The linearized temperature can be rewritten as:

$\begin{array}{c}T\left(t\right)=c_1\sin {\omega }_{0}t+c_2\cos {\omega }_{0}t+c_3\sin \left(\omega +{\omega }_0\right)t+c_4\cos \left(\omega +{\omega }_0\right)t\\ +c_5\sin \left(\omega -{\omega }_0\right)t+c_6\cos \left(\omega -{\omega }_0\right)t+c_7,\end{array}$ (3)

where

$\begin{array}{l}{c}_1=a_1,c_2=a_4,c_3=\frac{1}{2}\left(a_1{a}_3+a_4{a}_5\right),\\ c_4=\frac{1}{2}\left(a_4{a}_6-a_1{a}_2\right),c_5=\frac{1}{2}\left(a_4{a}_5-a_1{a}_3\right),\\ c_6=\frac{1}{2}\left(a_1{a}_2-a_4{a}_6\right),c_7=a_7.\end{array}$ (4)

The coefficients of the temperature $a_i$ can then be deduced as:

$\begin{array}{l}{a}_1=c_1,a_2=\frac{c_6-c_4}{c_1},a_3=\frac{c_3-c_5}{c_1},a_4=c_2,\\ a_5=\frac{c_3+c_5}{c_2},a_6=\frac{c_6+c_4}{c_2},a_7=c_7.\end{array}$ (5)

The origin of time is the first January at midnight (12 AM). At this time, the terrestrial globe is two days to the perihelion. Let $t_0$ be the time for the terrestrial globe to travel from its initial position (1^st January) to the perihelion (2^nd January). Equation (3) can then be rewritten as:

$\begin{array}{c}T\left(t\right)=c_1\sin {\omega }_0\left(t-t_0\right)+c_2\cos {\omega }_0\left(t-t_0\right)\\ +c_3\sin \left(\omega +{\omega }_0\right)\left(t-t_0\right)+c_4\cos \left(\omega +{\omega }_0\right)\left(t-t_0\right)\\ +c_5\sin \left(\omega -{\omega }_0\right)\left(t-t_0\right)+c_6\cos \left(\omega -{\omega }_0\right)\left(t-t_0\right)+c_7.\end{array}$ (6)

where $t_0=2$ days. The temperature oscillates with the three types of frequency: ${\omega }_0$, $\omega +{\omega }_0$ and $\omega -{\omega }_0$. The corresponding periods of oscillation are:

$T_0=\frac{2\pi }{{\omega }_0},$ (7)

$T_1=\frac{T_{0}T}{T_0+T},$ (8)

$T_2=\frac{T_{0}T}{T_0-T},$ (9)

respectively. The period of the temperature $T_{temp}$ is the common multiplier of the three periods, that is:

$T_{temp}=LCM\left(T_0,T_1,T_2\right)=LCM\left(T_0,\frac{T_0\cdot T}{T_0+T},\frac{T_0\cdot T}{T_0-T}\right).$ (10)

where $T$ is the period of rotation of the terrestrial globe around its polar axis. The Lower common multiplier of $T_0$, $T_1$ and $T_2$ can be written as:

$LCM\left(T_0,T_1,T_2\right)=m{T}_0=n\frac{T\cdot T_0}{T+T_0}=k\frac{T_0\cdot T}{T_0-T},$ (11)

where $m$, $n$ and $k$ are the smallest integer which satisfy Equation (11). Substituting $T$ and $T_0$ in Equation (11), we obtain:

$\frac{m}{n}=\frac{4}{1465}.$ (12)

and

$\frac{m}{k}=\frac{4}{1457}.$ (13)

The LCM is then obtained for $m=4$, $n=1465$ and $k=1457$. The LCM is then:

$LCM\left(T_0,T_1,T_2\right)=4T_0.$ (14)

where $T_1$ = 23h56mn04s and $T_2$ = 24h03mn57s. $T_1$ is slightly lower than the period of rotation of the terrestrial globe $T=24\text{h}$. The period $T_1$ corresponds exactly to the sidereal day measured from the Earth’s rotation nominal mean velocity. $T_2$ is slightly higher than the period of rotation of the terrestrial globe $T=24\text{h}$. The period of the temperature is then 4 years, that is:

$T_{temp}=4T_0$ (15)

We are now curious to know the differential equation satisfies by the seasonal temperature. The second derivative of Equation (6) leads to the following differential equation of the second order:

$\frac{{\partial }^{2}T\left(t\right)}{\partial t^2}+{\omega }_0^{2}T\left(t\right)={\omega }_0^2{c}_7-{\omega }^2{T}_{spin-orb}\left(t\right)-2\omega {\omega }_0\left(x_2\left(t\right)-x_3\left(t\right)\right),$ (16)

where

$x_2\left(t\right)=c_3\sin \left(\omega +{\omega }_0\right)\left(t-t_0\right)+c_4\cos \left(\omega +{\omega }_0\right)\left(t-t_0\right),$ (17)

and

$x_3\left(t\right)=c_5\sin \left(\omega -{\omega }_0\right)\left(t-t_0\right)+c_6\cos \left(\omega -{\omega }_0\right)\left(t-t_0\right).$ (18)

Equation (16) shows that the temperature at the near surface of the terrestrial globe oscillates with the Earth frequency of revolution. This oscillation is influenced externally by the spin-orbit coupling of the both rotation and revolution movements of the terrestrial globe. This coupling of the movements creates the instantaneous perturbation of the orbital temperature.

The data used were extracted from hourly meteorological database of Cameroon station P30 for the year 2008. If the number of experimental points used correspond to the number of hours in a year, it would be too big and the fluctuation that is the standard deviation would also increase. This is the reason why we have chosen only three values of the measured temperatures for each day of the year. Therefore, we build three vectors to record the daily temperature data at 9 AM, 12 AM and 3 PM. These three vectors were concatenated chronologically in one vector during the computation. The least square method was used to fit the curve of the temperature to the hourly data considered as experimental points. The residue $R$ is defined as:

$R=\underset{i=1}{\overset{N}{{\displaystyle \sum }}}{\left(T\left(t_i\right)-T_i\right)}^2,$ (19)

where $N$ is the number of experimental points $\left(T_i,t_i\right)$ used and $T_i$ is the measured temperature at the time $t_i$. The system of algebraic equations containing the unknown coefficients $c_i$ of the temperature is found using the consideration that the residue is minimum for each of the coefficient $c_i$ that is:

$\frac{\partial R}{\partial c_i}=0.$ (20)

We then obtain a system of 7 algebraic equations which is triangulated and solved using the Gauss pivot method. The standard deviation (SE) from any individual hourly measured temperature is obtained as:

$SE=\sqrt{\frac{R}{N}}.$ (21)

The standard deviation ( $\sigma$) from the annual mean temperature is obtained as:

$\sigma =\sqrt{\frac{{{\displaystyle \sum }}_{i=1}^N{\left(T\left(t_i\right)-\bar{T}\right)}^2}{N}}.$ (22)

where $\bar{T}$ is the annual mean temperature. The seasonal divisions of the studied locality were obtained using the orbital temperature. These seasonal divisions are obtained from the variability of the seasonal temperature over a year. When the seasonal temperature crosses the up-line temperature, the annual mean temperature and the down-line temperature, the transition times can be derived [10] .

Table 1 presents the coefficients of the temperature. Table 2 presents the transition dates between the different seasons and Table 3 presents the seasonal divisions. In Table 3 , the abbreviations correspond to the different season’s names, that is; ds: dry season, eds: ending dry season, srs: starting raining season, rs: raining season, ers: ending raining season, sds: starting dry season. Also, letters are used to indicate the beginning or the end of a season. The letters used are: D—December, A—April, M—May, J—June, O—October, N—November.

Figure 1 displays the seasonal temperature over a year. The horizontal lines represent the up-line temperature (temperature at the transition time $t_{12}$), the annual mean temperature and the down-line temperature (temperature at the transition time $t_{34}$) respectively. These transition times where plotted on all the figures. The daily oscillations due to the spin-orbit coupling are not visible here. The maximum and minimum temperatures observed over a year are $T_{\max }=30.10$, $T_{\min }=27.04$ respectively. The standard deviation from the annual mean temperature and the standard deviation from individual measured temperatures are $\sigma =1.07$, $SE=0.07$ respectively. Figure 2 displays the seasonal temperature from the beginning of the year up to the first transition time. This figure shows how the seasonal temperature oscillates around the orbital temperature. The daily oscillations due to the spin-orbit coupling are then visible here. Figure 3 displays the curve of the seasonal temperature in an interval corresponding to its period that

is 4 years. It seems like this curve has a period of one year. This is due to the fact that the spin-orbit temperature is weak compared to the orbital temperature. Therefore, the temperature can be studied in an interval of one year.