The Earth-atmosphere system (EAS) is continuously rotating in the Sun’s radiation field from which the entire EAS receives solar energy during each daily rotation. The input of solar energy into the EAS at a given location on the surface spanned by the top of the atmosphere (TOA) depends on the Earth’ heliocentric distance, $r$, the geocentric declination, ${\delta }_S$, and the local hour angle, $h$, of the Sun (e.g., [ 1 - 12 ]). The TOA may be interpreted as the height of the intervening atmospheric layer. Above this height, neither solar radiation nor infrared radiation is notably affected by gaseous or particulate matter. Generally, $r$, ${\delta }_S$ and $h$ depend on time, but on different time scales. While $h$ serves to describe the daily course of insolation at a given location between sunrise and sunset, $r$ and ${\delta }_S$ are necessary to capture, for instance, the seasonal variation and annual course of insolation. Since $r$ and ${\delta }_S$ are affected by the variation of the Earth’s orbit around the Sun due to smooth changes in the astronomical conditions caused by precession of both the equator and the ecliptic, nutation, changes in the obliquity (and long-term changes in eccentricity), it is indispensable to predict the astronomical elements $r$ and ${\delta }_S$ using trustworthy astronomical calculation methods.

Since the findings of Meech [ 1 ] and Wiener [ 2 , 3 ] that the global average of the daily mean solar irradiance at the TOA is equal to one quarter of the solar constant, later confirmed by Milankovitch [ 4 , 5 ], List [ 6 ], Fortak [ 13 ], Raschke et al. [ 14 ], Liou [ 9 ], Fu [ 10 ], Berger and Yin [ 15 ] and many others, has recently been disputed by experts of the German organization European Institute for Climate and Energy (EIKE), the goals of our study are: 1) to assess the prediction of insolation at the TOA for different time scales ranging from the diurnal course to the annual course and arbitrary periods of days in between, 2) to review the historical work on solar irradiance performed by Meech and Wiener and assess the accuracy of their findings using our results obtained for the year 1853 and the tropical years 1874/1875 and 2009/2010, and 3) to quantify the impact of smooth changes in the astronomical elements $r$ and ${\delta }_S$ on the daily mean irradiance over the annual course during the past four centuries by predicting the daily mean solar irradiance for the years 1610, 1650, 1710, 1750, 1810, 1850, 1910 and 1950 and comparing the results with those obtained for 2010. We also confirm the results of Kopp [ 16 ] for 2023.

Our predictions are based on the ICRS geocentric rectangular coordinates and their rates provided by the planetary and lunar ephemeris DE440 of the Jet propulsion Laboratory (JPL), California Institute of Technology (Park et al. [ 17 ]), hereafter called the JPL ephemeris DE440, and the adjustment of these data to the equator and the equinox of date (see Section 3) using the subroutines for aberration, frame bias, precession and nutation of the Naval Observatory Vector Astrometry Software (NOVAS), Version F3.1 (Kaplan et al. [ 18 , 19 ]), hereafter called the NOVAS F3.1 subroutines. Our predictions were performed for 90˚S to 90˚S N latitudes using an increment of 5˚, where a 10-minute time step was used. For 2009 and 2010, the value of the total solar irradiance (TSI) at 1AU, the so-called solar constant, of $S=1361\text{W}\cdot {\text{m}}^{-2}$ was chosen, in accord with Laue and Drummond [ 20 ], Raschke et al. [ 14 ], Kopp and Lean [ 21 ], and Kopp et al. [ 22 ]). For all other years considered in our study, we took the data from the reconstruction of the TSI for the past four centuries that are based on the “Community-Consensus TSI Composite” [ 23 ] and SATIRE-T model [ 24 ] (with modifications to fix spurious values prior to 1650).

Our paper is organized as follows: In Section 2, we give an overview regarding the historical work on solar irradiation. In Section 3, we discuss the astronomical aspects, including the state-of-the-art coordinate frame. The reconstruction of TSI for the past four centuries is briefly discussed in Section 4. In Sections 5 to 8, we discuss the solar irradiance from the diurnal course to the annual course and an arbitrary period of days in between, as well as the interannual, and seasonal variations of the solar irradiance during the past four centuries, all based on our simulations. We conclude on the impact of astronomical conditions on daily mean solar irradiance in the annual course in Section 9.

The solar irradiance reaching the TOA at a certain location is given by (e.g., [ 1 , 3 - 5 , 8 , 9 , 25 ])

$F_{s\downarrow }\left(\varphi ,\lambda \right)=F\cos {\Theta }_0$,(1)

where $\varphi$ is the latitude, $\lambda$ is the longitude, and $F$ is the TSI at the Earth’s actual heliocentric distance. The cosine of ${\Theta }_0$ occurs because only the component of the solar radiation flux density, $F$, that is perpendicular to the (infinitesimal) surface element is considered. This component is obtained using the scalar product between the vector $-F$ and the normal vector $n$ of this surface element (positively counted if $n$it is directed outwards), i.e., $-F\cdot n=-\left|F\right|\left|n\right|\cos \left(\pi -{\Theta }_0\right)=\left|F\right|\cos {\Theta }_0$, where $F=\left|F\right|$ denotes the magnitude of $F$, and $\left|n\right|=1$ is the magnitude of $n$. If the surface element is horizontally aligned as always considered here, ${\Theta }_0$ is the local zenith distance of the Sun’s center.

The TSI at the Earth’s heliocentric distance can be expressed by

$F={\left(\frac{r_S}{r}\right)}^2{E}_S$.(2)

Here, $r_S\cong 696300\text{km}$ is the radius [ 26 ] and $E_S$ the emittance of the Sun. This formula implies that the radiant power ( $=4\pi r_S^2{E}_S$) of the Sun is constant when the solar radiation is propagating through the space because of energy conservation principles in the absence of an intervening medium (e.g., [ 5 , 9 , 11 , 27 , 28 ]). In accord with Equation (2), we obtain for the Earth’s mean heliocentric distance, $r_0$, for which the so-called solar constant, $S$, is defined (e.g., [ 29 , 30 ])

$S={\left(\frac{r_S}{r_0}\right)}^2{E}_S$.(3)

Combining Equations (2) and (3) yields

$F={\left(\frac{r_0}{r}\right)}^{2}S=\frac{S}{{\rho }^2}$.(4)

where $\rho =r/r_0$ is the relative heliocentric distance. The quantity ${\left(r_0/r\right)}^2$ may be called the orbital effect. It does not vary more than 3.5 percent (e.g., [ 9 , 25 ]). Note that $r_0$ is often replaced by the semi-major axis $a$ (see also Equation (51)). We choose, however, $r_0=1\text{AU}$.

The mean angular velocity of the Earth’s rotation corresponds to ${\Omega }_E=2\pi /d_s\cong 7.2921\times {10}^{-5}{\text{s}}^{-1}$, where $d_s\cong 86164.09\text{s}$ is the mean sidereal day. Because this angular velocity of rotation is much larger than the angular velocity of the Earth’s revolution around the Sun ranging from about $2.060\times {10}^{-7}{\text{s}}^{-1}$ at perihelion and about $1.926\times {10}^{-7}{\text{s}}^{-1}$ at aphelion, the effect of the Earth’s orbital angular velocity is negligible ¹ . Therefore, in the case of the Earth, the function $\cos {\Theta }_0$ can be determined using rules of spherical trigonometry leading to (e.g., [ 1 , 3 - 6 , 8 - 11 , 25 , 29 ])

$\cos {\Theta }_0=\sin \varphi \sin {\delta }_S+\cos \varphi \cos {\delta }_S\cos h$.(5)

Here, ${\delta }_S$ and $h$ are again the geocentric declination and the local hour angle of the Sun, where $h=0$ defines the local solar noon that meets the criterion

$\cos {\Theta }_0=\sin \varphi \sin {\delta }_S+\cos \varphi \cos {\delta }_S=\cos \left(\varphi -{\delta }_S\right)$,(6)

and, hence, ${\Theta }_0=\varphi -{\delta }_S$. Note that $h$ counted from the meridian at local solar noon, addresses the rotation of the Earth around its polar axis, i.e., it serves to describe the diurnal variation of the insolation at the TOA at a specific location between sunrise, $h=-H$, and sunset, $h=H$, where $H$ is the so-called half-day, and $2H$ is the diurnal arc.

The Sun’s apparent geocentric declination is related to the apparent ecliptic longitude of the Sun, ${\lambda }_S$, counted from the dynamic vernal equinox of the northern hemisphere (NH), hereafter called the March equinox, by (see also Equation (32)) [ 3 , 4 , 5 , 8 , 35 ]

$\sin {\delta }_S=\sin {\lambda }_S\sin \epsilon$,(7)

where $\epsilon \cong 23˚26\text{'}18\text{'}\text{'}.42$ is the true obliquity of the ecliptic in 2025. (This means that the solar irradiance at the TOA is slightly affected by changes in the obliquity.) The apparent ecliptic longitude of the Sun differs from the true longitude of the Earth, $\lambda =\upsilon +\varpi$, by ${\lambda }_S=\lambda -180˚$. Here, $\upsilon$ is the true anomaly counted counterclockwise from the perihelion, $\varpi =\omega +{\Omega }_a$ is the longitude of the perihelion relative to the March equinox (see Figure 1 ), $\omega$ is the argument of the perihelion, and ${\Omega }_a$ is the longitude of the ascending node. Note that ${\delta }_S$ ranges from ${\delta }_S=-\epsilon$ (Tropic of Capricorn) at ${\lambda }_S=270˚$ to ${\delta }_S=\epsilon$ (Tropic of Cancer) at ${\lambda }_S=90˚$ (see Figure 2(b) ). Furthermore, for ${\lambda }_S=0˚$ and ${\lambda }_S=180˚$, the condition of the equinoxes, formula (7) provides ${\delta }_S=0$ [ 36 , 37 ].

According to Equations (1), (4) and (5), the TSI reaching the TOA at a certain location is given by (e.g., [ 1 , 3 - 5 , 8 - 10 , 25 ])

$F_{S\downarrow }\left(\varphi ,{\delta }_S,h,\rho \right)=\frac{S}{{\rho }^2}\left(\sin \varphi \sin {\delta }_S+\cos \varphi \cos {\delta }_S\cos h\right)$.(8)

Because $F_{S\downarrow }\left(\varphi ,{\delta }_S,h,\rho \right)$ is expressed in energy (J) per unit area (m²) and unit time (s), we may write

$F_{S\downarrow }\left(\varphi ,{\delta }_S,h,\rho \right)=\frac{\text{d}{W}_{S\downarrow }\left(\varphi ,{\delta }_S,h,\rho \right)}{\text{d}t}$,(9)

where $\text{d}{W}_{S\downarrow }\left(\varphi ,{\delta }_S,h,\rho \right)$ is the amount of solar energy that is flowing through the surface element during the time element dt (e.g., [ 1 - 6 , 9 ]). Consequently,

$\frac{\text{d}{W}_{S\downarrow }\left(\varphi ,{\delta }_S,h,\rho \right)}{\text{d}t}=\frac{S}{{\rho }^2}\left(\sin \varphi \sin {\delta }_S+\cos \varphi \cos {\delta }_S\cos h\right)$.(10)

We discuss the integration of this equation over different time scales in Sections 5 - 8. For readability, we omit the functional dependences expressed by $\left(\varphi ,{\delta }_S,h,\rho \right)$ hereafter.

Note that instead of $W_{S\downarrow }$, the normalized term $w_{S\downarrow }=W_{S\downarrow }/S$ may be used so that $\text{d}{w}_{S\downarrow }/\text{d}t=\left(\sin \varphi \sin {\delta }_S+\cos \varphi \cos {\delta }_S\cos h\right)/{\rho }^2$.

Inserting Equation (7) into Equations (5) and (10) yields [ 8 , 38 , 39 ]

$\cos {\Theta }_0=\sin \varphi \sin \epsilon \sin {\lambda }_S+\cos \varphi \sqrt{1-{\sin }^2\epsilon {\sin }^2{\lambda }_S}\cos h$(11)

and

$\frac{\text{d}{W}_{S\downarrow }}{\text{d}t}=\frac{S}{{\rho }^2}\left(\sin \varphi \sin \epsilon \sin {\lambda }_S+\cos \varphi \sqrt{1-{\sin }^2\epsilon {\sin }^2{\lambda }_S}\cos h\right)$.(12)

Alternatively, we may use $\cos {\delta }_S=\cos \left(\arcsin \left(\sin {\lambda }_S\sin \epsilon \right)\right)$ to obtain [ 35 ]

$\frac{\text{d}{W}_{S\downarrow }}{\text{d}t}=\frac{S}{{\rho }^2}\left(\sin \varphi \sin \epsilon \sin {\lambda }_S+\cos \varphi \cos \left(\arcsin \left(\sin \epsilon \sin {\lambda }_S\right)\right)\cos h\right)$.(13)

Equations (12) and (13) describe the long-term variation of the solar irradiation at a specific latitude as a function of the solar constant, the eccentricity, the obliquity, the Sun’s apparent ecliptic longitude, and the local hour angle. On the scales of multiple thousands of years (kyr), we have to pay attention to Milankovitch’s [ 4 , 5 ] astronomical theory of climatic variations that ranks as the most important achievement in the theory of climate in the 20^th century [ 11 , 41 ]. In accord with Berger [ 40 ], such long-term changes are denoted as climatic variations. Milankovitch’s astronomical theory regards the change of the eccentricity and the obliquity, and to precession and nutation phenomena due to the perturbations that Sun, Moon, and the principal planets of our solar system exert on the Earth’s orbit (e.g., [ 9 , 35 , 38 , 40 - 43 ]) ideally characterized by Equations (47) to (54). His theory plays a substantial role in the time series analysis of paleoclimate records (see, e.g., [ 43 , 44 ]). Because of these astronomical phenomena, the insolation at the TOA will vary during such long-term periods. Recently, Smulsky [ 35 ] presented a new astronomical theory of climatic variations that addresses three problems: the evolution of orbital motion, the evolution of the Earth’s rotational motion, and the evolution of the insolation controlled by the evolution of these motions.

According to Park et al. [ 17 ], the inertial coordinate frame of DE440 is connected to the International Celestial Reference System (ICRS). The current ICRS realization is achieved by Very Long Baseline Interferometry (VLBI) measurements of the positions of extragalactic radio sources (i.e., quasars) defined in the Third Realization of the International Celestial References Frame (ICRF3; Charlot et al. [ 45 ]), which is adopted by the International Astronomical Union (IAU). As pointed out by Park et al. [ 17 ], the orbits of the inner planets are tied to ICRF3 via VLBI measurements of the Mars-orbiting spacecraft (Konopliv et al. [ 46 ]) with respect to quasars with positions known in the ICRF. The ephemeris DE440 is valid for the years 1550 - 2650.

As shown by Figure 2(b) , the beginning of the astronomical seasons of the NH refers to the values of the Sun’s apparent declination at the apparent ecliptic longitudes: ${\lambda }_S=0˚$ (spring), ${\lambda }_S=90˚$ (summer), ${\lambda }_S=180˚$ (fall), and ${\lambda }_S=270˚$ (winter) [ 36 , 37 ]. In 1827, however, Fourier pointed out:

“Dans cette hypothèse du froid absolu de l’espace, s’il est possible de la concevoir, tous les effets de la chaleur, tels que nous les observons àla surface du globe, seraient dus àla présence du soleil. Les moindres variations de la distance de cet astre à la terre occasioneraient des changements très considérables dans les températures, l’excentricité de l’orbite terrestre donnerait naissance à diverses saisons.”

Translated from French into English:

“In this hypothesis of the absolute cold of space, if it is possible to conceive all the effects of heat that we observe on the surface of the globe, would be due to the presence of the Sun. The slightest variations in the distance of this star from the Earth would cause very considerable changes in temperatures, the eccentricity of the Earth’s orbit would give rise to various seasons.”

Fourier’s statements are misleading. The various astronomic seasons are caused by the Sun’s geocentric declination. It is well known, for instance, that the astronomic seasons of the NH differ from those of the southern hemisphere (SH) even for the same Earth’ heliocentric distance. The eccentricity mainly affects the variation of the TSI (see Equations (2) to (4) and Figure 2(d) ) from about 1317 W∙m⁻² at aphelion

( ${\lambda }_S\cong 104.2˚$), the maximum of the Earth’s heliocentric distance $r_a=a\left(1+e\right)\cong 1.521\times {10}^8\text{km}$, to about 1408 W∙m⁻² at perihelion ( ${\lambda }_S\cong 282.9˚$), the minimum of the Earth’s heliocentric distance $r_p=a\left(1-e\right)\cong 1.471\times {10}^8\text{km}$, where $a\cong 1.496\times {10}^8\text{km}$ and $e\cong 0.01671$ are the current semi-major axis

and the eccentricity, respectively. In addition, the eccentricity also affects the Earth’s orbit velocity that varies from about 29.29 km∙s⁻¹ at aphelion to about 30.29 km∙s⁻¹ at perihelion (see Figure 2(c) ) and, hence, the lengths of the astronomic seasons [ 15 ], i.e., March equinox-June solstice, 92.8 d; June solstice-September equinox, 93.6 d; September equinox-December solstice, 89.8 d; December solstice-March equinox, 89.0 d. This means that the summer half-year (spring plus summer) of the NH, is about 7.6 d longer than that of the SH (e.g., [ 2 , 3 , 5 , 15 ]).

The Earth’s heliocentric distance varies because of the orbital motion of the Earth around the Sun where, according to Kepler, the Earth’s orbit is nearly elliptical in shape with the Sun at one focus (see Figure 1 ). The mean plane spanned by the Earth’s orbit is called the ecliptic or the ecliptic plane, strictly formulated as follows (see Glossary of the IAU Division I Working GroupNomenclature for Fundamental Astronomy (NFA)): The ecliptic is the plane perpendicular to the mean heliocentric orbital angular momentum vector of the Earth-Moon barycenter in the Barycentric Celestial Reference System (BCRS). The ecliptic is also the apparent path of the Sun around the celestial sphere [ 47 ].

As mentioned before, the ecliptic is currently inclined by $\epsilon \cong 23˚26\text{'}19\text{'}\text{'}$ to the celestial equator, that is Earth’s equatorial plane projected on the celestial sphere. The ecliptic intersects the celestial equator at two points: The March equinox, $\Upsilon$, and the September equinox. The March equinox is at the ascending node of the ecliptic on the equator. It is the direction at which the Sun, in its annual apparent path around the Earth, crosses the equator from south to north. It is also referred to as “the First Point of Aries”. The inclination of the ecliptic plane to the plane of the equator, i.e., $\epsilon$, is the obliquity of the ecliptic [ 47 , 48 ].

Figure 2. (a) Earth’s heliocentric distance and, (b) Sun’s apparent geocentric declination, (c) Earth’s orbital velocity and (d) the total solar irradiance versus the Sun’s apparent ecliptic longitude computed for the tropical year 2009/2010 beginning with the March equinox, TDB = 2454910.9931 (March 20, 2009, 11:50 UT1). The required data were provided by the JPL ephemeris DE440 [ 17 ] and the NOVAS F3.1 subroutines [ 18 , 19 ]. A solar constant of $S=1361\text{W}\cdot {\text{m}}^{-2}$ was used.

As pointed out by Wiener [ 2 , 3 ], Lambert [ 49 ] was the first who dealt with this topic in expanded form, even by considering the average influence of the atmosphere and soil. Wiener also mentioned that Meech developed the formulas and calculated several tables [ 1 ]. Meech determined the annual irradiation for 1853 but avoided considering seasonal variations. The latitudinal and seasonal distribution of the normalized daily mean solar irradiance at the TOA for that year is illustrated in Figure 3 . Table 1 lists the results of Lambert and Meech used by Wiener for comparison with his own results. To assess these historical results, we added the results of our calculations for 1853 and the tropical years 1874/75 and 2009/10.

Ignoring the effects of the atmosphere, Wiener [ 2 , 3 ] computed the irradiation at the Earth’s surface for 17 equidistantly distributed days (with respect to $\Delta {\lambda }_S=22.5˚$) for the tropical year between March 20, 1874 and March 21, 1875, subdividing the northern and southern hemispheres at intervals of $\Delta \varphi =10˚$. Because he did not know the solar constant, he considered the normalized solar irradiance. His results are illustrated in Figure 4(a) . This latitudinal and seasonal distribution of the daily mean solar irradiance of the Earth without atmosphere, normalized by the solar constant, is nearly identical with that at the TOA because

the thickness of the intervening atmospheric layer of about ${\vartheta }_a\cong 150\text{km}$ would not notably change the amount of solar irradiance. At the distance at perihelion of $r=1.47098\times {10}^8\text{km}$ (in 2010), for instance, a change by ${\vartheta }_a\cong 150\text{km}$ would lead to ${\left(1-{\delta }_a/r\right)}^{-2}\cong 1.000002$. This means that the maximum of solar

irradiance at the subsolar point of about 1408 W∙m⁻² would be increased by 0.002 W∙m⁻². This increase is negligible because it is below the accuracy with which the TSI can be measured. Note that the daily average of an arbitrary quantity $\psi \left(\varphi ,\phi ,t\right)$ at a given location is defined by

$\bar{\psi }\left(\varphi ,\phi \right)=\frac{1}{d}{\displaystyle \underset{0}{\overset{d}{\int }}\psi \left(\varphi ,\phi ,t\right)\text{d}t}$.(14)

Milankovitch [ 5 ] weighted Wiener’s results with a solar constant of $S=2\text{cal}\cdot {\text{cm}}^{-2}\cdot {\min }^{-1}$. Later, List [ 6 ] re-scaled Milankovitch’s results by using a solar constant of $S=1.94\text{cal}\cdot {\text{cm}}^{-2}\cdot {\min }^{-1}$. List’s diagram of the latitudinal-annual distribution of the daily solar irradiance at the TOA has been used widely in the literature (e.g., [ 7 , 27 , 50 , 51 ]).

For assessing the accuracy of Wiener’s results, we considered the tropical year as well beginning on March 20, 1874, 00:00 UT1 ( $TDB=2405602.5$), where a subdivision of both hemispheres by $\Delta \varphi =5˚$ and a time step of 600 s were used. The Earth’s heliocentric distance and the Sun’s apparent geocentric declination, both time-dependent, were determined using again the data provided by the JPL ephemeris DE440 [ 17 ] and the NOVAS F3.1 subroutines [ 18 , 19 ]. Obviously, Wiener’s results agree well with our results, which is remarkable because Wiener did not have a computer. He derived his results using three different methods: 1) graphical, 2) mechanical quadrature, and 3) solution of elliptical integrals of first, second and third kind solved with the aid of Legendre’s tables. Only the results of the latter are listed in Table 1 .

Using the zonal average defined by (e.g., [ 7 , 52 - 55 ]),

$\hat{\psi }\left(\theta \right)=\frac{1}{2\pi }{\displaystyle \underset{0}{\overset{2\pi }{\int }}\psi \left(\theta ,\phi \right)\text{d}\phi }$,(15)

where $\psi \left(\theta ,\phi \right)$ is a field quantity like the daily solar irradiance and $\theta =\pi /2-\varphi$, the global average of $\langle \psi \rangle$ is given by (e.g., [ 7 , 52 , 53 , 55 - 58 ])

$\langle \psi \rangle =\frac{{\displaystyle \underset{\Omega }{\int }\psi \left(\theta ,\phi \right)\text{d}\Omega }}{{\displaystyle \underset{\Omega }{\int }\text{d}\Omega }}=\frac{1}{4\pi }{\displaystyle \underset{0}{\overset{2\pi }{\int }}{\displaystyle \underset{0}{\overset{\pi }{\int }}\psi \left(\theta ,\phi \right)\sin \theta \text{d}\theta \text{d}\phi }}=\frac{1}{2}{\displaystyle \underset{0}{\overset{\pi }{\int }}\hat{\psi }\left(\theta \right)\sin \theta \text{d}\theta }$.(16)

Here, $\Omega =4\pi$ is the solid angle of a sphere, and $\text{d}\Omega =\sin \theta \text{d}\theta \text{d}\phi$ is the differential solid angle, where $\theta$ and $\phi$ are the zenith and azimuthal angles in a right-handed spherical coordinate frame that serve to characterize the location; $\theta$ ranges from 0 (North Pole) to $\pi$ (South Pole), and $\phi$ ranges from 0 to $2\pi$, where $\phi =0$ may be considered to be equal to the prime meridian, i.e., the zero-longitude, about 100 m east of the transit circle at the Royal Observatory, Greenwich [ 47 ].

The zonal averages listed in Table 1 provide the following global averages of the solar irradiance normalized by the solar constant:

Lambert (1779): 0.24805

Meech (1857): 0.24995

Wiener (1879): 0.24972

This work for 1853: 0.24995

This work for 1874/1875: 0.24996

This work for 2009/2010: 0.24996

These results suggest that the global average of the daily mean solar irradiance at the TOA is equal to one quarter of the solar constant [ 1 , 3 - 5 ]. Tests for 2010 using an increment of $\Delta \varphi =2.5˚$ provide 0.25001.

As the Earth is not a sphere but an oblate spheroid and because of the obliquity, i.e., the tilt of the Earth’s rotational axis with respect to the normal vector, $n$, of the ecliptic plane pointing to the ecliptic pole (see Figure 1 ), mainly the gravitational forces of the Sun and the Moon cause a torque on the Earth’s equatorial bulge leading to a small temporal change in the angular momentum. This means that the assumption that the angular momentum in a central field is a conservative quantity is not exactly fulfilled. This torque tries to align the Earth’s rotational axis parallel to $n$ [ 64 , 65 ]. However, like in the case of a spinning toy top on which a torque is acting, the Earth’s rotational axis traces out a cone around the pole of the ecliptic (see Figure 1 ) in a cycle of about 26,000 years [ 66 ]. This behavior is called the luni-solar precession. Because the Sun and the Moon change their positions relative to each other, their gravitational forces also cause a nutation of the Earth’s rotational axis, where the principal period of nutation is 18.6 years [ 66 ]. This effect is much smaller in magnitude than the luni-solar precession. Nevertheless, both precession and nutation must be considered to calculate the Terrestrial Intermediate Origin (TIO) and the Celestial Intermediate Origin (CIO), the apparent geocentric position of the Sun, and the sidereal time at Greenwich or another location.

A closed elliptic orbit (ideally characterized by Equations (47) to (54)) requires that the gravitational potential reciprocally depends on $r$ (see the 2^nd term on the right-hand side of Equation (50)). Deviations from that due to the perturbations of the gravity field by other planets lead to an open orbit of a rosette-like shape (see Figure 9 ). The Earth’s orbit seems to move around the Sun, resulting in a precession of the perihelion. The precession of the orbital plane or the equator of a rotating body is a secular motion. In contrast, nutation is an oscillation of the rotation pole of a freely rotating body that is undergoing torque from external gravitational forces. Nutation of the Earth’s pole is specified in terms of components in obliquity and longitude [ 47 ]. According to the recommendation of the International Astronomic Union, Division I Working Group on Precession and the Ecliptic published by Hilton et al. [ 67 ], luni-solar precession and planetary precession should be replaced by the precession of the equator and the precession of the ecliptic for general use. Both precession phenomena are still subsumed under the notion of ‘general precession’. The combination of the general precession and the precession of the perihelion is called the climatic precession, and the related parameter $e\sin \varpi$ is called the climatic precession parameter. Figure 10 presents a sketch of the combined effect of these precession phenomena. Today, the North Pole tilts away from the Sun at perihelion during the southern summer. On the contrary, 11,000 years ago, the North Pole tilted towards the Sun at perihelion during the northern summer.

The speed of light is finite. It is about $c\cong 2.9879\times {10}^8\text{m}\cdot {\text{s}}^{-1}$ in vacuum. Thus, sunlight needs, at least, 492 s to travel to the Earth. It is called the light-time. Since the orbital velocity of the Earth is, at least, 29.3 km∙s⁻¹ (see Figure 2(c) ), the Earth would move in its orbit, at least, by 14,416 km during the light-time. Thus, it is unavoidable to consider the aberration of light due to the Earth’s motion.

According to Kaplan [ 68 ], there are a number of calculations that must be performed to obtain observables. These begin with astrometric reference data: a precomputed solar system ephemeris and, if a star is involved, a star catalog with positions and proper motions listed for a specified epoch. The computations account for the space motion of the object (star or planet), parallax (for a star) or light-time (for a planet), gravitational deflection of light, and the aberration of light due to the Earth’s motions. This means that the computation of the observable “solar irradiance at the TOA” should also account for precision, nutation, Earth’s rotation, polar motion, aberration of light etc. As pointed out by Kaplan [ 68 ], there are classical expressions for all these effects (except gravitational deflection), and relativity explicitly enters the procedure in only a few places, usually as added terms to the classical expressions and in the formulas that link the various time scales used. It has become common, then, to view this ensemble of calculations as being carried out entirely in a single reference system; or, two reference systems, barycentric and geocentric, that have parallel axes and differ only in the origin of coordinates (that is, they are connected by a Galilean transformation). For example, the coordinate system defined by the “equator and equinox of J2000.0”, can be thought of as either barycentric or geocentric. The relativistic effects then are interpreted simply as “corrections” to the classical result.

The March equinox plays an important role in astronomy. It is the point of origin for two different commonly used celestial coordinates: equatorial coordinates and ecliptic coordinates (see Figure 11 ). The line of intersection of the mean plane of the equator and the ecliptic defines the direction of the equinox. Using this direction as the origin, the right ascension, $\alpha$, is measured in the plane of the equator and the celestial (or ecliptic) longitude, $\lambda$, is measured in the plane of the ecliptic. Both right ascension and longitude are measured in the positive (or right-handed) sense. Like the hour angle, the right ascension is expressed in time from 0 h to 24 h. The declination, $\delta$, is measured from the equatorial plane, positive to the north, from 0˚ to 90˚, and the celestial latitude, $\beta$, is measured from the ecliptic plane, positive to the north, from 0˚ to 90˚ [ 48 ].

The vector $r$ from the origin, $O$, to a celestial object may be represented by rectangular coordinates ( $x,y,z$); for instance, in a Cartesian vector space by $r=xi+yj+zk$, where $i$, $j$, and $k$ are the unit vectors in the direction of $x$, $y$, and $z$. These unit vectors form a right-handed rectangular coordinate frame (aka right-handed trihedron). Coordinate frames with covariant and contravariant bases may also be used. The covariant and the contravariant basis vectors are defined (using Einstein’s summation convention)

by $q_i=\partial r/\partial q^i$ and $q^i=\partial r/\partial q_i$, $i=1,2,3$, respectively. Here, $q^i$ and $q_i$, $i=1,2,3$, are the contravariant and the covariant coordinates, respectively.

The position vector of the celestial object may also be represented by spherical coordinates, for instance, by $r=r\cos {\rm B}\cos \Gamma i+r\cos {\rm B}\sin \Gamma j+r\sin {\rm B}k$ in which the direction is specified by the longitudinal angle, $\Gamma$, in the $xy$-reference plane and the latitudinal angle, ${\rm B}$, from the reference plane to the position vector. Thus, the Cartesian coordinates are given by $x=r\cos {\rm B}\cos \Gamma$, $y=r\cos {\rm B}\sin \Gamma$, and $z=r\sin {\rm B}$. Therefore, using right ascension, $\alpha$, and declination, $\delta$, the equatorial coordinates can be expressed by [ 70 - 72 ] (see Figure 11 ).

$\begin{array}{lll}{x}_{eq}\hfill & =\hfill & r\cos \alpha \cos \delta \hfill \\ y_{eq}\hfill & =\hfill & r\sin \alpha \cos \delta \hfill \\ z_{eq}\hfill & =\hfill & r\sin \delta \hfill \end{array}$(17)

where the actual distance, $r$, is given by

$r=+\sqrt{x_{eq}^2+y_{eq}^2+z_{eq}^2}$.(18)

Since

$\frac{y_{eq}}{x_{eq}}=\tan \alpha$,(19)

the right ascension is given by

$\alpha =\arctan \left(\frac{y_{eq}}{x_{eq}}\right)=\text{atan}2\left(y_{eq},x_{eq}\right)$,(20)

where atan2 is the four-quadrant inverse tangent. Because of

$\frac{z_{eq}}{r}=\sin \delta \text{and}\frac{z_{eq}}{+\sqrt{x_{eq}^2+y_{eq}^2}}=\tan \delta$,(21)

the declination is given by

$\delta =\arcsin \left(\frac{z_{eq}}{r}\right)=\arctan \left(\frac{z_{eq}}{+\sqrt{x_{eq}^2+y_{eq}^2}}\right)$.(22)

As mentioned before, the equator and the ecliptic are moving because of the effects of perturbing forces on the rotation and motion of the Earth. Hence, the March equinox and the obliquity change with time. Therefore, these celestial coordinate frames must be carefully defined to relate to a standard frame that may be regarded as being fixed in space. Based on the geocentric rectangular coordinates provided by the JPL ephemeris DE440 [ 17 ], the right ascension, ${\alpha }_S$, and the declination, ${\delta }_S$, of the Sun can, in principle, be computed using Equations (20) and (22), respectively. However, first, an adjustment of the position vector $r$ for the aberration of light due to the Earth’s motion must be considered. Then, the reductions of the geocentric position $r_{GCRS}$ with respect to the Geocentric Celestial Reference System (GCRS) to a position $r_t$ with respect to the equator and the equinox of date for frame bias (GCRS to J2000.0), performed by matrix $B$, precession performed by matrix $P$, and nutation performed by matrix $N$ are required [ 68 , 72 ], i.e.,

$r_t=NPB{r}_{GCRS}=M{r}_{GCRS}$(23)

With $M=NPB$. These matrices are discussed, for instance, in the Astronomical Almanac for the Year 2023. As mentioned before, we used the NOVAS F3.1 subroutines for aberration, frame bias, precession and nutation (Kaplan et al. [ 18 , 19 ]) to calculate $r_t$.

The ecliptic coordinates $x_{ecl}$, $y_{ecl}$, and $z_{ecl}$ can be calculated similarly, but under consideration of the celestial longitude, $\lambda$, and the celestial latitude, $\beta$ (see Figure 11 ). The ecliptic coordinates and the equatorial coordinates are related to each other by

$\begin{array}{lll}{x}_{ecl}\hfill & =\hfill & x_{eq}\hfill \\ y_{ecl}\hfill & =\hfill & y_{eq}\cos \epsilon +z_{eq}\sin \epsilon \hfill \\ z_{ecl}\hfill & =\hfill & -y_{eq}\sin \epsilon +z_{eq}\cos \epsilon \hfill \end{array}$ (24)

Replacing $x_{eq}$, $y_{eq}$, and $z_{eq}$ according to Equation (17) gives for the ecliptic coordinates with respect to the equinox (either the mean or the true equinox) as the origin for the ecliptic longitude and latitude [ 70 , 72 ]

$\cos \beta \cos \lambda =\cos {\alpha }_e\cos \delta$,(25)

$\frac{y_{ecl}}{x_{ecl}}=\tan \lambda =\frac{\sin \delta \sin \epsilon +\sin {\alpha }_e\cos \delta \cos \epsilon }{\cos {\alpha }_e\cos \delta }$,(26)

leading to

$\cos \beta \sin \lambda =\sin \delta \sin \epsilon +\sin {\alpha }_e\cos \delta \cos \epsilon$,(27)

and

$\frac{z_{ecl}}{r}=\sin \beta =-\sin {\alpha }_e\cos \delta \sin \epsilon +\sin \delta \cos \epsilon$.(28)

We similarly obtain

$\frac{y_{eq}}{x_{eq}}=\tan {\alpha }_e=\frac{-\sin \beta \sin \epsilon +\cos \beta \cos \epsilon \sin \lambda }{\cos \beta \cos \lambda }$,(29)

leading to

$\sin {\alpha }_e\cos \delta =-\sin \beta \sin \epsilon +\cos \beta \cos \epsilon \sin \lambda$,(30)

and

$\frac{z_{eq}}{r}=\sin \delta =\sin \beta \cos \epsilon +\cos \beta \sin \epsilon \sin \lambda$.(31)

If we assume $\beta \approx 0$, Equation (31) provides [ 4 , 5 ]

$\sin \delta =\sin \lambda \sin \epsilon$(32)

on which Equation (7) is based. Note that $\lambda =\bar{\lambda }+\Delta \psi$, where $\bar{\lambda }$ is the mean longitude and $\Delta \psi$ is the nutation in the longitude. Furthermore, $\epsilon$ is either the mean or the true obliquity of the ecliptic, where $\bar{\epsilon }$ denotes the mean obliquity of date expressed by [ 17 , 74 ]

$\bar{\epsilon }=84381".448-46".815T-0".00059T^2+0".001813T^3$.(33)

The true obliquity of date is given by $\epsilon =\bar{\epsilon }+\Delta \epsilon$, where $\Delta \epsilon$ is the nutation in the obliquity. Both, $\Delta \psi$ and $\Delta \epsilon$ may be parameterized by $\Delta \psi =-19".1996\sin {\Omega }_{a,M}$ and $\Delta \epsilon =9".2025\cos {\Omega }_{a,M}$ [ 17 ], where ${\Omega }_{a,M}$ is the ascending node of the Moon’s orbit on the ecliptic given by [ 17 , 74 ]

${\Omega }_{a,M}=125˚02\text{'}40".280-1934˚08\text{'}10".549T+7".455T^2+0".008T^3$,(34)

where $T$ is the Barycentric Dynamical Time (TDB) in centuries with respect to the epoch J2000.0, i.e., $JD=2451545.0$ (January 1, 2000, 12:00 UT1), where the Julian century corresponds to 36525 d. However, we used $\Delta \psi$ and $\Delta \epsilon$ directly provided by the JPL ephemeris DE440. The differences between the parameterized and exactly calculated values are shown for 2010 in Figure 12 .

Figure 12. Nutation (a) in longitude and (b) in obliquity provided by the JPL ephemeris DE440 compared with the approximations $\Delta \psi =-19".1996\sin {\Omega }_{a,M}$ and $\Delta \epsilon =9".2025\cos {\Omega }_{a,M}$ mentioned by Park et al. [ 17 ].

The equinox right ascension is related to the intermediate right ascension, ${\alpha }_i$, by [ 72 ]

${\alpha }_e={\alpha }_i-E_o\left(T\right)$,(35)

where $E_o\left(T\right)=E_{prec}\left(T\right)-E_e\left(T\right)$ is the equation of the origins. Here, $E_{prec}\left(T\right)$ is the accumulated precession and $E_e\left(T\right)$ is the equation of the equinoxes representing the accumulated nutation. The precession portion is given in units of arcseconds by [ 76 ]

$E_{prec}\left(T\right)=-0".014506-4612".156534T-1".3915817T^2+0".00000044T^3+0".000029956T^4$.(36)

The equation of the equinoxes (expressed in arcseconds) is given by [ 68 , 76 ]

$\begin{array}{l}{E}_e\left(T\right)=\Delta \psi \cos \bar{\epsilon }+0.00264096\sin {\Omega }_{a,M}+0.00006352\sin 2{\Omega }_{a,M}\\ +0.00001175\sin \left(2F-2D+3{\Omega }_{a,M}\right)+0.00001121\sin \left(2F-2D+{\Omega }_{a,M}\right)\\ -0.00000455\sin \left(2F-2D+2{\Omega }_{a,M}\right)+0.00000202\sin \left(2F+3{\Omega }_{a,M}\right)\\ +0.00000198\sin \left(2F+{\Omega }_{a,M}\right)-0.00000172\sin 3{\Omega }_{a,M}-0.00000087T\sin {\Omega }_{a,M}.\end{array}$(37)

The fundamental luni-solar $F$ and $D$ can be found, for instance, in the Astronomical Almanac for the Year 2023, page B47. However, for our purposes, the equation of the equinoxes (expressed in seconds) can be approximated with sufficient accuracy by

$E_e\left(T\right)=\left(\Delta \psi \cos \bar{\epsilon }+0.002641\sin {\Omega }_{a,M}+0.000064\sin 2{\Omega }_{a,M}\right)/15$,(38)

The time can be obtained by use of the Earth Rotation Angle (ERA) $\theta$ (see Figure 13 ), i.e., the angle between the non-rotation origins defining the Celestial Intermediate Origin (CIO) and the Terrestrial Intermediate Origin (TIO) to realize the intermediate reference frame of epoch $t$, expressed as a function of UT1 by [ 68 , 76 ]

$\theta \left(D_U\right)=2\pi \left(0.7790572732640+1.00273781191135448D_U\right)$,(39)

where $D_U$ is the Julian UT1 date minus 2451545.0. We used the subroutine EROT of NOVAS F3.1 (Kaplan et al. [ 18 , 19 ]) to calculate ERA $\theta$. Using ERA, the Greenwich Apparent Sidereal Time (GAST) can be expressed by [ 76 ]

$GAST\left(D_U,T\right)=\theta \left(D_U\right)-E_o\left(T\right)$(40)

and the Greenwich Mean Sidereal Time (GMST) is given by

$GMST\left(D_U,T\right)=\theta \left(D_U\right)-E_{prec}\left(T\right)$(41)

so that

$GAST\left(D_U,T\right)=GMST\left(D_U,T\right)+E_e\left(T\right)$.(42)

To compute the CIO locator, $s$, the vector components of the Celestial Intermediate Pole (CIP), $x_{CIP}$ and $y_{CIP}$, are required. They can be determined from the matrix elements $x_{CIP}=M_{1,3}$ and $y_{CIP}=M_{2,3}$ (in our notation) so that

$s=-x_{CIP}\cdot y_{CIP}/2+\cdots$,(43)

where the dots represent lengthy series (for more details, see Capitaine et al. [ 77 ]). Tables for the series for $x_{CIP}$, $y_{CIP}$ and $s+x_{CIP}\cdot y_{CIP}/2$ are available only in electronic form, at the Strasbourg Astronomical Data Center (CDS) via anonymous ftp to cdsarc.u-strasbg.fr (130.79.128.5) or via http://cdsweb.u-strasbg.fr/cgi-bin/qcat?J/A+A/400/1145 .

Unless explicitly stated otherwise, we will refer to the Sun’s apparent geocentric right ascension and declination and the apparent ecliptic longitude and latitude.

Perihelion and aphelion can simply be determined using the first and second derivative tests [ 36 , 37 ]. In accord with Equation (18), the first derivative of the actual geocentric distance with respect to time is given by

$\frac{\text{d}r}{\text{d}t}=\frac{1}{r}\left(x_{eq}\frac{\text{d}{x}_{eq}}{\text{d}t}+y_{eq}\frac{\text{d}{y}_{eq}}{\text{d}t}+z_{eq}\frac{\text{d}{z}_{eq}}{\text{d}t}\right)$.(44)

As mentioned before, the JPL ephemeris DE440 provides the geocentric coordinates and their rates needed for computing this derivative. The optimum is given if the expression in the parentheses is equal to

zero (i.e., $\text{d}r/\text{d}t\cong 0$) [ 36 , 37 ]. The sign of the second derivative for the time of the optimum marks either the perihelion ( ${\text{d}}^{2}r/\text{d}{t}^2>0$) or the aphelion ( ${\text{d}}^{2}r/\text{d}{t}^2<0$). Because the ephemeris does not deliver ${\text{d}}^2{x}_{eq}/\text{d}{t}^2$, ${\text{d}}^2{y}_{eq}/\text{d}{t}^2$, and ${\text{d}}^2{z}_{eq}/\text{d}{t}^2$, we determined the second derivative numerically from the $\text{d}r/\text{d}t$

curve. Thus, Equation (44) can be used to determine the anomalistic year, i.e., the interval between successive passages of the Earth through the perihelion.

In accord with Equation (21), the Sun’s geocentric declination can be determined using

$\tan {\delta }_S=\frac{z_{eq}}{+\sqrt{x_{eq}^2+y_{eq}^2}}$.(45)

From Equation (45), we can infer the summer and winter solstice using the first and second derivative tests. The first derivative reads

$\frac{\text{d}{\delta }_S}{\text{d}t}={\cos }^2{\delta }_S\frac{\left(x_{eq}^2+y_{eq}^2\right)\frac{\text{d}{z}_{eq}}{\text{d}t}-z_{eq}\left(x_{eq}\frac{\text{d}{x}_{eq}}{\text{d}t}+y_{eq}\frac{\text{d}{y}_{eq}}{\text{d}t}\right)}{{\left(x_{eq}^2+y_{eq}^2\right)}^{3/2}}$.(46)

Optimum, minimum (winter solstice), and maximum (summer solstice) can be deduced using $\text{d}{\delta }_S/\text{d}t\cong 0$, ${\text{d}}^2{\delta }_S/\text{d}{t}^2>0$, and ${\text{d}}^2{\delta }_S/\text{d}{t}^2<0$, respectively. The $\text{d}{\delta }_S/\text{d}t$ curve serves to compute the second derivative numerically. If ${\text{d}}^2{\delta }_S/\text{d}{t}^2\cong 0$, a point of inflection may occur. This condition, however, is

only a necessary condition; the sufficient condition requires that the 2^nd derivatives in the neighborhood about this point have opposite signs. Figure 14 illustrates the Sun’s geocentric declination as well as the first and second derivatives, where the latter is multiplied by a factor of 20.

For assessing the results of our astronomical calculations, the results obtained for perihelion and aphelion for or around the years 1610, 1650, 1710, 1750, 1810, 1850, 1853, 1874, 1910, 1950, 2009, and 2010 are listed in Table 2 and Table 3 . These tables include the Earth’s heliocentric distance and the distance between the barycenter of the solar system and the Earth. In addition, the results obtained for the March and September equinoxes and the June and December solstices are listed in Table 4 and Table 5 , respectively. The perihelion moves from December 28, 1609, to January 3, 2010, leading to a shift of the beginning of the anomalistic year by about six days. In 1874, the Earth reached the perihelion twice. The aphelion moves from June 27, 1610, to July 6, 2010. Our results also confirm that the period from the March equinox to the subsequent September equinox is approximately 7.6 days longer than the period from this September equinox to the subsequent March equinox.

One may also consider Keplerian elements to compute the heliocentric distance

$r=\frac{p}{1+e\cos \upsilon }=\frac{a\left(1-e^2\right)}{1+e\cos \upsilon }$.(47)

Here, $p=L^2/\left(m\alpha \right)$, $e={\left\{1+2E{L}^2/\left(m{\alpha }^2\right)\right\}}^{1/2}$ is the eccentricity, $\upsilon$ is the true anomaly (see Figure 1), $\alpha ={\gamma }_{G}Mm$, where ${\gamma }_G$ is the gravitational constant, $M$ is the mass of the Sun, $m$ is the mass of the Earth, and $L=m{r}^2\text{d}\upsilon /\text{d}t=const.$ is the angular momentum considered as invariant with time, i.e., the angular momentum in a central field like Newton’s gravity field is a conservative quantity. The quantity $2p$ is called the latus rectum.

The Earth’s elliptic orbit around the Sun, characterized by Kepler’s first law that “the orbit of each planet is an ellipse and the Sun is at one of the two foci”, is a consequence of the state of energy in this central field expressed by [ 11 , 65 , 69 ]

$\frac{L^2}{m\alpha r}=1+{\left\{1+\frac{2E{L}^2}{m{\alpha }^2}\right\}}^{1/2}\cos \upsilon$,(48)

where

$E=T_{radial}+U_{eff}\left(r\right)$(49)

is the total energy, and

$U_{eff}\left(r\right)=L^2/\left(2m{r}^2\right)-\alpha /r$(50)

is the effective potential comprising the centrifugal potential and the gravitational potential [ 65 , 69 ], and $T_{radial}=m{\left(\text{d}r/\text{d}t\right)}^2/2$ is the radial kinetic energy (equal to zero for a circle). Equation (48) leads to Equation (47) if $p$ and $e$ are inserted. Choosing $\upsilon =0˚$ leads to $r_p=p/\left(1+e\right)$, and, hence, $r_p=a\left(1-e\right)$ the minimum of the Earth’ heliocentric distance at the perihelion. Choosing $\upsilon =180˚$ leads to $r_a=p/\left(1-e\right)$ and, hence, $r_a=a\left(1+e\right)$ the maximum of the Earth’ heliocentric distance at the aphelion. Thus, $p$ and $e$ are given by $p=a\left(1-e^2\right)$ and $e=\left(a^2-b^2\right)/a$ as already used in Equation (47). Therefore, the eccen-tricity may be expressed by $e=\left(r_a-r_p\right)/\left(r_a+r_p\right)=\left(r_a-r_p\right)/\left(2a\right)$ [ 35 ]. Furthermore, in accord with Equation (47), we define the relative heliocentric distance [ 4 , 5 , 35 ]

$\rho =\frac{r}{a}=\frac{1-e^2}{1+e\cos \upsilon }$.(51)

The mean heliocentric distance $r_0$ may be determined in accord with Kepler’s second law that “the radius vector drawn from the Sun’s center to the center of the planet sweeps out equal areas in equal times”.

The period $T$of one revolution of the Earth around the Sun is $T=2mA/L$, where $A=\pi ab=\pi a^2{\left(1-e^2\right)}^{1/2}$ is the area of the elliptic orbit, and $b=a{\left(1-e^2\right)}^{1/2}$ is the semi-minor axis.

Thus, we may write

$r^2\frac{\text{d}\upsilon }{\text{d}t}=\frac{2\pi }{T}{a}^2{\left(1-e^2\right)}^{1/2}$. (52)

Considering $a$ and $e$ as constant with time during this revolution, the integration of Equation (52) yields [ 9 , 11 ]

${\displaystyle \underset{0}{\overset{2\pi }{\int }}{r}^2\text{d}\upsilon }=\frac{2\pi }{T}{a}^2{\left(1-e^2\right)}^{1/2}{\displaystyle \underset{0}{\overset{T}{\int }}\text{d}t}=2\pi a^2{\left(1-e^2\right)}^{1/2}$ (53)

or

$r_0{}^2=\frac{1}{2\pi }{\displaystyle \underset{0}{\overset{2\pi }{\int }}{r}^2\text{d}\upsilon }=a^2{\left(1-e^2\right)}^{1/2}$. (54)

As the eccentricity is about $e\cong 0.01671$ (see Table 6 ), one may use $r_0=a{\left(1-e^2\right)}^{1/4}\cong a$. The deviation of $a$ from $r_0$ is about 10445 km.

For determining the true anomaly and, hence, the actual distance of the Earth, the transcendental equation for the eccentric anomaly called the Kepler equation [ 48 , 78 , 79 ],

$f\left(E\right)=E-e\sin E-M=0$,(55)

must be solved iteratively, for instance, by a Newton-Raphson method

$E^{\left(n+1\right)}=E^{\left(n\right)}-\frac{f\left(E^{\left(n\right)}\right)}{f^{\prime }\left(E^{\left(n\right)}\right)},n=0,1,2,\cdots$ (56)

where $f^{\prime }\left(E^{\left(n\right)}\right)=1-e\cos E^{\left(n\right)}$, $M=2\left(t-t_P\right)/U=L-\varpi$ is the mean anomaly, $U=2.978\times {10}^4\text{m}\cdot {\text{s}}^{-1}$ is the mean orbital velocity of the Earth, $t-t_P$ is the time since perihelion, $L$ is the mean longitude, and $\varpi$ is the longitude of the perihelion counted counterclockwise from the moving March equinox (see Figure 1). The procedure may be stopped if after k iteration steps the condition $\left|E^{\left(k\right)}-E^{\left(k-1\right)}\right|<{10}^{-6}\mathrm{deg}$ is fulfilled.

Even though the accuracy of the Kepler equation may be sufficient in several cases, we generally use the JPL ephemeris DE440 [ 17 ]. Note that the values of the Keplerian elements and their rates, with respect to the mean ecliptic and equinox of J2000.0, valid for the time interval 1800 AD - 2050 AD are listed in Table 8.10.2 by Standish and Williams [ 79 ] (see Table 6 ).

The heliocentric coordinates of the EMB in its orbital plane, $r_{hc}$, with the $x_{hc}$-axis aligned from the focus to the perihelion are then given by [ 79 ]:

$x_{hc}=r\cos \upsilon =a\left(\cos E-e\right)$(57)

and

$y_{hc}=r\sin \upsilon =a\sqrt{1-e^2}\sin E$(58)

with $z_{hc}=0$. The relation between the true anomaly and the eccentric anomaly is given by

$\tan \frac{\upsilon }{2}=\sqrt{\frac{1+e}{1-e}}\tan \frac{E}{2}$.(59)

Figure 15 shows the variation of the TSI at 1 AU, often called the solar constant, and the related

uncertainty during the period from November 1978 to April 2023. The results mainly follow the sunspot number, but solar activity also plays an important role. As illustrated by this figure, the solar activity was low in 2010, and the TSI was close to $S=1361\text{W}\cdot {\text{m}}^{-2}$. This fact is one of the reasons why Kramm et al. [ 53 ] considered 2010. The reconstruction of the TSI at 1 AU for the past 400 years is illustrated in Figure 16 . This reconstruction reflects both the Maunder minimum and the Dalton minimum of the sunspot numbers.

As mentioned before, the solar constant is the TSI at the Earth’s mean heliocentric distance $r_0$. However, the heliocentric distance of $r_0=1\text{AU}$ is considered due to the relatively small difference. At $TDB=\text{2455291}\text{.0764}$ (April 4, 2010, 13:50 UT1), $r_0=\text{1}\text{.00014990}$ corresponds to the Sun’s apparent geocentric position ${\alpha }_S={\text{00}}^{\text{h}}{\text{54}}^{\text{m}}{\text{04}}^{\text{s}}\text{.561}$ (right ascension), ${\delta }_S=5˚47\text{'}12".953$ (geocentric declination) and ${\lambda }_S\cong \text{14}˚\text{41}\text{'}\text{3}".\text{386}$ (ecliptic longitude), respectively (see Figure 2). The Earth’s heliocentric distance of $r_0=1\text{AU}$ is reached at $TDB=\text{2455290}\text{.5625}$ (April 4, 2010, 01:30 UT1) that corresponds to the Sun’s apparent geocentric position ${\alpha }_S={\text{00}}^{\text{h}}{\text{52}}^{\text{m}}{\text{11}}^{\text{s}}\text{.976}$, ${\delta }_S=\text{5}˚\text{35}\text{'}\text{27}".\text{332}$ and ${\lambda }_S=\text{14}˚\text{10}\text{'}\text{40}".\text{793}$, respectively. This means that the distance of 1 AU is reached near the March equinox. Because the March equinox plays a dominant role in astronomy, one may consider it as an additional reference point for the TSI. At the March equinox of 2010, the Earth’s heliocentric distance was about $r=\text{0}\text{.99594656}\text{AU}$ and, hence, $TSI\cong 1372\text{W}\cdot {\text{m}}^{-2}$.

In accord with Equation (10), the solar irradiation reaching the TOA at a given location during a certain period of a day, for instance, between sunrise and sunset is given by (e.g., [ 1 - 5 , 9 , 25 , 27 , 52 ])

$W_{S\downarrow }={\displaystyle \underset{t_r}{\overset{t_s}{\int }}\frac{S}{{\rho }^2}\left(\sin \varphi \sin {\delta }_S+\cos \varphi \cos {\delta }_S\cos h\right)\text{d}t}$,(60)

where $t_r$ and $t_s$ are the times of the location-dependent sunrise and sunset, respectively. As pointed out by Milankovitch [ 5 ], one must be aware of the discontinuity of radiation between sunset and subsequent sunrise, during which the considered element at the TOA is not irradiated. Thus, the integration between two arbitrary points of time, $t_1$ and $t_2$, of a day must fulfil the condition $t_r\le t_1<t_2\le t_s$. As the variation of the Earth’ heliocentric distance, $r$, and the Sun’s geocentric declination, ${\delta }_S$, during a day is negligible and may be replaced by their values at local solar noon, Equation (60) leads to

$W_{S\downarrow }=\frac{S}{{\rho }^2}\left(\sin \varphi \sin {\delta }_S{\displaystyle \underset{t_r}{\overset{t_s}{\int }}\text{d}t}+\cos \varphi \cos {\delta }_S{\displaystyle \underset{t_r}{\overset{t_s}{\int }}\cos h\text{d}t}\right)$.(61)

Because

$\frac{\text{d}W}{\text{d}t}=\frac{\text{d}W}{\text{d}h}\frac{\text{d}h}{\text{d}t}$,(62)

we may insert the mean angular velocity of the Earth’s rotation, $\Omega =\text{d}h/\text{d}t=2\pi /d_a$ (e.g., [ 1 , 3 , 7 - 9 ]),

into Equation (61), where $d_a$ is the local apparent solar day, counted from the local solar noon to the following local solar noon. Thus, Equation (61) may be written as

$W_{S\downarrow }=\frac{S}{{\rho }^2}\left(\sin \varphi \sin {\delta }_S{\displaystyle \underset{-H}{\overset{H}{\int }}\frac{\text{d}h}{\Omega }}+\cos \varphi \cos {\delta }_S{\displaystyle \underset{-H}{\overset{H}{\int }}\cos h\frac{\text{d}h}{\Omega }}\right)$,(63)

i.e., the integration is only performed from sunrise ( $-H$) to sunset ( $H$), where $H$ is the half-day.

The apparent solar time, $t_a$, varies during the Earth’s annual revolution around the Sun due to the eccentricity of the Earth’s orbit (see, e.g., Kepler’s second law), the obliquity of the ecliptic and small variations resulting from irregularities in the Earth’s rotation on its axis. Its difference from the mean solar time, $t$, can be determined by the so-called equation of time, $E_t$, by (e.g., [ 5 , 8 , 48 ])

$t_a=t+E_t$.(64)

Thus, we have

$\frac{\text{d}{t}_a}{\text{d}t}=1+\frac{\text{d}{E}_t}{\text{d}t}$.(65)

As $\left|\text{d}{E}_t/\text{d}t\right|\ll 1$, this term can be neglected so that $\text{d}{t}_a\cong \text{d}t$. Furthermore, $d_a$ varies with time during the Earth’s annual revolution around the Sun for the same reasons. The mean solar day, $d$, differs from $d_a$ by

$d_a=d+E_t=d\left(1+\frac{E_t}{d}\right)$.(66)

Figure 17. Variation in the equation of time (a) through the year and (b) in dependence of the Sun’s geocentric declination, the so-called analemmic curve (in accord with [ 36 , 37 , 66 , 73 ]).

As $\left|E_t/d\right|\ll 1$ (see Figure 17 ), we may use $d_a\cong d$. Note that the mean solar day is about 3^m56^s longer than the sidereal day. Furthermore, $E_t$ is independent of latitude but varies over the year and depends on the Sun’s geocentric declination (see Figure 17 ).

The half-day can be deduced using Equation (5) by setting $\cos {\Theta }_0=0$ (e.g., [ 1 , 3 , 5 , 9 , 11 , 15 ])

$H=\arccos \left(-\frac{\sin \varphi \sin {\delta }_S}{\cos \varphi \cos {\delta }_S}\right)=\arccos \left(-\tan \varphi \tan {\delta }_S\right)$,(67)

i.e., local sunrise and sunset mainly depend on $\varphi$ and ${\delta }_S$. This formula, however, is only valid if the half-day fulfills the condition $H<\pi$ [ 3 ] ( $\pi$ corresponds to 12 hours, i.e., 43200 seconds). The formula is invalid for those locations within the polar domes bounded by the polar circles. At these locations, the Sun does not set within 24 hours for which $\cos {\Theta }_0$ does not become zero [ 3 ]. Thus, the condition $H=\pi$ is that of the polar circle, i.e., $\varphi \cong -66˚33\text{'}42\text{'}\text{'}$(Antarctic circle) and $\varphi \cong 66˚33\text{'}42\text{'}\text{'}$ (Arctic circle). For $\varphi =0˚$ and for the condition of the equinoxes, ${\delta }_S=0˚$, one obtains $H=\pi /2$, i.e., the diurnal arc corresponds to 12 hours. Figure 18 illustrates the latitudinal-annual distribution of daylight for 2010.