The area of the sector formed by joining the tips of \(\textbf{r}_2\) and \(\textbf{r}_3\) around the orbit is \(B_1\). The area of the sector formed by joining the tips of \(\textbf{r}_1\) and \(\tex...The area of the sector formed by joining the tips of \(\textbf{r}_2\) and \(\textbf{r}_3\) around the orbit is \(B_1\). The area of the sector formed by joining the tips of \(\textbf{r}_1\) and \(\textbf{r}_2\) around the orbit is \(B_3\). Our aim is to use this approximate Equation to find approximate values for the heliocentric distances at the instants of the three observations, and then to refine them in order to satisfy the exact Equation 13.2.5.
I now make use of the sum of the sum-and-difference formulas from page 38 of chapter 3, namely \(\cos A \cos B = \frac{1}{2} (\cos S + \cos D)\) and \(\sin A \sin B = \frac{1}{2} (\cos D - \cos S ),\)...I now make use of the sum of the sum-and-difference formulas from page 38 of chapter 3, namely \(\cos A \cos B = \frac{1}{2} (\cos S + \cos D)\) and \(\sin A \sin B = \frac{1}{2} (\cos D - \cos S ),\) to obtain In these Equations we already know an approximate value for \(f_3\) (we’ll see how when we resume our numerical example); the unknowns in these Equations are \(R_3\), \(a\) and \(g_3\), and it is \(R_3\) that we are trying to find.
