Why Does Earth Rotate?

1. Introduction

Despite the fluctuations in Earth’s rotational speed being recognized factual for timescales from daily to centennial, the prevailing theory attributes these variations primarily to the conservation of the angular momentum [1] overlooking the complexity of Sun-Earth physics. It is widely accepted that the dynamics of the Equation of Time (EoT) arise from the shape of Earth’s orbit and the tilt of Earth’s rotational axis [2,3]. However, the analemmatic dynamics of the Sun meridian declination (SMD) and the EoT remains underexplored. The analysis on these pages builds on a previous study that assessed the dynamics of the SMD to estimate the latitudinal budget of natural beam irradiance on Earth [4]. The current work evaluates the combined dynamics (velocity, acceleration, net drive, etc.) and synchrony between the SMD and the EoT within a solar noon analemma, aiming to establish a connection to Earth’s rotational speed ($\mathit{\rho }$, km h⁻¹) across daily, monthly, and seasonal timescales.

When the SMD is plotted against the EoT it yields the solar noon analemma, a lemniscate describing the combined horizontal-vertical path of the mean-time noon Sun along the Gregorian year, as perceived from a specific location on Earth’s surface [5]. Following the SMD within a solar noon analemma, the mean-time Sun travels from north to south and backwards, depicting the analemma‘s y-coordinate or the Sun’s vertical path. Following the EoT within the solar noon analemma the mean-time Sun travels from east to west and backwards, depicting the analemma‘s x-coordinate or the Sun’s horizontal path.

The EoT establishes a rhythm for the daily divergence between the mean time and the solar time, specifically between the lengths of the mean day and the solar day [6]. Consequently, if the SMD is linked to the variation in Earth’s rotation along the planet’s orbit, then the EoT reflects not only fluctuations in the length of successive solar days but also variations in Earth’s rotational speed. This hypothesis might seem bold, but the Sun’s vertical path (SMD) and the Sun’s horizontal path (EoT or $\mathit{\rho }$) are synchronized in such a perfect harmony that evidence supports causal in lieu of casual connections.

If natural beam irradiance (NBI) is expected to align with the local meridian at mean time noon, but the EoT occurs to the left or right of the meridian, this deviation evidences a daily fluctuation in both the length of the solar day and Earth’s rotational speed. This proposal assumes the Sun-Earth physics is inherently linked to Earth’s rotation, even challenging the role of angular momentum as the primary driver of Earth’s rotational dynamics. Because the length of the solar day is tied to the Earth’s rotational speed [7], the light cone supplying NBI to the subsolar point [4] comprises the clock hand defining the only time frame Earth adheres to: the solar time. To comprehend the cause-effect relation between the EoT and the SMD, their interaction must be considered in isolation from any third factor. Earth does not revolve around distant stars nor does it adhere to civil time, thus the length of the day should neither be defined by distant stars, nor by a fixed 24-hour timeframe.

The inverse relationship between the length of the solar day and Earth’s rotational speed requires reversing the scale of the EoT for an intuitive analysis, as suggested in a previous study on fluctuations in the length of the planet’s sideral day [8]. The given inverse association motivated the analysis of the association between the Sun’s vertical path (SMD) and the Sun’s horizontal path (EoT) within a solar sundial noon analemma. In this context, negative values of the EoT correspond to periods when the rotational speed falls below the annual average and the mean-time Sun lags behind its mean velocity. Conversely, positive values of the EoT correspond to periods when the rotational speed exceeds the annual average and the mean-time Sun runs ahead of its mean velocity. Following a sundial noon analemma, Earth’s rotational speed must increase as the EoT advances from left to right and decrease as it progresses from right to left. Within a solar sundial noon analemma, the mean-time Sun travels from left to right during each of two trans-solstitial analemmatic phases, and from right to left during each of two trans-equinoctial analemmatic phases.

2. Materials and Methods

To analyze the joint vertical-horizontal path of the mean-time Sun, let $\mathit{\delta }$ denote the SMD and $\mathit{\delta }\mathbf{\prime }$ the EoT (both in arcdeg). Therefore, the joint function $\mathit{\delta }(\mathit{\delta }\mathbf{\prime })$ displays the solar sundial noon analemma, which illustrates the bidimensional path of the mean-time noon Sun throughout a Gregorian year. The records of $\mathit{\delta }$ and $\mathit{\delta }\mathbf{\prime }$ correspond to the vertical and horizontal angles from the mean-time noon Sun to the Equator ($\mathit{\delta }$) or to the local meridian ($\mathit{\delta }\mathbf{\prime }$), respectively. The difference in lengths between the mean day and the solar day (EoT) can be expressed in a variety of units including time (min), distance (km), or spheric coordinates (arcdeg). For the present analysis arcdeg was chosen, to match the units in which solar declination is usually stated. The EoT and the Sun’s horizontal-path ($\mathit{\delta }\mathbf{\prime }$) are referred to as synonyms in this document, while the same holds true for the SMD (true declination) and the Sun’s vertical-path ($\mathit{\delta }$).

2.1. Parameters of the Sun’s Vertical Path

The SMD ($\mathit{\delta }$, Equation (2)) was derived from the fractional year (Equation (1)) modified for solar noon, following the analemma’s geometric model [9]. Let $\mathit{\delta }$ denote the position of the Sun’s vertical path (SMD), which tracks the angle from the Equator to the position of the Sun throughout the year. To explore the dynamics of the Sun’s vertical path in greater depth, five additional parameters were considered: angular velocity ($\mathit{\omega }$, arcmin day^-1), angular acceleration ($\mathit{\alpha }$, arcsec day^-2), angular jerk ($\mathit{j}$, arcjerk day^-3), angular snap ($\mathit{\varsigma }$, arcsnap day^-4), and net drive ($\mathit{\zeta }$) of the SMD. The functions $\mathit{\omega }$, $\mathit{\alpha }$, $\mathit{j}$, and $\mathit{\varsigma }$ correspond to the first to fourth derivatives (Equations (3) to (6)) of the SMD ($\mathit{\delta }$) with respect to time ($\mathbf{t}$), same order. The net drive ($\mathit{\zeta },$Equation (7)) is defined by the sign of the product $\mathit{\omega }\mathit{\alpha }$, where $\mathit{\zeta }$ is stated as accelerative (speeding up) when the sign is positive or decelerative (slowing down) when negative.

$$\mathit{x}=[2\pi /365](\mathbf{t}-1)$$

(1)

$$\begin{gathered} \mathit{\delta }=[0.006918-0.399912cos(\mathit{x})+0.070257sin(\mathit{x})-0.006758cos(2\mathit{x})+ \\ 0.000907sin(2\mathit{x})-0.002697cos(3\mathit{x})+0.00148sin(3\mathit{x}\left)\right](180/\pi ) \end{gathered}$$

(2)

$$\begin{gathered} \mathit{\omega }=(d\mathit{\delta }/d\mathbf{t})=[2\pi /365]\left[0.399912sin\right(\mathit{x})+0.070257cos(\mathit{x})+0.013516sin(2\mathit{x})+ \\ 0.001814cos(2\mathit{x})+0.00809sin(3\mathit{x})+0.00444cos(3\mathit{x}\left)\right](\mathit{\pi }/3){(180/\pi )}^2 \end{gathered}$$

(3)

$$\begin{gathered} \mathit{\alpha }=(d\mathit{\omega }/d\mathbf{t})={[2\pi /365]}^2\left[0.399912\cos \left(\mathit{x}\right)-0.070257\sin \left(\mathit{x}\right)+0.027032\mathit{cos}\left(2\mathit{x}\right)- \\ 0.003628\mathit{sin}\left(2\mathit{x}\right)+0.024273\cos \left(3\mathit{x}\right)-0.01332\sin \left(3\mathit{x}\right)\right]{\left[\right(\mathit{\pi }/3\left)\right]}^2{(180/\pi )}^3 \end{gathered}$$

(4)

$$\begin{gathered} \mathit{j}=(d\mathit{\alpha }/d\mathbf{t})={[2\pi /365]}^3\left[-0.399912sin\left(\mathit{x}\right)-0.070257cos\left(\mathit{x}\right)-0.054064sin\left(2\mathit{x}\right)- \\ 0.007256cos\left(2\mathit{x}\right)-0.072819sin\left(3\mathit{x}\right)-0.03996cos\left(3\mathit{x}\right)\right]{\left[\right(\mathit{\pi }/3\left)\right]}^3{(180/\pi )}^4 \end{gathered}$$

(5)

$$\begin{gathered} \mathit{\varsigma }=(d\mathit{j}/d\mathbf{t})={[2\pi /365]}^4\left[-0.399912cos\left(\mathit{x}\right)+0.070257sin\left(\mathit{x}\right)-0.108128cos\left(2\mathit{x}\right)+ \\ 0.014512sin\left(2\mathit{x}\right)-0.218457cos\left(3\mathit{x}\right)+0.11988sin\left(3\mathit{x}\right)\right]{[\mathit{\pi }/3]}^4{(180/\pi )}^5 \end{gathered}$$

(6)

$$\mathit{\zeta }=sign\left(\mathit{\omega }\mathit{\alpha }\right)$$

(7)

where $\mathit{x}$ is the fractional year (radians), $\mathbf{t}$ is time within a year (days, 1 to 365).

2.2. Parameters of the Sun’s Horizontal Path

The EoT ($\mathit{\delta }\mathbf{\prime }$, Equation (8)) was derived from the fractional year (Equation (1)) modified for solar noon, following the geometric model [9] of the analemma. Let $\mathit{\delta }\mathbf{\prime }$ denote the position of the Sun’s horizontal path (EoT) which tracks the angle from the local meridian to the position of the Sun throughout the year. To explore the dynamics of the Sun’s horizontal path in greater depth, five additional parameters were analyzed: angular velocity ($\mathit{\omega }\mathbf{\prime }$, arcmin day^-1), angular acceleration ($\mathit{\alpha }\mathbf{\prime }$, arcsec day^-2), angular jerk ($\mathit{j}’$, arcjerk day^-3), angular snap ($\mathit{\varsigma }\mathbf{\prime }$, arcsnap day^-4), and net drive ($\mathit{\zeta }\mathbf{\prime }$) of the EoT ($\mathit{\delta }\mathbf{\prime }$). The functions ${\mathit{\omega }}^{\mathbf{\prime }}$, $\mathit{\alpha }\mathbf{\prime }$, $\mathit{j}\mathbf{\prime }$, and $\mathit{\varsigma }\mathbf{\prime }$ correspond to the first to fourth derivatives (Equations (9) to (12)) of the EoT with respect to time ($\mathbf{t}$), same order. The net drive (${\mathit{\zeta }}^{\mathbf{\prime }},$Equation (13)) is defined by the sign of the product ${\mathit{\omega }}^{\mathbf{\prime }}\mathit{\alpha }\mathbf{\prime }$, where $\mathit{\zeta }\mathbf{\prime }$ is stated as accelerative (speeding up) when the sign of the given product is positive or as decelerative (slowing down) when negative.

$$\begin{gathered} \delta \text{'}=EoT=-[0.0000075+0.001868cos(x)-0.032077sin(x)-0.014615cos(2x)- \\ 0.040849sin(2x\left)\right](180/\pi ) \end{gathered}$$

(8)

$$\begin{gathered} {\omega }^{\prime }=\left({\displaystyle \frac{d{\delta }^{\text{'}}}{d\mathrm{t}}}\right)=-[2\pi /365]\left[-0.001868\mathrm{s}\mathrm{i}\mathrm{n}\left(x\right)-0.032077\mathrm{c}\mathrm{o}\mathrm{s}\left(x\right)+0.02923sin\left(2x\right)- \\ 0.081698cos\left(2x\right)\right][\pi /3]{(180/\pi )}^2 \end{gathered}$$

(9)

$$\begin{gathered} \alpha \prime =(d\omega \prime /d\mathrm{t})=-{[2\pi /365]}^2\left[-0.001868\cos \left(x\right)+0.032077\mathit{sin}(x)+0.05846cos\left(2x\right)+ \\ 0.163396\mathit{sin}\left(2x\right)\right]{[\pi /3]}^2{(180/\pi )}^3 \end{gathered}$$

(10)

$$\begin{gathered} j^{\prime }=(d\alpha \prime /d\mathrm{t})=-{[2\pi /365]}^3\left[0.001868sin\right(x)+0.032077\mathit{cos}\left(x\right)-0.11692sin\left(2x\right)+ \\ 0.326792cos\left(2x\right)]{[\pi /3]}^3{(180/\pi )}^4 \end{gathered}$$

(11)

$$\begin{gathered} {\varsigma }^{\prime }=(dj\prime /d\mathrm{t})=-{[2\pi /365]}^4\left[0.001868\cos \left(x\right)-0.032077\sin (x)-0.233840cos\left(2x\right)- \\ 0.653584sin\left(2x\right)\right]{[\pi /3]}^4{(180/\pi )}^5 \end{gathered}$$

(12)

$$\zeta \prime =sign(\omega \prime \alpha \prime )$$

(13)

where $\mathit{x}$ is the fractional year (radians, Equation (1), $\mathbf{t}$ is the time within a year (days, 1 to 365). The minus sign of Equations (8) to (12) switch the EoT from an on-sky to a sundial noon analemma.

All signs of the horizontal path were arbitrarily reversed (Equations (8) to (12)) from those arising from the geometric model of solar declination, to produce the solar sundial noon analemma. Inverting the signs of the EoT allowed to intuitively associate negative records of $\mathit{\delta }\mathbf{\prime }$to moments where the Sun travels slower on its daily path, yielding longer solar days. Likewise, positive records of ${\mathit{\delta }}^{\mathbf{\prime }}$correspond to moments where the Sun travels faster on its daily path, yielding shorter solar days. Despite the ordinate axis of the sundial analemma being also reversable, such inversion was inappropriate for the planned discussion.

2.3. Adequate Units for the Path of the Mean-Time Noon Sun

When defining appropriate units to express the parameters of the Sun’s vertical path ($\mathit{\delta }$, $\mathit{\omega }$, $\mathit{\alpha }$, $\mathit{j}$, and $\mathit{\varsigma }$; Equations (2) to (6), and later those of the Sun’s horizontal path ($\mathit{\delta }\mathbf{\prime }$, $\mathit{\omega }\mathbf{\prime }$, $\mathit{\alpha }\mathbf{\prime }$, ${\mathit{j}}^{\mathbf{\prime }}$, and ${\mathit{\varsigma }}^{\mathbf{\prime }}$; Equations (8) to (12), the guiding fact was that every successive derivative shortened its range of occurrence by a factor of approximately $1/(\pi /3)$ radians (radians being the original unit), equivalent to $1/\left(\right(\pi /3\left)\right(180/\pi \left)\right)$ arcdeg. Accordingly, the kth time derivative of the Sun’s horizontal position shortened its range by${1/\left(\right(\pi /3)}^k{(180/\pi )}^k)$. In order to avoid working with too small numbers, every parameter was scaled up by the factor${(\pi /3)}^k{(180/\pi )}^{k+1}$, which is already included in Equations (2) to (6) and (8) to (12). The k superscript corresponds to the derivative order, whether for $\mathit{\delta }$ or $\mathit{\delta }\text{'}$, which becomes zero for Equations (2) and (8), whereas the +1 switches the units from radians to degrees.

After applying the factor, the numerical value of every successive derivative became scaled up by ${60}^k$ with respect to its integral; for instance, the fourth derivative was scaled by ${60}^4$. Consequently, it was only natural to down scale the units for all the derivatives [4]. The original unit was rad day^−k, whereas the units proposed here for position, velocity, acceleration, jerk, and snap are: arcdeg, arcmin day⁻¹, arcsec day⁻², arcjerk day⁻³, and arcsnap day⁻⁴, same order. After the factors were applied and the units assigned, the records of all the parameters of the SMD and the EoT fell within a similar range, which allows to display various parameters in a unique ordinate axis if required, despite their differing dimensions. Nonetheless, any comparison proposed between the parameters must clarify their scales. In order to establish the similarities between the dynamics of the SMD ($\mathit{\delta }$) and that of the EoT ($\mathit{\delta }\mathbf{\prime }$) analogous symbols and identical units were proposed for every corresponding parameter whether for the Sun’s vertical or horizontal path ($\mathit{\omega }$ and $\mathit{\omega }\mathbf{\prime }$, $\mathit{\alpha }$ and $\mathit{\alpha }\mathbf{\prime }$, etc.).

When the positional parameters ($\mathit{\delta }$and $\mathit{\delta }\mathbf{\prime }$) and their time derivatives were plotted against time, they displayed pseudo sinusoidal functions. Consequently, the classic terminology of sinusoidal curves is adopted from this point onward. In this context, the time interval in which a full cycle is completed is known as the period. A cycle can be further divided into two half-cycles, each framed between two zero-crossing points (ZCPs) two crests, or two troughs. When the ZCPs of the position ($\mathit{\delta }$and $\mathit{\delta }\mathbf{\prime }$) are chosen to divide a cycle into half-cycles, a crest occurs within a positive half-cycle, whereas a trough occurs within a negative half-cycle. Unlike the SMD and the EoT ($\mathit{\delta }$ and $\mathit{\delta }\mathbf{\prime }$), the crests and troughs of actual sinusoidal curves are equidistant from their ZCPs (the abscissa), and such distance is referred to as the cycle’s amplitude. For practical purposes, the annual cycle of the Sun’s horizontal path can be considered as a dual cycle —framed within a unique cycle of the Sun’s vertical path—, despite initial and final conditions of the EoT only converging by the end of the Gregorian year.

2.4. Sections of the EoT

The Sun’s vertical-path ($\mathit{\delta }$) was divided into two half-cycles by the ZCPs of $\mathit{\delta }$, which converge at the equinoxes to either the crest or through of $\mathit{\omega }$ —where a trough corresponds to a crest when $\mathit{\delta }$ points downward and $\mathit{\omega }$ is negative. Every half-cycle tracks the SMD within one hemisphere. Furthermore, dividing each half-cycle of the SMD by the solstitial ZCPs of $\mathit{\omega }$, yielded quarter-cycles which frame an entire season. Furthermore, the Sun’s horizontal-path ($\mathit{\delta }\mathbf{\prime }$) was divided into two cycles by the two troughs of $\mathit{\omega }\mathbf{\prime }$ and every cycle divided in two half-cycles by the crests of $\mathit{\omega }\mathbf{\prime }$. Finally, eight sections were produced by dividing every half-cycle into two sections by the ZCPs of $\mathit{\omega }\mathbf{\prime }$. Every resulting section comprised whether the first or the last of the two quarter-cycles conforming a season. Because the division was conducted by the key instances of $\mathit{\omega }\mathbf{\prime }$ rather than by time ranges, the resulting sections differ in duration. T

The Sun’s vertical path ($\mathit{\delta }$) takes ZCPs at the Equator, whereas the Sun’s horizontal-path ($\mathit{\delta }\mathbf{\prime }$) takes ZCPs at the local meridian. To analyze the annual dual cycle of the Sun’s horizontal path, the EoT was divided in eight sections by the four maxima and four ZCPs of $\mathit{\omega }\mathbf{\prime }$. The section boundaries can be either a crest and a ZCP of ${\mathit{\omega }}^{\mathbf{\prime }}$, or a ZCP and a trough of ${\mathit{\omega }}^{\mathbf{\prime }}$. Let the section names be early spring, late spring, early summer, late summer, early autumn, late autumn, early winter, and late winter. Let the section boundaries be denoted spring equinox, midspring, summer solstice, midsummer, autumn equinox, midautumn, winter solstice and midwinter.

To investigate the connection between the EoT and Earth’s rotation, the analemma was further divided into four phases: trans-equinoctial phases I and III and trans-solstitial phases II and IV. Each phase includes two consecutive sections of the EoT, while the numbers obey to the order in which they occur. Unlike the seasons of the SMD, which consist of two sections with opposing horizontal directions—the direction of the EoT—, each analemmatic phase consists of two consecutive sections whose horizontal direction is consistent both within and between. The direction of the EoT is identified by the sign of ${\mathit{\omega }}^{\mathbf{\prime }}$, where $\mathit{\delta }\mathbf{\prime }$ points right when the sign is positive or left when negative. Each analemmatic phase extends between two ZCPs of ${\mathit{\omega }}^{\mathbf{\prime }}$. The trans-equinoctial phase I occurs between the midwinter and midspring, and the trans-equinoctial phase III occurs between midsummer and midautumn. Therefore, the trans-equinoctial phase I encompasses late winter and early spring while the trans-equinoctial phase III includes late summer and early autumn. The trans-solstitial phase II occurs from midspring to midsummer and the trans-solstitial phase IV spans from midautumn to midwinter. Therefore, the trans-solstitial phase II encompasses late spring and early summer, while the trans-solstitial phase IV includes late autumn and early winter.

Each phase shows homogeneity regarding four aspects: (1) the sign of ${\mathit{\omega }}^{\mathbf{\prime }}$, and consequently (2) the direction of $\mathit{\delta }\mathbf{\prime }$; which inherits a consisting direction to both (3) Earth’s rotational speed and (4) to the length of the solar day. When only numerical data are available, the sign of ${\mathit{\omega }}^{\mathbf{\prime }}$is the sole indicator for the direction in which the Sun’s horizontal path occurs. The concept of phase within the EoT, was key for studying the dependance between the EoT, the length of the solar day, and Earth’s rotational speed.

2.5. Earth’s Rotational Speed and the EoT

Because the Earth orbits the Sun rather than the galaxy or distant stars, studies on Earth dynamics must prioritize Sun-Earth interactions. Given that Earth’s rotational speed ($\mathit{\rho }$) is discussed alongside the vertical ($\mathit{\omega }$) and horizontal ($\mathit{\omega }\mathbf{\prime }$) velocities of the mean-time noon Sun, the concept “speed” is reserved for the planet’s rotation. This is appropriate because the rotational speed is always positive.

This document defines solar day as the unique true day and assumes the Sun’s horizontal path is inherently tied to Earth’s rotation. On a hypothetical day when the EoT runs 16 min ahead of the meridian, this offset corresponds to a horizontal angle of +4 arcdeg beyond the local meridian, equivalent to 240 arcmin per day or 10 arcmin per hour compared to the mean day. This deviation indicates the Sun’s horizontal path extends 445 km above the mean day and the equatorial rotational speed is 18.5 km hour⁻¹ faster than the annual average speed. The EoT $(\mathit{\delta }\mathbf{\prime })$, must be either added to or subtracted from the average speed of rotation (1669.78 km h⁻¹). In fact, Earth’s rotational speed varies from 1653 to 1688 km h⁻¹ at the Equator (or 27.5 to 28.1 km min⁻¹). A fluctuation involving ±1 % throughout the year , equivalent to 852 km day⁻¹, 35.5 km h⁻¹, or 0.6 km min⁻¹ between the maximum and minimum speeds.

Because Earth’s linear rotational speed varies with latitude, it is convenient to derive an expression which accounts for this factor. The linear regressions in Equations (15) and (16) establish the association between Earth’s rotational speed and the EoT, for the Equator and the Tropic of Cancer, respectively, both derived from Equation (14). A general formula for Earth’s speed of rotation is also given in Equation (17), which considers the following steps: (1) assesses Earth’s circumference (km) at the desired latitude $\mathit{l}$ , which comprises a single path of the Sun around the planet, or the distance that NBI travels within a day, given by $2\pi rCos\mathit{l}$, (2) derives an equivalence between angular and linear distances by dividing Earth’s circumference by 360 arcdeg, (3) add the EoT to Earth’s circumference, both in km [10], and (4) divides the outcome by the 24 hours of a mean day (factor on the left of Equation (17)). The radius of Earth’s circumference and the Earth’s average speed used for the calculations were $r=40075km/2\pi$ and $\mathit{\rho }=(40075km/24h)$, respectively.

$${\rho }_i=\left(1669.78\right)(1+{\delta }^{\text{'}}/360)$$

(14)

$${\rho }_i=1669.78+4.63827\delta \prime$$

(15)

$${\rho }_i=1531.13+4.25770\delta \prime$$

(16)

$${\rho }_i=\left(1/24\right)\left(2\pi rCos\mathit{l}\right)(1+\delta \prime /360)$$

(17)

where ${\mathit{\rho }}_{\mathit{i}}$ is the Earth’s rotational speed on day $i$, $\mathit{l}$ is the site’s latitude and $\mathit{\delta }\mathbf{\prime }$ is the EoT on day $i$.

2.6. The Path of the Subsolar Point

Because the shortest distance between two points is a straight line and because light travels in a straight line, the subsolar point is the nearest spot from the Sun compared to any other location on Earth’s surface. Therefore, following the path of NBI across longitude might confirm the hypothesis that solar declination, equation of time and earth’s rotational speed are inherently linked.

In this context, we define the length ($\mathit{L}$) of the path spanned by NBI which is assessed as the length of Earth’s circumference at the latitude where the current sun meridian declination occurs, according to Equation (18). Then such varying parameter was compared with Earth’s rotational speed ($\mathit{\rho }$). Because L and $\mathit{\rho }$ are both defined as speeds, the rates at which they vary were explored as accelerations. Let ${\mathit{L}}_{\alpha }$ (Equation (19)) be the acceleration of Earth’s rotational speed.

$$\mathit{L}=2\pi rCos\left(\delta \right)$$

(18)

$${\mathbf{L}}_{\mathsf{\alpha }}=({\mathbf{L}}_{\mathrm{i}}-{\mathbf{L}}_{\mathrm{i}-1})/24$$

(19)

where $\mathbf{L}$ is the length of Earth’s circumference when the solar declination is $\mathsf{\delta }$, r is Earth’s radius, ${\mathit{L}}_{\alpha }$is the NBI acceleration, and ${\mathit{L}}_i-{\mathit{L}}_{i-1}$ are the circumference lengths for a particular day and the day preceeding.

3. Results and Discussion

3.1. The Sign Paradox

The sign of the parameters describing the Sun’s vertical path ($\mathit{\delta }$, $\mathit{\omega }$ and $\mathit{\alpha }$) arise because the Northern Hemisphere is the customary reference. Nonetheless, all the signs can be reversed without semantic consequence, because the Southern Hemisphere could also work as reference. Actually, the signs of SMD reverse naturally between hemispheres without interfering with the net drive of declination (speeding up or slowing down) of any season. Otherwise stated, the dynamics of the SMD would remain unaffected whether analyzed through austral instead of boreal seasons. Accordingly, whether a ZCP of $\mathit{\omega }$ occurs downward or upward is irrelevant from the standpoint of physics, which holds true for the ZCPs of $\mathit{\alpha }$. Hence, absolute values can be applied to $\mathit{\delta }$, $\mathit{\omega }$ and $\mathit{\alpha }$ when their association is assessed within hemisphere.

Unlike the signs of the parameters describing the Sun’s vertical path $(\mathit{\delta },\mathit{\omega }\mathrm{a}\mathrm{n}\mathrm{d}\mathit{\alpha }$…), those of the horizontal path $(\mathit{\delta }\mathbf{\prime },{\mathit{\omega }}^{\mathbf{\prime }}\mathrm{a}\mathrm{n}\mathrm{d}\mathit{\alpha }\mathbf{\prime }$…) cannot be reversed without consequence. For the Sun’s horizontal path ($\mathit{\delta }\mathbf{\prime }$), the meaning of a positive half-cycle opposes that of a negative half-cycle. The analyses undertaken in this work are based on a sundial meridional analemma, on which the direction of $\mathit{\delta }\mathbf{\prime }$ reverses from west to east and vice versa compared to a sky solar noon analemma. Reversing the signs is beneficial when relating the EoT to the Earth’s rotational speed or the length of the solar day. For instance, at the ZCPs of the EoT, the length of the solar day and Earth’s rotational speed converge to their annual averages (24 hours and 1669.78 km h⁻¹). A ZCP of ${\mathit{\delta }}^{\mathbf{\prime }}$ can occur both within a trans-equinoctial phase —where the rotational speed decreases progressively— and within a trans-solstitial phase —where the rotational speed is progressively increasing. Therefore, the former conforms a downward ZCP and the latter conforms an upward ZCP, which switch from positive to negative records or from negative to positive records, respectively.

Every time ${\mathit{\omega }}^{\mathbf{\prime }}$ switches signs at a phase’s boundary (midseason ZCPs of ${\mathit{\omega }}^{\mathbf{\prime }}$), Earth’s rotational speed switches from a growing to a decreasing streak, or vice versa, while the dynamics of the length of the solar day reverses direction simultaneously, varying inversely with Earth’s rotational speed.

3.2. Dynamics of the Sun Meridian Declination

The within season averages for position ($\mathit{\delta }$), angular velocity ($\mathit{\omega }$) acceleration ($\mathit{\alpha }$), jerk ($\mathit{j}$), snap ($\mathit{\varsigma }$) and net drive ($\mathit{\zeta }$) of the SMD are shown in Table 1. Given that the Sun meridian declination ($\mathit{\delta }$) defines the y-coordinate of the solar noon analemma as perceived from a given site on Earth’s surface, the annual cycle of SMD is referred through this document as the vertical path of the Sun ($\mathit{\delta }$). The period of the Sun’s vertical-path ($\mathit{\delta }$) extents for one year, which can be split into a positive half-cycle that spans the Northern Hemisphere and a negative half-cycle that spans the Southern Hemisphere. The crest and trough of $\mathit{\omega }$ converge with the equinoxes, whereas the ZCPs of $\mathit{\omega }$ concur with the solstices.

The functions $\mathit{\delta }\left(\mathit{\omega }\right)$ and $\mathit{\alpha }\left(\mathit{\omega }\right)$ approach circumference-like shapes. Accordingly, |$\mathit{\omega }$| and |$\mathit{\delta }$| vary inversely, so the highest |$\mathit{\omega }$| occurs at the lowest |$\mathit{\delta }$| and vice versa. An analogous association occurs between |$\mathit{\omega }$| and |$\mathit{\alpha }$|. Furthermore, |$\mathit{\alpha }$| and |$\mathit{\delta }$| vary directly and $\mathit{\alpha }\left(\mathit{\delta }\right)$ approaches a straight line. The factor ${[\mathsf{\pi }/3]}^{\mathrm{k}}{[180/\mathsf{\pi }]}^{\mathrm{k}}$, included in Equations (3) to (6), was applied because each successive derivative of $\mathit{\delta }$ occurred within 1/60th of the range spanned by its integral when all the parameters were expressed in radians. For instance, $\mathit{\alpha }$ and $\mathit{\varsigma }$ span 1/60² or 1/60⁴ compared to $\mathit{\delta }$’s range, respectively. The proposed units clarify the actual associations and indicate their differing scales. For instance, after switching units, the circle $\mathit{\delta }\left(\mathit{\omega }\right)$ associates $\mathit{\delta }$ in arcdeg with $\mathit{\omega }$ in arcmin day⁻¹.

After scaling the records up and assigning proper units, the resemblance between the within season averages is remarkable for each parameter. For instance, when signs are dismissed, the records occur within the interval 12 to 20. The weighted averages show a yearly equilibrium in the SMD dynamic-system, as the summatory of each individual parameter approaches zero. Although a deviation of 0.356 arcdeg in SMD is indicative of some strength not being considered. The average records shown in Table 1 must be interpreted with caution, given the that $\mathit{\omega }$, $\mathit{\alpha }$, $\mathit{j}$, and $\mathit{\varsigma }$ were scaled up.

In boreal spring, positive records of $\mathit{\delta }$ and $\mathit{\omega }$ coincide with a negative $\mathit{\alpha }$, which means the Sun decelerates as it departs from the Equator and approaches the Tropic of Cancer. In boreal summer, a positive $\mathit{\delta }$ converges to negative records of $\mathit{\omega }$ and $\mathit{\alpha }$, which indicates the apparent Sun accelerates as it departs from the Tropic of Cancer and approaches the Equator. In boreal autumn, negative records of $\mathit{\delta }$ and $\mathit{\omega }$ coincide with a positive $\mathit{\alpha }$, which means the Sun decelerates as it departs from the Equator and approaches the Tropic of Capricorn. In boreal winter, a negative $\mathit{\delta }$ converges with positive records of $\mathit{\omega }$ and $\mathit{\alpha }$, which indicates the apparent Sun accelerates as it departs from the Tropic of Capricorn and approaches the Equator.

Summarizing the Sun’s vertical path, the sign of $\mathit{\delta }$ can be dismissed and the net drive be stated as decelerative when the signs of $\mathit{\omega }$ and $\mathit{\alpha }$ diverge, or accelerative when the signs of the parameters $\mathit{\omega }$ and $\mathit{\alpha }$ coincide. As the sign of both $\mathit{\omega }$ and $\mathit{\alpha }$ are consistent within season, an identical net drive endures throughout each of the four seasons. Every particular $\mathit{\delta }$ holds characteristic records of $\mathit{\omega }$ and $\mathit{\alpha }$ within an hemisphere, regardless of whether such association occurs within a decelerative (spring or autumn) or an accelerative season (summer or winter), although differences do occur between hemispheres.

3.3. Dynamics of the Equation of Time

The position ($\mathit{\delta }\mathbf{\prime }$), angular velocity ($\mathit{\omega }\mathbf{\prime }$) acceleration ($\mathit{\alpha }\mathbf{\prime }$), jerk ($\mathit{j}\mathbf{\prime }$), snap ($\mathit{\varsigma }\mathbf{\prime }$) and net drive ($\mathit{\zeta }\mathbf{\prime }$) of the EoT, as well as the equatorial rotational speed of the Earth ($\mathit{\rho }$) are shown in Table 2. Given that the EoT ($\mathit{\delta }\mathbf{\prime }$) comprises the abscissa of the solar noon analemma as perceived from a given site on Earth’s surface, the annual cycle of EoT is referred through this document as the Sun’s horizontal path ($\mathit{\delta }\mathbf{\prime }$). Unlike the SMD, the EoT reaches ZCPs—through the local meridian—four times a year, therefore, the annual dynamics of the EoT was divided into two cycles whose periods last 173 and 192 days, for the Northern and Southern Hemisphere, respectively. Because the EoT was split by the key instances of $\mathit{\omega }\mathbf{\prime }$ rather than by those of $\mathit{\delta }$, the boundaries of the analemmatic cycles, phases and sections correspond to troughs, crests, and ZCPs of $\mathit{\omega }\mathbf{\prime }$.

Despite ZCPs of ${\mathit{\delta }}^{\mathbf{\prime }}$ being the natural boundaries to divide the EoT, a most intuitive analysis arises by splitting $\mathit{\delta }\mathbf{\prime }$ from the key moments of $\mathit{\omega }\mathbf{\prime }$. For instance, every half-cycle of $\mathit{\delta }\mathbf{\prime }$ becomes virtually synchronized with one of the four seasons of the year. Furthermore, dividing half-cycles (seasons) in quarter-cycles (sections) proved useful because the parameters switch sign and/or direction at midseason boundaries.

Dividing the EoT by the key instances of $\mathit{\omega }\mathbf{\prime }$ brought a number of advantages. Unlike the extrema of ${\mathit{\delta }}^{\mathbf{\prime }}$, the crests and troughs of $\mathit{\omega }\mathbf{\prime }$ occur at the solstices and near the equinoxes, respectively. Moreover, this approach allowed for the combined path $\mathit{\delta }\left({\mathit{\delta }}^{\mathbf{\prime }}\right)$ to be analyzed in sections whose horizontal net drives are consistent within but differing between. Thus, each cycle of the Sun’s horizontal path includes the path of the mean time noon Sun within one hemisphere, although the equinoctial boundaries denoted spring equinox and autumn equinox occur at +3 arcdeg of SMD rather than at the Equator. Accordingly, every season consists of two sections with opposing net drives and varying lengths.

The dynamics of the Sun’s horizontal path is analogous to that of the Sun’s vertical path. Thus, (1) the maxima and ZCPs of |$\mathit{\delta }\mathbf{\prime }$| and |$\mathit{\alpha }\mathbf{\prime }$| occur nearby, therefore they vary directly, (2) whereas $\mathit{\delta }\text{'}$ and $\mathit{\omega }\mathbf{\prime }$ vary inversely within every quarter-cycle of the EoT, (3) $\mathit{\omega }\mathbf{\prime }$ and $\mathit{\alpha }\mathbf{\prime }$ vary inversely, where the crests of |$\mathit{\alpha }\mathbf{\prime }$| nearly coincide with ZCPs of $\mathit{\omega }\mathbf{\prime }$ and the crests of |$\mathit{\omega }\mathbf{\prime }$| nearly meet the ZCPs of $\mathit{\alpha }\mathbf{\prime }$—so that |$\mathit{\omega }\mathbf{\prime }$| and |$\mathit{\alpha }\mathbf{\prime }$| cannot maximize together, and (4) initial and final conditions converge by the end of the Gregorian year, therefore the $\mathit{\delta }\mathbf{\prime }$ dual cycle repeats annually. Despite the synchrony between crests of $\mathit{\omega }\mathbf{\prime }$ and ZCPs of $\mathit{\alpha }\mathbf{\prime }$ or that between ZCP of $\mathit{\omega }\mathbf{\prime }$ and crest/trough of $\mathit{\delta }\mathbf{\prime }$ —where the EoT shifts directions, $\mathit{\alpha }\mathbf{\prime }$ and $\mathit{\delta }\mathbf{\prime }$ do not maximize nor they reach ZCPs together.

Regarding the net dive, a section of the EoT is always accelerative when the mean-time Sun approaches the local meridian or decelerative when the mean-time Sun departs from the local meridian, disregarding whether the direction of the actual motion goes to the right or left. As every section tagged early tracks a departure from the local meridian and every season tagged late tracks an approach to the local meridian, the four sections of the EoT labeled late are accelerative, whereas the four sections labeled early are decelerative. Thus, every season contains a decelerative and an accelerative section, in that order.

As the direction of $\mathit{\delta }\text{'}$ is not given by the values or signs of the EoT, the direction of the motion can be retrieved from the sign of $\mathit{\omega }\mathbf{\prime }$. A positive $\mathit{\omega }\mathbf{\prime }$ characterizes the trans-solstitial phases II and IV, where the mean-time noon Sun travels right. A negative $\mathit{\omega }\mathbf{\prime }$ characterizes the trans-equinoctial phases I and III, where the mean-time noon Sun travels left.

3.4. The Sun’s Combined Path at the Section Boundaries

During solstices or equinoxes $\mathit{\omega }\mathbf{\prime }$ reaches a maxima in perfect synchrony with a ZCP of $\mathit{\alpha }\mathbf{\prime }$, but such coincidence holds an opposite meaning. At solstices, crests of $\mathit{\omega }\mathbf{\prime }$ (3.288 or 6,99 arcmin day^-1) converge to downward ZCPs of $\mathit{\alpha }\mathbf{\prime }$, which occurs five days after or four days before an upward ZCP of $\mathit{\delta }\mathbf{\prime }$, for the summer or winter solstice, respectively. On the other hand, near the equinoctial troughs of $\mathit{\omega }\mathbf{\prime }$ (−7.786 or −5.65 arcmin day^-1 for the vernal or autumnal equinox, same order) meets upward ZCPs of $\mathit{\alpha }\mathbf{\prime }$, which occurs 18 days before (spring equinox) or 15 days after (autumn equinox) a downward ZCP of $\mathit{\delta }\mathbf{\prime }$, at characteristic $\mathit{\delta }$ records of 9.832 and 7.907 arcdeg, respectively.

At each of the four midseason boundaries, $\mathit{\omega }\mathbf{\prime }$ reaches a ZCP in perfect synchrony with an extrema of $\mathit{\delta }\mathbf{\prime }$. In seasons whose net drive of SMD is accelerative, a downward ZCP of ${\mathit{\omega }}^{\mathbf{\prime }}$ meets a crest of $\mathit{\delta }\mathbf{\prime }$, which takes place eight days before (midsummer) or six days after (midwinter) a trough of $\mathit{\alpha }\mathbf{\prime }$. In seasons whose net drive of SMD is decelerative, an upward ZCP of ${\mathit{\omega }}^{\mathbf{\prime }}$ meets a trough of $\mathit{\delta }\mathbf{\prime }$, which takes place five days after (midspring) or four days before a crest of $\mathit{\alpha }\mathbf{\prime }$. The midseason boundaries, defined by ZCPs of $\mathit{\omega }\mathbf{\prime }$, converge to a similar analemmatic $\mathit{\delta }$ whether for the Northern (−18.59 or −19.19 for midspring and midsummer) or Southern Hemisphere (−14.35 or −13.29 for midautumn and midwinter), but such $\mathit{\delta }$ differs between them. The given fact reinforces the strong association between the SMD and the EoT.

3.5. Combined Horizontal-Vertical Path of the Sun

In this document, the true solar declination $(\mathit{\delta }$) is referred to as the Sun’s vertical path, whose parameters are denoted $\mathit{\delta }$, $\mathit{\omega }$, $\mathit{\alpha }$ and $\mathit{\zeta }$. Likewise, the EoT $\left({\mathit{\delta }}^{\mathbf{\prime }}\right)$ is referred to as the Sun’s horizontal path, whose parameters are denoted $\mathit{\delta }\mathbf{\prime }$, $\mathit{\omega }\mathbf{\prime }$, $\mathit{\alpha }\mathbf{\prime }$ and $\mathit{\zeta }\mathbf{\prime }$. To explore the connection between the Sun’s vertical path (SMD) and the Sun’s horizontal path (EoT) along the year, their parameters were explored in the same timeline. Two cycles of the Sun’s horizontal path ($\mathit{\delta }\mathbf{\prime }$) occur in synchrony with a unique cycle of the Sun’s vertical path (SMD) along the year. A cycle of $\mathit{\delta }\mathbf{\prime }$ concurs with a half-cycle of the SMD in the Northern Hemisphere, which encompasses two seasons of the Gregorian year. In the horizontal path of the Sun, ${\mathit{\delta }}^{\mathbf{\prime }}$ switches left at midsummer or midwinter and switches right at midspring or midautumn, corresponding to the crests or troughs of the EoT, in the same order.

The direction of the mean time noon Sun regarding whether the EoT or the SMD is identified by the sign of the corresponding velocity. A positive record of $\mathit{\omega }$ indicates the mean-time Sun is moving toward the Tropic of Cancer (boreal direction), whereas a positive record of $\mathit{\omega }\mathbf{\prime }$indicates the mean-time Sun is heading right. The association between the SMD and the EoT is summarized in Figure 1, where every parameter of Sun’s vertical path is plotted against its corresponding parameter of the Sun’s horizontal path.

In Figure 1, every section denoted late is tagged grey to avoid overmarking. The ranges on which the velocity and acceleration of the SMD take place double those of the EoT, whereas jerk and snap vary within the same range for both the vertical and horizontal paths of the mean time noon Sun. Figure 1 indicates that the SMD and the EoT show a cause- effect association when each and every dynamic parameter is considered alone.

Regarding the net drive of the SMD, a season is accelerative when the mean time noon Sun approaches the Equator, or decelerative when the mean time noon Sun departs from the Equator. The net drive switches to decelerative when the SMD goes through a ZCP, where such ZCP signifies the equilibrium point and corresponds to the Equator. Every analemmatic section which departs de local meridian as well as every season (SMD) which departs from the Equator are decelerative. The signs of the parameters of the Sun’s horizontal path (EoT) occur in the very same order in either hemisphere. Although every parameter of the Sun’s vertical path (SMD) reverses signs between hemispheres, the net drives of the SMD occur in the exact same order for the sequence of analemmatic sections of either hemispheres. Therefore, the same outcome would have arisen whether a positive declination was assigned to the Southern Hemisphere, as there is no up or down in the universe.

Following the solar sundial noon analemma and disregarding some equinoctial deviance, one cycle of the EoT occur while the meridional Sun travels on one hemisphere, while a half-cycle of the EoT concurs to a season of the Gregorian year. Every half-cycle of the EoT consists of two sections whose resultant drives oppose. According to the reversed ${\mathsf{\delta }}^{\mathbf{\prime }}$, each half-cycle of $\mathit{\delta }\mathbf{\prime }$ that spans the left side of the solar sundial noon analemma (spring and autumn) has a decelerative section followed by an accelerative section, such contrasting drives switching at a midseason trough of $\mathit{\delta }\mathbf{\prime }$. Each half-cycles of ${\mathsf{\delta }}^{\mathbf{\prime }}$ that spans the right side of the solar noon sundial analemma (summer and winter) has an accelerative section followed by a decelerative section, such contrasting drives switching at a midseason crest of $\mathit{\delta }\mathbf{\prime }$. The second section of a season reverses the deviation from the local meridian caused by the first section; therefore, the Sun occurs near the local meridian around the end of the second section. The sign of the main three parameters (position, velocity and acceleration), whether for the Sun’s horizontal or vertical path, remains consistent within every section of the EoT.

The association between the SMD and the EoT becomes clear with the sundial noon analemma alone—even dismissing the vertical and horizontal velocities and accelerations. For instance, the Pearson correlations for the association between $\mathit{\delta }\mathbf{\prime }$ and $\mathit{\delta }$ yield $-$0.98 (P<0.001) for the trans-equinoctial phases and 0.90 (P<0.001) for the trans-solstitial phases. The association between the Sun’s horizontal and Sun’s vertical path is also noticed in the coordination of their resultant drives in phases I and III, or by their opposition in phases II and IV . The analysis of the independent and combined net drives is presented in Table 3.

The sign of $\mathit{\omega }$ was dismissed to assess the connection between ${\mathit{\omega }}^{\mathbf{\prime }}$ and |$\mathit{\omega }$|. The Pearson correlation indicates ${\mathit{\omega }}^{\mathbf{\prime }}$ and |$\mathit{\omega }$| are inversely correlated throughout the year. Because the association fluctuates with SMD the analemma was sliced in four segments of SMD in order to compare the strength of the association between ${\mathit{\omega }}^{\mathbf{\prime }}$ and |$\mathit{\omega }$| for the two sections occurring in the same interval of SMD —but opposing directions of declination—. For instance, the correlation coefficients relating ${\mathit{\omega }}^{\mathbf{\prime }}$ with |$\mathit{\omega }$| were similar between: late spring and early autumn (−0.968, −0.969), early spring and late summer (−0.999, −0.999), late winter and early autumn (−0.972, −0.978), or between early winter and late autumn (−0.963, −0.961). Moreover, a crest of ω’ meets a ZCP of ω at each solstice; whereas a trough of ω’ takes place near either equinoctial crests of $\mathit{\omega }$.

The sections within a trans-solstitial phase hold differing dynamics, the first section is characterized by an increasing ${\mathit{\omega }}^{\mathbf{\prime }}$ and a decreasing |$\mathit{\omega }$|, while the second section is characterized by a decreasing ${\mathit{\omega }}^{\mathbf{\prime }}$ and an increasing |$\mathit{\omega }$|. In the trans-equinoctial phases, ${\mathit{\omega }}^{\mathbf{\prime }}$ and |$\mathit{\omega }$| increase together as the combined direction of the mean-time Sun approaches simultaneously the Equator and the local meridian, but decrease together as the direction of the mean-time Sun departs at once from the Equator and the local meridian.

Despite the consistent increasing or decreasing pattern of α within season, $\mathit{\alpha }\mathbf{\prime }$ increases monotonically along the first section and decreases monotonically along the second section of every season, whereas the signs of both remain unchanged within season. The relationship between the horizontal and vertical accelerations, $\mathit{\alpha }\mathbf{\prime }$ and α, vary directly within the first section of every season but inversely within the second season. The Pearson correlation coefficients between $\mathit{\alpha }\mathbf{\prime }$ and α are virtually identical between the two sections of the same season, but their signs oppose. This fact remains true when comparing early spring to late spring (r = 0.96 or −0.93), early summer to late summer (r = 0.94 or −0.94), early autumn to late autumn (r =0.90 and −0.90), or early winter to late winter (r= 0.89 or −0.88). As the sign of α was dismissed for this analysis, the sign of the correlation coefficient obeys to changes in $\mathit{\alpha }\mathbf{\prime }$ between the two sections of the same season, because the direction of $\mathit{\alpha }\mathbf{\prime }$shifts at midseason.

A ZCP of $\mathit{\alpha }\mathbf{\prime }$ converges to the maxima of α at the solstices, while the extrema of $\mathit{\alpha }\mathbf{\prime }$ occur close to every midseason boundary at characteristic records of $\mathit{\delta }$, whereas a ZCP of $\mathit{\alpha }\mathbf{\prime }$ occurs near the vernal and autumnal equinoxes, at $\mathit{\delta }$ records of 2.928 and 2.714, respectively. Whether for the Sun’s vertical or horizontal path, an extrema of acceleration converges with a change in direction, a behavior characteristic of pendular motion.

Given that the local meridian and the Equator conform the equilibrium points for the EoT and SMD, respectively, every shift departing from the equilibrium point is decelerative and every shift approaching the equilibrium point is accelerative, whether for the Sun’s horizontal or vertical path. The net drive of a section becomes accelerative when the velocity is monotonically increasing or decelerative when the velocity is monotonically decreasing. An accelerative net drive can occur on either a positive or a negative direction, as long as the motion directions occurs towards the equilibrium point. Analogously, a decelerative net drive can occur on either a positive or a negative direction, as long as such direction departs from the equilibrium point. The direction of the actual motion can be retrieved from the sign of the velocity. This facts apply separately for the SMD or the EoT, disregarding of whether their net drives become coordinated or opposed.

3.6. Earth’s Speed of Rotation

Dividing the EoT in sections allowed for a close examination of the within season and interseason net drives. Nonetheless, to analyze the dynamics behind the length of the solar day and the Earth’s rotational speed, a more effective analysis arises by dividing the analemma in four phases according to their horizontal direction, where each phase encompasses two successive sections of the EoT. The key moments of Earth’s rotational speed within a sundial noon analemma are inherited from the EoT. The crests and troughs of $\mathit{\rho }$ correspond to the crests and troughs of $\mathit{\delta }$ —at ZCPs of $\mathit{\omega }\prime$. For instance, the midspring and midautumn troughs of the EoT correspond to troughs of $\mathit{\rho }$ (1665.3 and 1650.8 km h⁻¹), whereas the midsummer and midwinter crests of the EoT correspond to crests of $\mathit{\rho }$ (1677.4 and 1686.3 km h⁻¹).

Earth’s rotational speed accomplishes two phases of progressive decreases throughout the Gregorian year, denoted trans-equinoctial phases I and III. The phase I encompasses late winter and early spring, whereas the phase III encompasses late summer and early autumn. In the phases I and III , $\mathit{\rho }$ behaves monotonically decreasing from a crest to a trough of $\mathit{\delta }\mathbf{\prime }$ (both including a ZCPs of $\mathit{\omega }\mathbf{\prime }$). In the trans-equinoctial phases, $\mathit{\rho }$ decreases despite the net drives of the $\mathit{\delta }\mathbf{\prime }$ sections being accelerative before the near equinoctial trough of $\mathit{\omega }\mathbf{\prime }$ and decelerative after the near equinoctial trough of $\mathit{\omega }\mathbf{\prime }$, yielding first growing drops and then decreasing drops in $\mathit{\rho }$, respectively.

The net drives of the EoT and SMD become opposed along the trans-solstitial phases II and IV. In the trans-solstitial phase II, an accelerative $\mathit{\delta }\mathbf{\prime }$ and a decelerative $\mathit{\delta }$ characterize late spring, but both net drives reverse for early summer. In the trans-solstitial phase IV an accelerative $\mathit{\delta }\mathbf{\prime }$ and a decelerative $\mathit{\delta }$ characterize late autumn, but the net drives of the EoT and SMD reverse for early winter. Earth-Sun dynamics causes Earth’s rotational speed to increase during the trans-solstitial phases of the EoT, for SMD records exceeding the SMD ranges parenthetically specified above.

The Earth rotates below its average speed during most of spring and autumn, but above its average speed during most of summer and winter. The average rotational speed is reached only four times a year, either at the downward or upward ZCPs of the EoT ($\mathit{\delta }\prime$=0), on 16 Apr, 15 Jun, 2 Sept, and 26 Dec (days 106, 166, 245 and 360 of the year); whereas the $\mathit{\rho }$crests occur on 14 Feb and 28 Jul and the $\mathit{\rho }$ troughs fall on 15 May and 1 Nov. Accordingly, the trans-equinoctial phases I and II, where the Earth’s rotational speed decreases monotonically, span from 14 Feb to 15 May and from 28 Jul to 1 Nov, respectively; each lasting three months. Hence, the trans-solstitial phases II and IV, where $\mathit{\rho }$ increases monotonically, span from 1 Nov to 14 Feb, and from 15 May to 28 July.

Earth’s rotational speed accomplishes two phases of progressive increases throughout the Gregorian year, denoted trans-solstitial phases II and IV. The phase II encompasses late spring and early summer, and the phase IV encompasses late autumn and early winter. In the phases II and IV , $\mathit{\rho }$ behaves monotonically increasing from a trough to a crest of $\mathit{\delta }\mathbf{\prime }$ (both tagged by ZCPs of $\mathit{\omega }\mathbf{\prime }$). In the trans-solstitial phases, $\mathit{\rho }$ increases despite the net drives of the $\mathit{\delta }\mathbf{\prime }$ sections being accelerative before the solstice and decelerative after the solstice (going midway through an $\mathit{\alpha }\mathbf{\prime }$ ZCPs and a ${\mathit{\omega }}^{\mathbf{\prime }}$ crest), yielding first growing increments and then decreasing increments in $\mathit{\rho }$, respectively.

The net drives of EoT and SMD are coordinated throughout each trans-equinoctial phase. In the trans-equinoctial phase I, both exhibit a coordinated accelerative net drive in late winter ($\mathit{\delta }$ : −13.29, 3.08, range 16.3) but a coordinated decelerative net drive in early spring ($\mathit{\delta }$ : 3.08 to 18.59, range 15.5). In the trans-equinoctial phase II, both exhibit a coordinated accelerative net drive in late summer ($\mathit{\delta }$ : 19.19 to 2.57, range 16.2 ) but a coordinated decelerative net drive in early autumn ($\mathit{\delta }$ : 2.57 to −14.35, range 16.9).

As a simplified and practical conclusion, the EoT and the SMD exhibit coordinated net drives within the $\mathit{\delta }$ interval of −13 to 19 arcdeg, centered in $\mathit{\delta }$ = +3. Consequently, each of the analemmatic trans-equinoctial phase—each including two sections—spans approximately 16 arcdeg of SMD. According to the synchrony between the SMD and the EoT, the Sun influences significantly Earth’s rotation. To begin with, Earth’s rotational axis is a perpendicular projection to the Equator. Meanwhile the axis of the sunlight cone—extending from the Sun’s center to the subsolar point— also conforms a normal projection to the Earth’s rotational axis, a relationship that holds true throughout the year. The angular distance between the last vector and Earth’s Equator is known as SMD. Because the SMD is faultlessly synchronized with the four seasons along Earth’s orbit, the association here described between the Earth’s rotational speed and the SMD, may obey to the dynamic interaction between the SMD and Earth’s revolution. For instance, the Sun-Earth gravity imposes a torque which periodically forces Earth’s Equator into the ecliptic [11].

The increasing rotational speed of Earth characteristic of the trans-solstitial analemmatic phases II and IV at high SMDs, suggests that the angle at which the Sun reaches Earth modifies the Sun’s influence on Earth’s rotation. This perspective somehow implies NBI tags the axis of the Sun-Earth gravity, because it marks the shortest distance between the Sun and Earth’s surface, by landing on Earth’s surface at the subsolar point. The coordination in the net drives of the SMD and the EoT along the trans-equinoctial phases I and III of the solar sundial analemma suggests Earth resists rotation as NBI approaches the SMD +3 arcdeg. Conversely, the proximity of SMD to either the Tropic of Cancer or the Tropic of Capricorn enables a faster rotation. Thus, Earth’s rotational speed increases as the length of the parallel hosting the NBI is shorter, and decreases as the length of the parallel grows.

Although the center of mass-density controlling SMD lie in the Equator, the latitude +3 arcdeg conforms the equilibrium center for the association between the SMD and Earth’s rotational speed, as records of SMD above or below +3 arcdeg promote lower rate of change for $\mathit{\rho }$, or likewise, a lower velocity for $\mathit{\omega }\prime$. Because the $\mathit{\rho }$ dynamics differs between hemispheres, it can be hypothesized that the higher share of continental land of the Northern Hemisphere modifies the effect of SMD over Earth’s rotational speed. The midseason boundaries (midspring, midsummer, etc.) of the analemma, where $\mathit{\omega }\prime$=0, define the beginning and ending of the four analemmatic phases on which the dynamics of the EoT and $\mathit{\rho }$ progress in a consistent direction.

As Earth’s linear speed of rotation varies with latitude, rotational speed can be assessed by dividing Earth’s circumference of a particular latitude by the 24 hours in a mean day. At the Equator, Earth’s perimeter is 40,075 km, a distance spanned by the apparent Sun every solar day. Accordingly, Earth’s linear rotational speed averages 1669.78 km h⁻¹ at the Equator, or 27.8 km min⁻¹, where each degree of latitude comprehends 111.319 km. For the Tropics of Cancer and Capricorn, these parameters correspond to 1532 km h⁻¹, 25.53 km min⁻¹ and 102 km, respectively. The association between Earth’s linear rotational speed and NBI speed (the latter being the Earth’s circumference at the latitude holding NBI ,$\mathit{L}$) is displayed on Figure 2, first as a join function, then comparing the same parameters on a single timeline along the Gregorian year, and lastly contrasting their accelerations.

Whereas L represents the distance daily spanned by NBI across longitude, $\mathit{\rho }$ represents the ERS at the Equator, both parameters being characteristic of a full Earth’s spin. ERS ($\mathit{\rho }$) conforms a linear transformation of EoT with crests occurring in midsummer and midwinter, and troughs in midspring and midautumn. Because L defines the distance spanned by NBI within a day, L can be thought as the NBI speed across longitude (in km day⁻¹). Therefore, the shifts in L that occur between successive latitudes where the SMD lands can be read as the NBI acceleration (${\mathbf{L}}_{\mathsf{\alpha }}$). Expressing L in km h⁻¹ and ${\mathbf{L}}_{\mathsf{\alpha }}$ in km h⁻² allows for $\mathit{\rho }$ and NBI accelerations to be compared.

According to Figure 2a Earth’s rotational speed ($\mathit{\rho },$ ERS) varies on par with the length of the parallel holding NBI (L, or NBI speed, given in km day⁻¹). However, the association between theses speeds differs between the ascending and descending halves of SMD. The function L($\mathit{\rho }$) can be read as the intersection of two elliptical shapes, the one in green-blue corresponding to winter and spring (ascending SMD) and the one in yellow-brown corresponding to summer and autumn (descending SMD).

In Figure 2b, crests of NBI speed converge with the center of decreasing strakes in ERS near equinoxes, troughs of NBI speed meet the center of increasing stake in ERS at the solstices. Analogously, crests of ERS converge around the center of increasing strakes in NBI speed in midsummer and midwinter, while troughs in ERS meet the center of decreasing strakes in NBI speed in midspring and midautumn. Therefore, transcends the comparison between the NBI and ERS accelerations (${\mathbf{L}}_{\mathsf{\alpha }}$ and ${\mathsf{\rho }}_{\mathsf{\alpha }}$) proposed in Figure 2c,

The extrema of $\mathit{\rho }$ and ${\mathit{L}}_{\alpha }$ nearly coincide. The two maxima of $\mathit{\rho }$ occur at ${\mathit{L}}_{\alpha }$ records of 2.31 and 2.14 (midwinter and midsummer), whereas the two minima of $\mathit{\rho }$ occur at ${\mathit{L}}_{\alpha }$ records of −2.27 and −2.30 km hour⁻² (midspring and midautumn). During ascending SMD, the crest and trough of $\mathit{\rho }$ occur on the 9th or 7th day (respectively) following an extrema of ${\mathit{L}}_{\alpha }$ (2.44 and−2.33 km h⁻²). Whereas, during descending SMD the crest and trough of $\mathit{\rho }$ occur on the 10th or 9th day (respectively) preceding an extrema of ${\mathit{L}}_{\alpha }$ (2.31 and −2.42 km h⁻²).

A SMD of 16.5 (whether positive or negative) is closely related to the midseason thresholds for the directional shifts in the joint function comparing the accelerations of L and $\mathit{\rho }$. The four given extrema of ${\mathit{L}}_{\alpha }$ occur within a narrow range of records of SMD: at 16.47, −16.86, 16.64, and − 16.94 arcdeg, on days 35, 128, 219, 314 of the Gregorian year.

Figure (2c) shows a circular association between the SMD and the ERS through the NBI and ERS accelerations. ${\mathit{L}}_{\alpha }$ ranges -2.42 to 2.44 km h⁻², while ${\mathsf{\rho }}_{\alpha }$ ranges from -0.42 to 0.54. Crests in ${\mathsf{\rho }}_{\alpha }$ occur at ZCPs of ${\mathit{L}}_{\alpha }$ at the solstices, while troughs in ${\mathsf{\rho }}_{\alpha }$ occur near ZCPs of ${\mathit{L}}_{\alpha }$ around the equinoxes. The crests and troughs of ${\mathit{L}}_{\alpha }$ closely align with the SMD of 16.5 arcdeg at midseason points, on which the ${\mathsf{\rho }}_{\alpha }$ occurs below 0.2 km h⁻². For the association between EoT and SMD, the average SMD-latitude for the centers of the trans-equinoctial phases is +2.8 arcdeg, whereas the threshold of SMD-latitude at which the crests or troughs of ERS take place falls 2.7 or 2.4 arcdeg northward (−13.8 or 18.9) from the SMD-latitude where ${\mathbf{L}}_{\mathsf{\alpha }}$ maximizes, which signifies −16.5 for the Southern or 16.5 for the Northern Hemisphere. Therefore, the highest records of ${\mathsf{\rho }}_{\alpha }$ converge near ZCPs of ${\mathit{L}}_{\alpha }$, whereas the opposite holds true. When the cycle of solar declination is ascendant ${\mathsf{\rho }}_{\alpha }$ becomes zero on the 10th or 7th day after a maxima (crest for winter or trough for spring) of ${\mathit{L}}_{\alpha }$, but when the cycle of solar declination is descendant ${\mathsf{\rho }}_{\alpha }$ becomes zero on the 10th or 8th day preceding a maxima (crest for summer or trough for autumn) of ${\mathit{L}}_{\alpha }$.

By testing the role of the latitude at which SMD lands as explicative variable for ERS, we hypothesize that the Sun leads Earth’s rotation, so the latitude at which the helioterric gravity lands (NBI) must modify the magnitude of the given association. Being Earth a bulged sphere, its mass varies with latitude, and so does the distance between the planet’s core and surface. Therefore, the angle at which the helioterric line reaches the planet must modify the Sun’s influence on Earth’s revolution and totation. Earth’s rotational speed obeys both torque’s angle of forces leading such rotation and the distribution of Earth’s mass [12]. The directional shifts in ERS, LSD, or EoT are closely tied to a particular latitude: 16.5 arcdeg. But what is so unique about this latitude? When the difference in length between Earth’s circumferences of successive records the SMD are considered, defined as ${\mathbf{L}}_{\mathsf{\alpha }}$, the four maxima of ${\mathbf{L}}_{\mathsf{\alpha }}$ correspond to SMD records close to 16.5 arcdeg. While the circumference lengths are just a characteristic of a sphere, by considering only the $\mathbf{L}$ records that converge to the day values of solar declination, the speed of the subsolar point becomes tied to $\mathit{L}$ in the same way $\mathit{\rho }$ becomes tied to $\mathit{\delta }\mathbf{\prime }$.

4. Conclusion

Establishing the four maxima and the four ZCPs of the EoT velocity as section boundaries for the solar sundial noon analemma, unveiled that the Sun meridian declination (SMD) and the EoT hold a cause-effect interaction throughout the year. The combined dynamics can be summarized by their net drives. The vertical and horizontal dimensions of the solar sundial noon analemma (EoT and SMD) showed a coordinated net drive during two trans-equinoctial phases, within the SMD interval of −13 to 19 arcdeg —a range 32 arcdeg wide centered in +3 arcdeg of SMD—. Conversely, the EoT and the SMD showed opposing net drives during two trans-solstitial phases, outside the specified interval of SMD.

The dynamics of Earth’s rotational speed is closely linked to that of the EoT when examined in the context of a solar sundial noon analemma. Two trans-equinoctial analemmatic phases progress from right to left, characterized by a negative EoT velocity and a coordinated net drive between SMD and EoT, conveying successive decreases in Earth’s rotational speed and corresponding increases in the length of the solar day. Conversely, two trans-solstitial analemmatic phases progress from left to right, characterized by a positive EoT velocity and opposing net drives between SMD and EoT, conveying successive increases in Earth’s rotational speed and corresponding decreases in the length of the solar day.

The dynamics of the SMD, EoT and Earth’s rotational speed are synchronized with Earth’s orbit. Increments in the rotational speed occur when the orbit closely approaches or departs from the solstices, while reductions in the rotational speed occur when the orbit closely approaches or departs from the equinoxes. Because Earth’s rotational speed fluctuates by only $\pm 1\%$ throughout the year, and from the dynamical association discussed above: (1) the Sun causes the Earth to rotate as a consequence of Earth’s dynamics of revolution and declination. Furthermore, because the sections on which the rotational speed decreases are always near equinoctial, it follows that (2) the Earth becomes heavier to spin when the SMD is closer to +3 arcdeg of SMD but lighter to spin as the SMD approaches the Tropic of Cancer or Capricorn. Summarizing, the closer the SMD occurs to the latitude + 3 arcdeg, the slower the Earth rotates, which implies the Earth’s rotational speed varies with the latitude at which the sunlight cone reaches the planet.

Earth’s rotational speed and SMD are linked through NBI. The acceleration in the daily path of NBI —measured by daily shifts in the length of Earth’s circumference between successive latitudes where SMD occurs— varies inversely with the acceleration in Earth’s rotational speed.

5. Definitions

Amplitude. In sinusoidal curves, the amplitude of a function is the vertical distance from either the crest or the trough to the horizontal axis.

Angular velocity of SMD ($\mathit{\omega }=d\mathit{\delta }/dt,$arcmin day^-1). The first-order time derivative of SMD, within a solar meridional analemma, which describes the daily rate of change for SMD.

Angular acceleration of SMD ($\mathit{\alpha }=d\mathit{\omega }/dt,$arcsec day^-2). The second-order time derivative of the SMD within a solar meridional analemma, which quantifies the rate of change for the velocity of SMD.

Angular jerk of SMD ($\mathit{j}=d\mathit{\alpha }/dt,$arcjerk day^-3). The third-order time derivative of SMD, which indicates the rate at which the acceleration of the SMD changes.

Angular snap of SMD ($\mathit{\varsigma }=d\mathit{\alpha }/dt,$arcsnap day^-4). The fourth-order time derivative of SMD, which specifies the rate at which the jerk of SMD occurs.

Angular velocity of the EoT ($\mathit{\omega }\mathbf{\prime }=d\delta \text{'}/dt,$arcmin day⁻¹). The first-order time derivative of the EoT within a solar meridional analemma, which describes the rate of change for EoT.

Angular acceleration of the EoT ($\mathit{\alpha }\mathbf{\prime }=d\mathit{\omega }\mathbf{\prime }/dt,$arcsec day^-2). The second-order time derivative of the EoT within a solar meridional analemma, which quantifies the rate of change for the velocity of EoT.

Angular jerk of the EoT ($\mathit{j}\mathbf{\prime }=d\mathit{\alpha }\mathbf{\prime }/dt,$arcjerk day^-3). The third-order time derivative of the EoT, which indicates the rate at which the acceleration of EoT changes.

Angular snap of the EoT ($\mathit{\varsigma }\mathbf{\prime }=d\mathit{j}\mathbf{\prime }/dt,$arcsnap day^-4). The fourth-order time derivative of the EoT, which accounts for the rate at which the jerk of EoT occurs.

arcjerk: A unit required to undertake this analysis. 1 arcjerk is equivalent to 1/60³ arcdeg, 1/60² arcmin, 1/60 arcsec or 60 arcsnap. The unit was proposed after using the factor ${[\mathit{\pi }/3]}^{\mathit{k}}{[180/\pi ]}^{k+1}=[180/\pi ]{60}^3$ for the third order derivative of position, whether for the EoT or for the SMD.

arcsnap: A unit required to undertake this analysis. 1 arcsnap is equivalent to 1/60⁴ arcdeg, 1/60³ arcmin, 1/60² arcsec and 1/60 arcjerk. The unit was proposed after using the factor ${[\mathsf{\pi }/3]}^{\mathrm{k}}{[180/\mathsf{\pi }]}^{\mathrm{k}+1}=[180/\pi ]{60}^4$for the fourth-order derivative of position, whether for the EoT or for the SMD.

Crest. A maximum of a function where the direction of the motion switches from upward to downward, (or from right to left). The slope of a function becomes zero at the crest.

Cycle. Repetitive loop of a sinusoidal curve. The cycle is completed when final and initial conditions converge, connecting one cycle to the next. A cycle can be split into half-cycles and/or quarter-cycles, following fixed instances of the position or following a parameter derived from the position.

Equation of Time, or EoT ($\delta \text{'}$). Also referred here to (in this document) as the Sun’s horizontal path. , The EoT , when used together with SMD tracks the daily discrepancies between the true solar time and the mean-time throughout a Gregorian year. The EoT locates the longitude at which NBI lands at the mean-time noon when SMD converges the site’s latitude.

Light cone, or sunbeam cone: A figurative conic section interrupted by two circular planes, the larger plane corresponds to the solar disk along its full two-dimensional shape on the sky, while the smaller plane is the lumbra.

Lumbra. A circular area drawn when the light-cone lands on Earth’s surface, such entire area holding natural beam irradiance (NBI) simultaneously.

Net drive. The net drive, or resultant drive, associates the sign of the velocity on which a motion points (which is the direction of the motion) and the sign of its acceleration (which indicates the direction of the strength controlling the motion). Only two kinds of resultant drive can take place within a one-dimensional displacement: accelerative (speeding-up) when $\mathit{\alpha }$ and $\mathit{\omega }$ hold the same sign, or decelerative (slowing-down) when those sings oppose.

Natural beam irradiance (NBI). The amount of solar irradiance (Nm⁻²) delivered as a normal projection from the solar ring to the lumbra. NBI lands on a site when the Sun meridian declination converges the site’s latitude and the Sun aligns with the local meridian.

Period. The time it takes for an entire cycle to be completed.

Phase of EoT. Each of 4 segments defined here for the EoT within a solar sundial noon analemma. A phase always starts and finishes at a zero-crossing point of velocity for the EoT. This concept was proposed to link the EoT to Earth’s rotational speed.

Pseudo-sinusoidal function. A graph resembling a sinusoidal function (derived from a sine or cosine function), but whose amplitudes, periods and zero crossing points fail to occur on a neat rhythm. The pseudo-sinusoidal curve may be described by a combined sine-cosine function.

Solar meridional analemma ($\delta \text{'},\delta$). A function built by plotting SMD over the EoT. For any given site on Earth’s surface, the solar meridional analemma displays the combined vertical-horizontal path of the mean-time Sun for every day within a Gregorian year. By taking the Sun’s position at mean-time noon, the latitude of the site becomes the reference point, whereas SMD accounts for the true declination.

Solar sundial analemma: When the sign of the EoT is switched, the analemma becomes reversed regarding the horizontal axis, conforming a sundial analemma (as opposed to an analemma recorded by pictures on the sky). The name comes from the way in which this analemma can be recorded on land by a gnomon arm. In the present document, the analemma is not reversed regarding SMD, because reversing the EoT suffices the proposed analysis.

Sun meridian declination, or SMD ($\delta$). Also referred here as the Sun’s vertical path, accounts for the true declination. The SMD is the angle between the Sun and Earth’s Equator and locates the latitude at which the NBI lands on Earth’s surface at mean time-noon, for every day of a Gregorian year.

Trough. A minimum (negative extrema) of a function, marks the point where the direction of the motion switches from downward to upward, or from left to right. The slope of a function becomes zero at the trough.

Zero crossing point (ZCP). When a function oscillates around zero, a ZCP is the point where the curve crosses the abscissa. An upward ZCP occurs when the curve crosses the horizontal axis from negative to positive records, while a downward ZCP occurs when the function crosses the horizontal axis from positive to negative records.