13.3: Sectors
( \newcommand{\kernel}{\mathrm{null}\,}\)
Figure XIII.2 shows a portion of an elliptic (or other conic section) orbit, and it shows the radii vectores of the planet’s position at instants of time t1, t2 and t3.
FIGURE XIII.2
The notation I am going to use is as follows:
- The area of the sector formed by joining the tips of r2 and r3 around the orbit is B1.
- The area of the sector formed by joining the tips of r3 and r1 around the orbit is B2.
- The area of the sector formed by joining the tips of r1 and r2 around the orbit is B3.
- The time interval t3−t2 is τ1.
- The time interval t3−t1 is τ2.
- The time interval t2−t1 is τ3.
Provided the arc is fairly small, then to a good approximation (in other words we can approximate the sectors by triangles), we have
B2r2≈B1r1+B3r3.
That is,
r2≈b1r1+b3r3,
where
b1=B1/B2
and
b3=B3/B2
The coefficients b1 and b3 are the sector ratios, and the coefficients a1 and a3 are the triangle ratios.
By Kepler’s second law, the sector areas are proportional to the time intervals.
That is b1=τ1/τ2
and b3=τ3/τ2.
Thus the coefficients in Equation ??? are known. 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. We shall embark upon our attempt to do this in Section 13.6, but we should first look at the following three sections.