Solar global irradiance from actinometric degree data for Montsouris (Paris) 1873–1877

Abstract

In the past, long-term recordings of solar radiation energy were not commonly conducted. However, several observations using the Arago-Davy actinometer were made in different parts of the world during the nineteenth century. In this paper, we propose a method to convert actinometric degree data into information on global solar irradiance on a horizontal surface. We utilized hourly actinometric degree data from the Montsouris Observatory in Paris recorded between 1873 and 1877. Three models were tested for estimating solar global irradiance at ground level from actinometric degrees. Despite the quality of the solar irradiance data provided by the Twentieth Century Reanalysis Project version 3 (20CRv3) was subjected to criticisms, these data are used here as a reference, taking into account the lack of similar data for the nineteenth century. One of the main challenges in this study is the fact that the solar irradiance incident on the Arago-Davy actinometer is not global solar irradiance on horizontal surface, ${G}_{H}$, a quantity which is usually measured and recorded. To address this issue, a proxy value of ${G}_{H}$, named ${G}_{H}^{+}$, is defined by using the first sub-model of the Ferrel-Pouillet model (named FP1). The analysis showed that ${G}_{H}^{+}$ is suitable for estimating both hourly and daily averaged values of the global solar irradiance ${G}_{H}$. However, the model FP1 requires two input parameters, namely the actinometric degree $D$ and the bright-bulb thermometer temperature ${t}_{{\mathrm{bright}}-{\mathrm{bulb}}}$, while most long-term observations on the globe have data only for $D$. This makes the FP1 model less useful in these cases. To address this, a method was proposed that relies only on $D$ as an input parameter. This method provides hourly and daily averaged irradiance data that are similar to those obtained using the FP1 model with two input parameters. The accuracy of this method is expected to remain the same in colder climates but decrease in warmer climates. The proposed method also provides credible lower and upper bounds for the interval of variation of monthly averaged irradiance. Additionally, at the level of monthly averaged solar irradiance data, the proposed method and the 20CRv3 project seem to support each other.

Appendices

Appendix 1

1.1 Davy-Pouillet (DP) model

The theory of the Davy-Pouillet (DP) model, well explained by Besson (1928), is based on the Bouguer-like law Eq. (2) by taking into account that $A$ is the difference of temperature between the conjugated thermometers on the top of the atmosphere (subscript 0):

$$A\equiv {t}_{{\mathrm{black}}-{\mathrm{bulb}},0}-{t}_{{\mathrm{bright}}-{\mathrm{bulb}},0}$$

(A1)

Using Eqs. (2) and (A1) yields:

$${t}_{{\mathrm{black}}-{\mathrm{bulb}}}-{t}_{{\mathrm{bright}}-{\mathrm{bulb}}}=\left({t}_{{\mathrm{black}}-{\mathrm{bulb}},0}-{t}_{{\mathrm{bright}}-{\mathrm{bulb}},0}\right){\tau }_{atm}^{m}$$

(A2)

Marié-Davy has made nine observation during 1873 and 1874 in days with very clear sky and obtained a series of values for the difference ${t}_{{\mathrm{black}}-{\mathrm{bulb}}}-{t}_{{\mathrm{bright}}-{\mathrm{bulb}}}$. Taking into account that the values of $m$ may be computed for each of the nine observations by using astronomic relations, a series of nine equations in the unknowns ${t}_{{\mathrm{black}}-{\mathrm{bulb}},0}-{t}_{{\mathrm{bright}}-{\mathrm{bulb}},0}$ and ${\tau }_{atm}$ have been obtained. These unknowns have been fitted to the data and the results were (Marié-Davy 1888, p. 234):

$${t}_{{\mathrm{black}}-{\mathrm{bulb}},0}-{t}_{{\mathrm{bright}}-{\mathrm{bulb}},0}={17}^{^\circ }C$$

(A3a)

$$\tau =0.875$$

(A3b)

Marié-Davy has translated the temperatures indicated by the actinometer into the actinometric degree $D$, which is defined (in percent) by the fraction of the temperature difference ${t}_{{\mathrm{black}}-{\mathrm{bulb}}}-{t}_{{\mathrm{bright}}-{\mathrm{bulb}}}$ associated with ${t}_{{\mathrm{black}}-{\mathrm{bulb}},0}-{t}_{{\mathrm{bright}}-{\mathrm{bulb}},0}$:

$$D\left(\%\right)\equiv \frac{{t}_{{\mathrm{black}}-{\mathrm{bulb}}}-{t}_{{\mathrm{bright}}-{\mathrm{bulb}}}}{{t}_{{\mathrm{black}}-{\mathrm{bulb}},0}-{t}_{{\mathrm{bright}}-{\mathrm{bulb}},0}}\times 100$$

(A4)

For the instrument used at Montsouris the constant $100/\left({t}_{{\mathrm{black}}-{\mathrm{bulb}},0}-{t}_{{\mathrm{bright}}-{\mathrm{bulb}},0}\right)$ has the value $100/17$, or 5.88. Therefore, from Eq. (A4), one finds for the Montsouris actinometer:

$$D\left(\%\right)=5.88\left({t}_{{\mathrm{black}}-{\mathrm{bulb}}}-{t}_{{\mathrm{bright}}-{\mathrm{bulb}}}\right)$$

(A5)

Notice that for an Arago-Davy actinometer other than the Montsouris instrument, which indicates the temperatures $t{^{\prime}}_{{\mathrm{black}}-{\mathrm{bulb}}}$ and $t{^{\prime}}_{{\mathrm{bright}}-{\mathrm{bulb}}}$ when the Montsouris instrument indicates ${t}_{{\mathrm{black}}-{\mathrm{bulb}}}$ and ${t}_{{\mathrm{bright}}-{\mathrm{bulb}}}$, respectively, the multiplying constant is

$$C=5.88\frac{{t}_{{\mathrm{black}}-{\mathrm{bulb}}}-{t}_{{\mathrm{bright}}-{\mathrm{bulb}}}}{t{^{\prime}}_{{\mathrm{black}}-{\mathrm{bulb}}}-t{^{\prime}}_{{\mathrm{bright}}-{\mathrm{bulb}}}}$$

(A6)

Therefore, calibrating another Arago-Davy actinometer requires several simultaneous measurements with that actinometer and the Montsouris actinometer, respectively, using the Eq. (A6), and computing the average of the results obtained. Computation of the actinometric degree $D\left(\%\right)$ for that actinometer is based on Eq. (A5) by replacing 5.88 with the value $C$ given by Eq. (A6).

Extracting the temperature difference ${t}_{{\mathrm{black}}-{\mathrm{bulb}}}-{t}_{{\mathrm{bright}}-{\mathrm{bulb}}}$ from Eq. (A5) and replacing it in Eq. (2), one finds:

$$\frac{D\left(\%\right)}{5.88}=A{\tau }^{m}$$

(A7)

From Eqs. (A7), (A1), and (A3a), one finds:

$$D\left(\%\right)=100{\tau }_{atm}^{m}$$

(A8)

Eliminating ${\tau }^{m}$ between Eq. (A8) and Eq. (4) used by Pouillet (1838), one finds the direct solar irradiance on a surface perpendicular on the direction Sun-Earth:

$${G}_{dir,N}=\frac{D\left(\%\right)}{100}{I}_{sc}$$

(A9)

which is the form of the Davy-Pouillet model used in Table 1.

Appendix 2

2.1 Ferrel-Pouillet (FP) model

Next, we will provide a detailed description of the Ferrel-Pouillet model of the Arago-Davy actinometer. While the first part of the model was originally proposed by Pouillet (1838), it was Ferrel who applied Pouillet’s theory to the Arago-Davy actinometer. It is worth noting that the derivation of this model is of interest in its own right since it does not rely on the Stefan-Boltzmann radiation law, which was not yet known when Pouillet conducted his work. Here we present the model based on Ferrel’s (1886) explanation, using modern terminology and notation.

The following relationships involve dimensional numerical constants. They correspond to temperatures expressed in degree Celsius, time in minutes, length in centimeters, and heat in calories.

We assume that between the thermometric bulb (body B, temperature ${t}_{\mathrm{bulb}}$) and the wall of the enclosure (body B’, temperature ${t}_{\mathrm{encl}}$), there is empty space (vacuum). We assume that ${t}_{\mathrm{bulb}}\ge {t}_{\mathrm{encl}}$. Both the bulb and the enclosure walls emit radiation. The energy radiated by the bulb is incident on the enclosure walls, being (in part) reflected, transmitted, or absorbed. The energy radiated by the enclosure wall is in part absorbed by the bulb. Energy conservation states that the bulb temperature changes in time as follows:

$${C}_{\mathrm{bulb}}\frac{{dt}_{\mathrm{bulb}}}{d\tau }={F}_{{\mathrm{bulb}},net}$$

(B1)

where ${C}_{\mathrm{bulb}}$ is bulb heat capacity, $\tau$ is time while ${F}_{{\mathrm{bulb}},net}$ is the net energy flux exchanged between the bulb and the enclosure walls given by:

$${F}_{{\mathrm{bulb}},net}={F}_{\mathrm{bulb}}-{F}_{{\mathrm{encl}}-{\mathrm{bulb}}}$$

(B2)

where ${F}_{\mathrm{bulb}}$ is the energy flux radiated in all directions by the bulb and ${F}_{{\mathrm{encl}}-{\mathrm{bulb}}}$ is the energy flux radiated by the enclosure wall and absorbed by the bulb. Notice that the temperatures of the bulb and enclosure are relatively small, and therefore, the radiation emitted by these bodies are mainly in the IR portion of the spectrum.

Generally, the energy flux radiated by a body is given by Eq. (1) of Ferrel (1886):

$$F=\int\limits_{s,\sigma}K\varepsilon\left(i\right)cosi\psi\left(i\right)\phi\left(t\right)\mathrm{dsd}\sigma$$

(B3)

where $K$ is a constant corresponding to the case of a blackbody, $\varepsilon \left(i\right)$ is the emittance of the body, $i$ is the emission angle with respect to the normal to the body surface while integration is taken over the whole surface of the body (infinitesimal element $ds$) and over the whole emission solid angle (infinitesimal element $d\sigma$ with vertex at $ds$). In Eq. (B3), the function $\psi \left(i\right)$, which generally differs little from unity, takes into account the variation in respect to the law of cosines while $\phi \left(t\right)$ is a function of temperature as described by the law of radiation to be used. Usage of the general Eq. (B3) yields the following results (Ferrel 1886, p. 91):

$${F}_{\mathrm{bulb}}=-K\phi \left({t}_{\mathrm{bulb}}\right)\underset{\sigma }{\int }{\varepsilon }_{{\mathrm{bulb}},IR}\left(i\right)\psi \left(i\right)cosid\sigma \underset{s}{\int }ds$$

(B4)

$$F_{\mathrm{encl}-\mathrm{bulb}}=K\phi\left(t_{\mathrm{encl}}\right)\int_{\sigma'}\alpha_{bulb,IR}\left(i'\right)\psi'\left(i'\right)\;d\sigma'\int_{s'}\varepsilon_{encl,IR}\left(i'\right)\psi'\left(i'\right)ds'$$

(B5)

where ${\varepsilon }_{{\mathrm{bulb}},IR}$, ${\alpha }_{{\mathrm{bulb}},IR}$, and ${\varepsilon }_{{\mathrm{encl}},IR}$ are the emittance and absorbance of the bulb and the emittance of the enclosure wall, respectively. In Eq. (B4), integration with regard to $d\sigma$ should be performed over the whole solid angle in which radiation from $ds$ is emitted. For the spherical bulb, this solid angle is a hemisphere. The integration over $ds$ must include the whole surface of the bulb. The prime (‘) in Eq. (B5) shows that integration is performed over the surface of the enclosure wall and over the solid angle subtended by the bulb when seen from different parts of the enclosure wall. Therefore, the integral with regard to ${d\sigma }^{^{\prime}}$ must be taken for each element ${ds}^{^{\prime}}$ of the enclosure wall, through the solid angle subtended by the bulb, and with regard to ${ds}^{^{\prime}}$ so as to include all the elements of the enclosure surface which emit radiation to the bulb.

Computations needed in Eqs. (B4) and (B5) may be performed by using several reasonable approximations appropriate for the specific geometry configuration, namely a spherical bulb and an enclosure completely surrounding the bulb. Details of computations are given by Ferrel (1886, p. 91–95). The result is (see Eq. (16) of Ferrel (1886)):

$${F}_{\mathrm{bulb}}=-K\pi {\overline{\varepsilon }}_{{\mathrm{bulb}},IR}{s}_{\mathrm{bulb}}\phi \left({t}_{\mathrm{bulb}}\right)$$

(B6)

$${F}_{{\mathrm{encl}}-{\mathrm{bulb}}}=K{\overline{\alpha }}_{{\mathrm{bulb}},IR}{\overline{\varepsilon }}_{{\mathrm{encl}},IR}\Psi {QS}_{\mathrm{bulb}}\phi \left({t}_{\mathrm{encl}}\right)$$

(B7)

where ${\overline{\varepsilon }}_{{\mathrm{bulb}},IR}$, ${\overline{\alpha }}_{{\mathrm{bulb}},IR}$, and ${\overline{\varepsilon }}_{{\mathrm{encl}},IR}$ are the emittance and absorbance of the bulb and the emittance of the enclosure wall, respectively, averaged over radiation wavelength spectrum and over all incidence angles, ${s}_{\mathrm{bulb}}$ and ${S}_{\mathrm{bulb}}$ are the bulb surface and the section area of the bulb intercepting the greatest amount of radiation energy from the enclosure wall, respectively. For the specific geometry configuration considered here and taking into account that the sun rays are almost parallel, the functions $\Psi$ and $Q$ entering Eq. (B7) are well approximated by $Q=1$ and $\Psi =4\pi$. From Eqs. (B2), (B6), and (B7), we find:

$${F}_{{\mathrm{bulb}},net}=-K\left[\pi {\overline{\varepsilon }}_{{\mathrm{bulb}},IR}{s}_{\mathrm{bulb}}\phi \left({t}_{\mathrm{bulb}}\right)-4\pi {\overline{\alpha }}_{{\mathrm{bulb}},IR}{\overline{\varepsilon }}_{{\mathrm{encl}},IR}{S}_{\mathrm{bulb}}\phi \left({t}_{\mathrm{encl}}\right)\right]$$

(B8)

The enclosure wall emits radiation towards the bulb and towards the environment. All the radiation emitted by the enclosure wall towards the bulb is finally absorbed, due to the multiple reflections and absorptions between wall enclosure and bulb. Therefore, ${\overline{\varepsilon }}_{{\mathrm{encl}},IR}=1$. Also,

$${s}_{\mathrm{bulb}}=4{S}_{\mathrm{bulb}}=4\pi {R}_{\mathrm{bulb}}^{2}$$

(B9)

where ${R}_{\mathrm{bulb}}$ is the radius of the spherical bulb. Then, usage of Eqs. (B1) and (B8) yields the time variation of bulb temperature:

$$-\frac{{dt}_{\mathrm{bulb}}}{d\tau }=\frac{\pi {s}_{\mathrm{bulb}}K}{{C}_{\mathrm{bulb}}}\left[{\overline{\varepsilon }}_{{\mathrm{bulb}},IR}\phi \left({t}_{\mathrm{bulb}}\right)-{\overline{\alpha }}_{{\mathrm{bulb}},IR}\phi \left({t}_{\mathrm{encl}}\right)\right]$$

(B10)

Equation (B10) is Eq. (22) of Ferrel (1886, p. 99).

The function $\phi \left(t\right)$ in Eq. (B10) is still to be specified. The models proposed by Pouillet (1838) and Ferrel (1886) used the form of $\phi \left(t\right)$ as specified by the law of Dulong and Petit of radiation cooling. This law has been published by Dulong and Petit (1817), and has been extensively used by researchers before Stefan’s law has been published in 1879. A brief summary follows. Assume a body in vacuo completely surrounded by an enclosure. The temperature of the body is higher than that of the enclosure. Radiation is transferred from the body to the enclosure and the body cools down until its temperatures become equal with that of the enclosure. Then, the rate of radiation energy transferred from the body to the enclosure is exactly equal to that transferred from the enclosure to the body. The experimental equipment used by Dulong and Petit to study cooling in vacuum is described in Dulong and Petit (1817, p. 245). The experiments involved a spherical thermometric bulb of 6 or 2 cm in diameter, placed in vacuo in a larger enclosure surrounded by water kept at constant temperature 0 °C (see Fig. 2 of Dulong and Petit (1817)). A set of experiments has been performed, starting with the following temperatures ${t}_{\mathrm{encl}}$ of the enclosure: 0 °C, 20 °C, 40 °C, 60 °C, 80 °C. The cooling rates of the thermometric bulb have been determined each time. The conclusion was that the rate of cooling is well represented by the following relationship, which constitutes the Dulong and Petit radiation cooling law:

$$-\frac{{dt}_{\mathrm{bulb}}}{d\tau }=2.037\left({\mu }^{{t}_{\mathrm{bulb}}}-{\mu }^{{t}_{\mathrm{encl}}}\right)$$

(B11)

where $\mu =1.0077$. The constant 2.037 in Eq. (B11) corresponds to a cooling process starting with the initial temperature ${t}_{\mathrm{encl}}={0}^{^\circ }C$ while Eq. (B11) is valid for values of the bulb temperature ${t}_{\mathrm{bulb}}$ lower than 240 °C.

Equation (B11) for the rate of cooling in vacuum is given on p. 252 of Dulong and Petit (1817). Dulong and Petit stated that their relationship applies for any body in a temperature interval of 300 °C. After about 50 years, the results obtained by using Dulong and Petit law have been compared with results predicted by Stefan law. It has been shown that the parameter $\mu$ of the Dulong-Petit theory depends on temperature, ranging between 1.0069 and 1.0082 and that the value $\mu =1.0077$ is more appropriate for larger temperature values (Ferrel 1889).

Assuming ${\overline{\varepsilon }}_{{\mathrm{bulb}},IR}={\overline{\alpha }}_{{\mathrm{bulb}},IR}$ and comparing Eqs. (B10) and (B11), we get the form of $\phi \left(t\right)$:

$$\phi \left(t\right)={\mu }^{t}$$

(B12)

and the additional condition:

$$B\equiv \pi K=\frac{2.037{C}_{\mathrm{bulb}}}{{s}_{\mathrm{bulb}}{\overline{\varepsilon }}_{{\mathrm{bulb}},IR}}$$

(B13)

Pouillet determined the value of $B$ in Eq. (B13) for the bulbs used in the experiments of Dulong and Petit, by knowing the bulb surface area ${s}_{\mathrm{bulb}}$, the heat capacity ${C}_{\mathrm{bulb}}$, from the known specific heat and mass density of bulb glass and mercury inside the bulb and assuming the value ${\overline{\varepsilon }}_{{\mathrm{bulb}},IR}=0.8$:

$$B=1.146cal/\left({}^{^\circ }C{cm}^{2}\right)$$

(B14)

The value is only approximate taking into account the uncertainty with regard to the exact value of ${\overline{\varepsilon }}_{{\mathrm{bulb}},IR}$. Ferrel made an analysis of more recent results and concluded that the value of ${\overline{\varepsilon }}_{{\mathrm{bulb}},IR}$ adopted by Pouillet is underestimated and the value 0.83 is more realistic. This yields a $B$ slightly lower than (F13). However, Ferrel used (F13) in his calculations.

Using (Eqs. (B10), (B13), and ${\overline{\varepsilon }}_{{\mathrm{bulb}},IR}={\overline{\alpha }}_{{\mathrm{bulb}},IR}$, one finds:

$$-\frac{{dt}_{\mathrm{bulb}}}{d\tau }=\frac{{Bs}_{\mathrm{bulb}}{\overline{\varepsilon }}_{{\mathrm{bulb}},IR}}{{C}_{\mathrm{bulb}}}\left[{\mu }^{{t}_{\mathrm{bulb}}}-{\mu }^{{t}_{\mathrm{encl}}}\right]$$

(B15)

Equation (B15) was first obtained in 1838 by Pouillet under the assumption that ${\overline{\varepsilon }}_{{\mathrm{bulb}},IR}={\overline{\alpha }}_{{\mathrm{bulb}},IR}$ (Pouillet 1838, p. 15). Pouillet commented on this last equality in Note 1 on page 43 of (Pouillet 1838).

The solar energy flux incident on the spherical bulb is ${S}_{\mathrm{bulb}}G$ where $G$ is solar incident irradiance. Then, the flux of solar energy absorbed by the bulb, ${F}_{{\mathrm{bulb}},{\mathrm{solar}}}$, is given by:

$${F}_{{\mathrm{bulb}},{\mathrm{solar}}}={\overline{\alpha }}_{{\mathrm{bulb}},{\mathrm{solar}}}{S}_{\mathrm{bulb}}G$$

(B16)

where ${\overline{\alpha }}_{{\mathrm{bulb}},{\mathrm{solar}}}$ is the absorptance of the bulb, averaged over the whole spectrum of solar radiation wavelengths and over the solid angle subtended by the sun. When solar radiation is incident on the bulb, the solar energy flux absorbed by the bulb, ${F}_{{\mathrm{bulb}},{\mathrm{solar}}}$, is added to the net energy flux exchanged between the bulb and the enclosure walls, ${F}_{{\mathrm{bulb}},net}$, and the energy conservation Eq. (B1) turns into

$${C}_{\mathrm{bulb}}\frac{{dt}_{\mathrm{bulb}}}{d\tau }={F}_{{\mathrm{bulb}},net}+{F}_{{\mathrm{bulb}},{\mathrm{solar}}}$$

(B17)

Usage of Eqs. (B17), (B16), and (B8) and the assumption ${\overline{\varepsilon }}_{\mathrm{bulb}}={\overline{\alpha }}_{\mathrm{bulb}}$ yields

$$-{C}_{\mathrm{bulb}}\frac{{dt}_{\mathrm{bulb}}}{d\tau }={Bs}_{\mathrm{bulb}}{\overline{\varepsilon }}_{{\mathrm{bulb}},IR}\left({\mu }^{{t}_{\mathrm{bulb}}}-{\mu }^{{t}_{\mathrm{encl}}}\right)+{\overline{\alpha }}_{{\mathrm{bulb}},{\mathrm{solar}}}{S}_{\mathrm{bulb}}G$$

(B18)

For time intervals when solar incident irradiance can be regarded as constant the l.h.s. of Eq. (B18) sensibly vanishes and from Eq. (B13), one finds:

$${\rho }_{\mathrm{bulb}}G=B\left({\mu }^{{t}_{\mathrm{bulb}}}-{\mu }^{{t}_{\mathrm{encl}}}\right)$$

(B19)

where:

$${\rho }_{\mathrm{bulb}}\equiv \frac{{S}_{b}{\overline{\alpha }}_{{\mathrm{bulb}},{\mathrm{solar}}}}{{s}_{b}{\overline{\varepsilon }}_{{\mathrm{bulb}},IR}}$$

(B20)

Notice that vanishing solar irradiance ($G=0$) requires from Eq. (B19) that ${t}_{\mathrm{bulb}}={t}_{\mathrm{encl}}$. However, there is experimental evidence that during night ${t}_{\mathrm{bulb}}-{t}_{\mathrm{encl}}<0$. Therefore, a more appropriate steady state form of Eq. (B18) is:

$${\rho }_{\mathrm{bulb}}G=B{\mu }^{{t}_{\mathrm{bulb}}}-\left(1-m\right)B{\mu }^{{t}_{\mathrm{encl}}}$$

(B21)

where $m$ is a small correction factor, of the order of 0.0114 (Ferrel 1886, p. 373).

Equation (B21) applies for both bulbs of the Arago-Davy actinometer. However, the values of the parameter ${\rho }_{\mathrm{bulb}}$, defined by Eq. (B20), are different for the two bulbs since ${\overline{\varepsilon }}_{{\mathrm{bulb}},IR}$ is different in the two cases. Equation (B21) is used twice, for the black bulb thermometer (of temperature ${t}_{{\mathrm{black}}-{\mathrm{bulb}}}$) and the bright-bulb thermometer (of temperature ${t}_{{\mathrm{bright}}-{\mathrm{bulb}}}$, respectively:

$${\rho }_{{\mathrm{black}}-{\mathrm{bulb}}}G=B{\mu }^{{t}_{{\mathrm{black}}-{\mathrm{bulb}}}}-\left(1-m\right)B{\mu }^{{t}_{\mathrm{encl}}}$$

(B22)

$${\rho }_{{\mathrm{bright}}-{\mathrm{bulb}}}G=B{\mu }^{{t}_{{\mathrm{bright}}-{\mathrm{bulb}}}}-\left(1-m\right)B{\mu }^{{t}_{\mathrm{encl}}}$$

(B23)

Removing $B{\mu }^{{t}_{\mathrm{encl}}}$ between Eqs. (B22) and (B23) yields:

$$G=\frac{B}{{\uprho }_{{\mathrm{black}}-{\mathrm{bulb}}}-{\uprho }_{{\mathrm{bright}}-{\mathrm{bulb}}}}{\upmu }^{{t}_{{\mathrm{bright}}-{\mathrm{bulb}}}}\left({\upmu }^{{t}_{{\mathrm{black}}-{\mathrm{bulb}}}-{t}_{{\mathrm{bright}}-{\mathrm{bulb}}}}-1\right)$$

(B24)

Removing $G$ between Eqs. (B22) and (B23) yields:

$${\mu }^{{t}_{\mathrm{encl}}}={c}^{^{\prime}}{\mu }^{{t}_{{\mathrm{bright}}-{\mathrm{bulb}}}}+\left(1-{c}^{^{\prime}}\right){\mu }^{{t}_{{\mathrm{black}}-{\mathrm{bulb}}}}$$

(B25)

where

$${c}^{^{\prime}}\equiv \frac{{\rho }_{{\mathrm{black}}-{\mathrm{bulb}}}}{\left({\rho }_{{\mathrm{black}}-{\mathrm{bulb}}}-{\rho }_{{\mathrm{bright}}-{\mathrm{bulb}}}\right)\left(1-m\right)}$$

(B26)

When the black bulb is considered,${\overline{\alpha }}_{{\mathrm{black}}-{\mathrm{bulb}},{\mathrm{solar}}}={\overline{\varepsilon }}_{{\mathrm{black}}-{\mathrm{bulb}},IR}=1$. Then, usage of Eqs. (B21) and (B9) yields:

$${\rho }_{{\mathrm{black}}-{\mathrm{bulb}}}=\frac{1}{4}$$

(B27)

The following calibration factor is defined:

$${c}_{\mathrm{calib}}={c}^{^{\prime}}\left(1-m\right)$$

(B28)

Usage of Eqs. (B24), (B26), (B27), and (B28) yields:

$$G=4{Bc}_{\mathrm{calib}}{\mu }^{{t}_{{\mathrm{bright}}-{\mathrm{bulb}}}}\left({\mu }^{{t}_{{\mathrm{black}}-{\mathrm{bulb}}}-{t}_{{\mathrm{bright}}-{\mathrm{bulb}}}}-1\right)$$

(B29)

Equation (B29) is precisely the expression of the Ferrel-Pouillet model given in Table 1 of the paper. The conversion factor ${c}_{\mathrm{conv}}$ appearing in Table 2 is needed since $G$ in Eq. (B29) is expressed in $cal/\left({cm}^{2}min\right)$ while the SI units of $G$ in Table 1 are $W/{m}^{2}$. Notice that if Eq. (B14) is used, the value $4B$ is 4.584. This value is approximate but perhaps not much in error. Also, it is expected that the value of $\mu$ would be somewhat different for a perfect vacuum (Ferrel 1886).

Estimation of the value of the calibration factor ${c}_{\mathrm{calib}}$ has been made experimentally by Ferrel (1884). Measurements have been performed at Washington, D.C. on 24th of March 1883 by using two pairs of conjugate bright and black bulb thermometers of the Hicks patent, Nos 194 and 191 black, with 191 and 193 bright, respectively. A series of observations have been made, obtaining a series of values for ${t}_{{\mathrm{black}}-{\mathrm{bulb}}}$, ${t}_{{\mathrm{bright}}-{\mathrm{bulb}}}$, and ${t}_{\mathrm{encl}}$. These values have been replaced in Eqs. (B25) and (B28) (with $m=0.0114$) by obtaining a series of equations in the unknown ${c}_{\mathrm{calib}}$. The value of ${c}_{\mathrm{calib}}$ has been determined from this series of equations by the method of least squares (Ferrel 1886, p 374). An average value ${c}_{\mathrm{calib}}=1.482$ has been obtained (Ferrel 1884, p. 42). This value is used in the Ferrel-Pouillet model FP1 of Table 2.

The value of the calibration factor ${c}_{\mathrm{calib}}$ may be determined theoretically by using Eqs. (B26), (B27), and (B28). Bright-bulb and black bulb mercurial thermometers are heterogeneous bodies. The mercury and the glass have different reflection, absorption, and emission properties in respect with short wavelengths solar radiation and long wavelengths radiation they are emitting, respectively. The absorption and emission properties of the bulb, as a whole, depend on the interplay between these specific properties (Ferrel 1885). In case of the bright-bulb thermometer, the effect of both the glass wall and the mercury should be taken into account when estimating the averaged values of the solar energy absorptance ${\overline{\alpha }}_{{\mathrm{bright}}-{\mathrm{bulb}},{\mathrm{solar}}}$ and infrared emittance ${\overline{\varepsilon }}_{{\mathrm{bright}}-{\mathrm{bulb}},IR}$. A rough analysis by Ferrel yields ${\overline{\alpha }}_{{\mathrm{bright}}-{\mathrm{bulb}},{\mathrm{solar}}}=0.16$ and ${\overline{\varepsilon }}_{{\mathrm{bright}}-{\mathrm{bulb}},IR}=0.5$ (Ferrel 1886, p. 129). With these values and neglecting the small value of $m$, we find ${c}_{\mathrm{calib}}=1.471$. This value is close to the average experimental value 1.482 obtained by Ferrel. For two different values of ${\overline{\varepsilon }}_{{\mathrm{bright}}-{\mathrm{bulb}},IR}$ around 0.5, namely 0.8 and 0.45, one finds the calibration factor values 1.250 and 1.563, respectively. These values are close to the experimental values 1.265 and 1.526, respectively, obtained by Ferrel (1884, p. 44). Since the last two experimental values of the calibration factor are smaller and larger than the average value, they are used in Table 1 for the models FP2 and FP3, respectively.

Appendix 3

3.1 Computation of zenith angle

The cosine of the zenith angle $z$ is computed by:

$$cosz=sin\phi sin\delta cos\phi +cos\phi cos\delta cos\omega$$

(C1)

where $\phi$ is the latitude of the place, $\delta$ is astronomic declination, and $\omega$ is the hour angle. The astronomic declination, in degrees, is computed with (Russo 1978):

$$\delta =\left({23}^{^\circ }+\frac{{27}^{^\circ }}{60}\right)cosM$$

(C2)

where $M$, in degrees, is given by:

$$M=\frac{{180}^{^\circ }}{186}\left(a+n+\frac{t}{24}\right)$$

(C3)

In Eq. (C3), $n$ is the number of the day in the year ($n=1$ at January 1st), $t$, in hours, is the solar time while $a=12$ and 13 for a leap year and an ordinary year, respectively. The hour angle in Eq. (C1) is given, in degrees, by:

$$\omega ={15}^{^\circ }\left(12-t\right)$$

(C4)

In computations, the solar time $t$ entering Eqs. (C3) and (C4) have been directly replaced by the local time. This means that the usual corrections taking into account the equation of time and the small difference of longitude of the location in respect with the standard meridian of the time zone were not performed. This is justified by the fact that observations made with Arago-Davy actinometers are relevant for intervals usually longer than 1 h, and therefore, the small corrections are absorbed in the averaging process.