## 2 Theory and Method  
### The Observation of the Sun and a Solar Eclipse  
In general, a GMR can be used to observe the T\({}_{\rm B}\) of the atmosphere. When the antenna is pointed at a certain elevation angle to the sky, the antenna temperature with only the clear sky in the beam can be estimated by [23]:  
\[T_{sky}(\theta)=T_{bg}e^{-\tau(\theta)}+T_{m}\Big{[}1-e^{-\tau(\theta)}\Big{]}, \tag{1}\]  
where \(\theta\) is the observation elevation angle of GMR, \(T_{m}\) is the atmospheric effective temperature and \(T_{bg}\) is the cosmic background T\({}_{\rm B}\) (\(T_{bg}\) = 2.75 K). \(\tau(\theta)\) is the atmospheric attenuation at the elevation angle.  
When a GMR is used to observe the sun, the T\({}_{\rm B}\) measured by the antenna depends on the average solar radiation and the ratio of the solar solid angle to the antenna solid angle and the T\({}_{\rm B}\) observation is inversely proportional to the solar solid angle [24]. During the solar eclipse, when the antenna is pointed to the sun, the antenna temperature \(T_{sm}(\theta)\) received from both the sun and the moon in the antenna beam would be:  
\[T_{sm}(\theta)=T_{bg}e^{-\tau(\theta)}+T_{m}\Big{[}1-e^{-\tau(\theta)}\Big{]} +\frac{\Omega_{s}-\Omega_{0}}{\Omega_{A}}\overline{T}_{sun}e^{-\tau(\theta) }+\frac{\Omega_{m}}{\Omega_{A}}\overline{T}_{moon}e^{-\tau(\theta)}, \tag{2}\]  
where \(\overline{T}_{sun}\) and \(\overline{T}_{moon}\) are the average T\({}_{\rm B}\) for the sun and moon, \(\Omega_{m}\), \(\Omega_{s}\) and \(\Omega_{A}\) are the solid angles of the moon, the sun and the antenna beam, respectively. \(\Omega_{0}\) is the solid angle of the sun shaded by the moon. \(\Omega_{0}\) equals zero when there is no eclipse and equals the solar solid angle when there is a total solar eclipse. The value of \(\Omega_{0}\) is between (0, \(\Omega_{s}\)) for annular or partial eclipse cases.  
During observation of the sun, the T\({}_{\rm B}\) increment measured by the GMR can be obtained by subtracting (1) from (2),  
\[\Delta T^{\prime}_{sun}(\theta)=[\ \frac{\Omega_{s}-\Omega_{0}}{\Omega_{A}} \overline{T}_{sun}+\frac{\Omega_{m}}{\Omega_{A}}\overline{T}_{moon}]e^{-\tau( \theta)}. \tag{3}\]  
According to the Equations (1)-(3), when the atmospheric distribution is homogeneous, we can obtain the solar T\({}_{\rm B}\) increment by observing the radiation of the sky without the sun and the direction of the sun at same elevation (the solar elevation). When the GMR observes the solar radiation, because of the atmospheric attenuation, the solar radiant energy reaching the earth is reduced. During the observation, the atmospheric attenuation is a key factor and it must be calculated.  
The atmospheric attenuation \(\tau(\theta)\) can be calculated by using elevation scanning of the sky [23]. After atmospheric opacity calibration, one obtains:  
\[\Delta T_{sun}=\frac{\Omega_{s}-\Omega_{0}}{\Omega_{A}}\overline{T}_{sun}+ \frac{\Omega_{m}}{\Omega_{A}}\overline{T}_{moon}, \tag{4}\]  
where \(\Delta T_{sun}\) is the power from the sun arriving at the antenna without atmospheric attenuation. Considering that \(\overline{T}_{sun}\gg\overline{T}_{moon}\), i.e., \(\overline{T}_{moon}\) = 0 in Equation (4), one has:  
\[\Delta T_{sun}=\frac{\Omega_{s}-\Omega_{0}}{\Omega_{A}}\overline{T}_{sun}. \tag{5}\]  
This is the simplified formula for estimating the T\({}_{\rm B}\) increment of the sun reaching the antenna during the solar eclipse.