Planets, springs and pendulums

Seeing connections between different areas of physics is a good way to teach physics. In the orbit of a planet, there is a continuous interchange between gravitational potential energy and kinetic energy with the sum being constant. This is essentially the same physics as a mass on the end of a spring, or a pendulum. In this paper, equivalent spring constants are calculated for planetary orbits and the pendulum equation used to derive Kepler’s third law.


Introduction
There are many surprising connections between different areas of physics.For example, Oks (2019) makes connections between the universe and atomic physics.Similarities and analogies are powerful techniques in teaching any subject.This paper describes fundamental connections between the periodic motions of planetary orbits, pendulums, and springs.

Relating celestial and terrestrial physics
Although the orbits of the planets are close to circular (Hughes 2024) throughout their orbits the orbital speed and the distance between the planet and Sun is continuously changing.A planet moves fastest at perihelion and slowest at aphelion.In the rotating reference frame of a line joining the centres of the planet and Sun, the planet would appear to move towards and away from the Sun like a mass on the end of a spring.
Figure 1 shows a plot of the distance of the Earth from the Sun relative to 1 AU throughout a complete orbit starting close to the southern autumnal equinox.The values were collected using Stellarium, freely available planetarium software (www.stellarium.org).(Data in supplementary material).The plot clearly shows that the distance between the Earth and Sun is a sinusoidal function of time.The variation in the distance and speed means there is an interchange between potential energy (PE) and kinetic energy (KE).PE varies with radial distance from the Sun and KE varies with tangential velocity.
As a planet descends to perihelion, it speeds up due to the conversion of PE into KE and as it ascends from perihelion to aphelion, KE is converted into PE.The total amount of energy is constant.This suggests the analogy of a mass on the end of a spring or of a pendulum (figure 2).The diagram is not to scale.On the scale of the solar system, the curvature of the path of a planetary pendulum is close to flat and at right angles to the tangent of the orbit.
Equation (1) relates the change in PE and KE over one orbit.
where r a and r p are respectively the aphelion and perihelion distance, v a the orbital velocity at aphelion and v p the orbital velocity at perihelion, assumed to be the minimum and maximum orbital velocities, m 1 the mass of the Sun, m 2 the mass of the planet, G the gravitational constant, ∆PE the change in gravitational PE and ∆KE the change in KE.Note that since the sum of a sinusoidal wave is zero over one cycle, the 'average' orbit of a celestial object is a circle.Table 1 shows the ∆PE and ∆KE for the planets calculated as in equation ( 1) and the percentage difference between the two values.The percentage difference is less than 1% for all the planets save Neptune which could be due to the influence of Pluto on the orbit of Neptune.

Springs
The elastic PE stored in a spring is given by PE = 1 2 kx 2 .When applied to a planet, x is the radial displacement of the planet from a perfectly circular orbit.Analysing planetary motion by modelling it simply as a mass on the end of a spring that is aligned radially with the orbit, a planet has the maximum KE as it passes through the mid-point between aphelion and perihelion, which is equal to the 'elastic PE', enabling the 'spring constant' of the orbit of a planet to be calculated.We can write, For the orbit of the Earth, x is half the distance between aphelion and perihelion (2.505 × 10 9 m), m the mass of the Earth (5.972 × 10 24 kg), and v the radial velocity of the Earth at an equinox (i.e. a mid-point between aphelion and perihelion) The equation of radial motion is, where r is the radial distance of the Earth from the 1 AU orbit, A, the amplitude taken as half the distance between perihelion and aphelion, 2.505 × 10 9 m, ω the angular frequency (2π /T) where T is the orbital period of the Earth, and t, the time taken for the Earth to reach a certain position in the orbit relative to the southern autumnal equinox.
The maximum radial velocity can be found by differentiating equation ( 3) ) .
To find the velocity at an equinox, mid-way between apogee and perigee when the Earth is moving at maximum velocity, we set the bracket to zero, and obtain, This gives a spring constant for the Earth's orbit of As a cross-check the spring constant can also be calculated from the mass of the Earth and period of the orbit around the Sun Arriving at the same value via two different methods gives confidence that the physics is correct.

Pendulums
The period (T) of a pendulum is given by T = 2π √ l/g, where, l is the length of the string and g the acceleration due to gravity.In the pendulum model shown in figure 2, the Earth is suspended from a 'sky hook' 1 AU tangential to the orbit, with a 'virtual Sun' 1 AU in the other direction.The virtual Sun produces an acceleration due to gravity equal to the acceleration due to gravity at the position of the Earth's orbit.The period of the pendulum is one year.
As a cross check, we can calculate the period of a pendulum with a line length equal to 1 AU in the gravitational field of the Sun at 1 AU.Using a value of 1.495978707 × 10 11 m for one AU and 1.989 × 10 30 kg for the mass of the Sun we arrive at a period equal to 0.999 92 years, within 0.01% of one year.The swing angle of the Earth pendulum is less than 1 • , and therefore the restoring force is proportional to displacement as required for simple harmonic motion.
Kepler's third law can also be derived from the planetary pendulum analogy by inserting the expression for the acceleration due to the gravity of the Sun into the pendulum equation.
M is the solar mass.More properly, the mass of a planet should be added to the mass of the Sun to account for the pull of the Earth on the Sun, Figure 3 shows the spring constants for the eight planets.The plot reveals an interesting pattern.The spring constants of Mercury and Earth, and Venus and Jupiter, appear to be paired.The spring constant of the other four planets, Mars, Saturn, Uranus and Neptune, are all very small.The eccentricity of the Earth's orbit changes over thousands of years which affects the Earth's climate (Milankovitch cycles).The dominant cycle has a period of 405 000 years due to the combined gravitational influence of Venus and Jupiter, which has been stable for hundreds of millions of years (Kent et al 2018).
An interesting question is whether this stability is related to Venus and Jupiter having the highest virtual spring constant of the planets in the solar system or is just a coincidence.This approach could potentially be useful for the search for habitable exoplanets (Horner et al 2020).
The spring and pendulum analogies discussed in this paper are also applicable to binary star systems, stars orbiting black holes and galaxies orbiting the centre of a galactic cluster.

Figure 1 .
Figure 1.A plot of Earth-Sun distance data obtained from Stellarium.The start of the plot is the 2 April 2024, at the southern autumnal equinox, with the Earth at 1.000 AU from the Sun.Each data point is advanced 12 d, except for the last two points-day 360 and 365.Distances are relative to 1 AU.

Figure 2 .
Figure 2. The orbit of a planet is modelled by a mass on the end of a spring and a pendulum.(Diagram not to scale).

Figure 3 .
Figure 3. Equivalent spring constants of the planets.