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I mathematically proved that the laws of the spring apply to the law of gravity, and I was able to determine the speed of the planet mathematically, the period it takes to complete a complete revolution around the sun, and the force with which the planet moves through the laws of the spring. This confirms that the force of gravity between the planet and the sun is in the form of a spring, which explains the following: for example, when the Earth is at the closest possible point to the sun, the Earth moves away from the sun, even though the force of gravity is as great as possible. This behavior of maximum compression of the spring is called the maximum compression gravitational force. It also explains that when Earth is at the furthest possible point from the sun, it approaches the sun again, even though the force of gravity is as small as possible. This behavior of the maximum expansion of the spring is called the maximum expansion of the gravitational force.

Introduction

The solar system is a vast and complex system held together by gravity. The Sun, which contains 99.8% of the solar system’s mass, is the center of this gravitational pull. Planets, moons, asteroids, comets, and dust all orbit the sun according to gravity laws. Gravity is the force of attraction between two mass objects. The greater an object’s mass, the greater its gravitational pull. The Sun has a very large mass, and thus, its gravitational pull is very strong. This is why planets orbit the sun in nearly circular paths. Planets also have their own gravitational pull. This is why the Moon orbits the planets. For example, the Moon orbits the Earth because of its gravitational pull. Solar power systems are constantly evolving. The planets slowly moved away from the sun, and the moon slowly moved away from the planets. This is because of the planets’ force and the moon’s gravity. Gravity is also responsible for the formation of stars and galaxies. When a large cloud of gas and dust collapses under gravity, it forms a star. The star’s gravity then pulls more gas and dust, which eventually forms planets. Without gravity, the solar system cannot exist. The planets would not orbit the sun, the moons would not orbit the planets, and stars and galaxies would not form. Gravity is a fundamental force in nature that supports life. Gravity is a fascinating and mysterious phenomenon. We are still learning how it works, but we know it is essential for the solar system and the universe’s existence. See the NASA data in Table I [1].

Mercury Venus Earth Moon Mars Jupiter Saturn Uranus Neptune Pluto
Mass (1024 kg) 0.330 4.87 5.97 0.073 0.642 1898 568 86.8 102 0.0130
Distance from sun (106 km) 57.9 108.2 149.6 0.384 228.0 778.5 1432.0 2867.0 4515.0 5906.4
Perihelion (106 km) 46.0 107.5 147.1 0.363 206.7 740.6 1357.6 2732.7 4471.1 4436.8
Aphelion (106 km) 69.8 108.9 152.1 0.406 249.3 816.4 1506.5 3001.4 4558.9 7375.9
Orbital period (days) 88.0 224.7 365.2 27.3 687.0 4331 10,747 30,589 59,800 90,560
Orbital velocity (km/s) 47.4 35.0 29.8 1 24.1 13.1 9.7 6.8 5.4 4.7
Table I. NASA Data

Methods

The Earth revolves around the sun in a complete cycle during a specific period in a fixed orbit. This cycle was repeated continuously, as shown in Fig. 1.

Fig. 1. Earth's orbit around the sun.

As we study the Earth’s movement around the sun, we find that it is mathematically similar to the spring’s movement. When the earth reaches its maximum distance from the sun, 152 × 106 km, it represents the spring’s expansion. Then, once the earth reaches the closest distance to the sun at 147.1 × 106 km, it represents the spring’s contraction. This movement is repeated by the earth moving away and approaching the sun, as if the spring were expanding and contracting, as shown in Fig. 2.

Fig. 2. The maximum and minimum distance to earth.

Let’s calculate the spring constant k using F=kx, [2] but we need to find the value of F force of gravity using Newton’s law F=Gm1m2d2.

G = 6.673 × 10 11 m 3 / kg   s 2 ; S u n   m 1 = 1.988 × 10 30   kg ; E a r t h   m 2 = 5.97 × 10 24   kg

First, calculate F when the earth is in the closest place to the sun, which is d = 147.1 × 106 km.

F = 6.673 × 10 11 1.988 × 10 33 × 5.97 × 10 24 ( 147.1 × 10 9 ) 2
F = 3.66 × 10 22 N

To calculate the spring constant k, we use the following formula F=kx:

3.66 × 10 22 = k × 147.1 × 10 9
k = 24.88 × 10 10 N / m

Secondly, the calculation of F when the earth is in the farthest place from the sun which is d = 152.1 × 106 km.

F = 6.673 × 10 11 1.988 × 10 33 × 5.97 × 10 24 ( 152.1 × 10 9 ) 2
F = 3.423 × 10 22 N

To calculate the spring constant k, we use the following formula F=kx:

3.423 × 10 22 = k × 152.1 × 10 9
k = 22.51 × 10 10 N / m

The rest of the solar system is shown in Table II. The next step is to prove the validity of the k value for the entire solar system.

Mercury Venus Earth Moon Mars
Mass (1024 kg) 0.330 4.87 5.97 0.073 0.642
Distance from Sun (106 km) 57.9 108.2 149.6 0.384 228.0
Perihelion (106 km) 46.0 107.5 147.1 0.363 206.7
Aphelion (106 km) 69.8 108.9 152.1 0.406 249.3
F force of gravity Perihelion (N) 2.069 × 1022 5.59 × 1022 3.66 × 1022 2.2 × 1020 1.99 × 1021
F force of gravity Aphelion (N) 8.98 × 1021 5.44 × 1022 3.424 × 1022 1.7 × 1020 1.37 × 1021
k Perihelion (N/m) 44.98 × 1010 52 × 1010 24.88 × 1010 60.79 × 1010 0.9643 × 1010
k Aphelion (N/m) 12.87 × 1010 50 × 1010 22.51 × 1010 43.45 × 1010 0.5496 × 1010
Jupiter Saturn Uranus Neptune Pluto
Mass (1024 kg) 1898 568 86.8 102 0.0130
Distance from Sun (106 km) 778.5 1432.0 2867.0 4515.0 5906.4
Perihelion (106 km) 740.6 1357.6 2732.7 4471.1 4436.8
Aphelion (106 km) 816.4 1506.5 3001.4 4558.9 7375.9
F force of gravity Perihelion (N) 4.59 × 1023 4.08 × 1022 1.54 × 1021 6.76 × 1020 8.76 × 1016
F force of gravity Aphelion (N) 3.77 × 1023 3.32 × 1022 1.27 × 1021 6.51 × 1020 3.1 × 1016
k Perihelion (N/m) 61.98 × 1010 3.01 × 1010 5.6 × 108 1.5 × 108 1.97 × 104
k Aphelion (N/m) 46.27 × 1010 2.2 × 1010 4.2 × 108 1.4 × 108 4.29 × 103
Table II. k Constant

Proofing 1

The speed at which the planets of the solar system rotate around the sun is known. According to NASA data, Earth’s speed around the sun is 29.8 km/s. Spring movement can be used to calculate the speed of the attached body.

The following law v=Ak/m [2] uses the spring constant to calculate the speed. For Earth at A = 147.1 × 109 m, k = 24.88 × 1010 N/m, m = 5.97 × 1024 kg.

v p r e i h e l i o n = 147.1 × 10 9 24.88 × 10 10 5.97 × 10 24
v p r e i h e l i o n = 30.03  km / s
Earth at  A = 152.1 × 10 9 m , k = 22.51 × 10 10 N / m , m = 5.97 × 10 24 kg .
v a p h e l i o n = 152.1 × 10 9 22.51 × 10 10 5.97 × 10 24
v a p h e l i o n = 29.53  km / s
v a v e r a g e = 30.03 + 29.53 2 = 29.784  km / s

Using the spring law, we calculated the spring constant, which corresponds to the observed speed of Earth. This proves our theory that a force other than gravity exists. Table III lists all members of the solar system. The spring constant in this table accurately calculates the speed of the planet’s flow, which is comparable to NASA’s calculations.

Mercury Venus Earth Moon Mars
NASA orbital velocity (km/s) 47.4 35.0 29.8 1 24.1
k Perihelion (N/m) 44.98 × 1010 52 × 1010 24.88 × 1010 60.79 × 1010 0.9643 × 1010
k Aphelion (N/m) 12.87 × 1010 50 × 1010 22.51 × 1010 43.45 × 1010 0.5496 × 1010
Velocity Perihelion related to k (km/s) 53.70 35.12 30.03 1.04 25.33
Velocity Aphelion related to k (km/s) 43.60 34.90 29.53 0.99 23.06
Velocity average related to k (km/s) 48.65 35.015 29.784 1.019 24.2
Jupiter Saturn Uranus Neptune Pluto
NASA orbital velocity (km/s) 13.1 9.7 6.8 5.4 4.7
k Perihelion (N/m) 61.98 × 1010 3.01 × 1010 5.6 × 108 1.5 × 108 1.97 × 104
k Aphelion (N/m) 46.27 × 1010 2.2 × 1010 4.2 × 108 1.4 × 108 4.29 × 103
Velocity Perihelion related to k (km/s) 11.56 9.89 6.96 5.44 5.46
Velocity Aphelion related to k (km/s) 12.74 9.38 6.64 5.39 4.24
Velocity average related to k (km/s) 12.15 9.63 6.8 5.42 4.85
Table III. Velocity

Proofing 2

We know that the Earth’s rotation period around the sun is 365.2 days, according to NASA data. According to this law, T=2πm/k [2] we can calculate the duration of the Earth’s rotation around the sun in terms of the spring constant.

If k=24.88×1010N/m,m=5.97×1024kg,

T p e r i h e l i o n = 2 π 5.97 × 10 24 24.88 × 10 10
T p e r i h e l i o n = 30778102.8 s = 356.23 d a y s

If k=22.51×1010N/m,m=5.97×1024kg,

T a p h e l i o n = 2 π 5.97 × 10 24 22.51 × 10 10
T a p h e l i o n = 32357822.35 s = 374.51 d a y s
T a v e r a g e = 356.23 + 374.51 2 = 365.37 d a y s

The spring constant was proven to be valid by determining how long the Earth takes to complete an entire cycle around the sun. We performed this for all solar system members in Table IV. In comparison with NASA data on planets far from the sun, there was an error rate of no more than 2.5%.

Mercury Venus Earth Moon Mars
NASA orbital period (days) 88.0 224.7 365.2 27.3 687.0
k Perihelion (N/m) 44.98 × 1010 52 × 1010 24.88 × 1010 60.79 × 1010 0.9643 × 1010
k Aphelion (N/m) 12.87 × 1010 50 × 1010 22.51 × 1010 43.45 × 1010 0.5496 × 1010
T Perihelion period (days) 62.28 222.55 356.23 25.20 593.37
T Aphelion period (days) 116.44 226.95 374.51 29.80 785.97
T average period (days) 89.36 224.75 365.37 27.50 689.67
Jupiter Saturn Uranus Neptune Pluto
NASA orbital period (days) 4331 10,747 30,589 59,800 90,560
k Perihelion (N/m) 61.98 × 1010 3.01 × 1010 5.6 × 108 1.5 × 108 1.97 × 104
k Aphelion (N/m) 46.27 × 1010 2.2 × 1010 4.2 × 108 1.4 × 108 4.29 × 103
T Perihelion period (days) 4,024.28 9,989.80 28,630.70 59,968.14 59,075.13
T Aphelion period (days) 4,657.62 11,685.00 33,059.88 62,072.92 126,592.84
T average period (days) 4,340.95 10,837.40 30,845.29 61,020.53 92,833.99
Table IV. Period

Energy Stored in the Spring

In springs, the equilibrium position is indicated by x = 0.00 m, which is the position where the energy contained in the spring is zero. In a spring, U=1/2kx2 [2] is the potential energy stored when it’s stretched or compressed at a distance x. If the spring constant of the earth is k = 24.88 × 1010 N/m and x = 147.1 × 109 m, then,

U p e r i h e l i o n = 1 2 24.88 × 10 10 × ( 147.1 × 10 9 ) 2
U p e r i h e l i o n = 2.69 × 10 33 J

If the spring constant of the earth is k = 22.51 × 1010 N/m and x = 152.1 × 109 m, then,

U a p h e l i o n = 1 2 22.51 × 10 10 × ( 152.1 × 10 9 ) 2
U a p h e l i o n = 2.6 × 10 33 J
U a v e r a g e = ( 2.69 × 10 33 ) + ( 2.6 × 10 33 ) 2 = 2.64 × 10 33 J

By using the earth’s spring constant, we calculated the energy stored in the spring. Table V shows all the planets in our solar system.

Mercury Venus Earth Moon Mars
Perihelion (106 km) 46.0 107.5 147.1 0.363 206.7
Aphelion (106 km) 69.8 108.9 152.1 0.406 249.3
k Perihelion (N/m) 44.98 × 1010 52 × 1010 24.88 × 1010 60.79 × 1010 0.9643 × 1010
k Aphelion (N/m) 12.87 × 1010 50 × 1010 22.51 × 1010 43.45 × 1010 0.5496 × 1010
Energy U Perihelion (J) 4.75 × 1032 3.0 × 1033 2.69 × 1033 4.0 × 1028 2.059 × 1032
Energy U Aphelion (J) 3.13 × 1032 2.96 × 1033 2.6 × 1033 3.58 × 1028 1.07 × 1032
Energy U average (J) 3.94 × 1032 2.98 × 1033 2.64 × 1033 3.79 × 1028 1.564 × 1032
Jupiter Saturn Uranus Neptune Pluto
Perihelion (106 km) 740.6 1357.6 2732.7 4471.1 4436.8
Aphelion (106 km) 816.4 1506.5 3001.4 4558.9 7375.9
k Perihelion (N/m) 61.98 × 1010 3.01 × 1010 5.6 × 108 1.5 × 108 1.97 × 104
k Aphelion (N/m) 46.27 × 1010 2.2 × 1010 4.2 × 108 1.4 × 108 4.29 × 103
Energy U Perihelion (J) 1.699 × 1035 2.77 × 1034 2.09 × 1033 1.499 × 1033 1.938 × 1029
Energy U Aphelion (J) 1.54 × 1035 2.49 × 1034 1.89 × 1033 1.45 × 1033 1.166 × 1029
Energy U average (J) 1.619 × 1035 2.63 × 1034 1.99 × 1033 1.47 × 1033 1.55 × 1029
Table V. Energy Stored Related to k Constant

Proofing Energy Stored in the Spring

Based on the planet’s mass and velocity, we can calculate the kinetic energy of the planet using the law of KE=1/2mv2 [2] see Table VI. Earth example:

K E = 1 2 5.97 × 10 24 × ( 29800 ) 2
K E = 2.65 × 10 33 J
Mercury Venus Earth Moon Mars Jupiter Saturn Uranus Neptune Pluto
Mass (1024 kg) 0.330 4.87 5.97 0.073 0.642 1898 568 86.8 102 0.0130
Orbital velocity (km/s) 47.4 35.0 29.8 1 24.1 13.1 9.7 6.8 5.4 4.7
Energy KE (J) 3.7 × 1032 2.98 × 1033 2.65 × 1033 3.65 × 1028 1.86 × 1032 1.628 × 1035 2.67 × 1034 2.0 × 1033 1.487 × 1033 1.435 × 1029
Table VI. Energy Related to Mass and Velocity

The energy oscillates between the kinetic energy of the mass and the potential energy stored in the spring. The energy oscillates back and forth between the kinetic energy and potential, changing completely from one form of energy to the other as the system oscillates. The motion starts with all energy stored in the spring as the elastic potential energy. As the planet moves, the elastic potential energy is converted into kinetic energy, which becomes entirely kinetic energy at the aphelion position. Some kinetic energy results in Table VI are quite close to those in Table V, which is the energy stored in the spring. If we apply the velocities found in Table III, we obtain results that are identical to those found in Table V (see Table VII).

Mercury Venus Earth Moon Mars Jupiter Saturn Uranus Neptune Pluto
Mass (1024 kg) 0.330 4.87 5.97 0.073 0.642 1898 568 86.8 102 0.0130
Velocity avearge related to k (km/s) 48.65 35.015 29.784 1.019 24.2 12.15 9.63 6.8 5.42 4.85
Energy KE (J) 3.9 × 1032 2.98 × 1033 2.64 × 10 3.79 × 1028 1.879 × 1032 1.4 × 1035 2.63 × 1034 2.0 × 1033 1.49 × 1033 1.43 × 1029
Table VII. Energy Stored Related to Our Velocity Calculation

This confirms the connection between the planet and the sun, which is the center of the solar system. This confirms that the Spring laws apply to this force.

Conclusion

In a polyatomic gas, we may choose to think of the molecule as consisting of a number of masses (atoms) connected by springs [3]. Mathematically, the gravity between the planet and the center around which it rotates behaves like a spring. This is because the force of gravity between two objects is linked to the product of their masses. It is inversely proportional to the square of the distance between them. This is the same relationship as Hooke’s law, which describes the force of a spring as proportional to the distance it is stretched or compressed. The spring constant measures spring stiffness. It is defined as the force required to stretch or compress the spring by a unit distance.

The force of gravity between a planet and the sun can be determined as the spring constant using the law F=kx.

The orbital speed is the speed at which the planet travels around the sun. This can be determined using the law v=Ak/m.

The orbital period is the time it takes for the planet to complete one revolution around the sun. This can be determined using the law T=2πm/k.

The potential energy of a planet is the energy it has due to its position in the sun’s gravitational field. The potential energy is stored in spring. When the planet moves away from the sun, potential energy is released and converted into kinetic energy. Kinetic energy can be determined using the law U=1/2kx2.

Spring laws can also be used to explain galaxies and other astronomical objects’ motion.

References

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