Transport in Astrophysics: V. On the Red Sun at Horizon

Lorenzo Zaninetti

Physics Department, University of Turin, Turin, Italy.

DOI: 10.4236/ijaa.2023.134017                

Abstract

In order to simulate the red sun at the horizon we need to evaluate the average density of matter along a line of sight characterized by a given elevation angle. The decrease in frequency or the increase in wavelength of the light is modeled by the Bouguer-Beer-Lambert law and as a consequence, all the Planck spectrum is shifted toward lower frequencies or longer wavelengths.

Keywords

Solar System, General, Earth

1. Introduction

The red color of the sun at the horizon raises some astrophysical questions that should be solved:

1) What is the mechanism that changes the color of the sun from yellow to red?

2) What is the role of the declination angle in this change of color?

The solar spectrum at the top of the atmosphere is here assumed to be modeled by the Planck distribution which dates back to 1901 [1] . The Planck distribution is an active field of research and we select some topics under discussion: A careful examination of the maximum in emissivity in the following domains: frequencies, wavelength and wave number [2] ; a parametrization of the peaks in emissivity with the Lambert function [3] ; a parametrization of the fraction of the total power emitted by a blackbody in a given spectral band with the polylogarithm function [4] ; a semi-analytical method to calculate the total radiance received from a black body between two frequencies [5] ; a careful analysis of the Planck distribution in order to explain three features of the Solar photons: refrangibility, heat effect and chemical effect [6] . In order to answer the posed questions, we review in Section 2 the existing data on the density of the atmosphere and we analyze in Section 3 the behavior of the density of matter along three lines of sight: horizon, zenith and variable declination angle. The losses in energy along the line of sight are evaluated in the framework of a blackbody distribution for frequencies, see Section 4, and a blackbody distribution for wavelengths, see Section 5. Section 6 derives the temperature of the sun’s spectrum for the air mass zero (AM0).

2. The Density Profile of the Atmosphere

We assume that the atmosphere has a density of the type

$\rho \left(r\right)=C_{atm}\ast \exp \left(-\frac{r}{s}\right)\mathrm{,}$ (1)

where r is the altitude above the sea level, $C_{atm}$ and s are two numerical parameters to be found from the available data. In order to find the above parameters we processed the data of the U.S. standard atmosphere as reported in https://www.engineeringtoolbox.com/. Conversely a ray of the sun in the travel from the top of the atmosphere interacts with a growing density of air

$\rho \left(r\right)=C_{out}\ast \exp \left(\frac{r}{s}\right)\mathrm{,}$ (2)

where $C_{out}$ is a constant and r varies between 0 at the top of the atmosphere and t, the thickness of the atmosphere. In both cases r is evaluated along a line which crosses the center of Earth and Table 1 reports the numerical values for the atmosphere here adopted. A first application is the average density of the atmosphere evaluated along the two directions of a radial line which crosses the center of Earth

$\overline{{\rho }_{zenith}}=\frac{{\displaystyle {\int }_0^t{C}_{atm}\ast \exp \left(-\frac{r}{s}\right)\text{d}r}}{t}=\frac{{\displaystyle {\int }_0^t{C}_{out}\ast \exp \left(\frac{r}{s}\right)\text{d}r}}{t}=0.127892\frac{\text{kg}}{{\text{m}}^{\text{3}}}\mathrm{.}$ (3)

The above result means that the average density at zenith is equal in both directions.

Table 1. Numerical values for the parameters in SI.

3. The Involved Geometry

We start with the observer situated on the Earth’s surface at the center of an X-Y frame with coordinates (0, 0). A first line represents the line of sight of the sun

$y=\tan \left(\theta \right)x\mathrm{,}$ (4)

where the angle $\theta$ in rad grows in the counterclockwise direction. The sun at

the horizon has an angle $\theta =0$ and sun at the zenith has an angle $\theta =\frac{\pi }{2}$ ; in astronomy $\theta$ is named elevation angle.

A second line

$y=\tan \left(\varphi \right)x-a\mathrm{,}$ (5)

crosses the center of the Earth (0, −a) with a representing the Earth’s radius, see the numerical value in Table 1, and $\varphi$ a counter-clockwise angle in rad. The two lines are reported in Figure 1. A first circle represents the Earth’s surface

${\left(y+a\right)}^2+x^2=a^2\mathrm{,}$ (6)

and a second circle represents the end of the atmosphere

${\left(y+a\right)}^2+x^2={\left(a+t\right)}^2\mathrm{.}$ (7)

We now analyze two directions of sight, zenith and horizon, and then the line of sight as function of the angle of sight $\theta$ .

3.1. Two Directions

We limit ourselves to the zenith, $\theta =\frac{\pi }{2}$ , and to the horizon, $\theta =0$ , in order to

Figure 1. The first green line of sight with gradient equal to $\tan \left(\theta \right)$ and the second gray line which connects the center of Earth to the atmosphere, gradient equal to $\tan \left(\varphi \right)$ .

obtain first results in a simple way. The intersection of the line $y=0$ with the second circle (7), $x_{\max }$ is at

$x_{\max }=\sqrt{2at+t^2},$ (8)

which means that at the horizon the line of sight is ≈12.6 bigger in respect to the zenith. A way to parameterize the second line, see Equation (5), which crosses the center of the Earth is

$y=\frac{a\left(x-x_h\right)}{x_h}\mathrm{,}$ (9)

where $x_h$ is the intercept with the line $y=0$ which varies between 0 and $x_{\max }$ , see Equation (8). The distance as going from outside to inside in the atmosphere along the above line to the line $y=0$ is function of $x_h$

$\Delta r=t-\sqrt{{\left(x_h-\frac{x_{h}a}{\sqrt{a^2+x_h^2}}\right)}^2+\frac{a^2{\left(-\sqrt{a^2+x_h^2}+a\right)}^2}{a^2+x_h^2}}.$ (10)

When $x_h=x_{\max }$ we have $\Delta r=0$ and when $x_h=0$ we have $\Delta r=t$ ; Figure 2 reports an example of evaluation of $\Delta r$ .

A second application is the average density of the atmosphere evaluated along the line of sight for the horizon which is

$\overline{{\rho }_{horizon}}=\frac{1}{\sqrt{2at+t^2}}{\displaystyle {\int }_0^{\sqrt{2at+t^2}}}{C}_{out}{\text{e}}^{\frac{t-\sqrt{{\left(x-\frac{xa}{\sqrt{a^2+x^2}}\right)}^2+\frac{a^2{\left(-\sqrt{a^2+x^2}+a\right)}^2}{a^2+x^2}}}{s}}=0.380299\frac{\text{kg}}{{\text{m}}^{\text{3}}}\mathrm{.}$ (11)

Figure 2. First circle in red, second circle in blue, line which connect the center of Earth to y = 0 in gray and thick line which represents the distance on which to evaluate the density in green.

The above numerical result allows to say that the average density along the line of sight for the horizon is ≈2.97 time bigger than that along the line of sight of the zenith. The line of sight along the zenith direction is $l_{zenith}=t$ and along the horizon direction is $l_{horizon}=x_{\max }$ .

3.2. Variable Angle of Sight

The angle $\theta$ characterizes the line of sight, see Equation (4), and the intersection of the above line with the second circle, see Equation (7), ( $x_{s\mathrm{,2}}$ , $y_{s\mathrm{,2}}$ ), is at

$x_{s,2}=\left(\sqrt{\left({\sin }^2\left(\theta \right)a^2+2at+t^2\right){\sec }^2\left(\theta \right)}\cos \left(\theta \right)-\sin \left(\theta \right)a\right)\cos \left(\theta \right)$ (12)

$y_{s\mathrm{,2}}=\sqrt{{\sin }^2\left(\theta \right)a^2+2at+t^2}\sin \left(\theta \right)-{\sin }^2\left(\theta \right)a\mathrm{.}$ (13)

The second line which crosses the center of the Earth, see Equation (5), has a minimum angle $\varphi$

${\varphi }_{\min }=\arctan \left(\frac{\sqrt{{\sin }^2\left(\theta \right)a^2+2at+t^2}\tan \left(\theta \right)+a\cos \left(\theta \right)}{-\sin \left(\theta \right)a+\sqrt{{\sin }^2\left(\theta \right)a^2+2at+t^2}}\right),$ (14)

which means that the range of $\varphi$ is $\left[{\varphi }_{\min },\frac{\pi }{2}\right]$ and Figure 3 reports the value of ${\varphi }_{\min }$ as function of the angle $\theta$ in degree.

The intersection, ( $x_{s\mathrm{,}e}$ , $y_{s\mathrm{,}e}$ ), between the line which represents the line of sight, see Equation (4), and that one which represents the radial direction in respect to the center of Earth, see Equation (5), is at

$x_{s\mathrm{,}e}=-\frac{a}{\tan \left(\theta \right)-\tan \left(\varphi \right)}$ (15)

Figure 3. The value of ${\varphi }_{\min }$ as function of the angle $\theta$ in degree, parameters as in Table 1.

$y_{s\mathrm{,}e}=-\frac{\tan \left(\theta \right)a}{\tan \left(\theta \right)-\tan \left(\varphi \right)}\mathrm{.}$ (16)

The intersection, ( $x_{e\mathrm{,}c}$ , $y_{e\mathrm{,}c}$ ), between the line which represents the radial direction in respect to the center of Earth, see Equation (5), and the second circle, see Equation (7), is at

$x_{e\mathrm{,}c}=\frac{\sqrt{\left(2at+t^2\right){\sec }^2\left(\varphi \right)+{\tan }^2\left(\varphi \right)a^2+a^2}}{{\tan }^2\left(\varphi \right)+1}$ (17)

$y_{e\mathrm{,}c}=\frac{\tan \left(\varphi \right)\sqrt{\left(2at+t^2\right){\sec }^2\left(\varphi \right)+{\tan }^2\left(\varphi \right)a^2+a^2}}{{\tan }^2\left(\varphi \right)+1}-a\mathrm{.}$ (18)

As a consequence of the above intersection the distance $\Delta r$ as evaluated on line crossing the Earth from outside the atmosphere to the intersection of the two lines is

$\Delta r=\sqrt{\frac{{\left({\tan }^2\left(\varphi \right)a-\tan \left(\varphi \right)A+A\tan \left(\theta \right)+a\right)}^2}{\left({\tan }^2\left(\varphi \right)+1\right){\left(\tan \left(\varphi \right)-\tan \left(\theta \right)\right)}^2}},$ (19)

where

$A=\sqrt{{\left(a+t\right)}^2{\sec }^2\left(\varphi \right)}\mathrm{.}$ (20)

We are now ready to evaluate the average density of the atmosphere along the line of sight $\theta$ which is defined as

$\overline{\rho \left(\theta \right)}=\frac{{\displaystyle {\int }_{{\varphi }_{\min }}^{\frac{\pi }{2}}}{C}_{out}{\text{e}}^{\frac{\sqrt{\frac{{\left(\left(-\tan \left(\varphi \right)+\tan \left(\theta \right)\right)\sqrt{{\left(a+t\right)}^2{\sec }^2\left(\varphi \right)}+\left({\tan }^2\left(\varphi \right)+1\right)a\right)}^2}{\left({\tan }^2\left(\varphi \right)+1\right){\left(\tan \left(\varphi \right)-\tan \left(\theta \right)\right)}^2}}}{s}}\text{d}\varphi }{\frac{\pi }{2}-{\varphi }_{\min }}\mathrm{,}$ (21)

whith ${\varphi }_{\min }$ as given by Equation (14). The above integral does not have an analytical solution and therefore we introduce the following fit for the numerical integration

$\overline{\rho \left(\theta \right)}\approx 0.131733+0.244471{\text{e}}^{-0.37047\theta \left(\text{degree}\right)}\mathrm{,}$ (22)

see Figure 4. The percentage error of the fit, $\delta$, is $\delta =1.07\%$ at $\theta =0$ degree and $\delta =3\%$ at $\theta =90$ degree.

4. The Planck Distribution in Frequencies

The Planck distribution for the spectral radiance [1] [7] , $B_{\nu }$ , is

$B_{\nu }=\frac{2h{\nu }^3}{c^2\left({\text{e}}^{\frac{h\nu }{kT}}-1\right)}\frac{\text{J}}{\text{s}\cdot {\text{m}}^{\text{2}}\cdot \text{sr}\cdot \text{Hz}}\mathrm{,}$ (23)

where $\nu$ is the frequency, c the light velocity, h the Planck constant, T the temperature and k the Boltzmann constant. The numerical values of the above physical constants are reported in Table 2 in SI units [8] .

Figure 4. The value of the average density of the atmosphere as function of the angle $\theta$ in degree, parameters as in Table 1. The red full line represents the fit as given by Equation (22) and the blue square the numerical integration.

Table 2. Numerical values for the parameters connected with the Planck distribution in SI.

The spectral radiance has a maximum, ${\nu }_{\max }$ , at

${\nu }_{\max }=\frac{kT\left(W\left(-3{\text{e}}^{-3}\right)+3\right)}{h}$ (24)

where W is the Lambert W function, after [9] ; with the data of Table 2.

${\nu }_{\max }=3.3933\mp {10}^{14}\text{Hz}$ which is in the near infrared region. We now analyze how is possible to decrease the frequency of the sun at zenith characterized by the yellow color with frequency ${\nu }_{yellow}$ to the red frequency, ${\nu }_{red}$ , at horizon, see Table 2 for the numerical values. The ratio $\frac{{\nu }_{red}}{{\nu }_{yellow}}=0.8461$ , see parameters in Table 2, is the target of the following simulation. The Bouguer-Beer-Lambert law [10] [11] [12] is very useful in spectrophotometry

$I\left(d\right)=I_0\exp -\alpha d\mathrm{,}$ (25)

where $I_0$ is the initial intensity of the light, I is the intensity of the light after traveling the distance d and $\alpha$ is a coefficient of absorption expressed in unit neper, see formula (1) in [13] . The parameter $\alpha$ can be expressed as

$\alpha =c_a{c}_b\mathrm{,}$ (26)

where $c_b$ is a coefficient of attenuation and $c_a$ the concentration. Here we assume that the energy E of a photon which travels the atmosphere decreases according to the following ODE

$\frac{\text{d}}{\text{d}x}E\left(x\right)=-\mu {\rho }_{ave}E\left(x\right),$ (27)

where ${\rho }_{ave}$ is the averaged density of matter in kg/m³ and $\mu$ the attenuation coefficient for energy in m²/kg. A usual assumption is $E=h\nu$ which produces the following ODE in frequency

$\frac{\text{d}}{\text{d}x}\nu \left(x\right)=-\mu {\rho }_{ave}\nu \left(x\right)\mathrm{,}$ (28)

see formula (5) in [14] . The above ODE is solved assuming the initial condition $\nu \left(0\right)={\nu }_0$

$\nu \left(x\right)={\nu }_0{\text{e}}^{-\mu {\rho }_{ave}x}\mathrm{.}$ (29)

Figure 5 reports the decrease in frequency at the zenith which is minimum and Figure 6 reports the decrease in frequency at the zenith which covers the yellow-red transition.

Figure 5. The decrease in frequency as function of the distance at the zenith, ( $\theta =\frac{\pi }{2}$ ), parameters as in Table 1.

Figure 6. The decrease in frequency as function of the distance at horizon, ( $\theta =0$ ), parameters as in Table 1.

We are now ready to display the evolution of the spectral radiance adopting the following rules:

1) We select an initial frequency at the top of the atmosphere.

2) We evaluate the spectral radiance for the initial frequency.

3) We evaluate the final frequency in a given direction with formula (29) adopting the appropriate density and line of sight which are functions of the angle $\theta$ .

4) The spectral radiance of the final frequency is that of the initial frequency evaluated at point [2] .

As an example, Figure 7 reports the overall spectral radiance in three directions of sight.

5. The Planck Distribution in Wavelengths

The Planck distribution in wavelengths is

$B_{\lambda }=\frac{2h{c}^2}{{\lambda }^5\left({\text{e}}^{\frac{hc}{\lambda kT}}-1\right)}\mathrm{,}$ (30)

and the maximum, ${\lambda }_{\max }$ , is at

${\lambda }_{\max }=\frac{hc}{Tk\left(W\left(-5{\text{e}}^{-5}\right)+5\right)}\mathrm{.}$ (31)

Figure 7. The modified spectral radiance at the horizon ( $\theta =0$ degree), red line, at $\theta =3$ degree, orange line and at the zenith ( $\theta =0$ degree), yellow line.

As a practical application at the temperature here adopted of 5772 K the maximum of the radiance is at 502.039 × 10⁻⁹ m or 502.039 nm. We start from Equation (28) in order to obtain the ODE for the increase in wavelength

$\frac{\text{d}}{\text{d}x}\lambda \left(x\right)=\lambda \left(x\right)\mu {\rho }_{ave}\mathrm{.}$ (32)

The above ODE is solved assuming the initial condition $\lambda \left(0\right)={\lambda }_0$

$\lambda \left(x\right)={\lambda }_0{\text{e}}^{\mu {\rho }_{ave}x}\mathrm{.}$ (33)

We are now ready to display the evolution of the spectrum adopting the following rules:

1) We select an initial wavelength at the top of the atmosphere.

2) We evaluate the spectral radiance for the initial wavelength.

3) We evaluate the final wavelength in a given direction with formula (33) adopting the appropriate density and line of sight which are functions of the angle $\theta$ .

4) The spectral radiance of the final wavelength is that of the initial wavelength evaluated at point [2] .

As an example Figure 8 reports the overall spectrum on three directions of sight.

6. Temperature Determination

The solar cell measurements require an accurate knowledge of the solar spectrum at the top of the atmosphere, as an example at 35 km, which is called air mass zero (AM0) [15] . The data of AMO can be found, as an example, digitizing ASTM E-490 on a search engine, and are displayed as a green point in Figure 9. We fit the AM0 data with the following function

Figure 8. The theoretical spectrum at the horizon ( $\theta =0$ degree), red line, at $\theta =3$ degree, orange line and at the zenith ( $\theta =0$ degree), yellow line, the wavelengths are expressed in nm.

Figure 9. Spectral radiance versus wavelength in nm for AM0, green points, and theoretical fit, red line.

$f\left(\lambda \mathrm{,}T\right)=B\times \frac{2h{c}^2}{{\lambda }^5\left({\text{e}}^{\frac{hc}{\lambda kT}}-1\right)}\mathrm{,}$ (34)

where B is a constant which allows to match the data. The temperature and the constant can be found through the Levenberg-Marquardt method (subroutine MRQMIN in [16] ) and turn out to be $T=5467.49\text{K}$ and $A=8.9033\times {10}^{-14}$ . The above value of temperature has a percent error of 5.27% on the value here adopted of 5772 K.

7. Conclusions

The average density of the atmosphere

The evaluation of the average density of the atmosphere is simple along a radial direction which crosses the center of Earth but complicated along a given elevation angle, see Equation (22) for a numerical approximation.

The decrease in frequencies

The ODE for the decrease in frequency due to the Bouguer-Beer-Lambert law, see Equation (28), allows the yellow-red transition for the sun at the horizon. Due to the fact that the initial value of the spectral radiance is maintained, all the spectrum is shifted toward lower frequencies, see as an example Figure 7.

The increase in wavelengths

The ODE for the increase in wavelengths, see Equation (33), explains the yellow-red transition. Also here the initial value of the spectral radiance is maintained and as a consequence all the spectrum is shifted toward bigger wavelengths, see as an example Figure 8.

Conflicts of Interest

The author declares no conflicts of interest regarding the publication of this paper.

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