# A New Model of Solar Illumination of Earth’s Atmosphere during Night-Time

^{*}

## Abstract

**:**

## 1. Introduction

^{2}of an ionospheric section, measured in TEC Unit (1 TEC Unit = 10

^{16}electrons/m

^{2}), and can be determined by ground-based (e.g., from ionosondes, Global Navigation Satellite Systems receivers [13]) and space-based (e.g., FORMOSAT/COSMIC series [14,15], DEMETER [16]) measurements.

## 2. Materials and Methods

#### 2.1. Solar Terminator Height Determination

_{0}the axis of rotation of the Earth.

- H is the terminal point of the ellipsoidal height h;
- $N=\frac{a}{\sqrt{1-{e}^{2}\xb7si{n}^{2}\theta}}$ is the length of the straight line coinciding with the ellipsoid’s normal n connecting the point P and the point of intersection between n and Z″ axis P
_{0}(length $\overline{P{P}_{0}}$ in Figure 1); - e
^{2}and a, respectively equal to 0.00669438 and to 6,378,137 m, are the ellipsoidal parameters first eccentricity squared and semi-major axis of the reference ellipsoid used (WGS84); - θ and φ are, respectively, the geodetic latitude and the longitude.

_{H}to determine h. To do this, it is possible to exploit the fact that (given the assumption of solar rays all parallel to each other) the projection on the XZ plane of the point H is a point H

_{xz}given by the intersection between the projection on the XZ plane of n (n

_{xz}) and the ellipsoid E (see Figure 2). It will, then, be necessary to determine the equation of the two geometric figures (n and E) and their intersection at the point H

_{xz}.

_{xz}is a straight line passing through the projections on the XZ plane of the points P and P

_{0}. The coordinates of the points P and P

_{0}in the X″Y″Z″ reference system are given, again, by the well-known equations of the ellipsoid’s normal provided by geodesy:

_{0}in XYZ are given by the following expressions:

_{xz}passing through the points P e P

_{0}(and through the point H

_{xz}; see Figure 2):

_{E}as a function of Z

_{E}on the XZ plane (for Y = 0). Again exploiting the rotation equations of the reference system, in XYZ, after some substitutions, we have:

_{n}(Z

_{n}) and X

_{E}(Z

_{E}), (38) and (37), it is possible to determine their point of intersection Z

_{H}(X

_{H}) by equating them:

- Solar declination angle δ;
- Local hour angle from sunset λ;
- Solar elevation angle α.

#### 2.1.1. Solar Declination Angle: δ

#### 2.1.2. Equation of Time: ET (min)

#### 2.1.3. Local Hour Angle from Sunset λ

- HD (min/UTC), the hour of the day hh:mm:ss UTC of point P expressed in minutes, is calculated:

- 2.
- TST (min/UTC), the true solar time of point P expressed in minutes, is obtained as follows:

- 3.
- LHA (°), the local hour angle of point P expressed in degrees, is given by:

- 4.
- λ (°), local hour angle from sunset expressed in degrees, then, is:

#### 2.1.4. Solar Elevation Angle α

#### 2.2. STH–TEC Correlation Analysis

#### Collection of TEC Data Samples

## 3. Results

#### 3.1. STH as Function of Time

#### 3.2. STH–TEC Correlation Analysis

## 4. Discussion

## Supplementary Materials

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Solar illumination model of the Earth’s ellipsoid at the time of the equinox and representation of the ellipsoidal height of the solar terminator (Solar Terminator Height) h and of some of the related parameters.

**Figure 2.**Solar illumination model of the Earth’s ellipsoid at the time of the Northern Hemisphere winter solstice and representation of the ellipsoidal height of the solar terminator (Solar Terminator Height) h, of the declination angle δ and of some of the related parameters.

**Figure 3.**TEC measurements collected by the GPS L’Aquila receiver (Central Italy) on 22 March 2016 showing the fairly standard behavior of TEC at mid-latitudes (in quiet geomagnetic activity conditions).

**Figure 4.**Examples of time-dependent STH determination by using the proposed model. The two figures show the trend of STH as a function of time over an entire year (2000) in tropical latitudes. Figure (

**a**) represents STH in the Northern Hemisphere. Figure (

**b**) represents STH in the Southern Hemisphere. Note that, in the two STH peaks—corresponding to the solar midnight of the two days of the year in which the latitude θ is closest to −δ (declination)—small variations in computation time or longitude are enough to ensure that each of the two STH peaks varies considerably (in fact, if it were θ = −δ, when SZA = −180, it would be STH = ∞). Therefore, precise longitude values were chosen, such as to better emphasize such an aspect.

**Figure 5.**Examples of time-dependent STH determination by using the proposed model. The two figures show the trend of STH as a function of time over an entire year (2000) in temperate latitudes. Figure (

**a**) represents STH in the Northern Hemisphere. Figure (

**b**) represents STH in the Southern Hemisphere.

**Figure 6.**Examples of time-dependent STH determination by using the proposed model. The two figures show the trend of STH as a function of time over an entire year (2000) in polar latitudes. Figure (

**a**) represents STH in the Northern Hemisphere. Figure (

**b**) represents STH in the Southern Hemisphere.

**Figure 7.**The 12 graphs show the TEC–STH scatter plots and the related linear regression line of 20 years (2000–2019) of data grouped by month, with solar activity level between 100 and 120 s.f.u., under quiet geomagnetic conditions and in the time interval between sunset and solar midnight. The linear correlation coefficient, the root mean squared error and the daily mean Time Coverage (TC) of the month are shown at the top of the graphs.

**Figure 8.**The 6 graphs show the TEC–STH scatter plots and the related log-linear regression line of 20 years (2000–2019) of data grouped by month, for the period October–March, with solar activity level between 100 and 120 s.f.u., under quiet geomagnetic conditions and in the time interval between sunset and solar midnight. The log-linear correlation coefficient, the root mean squared error and the daily mean Time Coverage (TC) of the month are shown at the top of the graphs.

**Table 1.**Solar activity levels of F10.7 index in solar flux unit (s.f.u.) in which the STH–TEC linear regression graphs are grouped.

Solar Activity Level | Range 1 | Range 2 | Range 3 | Range 4 | Range 5 | Range 6 |
---|---|---|---|---|---|---|

F10.7 (s.f.u.) | 0–80 | 80–100 | 100–120 | 120–140 | 140–160 | 160–Inf. |

**Table 2.**TEC–STH correlation coefficient values for the sunset–solar midnight period obtained in each of the 12 months of the year in each of the 6 predetermined solar activity ranges and for both linear (Lin) and log-linear (Log-Lin) correlation. The values in bold are the maximums in the comparison between Lin and Log-Lin.

Linear and Log-Linear TEC–STH Correlation Coefficients Comparison | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

Month | F10.7 < 80 | 80 < F10.7 < 100 | 100 < F10.7 < 120 | 120 < F10.7 < 140 | 140 < F10.7 < 160 | F10.7 > 160 | ||||||

Lin | Log-Lin | Lin | Log-Lin | Lin | Log-Lin | Lin | Log-Lin | Lin | Log-Lin | Lin | Log-Lin | |

1 | −0.38 | −0.45 | −0.55 | –0.67 | –0.48 | –0.65 | –0.60 | –0.78 | –0.60 | –0.79 | –0.45 | –0.63 |

2 | –0.45 | –0.50 | –0.64 | –0.68 | –0.58 | –0.69 | –0.71 | –0.82 | –0.69 | –0.81 | –0.47 | –0.53 |

3 | –0.49 | –0.50 | –0.66 | –0.64 | –0.70 | –0.74 | –0.70 | –0.72 | –0.71 | –0.71 | –0.67 | –0.66 |

4 | –0.66 | –0.56 | –0.67 | –0.60 | –0.66 | –0.57 | –0.74 | –0.69 | –0.71 | –0.68 | –0.59 | –0.63 |

5 | –0.66 | –0.53 | –0.61 | –0.49 | –0.72 | –0.58 | –0.78 | –0.67 | –0.65 | –0.56 | –0.77 | –0.70 |

6 | –0.69 | –0.55 | –0.60 | –0.50 | –0.63 | –0.52 | –0.65 | –0.56 | –0.72 | –0.61 | –0.62 | –0.52 |

7 | –0.70 | –0.58 | –0.66 | –0.53 | –0.77 | –0.63 | –0.76 | –0.62 | –0.76 | –0.64 | –0.63 | –0.53 |

8 | –0.64 | –0.53 | –0.67 | –0.57 | –0.80 | –0.69 | –0.79 | –0.67 | –0.78 | –0.65 | –0.75 | –0.63 |

9 | –0.54 | –0.52 | –0.65 | –0.63 | –0.79 | –0.78 | –0.61 | –0.66 | –0.75 | –0.76 | –0.64 | –0.65 |

10 | –0.44 | –0.42 | –0.67 | –0.58 | –0.66 | –0.74 | –0.62 | –0.73 | –0.69 | –0.83 | –0.59 | –0.70 |

11 | –0.42 | –0.44 | –0.58 | –0.60 | –0.52 | –0.63 | –0.59 | –0.74 | –0.65 | –0.82 | –0.55 | –0.75 |

12 | –0.39 | –0.45 | –0.47 | –0.60 | –0.64 | –0.78 | –0.57 | –0.75 | –0.62 | –0.82 | –0.54 | –0.76 |

**Table 3.**Averages of the TEC–STH correlation coefficients of the periods May–August (linear) and November–February (log-linear), in each of the 6 predetermined solar activity ranges.

Seasonal Means of The TEC–STH Correlation Coefficients | ||||||
---|---|---|---|---|---|---|

F10.7 < 80 | 80 < F10.7 < 100 | 100 < F10.7 < 120 | 120 < F10.7 < 140 | 140 < F10.7 < 160 | F10.7 > 160 | |

Lin (May–Aug) | −0.67 | −0.64 | −0.73 | −0.74 | −0.73 | −0.69 |

Log-Lin (Nov–Feb) | −0.46 | −0.64 | −0.69 | −0.77 | −0.81 | −0.67 |

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Colonna, R.; Tramutoli, V. A New Model of Solar Illumination of Earth’s Atmosphere during Night-Time. *Earth* **2021**, *2*, 191-207.
https://doi.org/10.3390/earth2020012

**AMA Style**

Colonna R, Tramutoli V. A New Model of Solar Illumination of Earth’s Atmosphere during Night-Time. *Earth*. 2021; 2(2):191-207.
https://doi.org/10.3390/earth2020012

**Chicago/Turabian Style**

Colonna, Roberto, and Valerio Tramutoli. 2021. "A New Model of Solar Illumination of Earth’s Atmosphere during Night-Time" *Earth* 2, no. 2: 191-207.
https://doi.org/10.3390/earth2020012