in the temperature of radiation ΔT_r. As a result of the calculation, we will get the contribution of the energy of civilization to the temperature increase, which is less than the observed global warming; the difference between the latter and calculated values is determined by the effect of increasing concentration of carbon dioxide in the atmosphere.

To calculate, it is necessary to convert the annual energy production E_i into an increase in the average power of civilization per m² of the planet’s surface, ε i . Obviously, ε i = E i / ( ( 5.616 × 10 6 ) × ( 5.1 × 10 14 ) ) , where 5.616 × 10⁶ is the number of seconds per a year, and 5.1 × 10¹⁴ is the surface area of the Earth (m²). Then it follows from Equation (6) that the energy of civilization for the i-th year will cause the following increase in temperature:

Δ T i = P s + ∑ i ε i σ 4 − P s + ∑ i − 1 ε i σ 4 , (7)

where ∑ i ε i and ∑ i − 1 ε i are the sums of the energies of civilization (SEC), for i и (i − 1) years, respectively.

The civilization’s energy began to increase noticeably only after the beginning of the industrial era, i.e. after ~1800 year, whereas before it was close to zero. Strictly speaking, both expressions on the right side of the equation (7) should include not P_s, but the value (P_s + α), where α is the additional energy of civilization before the beginning of the period of interest to us. However, since α ≪ P s and due to the subtraction operation, the contribution of α is insignificant and can be neglected. Adding up the annual temperature rises, after some algebra, we get that the total energy of civilization for the period from i₁ to i₂ year will cause the following warming, ∆T:

Δ T = P s + ∑ i ε i σ 4 − P s σ 4 (8)

Here ∑ i ε i is SEC from i₁ to i₂. Since the energy of civilization is much less than the power of solar radiation, ∑ i ε i ≪ P s , we can generate a power series expansion for Equation (8) about the point ∑ i ε i and limit ourself to the first term:

Δ T = 1 σ 4 ∑ i ε i ( P s + ω ) 3 / 4 (9)

If we (a) determine the proportionality coefficient, γ = 1 σ 4 , and (b) substitute

Equation (2) into it, we obtain an expression for γ in terms of basic physical constants:

γ = 1 4 π 5 / 4 ( 15 2 ) 1 / 4 ( c 2 h 3 k 4 ) 1 / 4 = 16.2008   K ⋅ m 0.5 ⋅ W − 0.25 (10)

and come to the final expression:

Δ T ≈ 16.2008 ∑ i ε i P s 3 / 4 (11)

Equation (11) is a very good approximation to the exact expression.

Table 1 presents (a) the values of the SEC based on the annually published data on energy production and (b) the calculated contributions of the energy of civilization to the overall warming. The data is presented for three arbitrary time periods.

The radiation surface coincides with the thin surface layer of the biosphere, in

^а [16] ; ^b [17] ; ^c [18] .

which human activity is concentrated. Comparison of calculated values of ΔT (Equation (11)) and experimental values of global warming, ΔT_exp, shows that the energy generated by humanity can cause up to half the magnitude of global warming. In the future, with the growth of energy production, this share may increase even more. Therefore, coordinated measures to limit energy consumption are required to mitigate the effects of warming. The table also reveals that the SEC power, 1.40 W/m², for 70 years from 1950 to 2020 is equivalent to more than 0.5% of the solar radiation power on the earth’s surface.

If to use a metaphor, the contribution of civilization to global warming can be compared to the action of a heater that heats a closed room as people living in it acquire the necessary knowledge and begin to use energy sources stored in the same room.

It should be noted some important points in assessing the contribution of the energy of civilization. The latter is the sum of all types of energy produced by mankind regardless of sources, i.e., it includes renewable and non-renewable, carbon-free and organic energy sources, that is, all sources without exception. This means, for example, that one kilowatt of energy received from a wind turbine, coal or nuclear power plant, etc. cause the same thermal pollution. Here we compare the energy sources only from the point of view of their impact on thermal pollution, leaving aside other environmental effects. Even if it were possible to switch to carbon-free energy and stop the temperature rise due to the greenhouse effect, the heating of atmosphere due to the energy of civilization will continue and even increase if measures are not taken to limit energy production.