**4. Einstein's equation and the nuclear fusion**

Nuclear fusion is a nuclear reaction that causes two or more nuclei to collide at very high speeds and merge to form a new type of atomic nucleus. Sometimes, the energy needed to initiate this process is provided by a very high "working" pressure (e.g., inside the stars this pressure is provided by the gravity determined by their mass).

By fusion (the union of nuclei) easier than iron, more energy than is necessary is produced, in order to form bonds for the newly formed nucleus. As a result, by the fusion of "light" nuclei, energy can be obtained.

**Note**: For fusion of nuclei heavier than iron, in order to achieve the necessary connections to maintain the cohesion of new nuclei, external energy contribution (consumption) is required.

**Figure 6** shows a scheme of the fusion reaction between a deuterium atom (H2 ) and a tritium atom (H3 ), after which a helium (He4 ) atom is formed. From the reaction also results a neutron.

The amount of energy produced is 17.59 MeV = 2.8·10<sup>−</sup>12 J, which is in agreement with Einstein's equation, considering the equivalent loss of mass as a result of the fusion reaction [18–20].

The only application of the artificially produced fusion is the hydrogen bomb. **Figure 7** shows the explosion of the first hydrogen bomb, whose code name was "Ivy Mike" (November 1, 1952).

**71**

**Figure 6.**

**Figure 7.**

*tritium\_fusion.svg).*

Experimental Reactor (ITER) [21].

*Einstein's Equation in Nuclear and Solar Energy DOI: http://dx.doi.org/10.5772/intechopen.90574*

One of the most important scientific research projects, which aim to obtain energy through fusion, for peaceful use, is the International Thermonuclear

*Scheme of the fusion reaction between deuterium and tritium (https://en.wikipedia.org/wiki/File:Deuterium-*

countries, the USA, Japan, Russia, China, South Korea, and India.

*Explosion of the first hydrogen bomb (https://en.wikipedia.org/wiki/File:IvyMike2.jpg).*

**Figure 8** presents the small-scale model of the ITER fusion reactor.

will consume approx. 50 MW to ensure its own energy consumption.

The project is carried out in collaboration with many countries: European Union

The ITER fusion reactor was designed to produce 500 MW of final energy and

The construction of the ITER complex began in 2013, with the cost of construction now reaching \$ 16 billion USD, almost three times more than originally expected. The infrastructure is expected to be completed in 2025, the commissioning of the reactor to be completed in the same year, as well. Plasma experiments should start in 2025, and deuterium-tritium fusion experiments should begin in 2027.

*Einstein's Equation in Nuclear and Solar Energy DOI: http://dx.doi.org/10.5772/intechopen.90574*

#### **Figure 6.**

*Thermodynamics and Energy Engineering*

○ A list of the longest ships in the world is available on the Internet: http://

○ The inscription on the main deck was made by the aircraft carrier's crew to

○ The aircraft carrier USS Enterprise was decommissioned due to the long period of operation of the nuclear propulsion system and due to the equipment on the main deck, which allowed the radar position to be detected on the aircraft carrier. The aircraft carriers of the "Nimitz" class are of "stealth"

**Figure 5** presents the scheme of operation for the nuclear propulsion system of

Nuclear fusion is a nuclear reaction that causes two or more nuclei to collide at very high speeds and merge to form a new type of atomic nucleus. Sometimes, the energy needed to initiate this process is provided by a very high "working" pressure (e.g., inside the stars this pressure is provided by the gravity determined by their

By fusion (the union of nuclei) easier than iron, more energy than is necessary is produced, in order to form bonds for the newly formed nucleus. As a result, by the

**Note**: For fusion of nuclei heavier than iron, in order to achieve the necessary connections to maintain the cohesion of new nuclei, external energy contribution

**Figure 6** shows a scheme of the fusion reaction between a deuterium atom (H2

The amount of energy produced is 17.59 MeV = 2.8·10<sup>−</sup>12 J, which is in agreement with Einstein's equation, considering the equivalent loss of mass as a result of the

The only application of the artificially produced fusion is the hydrogen bomb. **Figure 7** shows the explosion of the first hydrogen bomb, whose code name was

), after which a helium (He4

)

) atom is formed. From the reac-

en.wikipedia.org/wiki/List\_of\_longest\_naval\_ships.

*Scheme of the nuclear propulsion system of ships and submarines (http://www.subadventures.net/*

mark 40 years of naval nuclear propulsion.

type (hidden or not detectable on the radar).

**4. Einstein's equation and the nuclear fusion**

fusion of "light" nuclei, energy can be obtained.

ships and submarines.

(consumption) is required.

tion also results a neutron.

and a tritium atom (H3

fusion reaction [18–20].

"Ivy Mike" (November 1, 1952).

**70**

mass).

**Figure 5.**

*Sub\_04\_719\_files/image018.jpg).*

*Scheme of the fusion reaction between deuterium and tritium (https://en.wikipedia.org/wiki/File:Deuteriumtritium\_fusion.svg).*

#### **Figure 7.**

*Explosion of the first hydrogen bomb (https://en.wikipedia.org/wiki/File:IvyMike2.jpg).*

One of the most important scientific research projects, which aim to obtain energy through fusion, for peaceful use, is the International Thermonuclear Experimental Reactor (ITER) [21].

The project is carried out in collaboration with many countries: European Union countries, the USA, Japan, Russia, China, South Korea, and India.

**Figure 8** presents the small-scale model of the ITER fusion reactor.

The ITER fusion reactor was designed to produce 500 MW of final energy and will consume approx. 50 MW to ensure its own energy consumption.

The construction of the ITER complex began in 2013, with the cost of construction now reaching \$ 16 billion USD, almost three times more than originally expected.

The infrastructure is expected to be completed in 2025, the commissioning of the reactor to be completed in the same year, as well. Plasma experiments should start in 2025, and deuterium-tritium fusion experiments should begin in 2027.

#### **Figure 8.**

*Small-scale model of ITER (https://upload.wikimedia.org/wikipedia/commons/thumb/7/75/ITER\_Exhibit\_% 2801810402%29\_%2812219071813%29\_%28cropped%29.jpg/250px-ITER\_Exhibit\_%2801810402%29\_%28122190 71813%29\_%28cropped%29.jpg).*

The project contributes to the implementation of the results obtained in decades of research, in an experimental installation, which will allow the transition to a commercial installation.

### **5. Einstein's equation and the solar energy**

The most representative example of fusion is represented by the reactions inside the Sun.

The Sun represents the energy source of the Earth [22], contributing to maintain the planet's temperature well above the value of almost 0 K, encountered in the interplanetary space and is the only source of energy capable to sustain life on Earth [23].

The Sun can be considered as a sphere with a diameter of approximatively 1.4 million km, more precisely 1.39 × 109 m [24], at a distance of approx. 150 million km from the Earth, 1.5 × 1011 m [24]. This distance is so high that two straight lines that start from one point on the Earth's surface to two diametrically opposite points on the solar disk form an angle of approximately half a degree. Under these conditions, although solar radiation is emitted in all directions, it can be considered that the solar rays that reach the Earth's surface are parallel [25].

In the core of the Sun, continuous nuclear fusion reactions occur, by which hydrogen is converted into helium. Currently, the mass composition of the Sun is approx. 71% hydrogen, 27.1% helium, 0.97% oxygen, and other elements in lower concentrations [26].

The rate of conversion of hydrogen into helium is approx. 4.26 million tons per second [27, 28]. This flow of substance is continuously transformed into energy. It is estimated that at this rate, in the next 10 million years, approximatively 1% of the current amount of hydrogen will be consumed, so there is no imminent danger of depletion of the Sun's energy source. The lifetime of the Sun is estimated at approximatively 4–5 billion years.

Considering the mass flow of solar substance that is consumed continuously turning into energy ṁ = 4.26 million t/s = 4.26·109 kg/s, the thermal power of the solar radiation emitted as a result of this process (P) can be calculated starting from the famous equation of Einstein for energy calculation (E):

**73**

*Einstein's Equation in Nuclear and Solar Energy DOI: http://dx.doi.org/10.5772/intechopen.90574*

where c = 300,000 km/s = 3·108

where SS = 6.08·1012 km2

Replacing it, we obtain:

Sun, we obtain:

relation:

Cernavodă.

determined as:

E = m · c<sup>2</sup>

[J];P = ṁ ⋅ c<sup>2</sup>

Substituting the relation of the thermal power of the radiation emitted by the

The specific power of the radiation emitted by the Sun (PS), representing the power of the radiation emitted by the surface unit, can be calculated with the

PS = P/ SS [W/ m2

PS = 38.34 · 1025/6.08 · 10<sup>18</sup> = 63.059 · 10<sup>6</sup> W/ m<sup>2</sup> = 63.059 MW/ m<sup>2</sup>

words, every square meter of the Sun's surface emits energy characterized by a thermal power approximately equivalent to one tenth of the power of a reactor from

PS = σ · T<sup>4</sup>

T = 4 √

Since the Sun emits radiation over all wavelengths, it can be considered an absolute black body [29, 30], and the power emitted in the unit of time, on the surface unit, by an absolute black body (PS) depends only on its temperature and can be

Using the above relationship, the value of the Sun's surface temperature can be

\_ \_ PS

This value corresponds to that indicated by most bibliographic sources, which

It can be considered that the solar radiation is emitted uniformly in all directions and can be found throughout the solar system. The intensity of the available solar radiation due to this mechanism obviously depends on the distance to the Sun, and

The core temperature of the Sun is estimated to vary between (8 and 40)·106

[W/ m2

[W/m2

K4

<sup>σ</sup> [K] (7)

For comparison, it is mentioned that the maximum power developed by the Renault engine K7M (1.6 MPI), which equips some models of the Renault Group cars, is 64 kW, at a maximum speed of 5500 rpm. Thus, the specific power of the radiation emitted by the Sun (PS) is approximately equivalent to that of 1000 engines that equip these cars, which operate at maximum speed. Considering that the length of a car is 4.25 m, those 1000 cars placed one after the other, in a straight line, "bar to bar" would stretch 4.25 km. Also for comparison, the net power of a nuclear reactor from Cernavodă (655 MW) represents about 10 times more than

= 6.08·1018 m2

the specific power of the radiation emitted by the Sun (63 MW/m<sup>2</sup>

calculated according to Boltzmann's law, with the relation:

where σ is Boltzmann's constant: σ = 5.67·10<sup>−</sup><sup>8</sup>

Replacing it, we obtain T = 5774 K ≈ 5500°C.

also confirms that all undertaken calculations are correct.

temperature of the black body (of the Sun) [K].

m/s is the speed of light.

P = 4.26 · 10<sup>9</sup> · 3<sup>2</sup> · 108·2 = 38.34 · 10<sup>25</sup> W. (3)

[W]. (2)

]. (4)

. (5)

). In other

]. (6)

]; T is the absolute

K [14].

is the total surface area of the Sun.

*Einstein's Equation in Nuclear and Solar Energy DOI: http://dx.doi.org/10.5772/intechopen.90574*

*Thermodynamics and Energy Engineering*

commercial installation.

*71813%29\_%28cropped%29.jpg).*

concentrations [26].

matively 4–5 billion years.

the Sun.

**Figure 8.**

**5. Einstein's equation and the solar energy**

1.4 million km, more precisely 1.39 × 109

the solar rays that reach the Earth's surface are parallel [25].

turning into energy ṁ = 4.26 million t/s = 4.26·109

the famous equation of Einstein for energy calculation (E):

The project contributes to the implementation of the results obtained in decades

*Small-scale model of ITER (https://upload.wikimedia.org/wikipedia/commons/thumb/7/75/ITER\_Exhibit\_% 2801810402%29\_%2812219071813%29\_%28cropped%29.jpg/250px-ITER\_Exhibit\_%2801810402%29\_%28122190*

The most representative example of fusion is represented by the reactions inside

The Sun represents the energy source of the Earth [22], contributing to maintain the planet's temperature well above the value of almost 0 K, encountered in the interplanetary space and is the only source of energy capable to sustain life on Earth [23]. The Sun can be considered as a sphere with a diameter of approximatively

km from the Earth, 1.5 × 1011 m [24]. This distance is so high that two straight lines that start from one point on the Earth's surface to two diametrically opposite points on the solar disk form an angle of approximately half a degree. Under these conditions, although solar radiation is emitted in all directions, it can be considered that

In the core of the Sun, continuous nuclear fusion reactions occur, by which hydrogen is converted into helium. Currently, the mass composition of the Sun is approx. 71% hydrogen, 27.1% helium, 0.97% oxygen, and other elements in lower

The rate of conversion of hydrogen into helium is approx. 4.26 million tons per second [27, 28]. This flow of substance is continuously transformed into energy. It is estimated that at this rate, in the next 10 million years, approximatively 1% of the current amount of hydrogen will be consumed, so there is no imminent danger of depletion of the Sun's energy source. The lifetime of the Sun is estimated at approxi-

Considering the mass flow of solar substance that is consumed continuously

solar radiation emitted as a result of this process (P) can be calculated starting from

m [24], at a distance of approx. 150 million

kg/s, the thermal power of the

of research, in an experimental installation, which will allow the transition to a

**72**

$$\mathbf{E} = \mathbf{m} \cdot \mathbf{c}^2 \text{ [J]}; \mathbf{P} = \dot{\mathbf{m}} \cdot \mathbf{c}^2 \text{ [W]}.\tag{2}$$

where c = 300,000 km/s = 3·108 m/s is the speed of light.

Substituting the relation of the thermal power of the radiation emitted by the Sun, we obtain:

$$\text{P} = \text{4.26} \cdot \text{10}^{9} \cdot \text{3}^{2} \cdot \text{10}^{8.2} = \text{38.34} \cdot \text{10}^{25} \text{W}. \tag{3}$$

The specific power of the radiation emitted by the Sun (PS), representing the power of the radiation emitted by the surface unit, can be calculated with the relation:

$$\mathbf{P\_S = P/S\_S \left[W/m^2\right]}.\tag{4}$$

where SS = 6.08·1012 km2 = 6.08·1018 m2 is the total surface area of the Sun. Replacing it, we obtain:

$$P\_{\mathbb{S}} = 38.34 \cdot 10^{25} / 6.08 \cdot 10^{18} = 63.059 \cdot 10^{6} \,\mathrm{W/m}^2 = 63.059 \,\mathrm{MW/m}^2. \tag{5}$$

For comparison, it is mentioned that the maximum power developed by the Renault engine K7M (1.6 MPI), which equips some models of the Renault Group cars, is 64 kW, at a maximum speed of 5500 rpm. Thus, the specific power of the radiation emitted by the Sun (PS) is approximately equivalent to that of 1000 engines that equip these cars, which operate at maximum speed. Considering that the length of a car is 4.25 m, those 1000 cars placed one after the other, in a straight line, "bar to bar" would stretch 4.25 km. Also for comparison, the net power of a nuclear reactor from Cernavodă (655 MW) represents about 10 times more than the specific power of the radiation emitted by the Sun (63 MW/m<sup>2</sup> ). In other words, every square meter of the Sun's surface emits energy characterized by a thermal power approximately equivalent to one tenth of the power of a reactor from Cernavodă.

Since the Sun emits radiation over all wavelengths, it can be considered an absolute black body [29, 30], and the power emitted in the unit of time, on the surface unit, by an absolute black body (PS) depends only on its temperature and can be calculated according to Boltzmann's law, with the relation:

$$\mathbf{P\_S = \sigma \cdot T^4 \left[W/m^2\right]}.\tag{6}$$

where σ is Boltzmann's constant: σ = 5.67·10<sup>−</sup><sup>8</sup> [W/m2 K4 ]; T is the absolute temperature of the black body (of the Sun) [K].

Using the above relationship, the value of the Sun's surface temperature can be determined as: \_

$$\mathbf{T} = \sqrt[4]{\frac{\mathbf{P}\_S}{\sigma}} \quad \text{[K]} \tag{7}$$

Replacing it, we obtain T = 5774 K ≈ 5500°C.

This value corresponds to that indicated by most bibliographic sources, which also confirms that all undertaken calculations are correct.

The core temperature of the Sun is estimated to vary between (8 and 40)·106 K [14]. It can be considered that the solar radiation is emitted uniformly in all directions and can be found throughout the solar system. The intensity of the available solar radiation due to this mechanism obviously depends on the distance to the Sun, and

the thermal power of the solar radiation is evenly distributed on spherical surfaces, with the Sun in the center.

On these considerations, the thermal power of the radiation emitted by the Sun (P = 38.34·1025 W) can be calculated with the relation:

$$\mathbf{P} = \mathbf{I}\_{\mathbf{S}} \cdot \mathbf{S}\_{\mathbf{S}} \text{ [W]}.\tag{8}$$

where IS [W/m2 ] is the intensity of radiation available on the surface unit of a sphere with the Sun in the center; SS [m2 ] is the surface of the sphere on which the intensity of the solar radiation is calculated.

By using the relation presented above, the intensity of the solar radiation related to the surface unit of a sphere having the Sun in the center (IS) can be calculated with the relation:

$$\mathbf{I}\_{\rm S} = \mathbf{P} / \mathbf{S}\_{\rm S} \left[ \mathbf{W} / \mathbf{m}^2 \right]. \tag{9}$$

where SS = 4·π·D2 [m2 ].

Replacing it in the previous relationship, we obtain:

$$\mathbf{I}\_{\rm S} = \mathbf{P}/(\mathbf{4} \cdot \boldsymbol{\pi} \cdot \mathbf{D}^2). \tag{10}$$

**75**

**Author details**

and Mugur C. Balan\*

*Einstein's Equation in Nuclear and Solar Energy DOI: http://dx.doi.org/10.5772/intechopen.90574*

the famous equation of Einstein (E = m·c2

Even if nuclear and solar energies seem to be different domains, the study proved that fission, fusion, and solar energy can be connected and have in common

). Both in fission and fusion, the mass varies during the reactions, and it was highlighted that the mass variation and the released energy are related by the equation

The same equation was also applied to the mass flow of solar substance that is continuously consumed in the solar fusion reactions, and starting from this point, it was possible to calculate important parameters such as the energy and the power

Following this new approach, it was possible to determine the temperature of the sun's surface and the solar constant, both being in agreement with the values

It can be concluded that fission, fusion, and solar energy are linked together by

Ancuta M. Magurean, Octavian G. Pop, Adrian G. Pocola, Alexandru Serban

© 2019 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

Technical University of Cluj-Napoca, Cluj-Napoca, Romania

\*Address all correspondence to: mugur.balan@termo.utcluj.ro

provided the original work is properly cited.

**6. Conclusions**

of Einstein.

emitted by the sun.

provided in literature.

the equation of Einstein.

Thus, the intensity of the available solar radiation at the upper limit of the Earth's atmosphere can be calculated using the previous relation, considering that D is the distance between the Earth and Sun and D = 149,597,871 km = 1.496·108 km = 1.496·1011 m:

$$\mathbf{I}\_{\mathrm{S}} = \mathbf{3}8.34 \cdot \mathbf{10}^{\mathrm{25}} / \left( 4 \cdot \pi \cdot \mathbf{1.1496}^{\mathrm{2}} \cdot \mathbf{10}^{\mathrm{112}} \right) = \mathbf{1.364} \cdot \mathbf{10}^{\mathrm{3}} \, \mathrm{W/m}^2. \tag{11}$$

The intensity of the available solar radiation at the upper limit of the Earth's atmosphere is referred to as the solar constant [31].

The value of the solar constant calculated previously corresponds to the value adopted by the World Radiation Center, of 1367 W/m2 . This value is also reported by numerous bibliographic sources. The value of the solar constant, which is determined by measurements undertaken by satellites, underwent several corrections over time, as can be seen in **Table 1**.

The value of the available solar radiation at the upper limit of the terrestrial atmosphere suffers throughout the year small variations of approx. ± 3%, mainly due to fluctuations in the distance between the Earth and the Sun [24].


#### **Table 1.**

*Accepted values over time for the solar constant.*
