## 2 Theory and Method
### The Effect of Earth-Sun Distance for the Solar Observation  
From Equation (5), when there is no eclipse and the antenna is centered directly towards the sun, the T\({}_{\rm B}\) that antenna receives from the sun without atmospheric attenuation can be given as:  
\[\Delta T_{sun}=\frac{\Omega_{s}}{\Omega_{A}}\overline{T}_{sun}. \tag{6}\]  
The solar solid angle would be derived as follows:  
\[\Omega_{s}=2\pi[1-\cos(\theta_{s})], \tag{7}\]  
where \(\theta_{s}\) is the angular radius of the sun. For a small angle \(\theta_{s}\), \(\cos(\theta_{s})\) can be approximated as follows:  
\[\cos(\theta_{s})\approx 1-\frac{1}{2}{\theta_{s}}^{2}, \tag{8}\]  
and \(\theta_{s}\) can be given by:  
\[\theta_{s}\approx\frac{r}{d}, \tag{9}\]  
where \(r\) is the radius of the sun and \(d\) is the distance from the sun to the earth. Then the solid angle subtended by the cone given by Equation (7) would be approximately as follows:  
\[\Omega_{s}=\pi\frac{r^{2}}{d^{2}}. \tag{10}\]  
Then, Equation (6) becomes:  
\[\Delta T_{sun}=\frac{\pi}{\Omega_{A}}\frac{r^{2}}{d^{2}}\overline{T}_{sun}. \tag{11}\]  
According to Equation (11), the T\({}_{\rm B}\) increment is inversely proportional to the square of the distance between the sun and the earth. Since the earth's orbit around the sun is an elliptical and the sun is at one focus of the ellipse, the Earth-Sun distance varies by about 3.4% and the intensity of the solar radiation is 6.5% throughout the year [25]. This is to say that the T\({}_{\rm B}\) observation is related to the Earth-Sun distance. The variation of the Earth-Sun distance affects the ratio R of the solar solid angle to the antenna solid angle. If the beam width is known, the ratio R can be calculated accurately. However, as the antenna beamwidth measurement is usually carried out in a microwave anechoic chamber, it is complex and expensive. That being said, antenna pattern measurements with the sun as a signal are widely used by active and passive microwave instruments in meteorology [1; 17]. The sun can be assumed to be a point source for the antenna. During the scanning of the sun, the received solar power can be approximated using a Gaussian function. Assuming a Gaussian function for the antenna pattern, the ratio can be approximated as follows [26; 27]:  
\[R=\frac{\Omega_{s}}{\Omega_{A}}=1-\exp\Bigl{[}-4\ln 2(\theta_{s}/\theta_{A})^{2 }\Bigr{]}. \tag{12}\]  
Taking into account that the beamwidth of the GMR was less than 5\({}^{\circ}\), we simulated the relationship between the sun filled-beam factor and antenna beamwidth, as shown in Figure 1. The beamwidths of the GMR were around 3-5\({}^{\circ}\) at the K-band, and the sun filled-beam factor was less than 0.02. The orbit of the earth is an ellipse and the ratio \(R\) is depends on the Earth-Sun distance.  
Utilizing Equation (6), we can accurately estimate the T\({}_{\rm B}\) increment. For example, when the antenna beamwidth is 4.5\({}^{\circ}\), the calculation results show that the variation of the T\({}_{\rm B}\) increment is about 3 K throughout the year, and the variation is more obvious with a narrower beamwidth. The sensitivity for most GMRs for meteorological applications consists of less than 0.3 K in 1 s of integration time and therefore it is certain that a radiometer is capable of precise measurements of the annual variation of T\({}_{\rm B}\) increment.  
During observation, we can ignore the variation of the T\({}_{\rm B}\) increment due to sun movements in 1 s of integration time.