# Appendix

By

Dr Lim Zhi Han

,

Koh Zhi Yong

,

Gerald Kang Joon Kiat

●

Published 2022-09-10

The figure above shows the schematic of the transit of Venus as observed from two different locations on Earth. All lengths depicted on the figure are measured in terms of visual angles and equivalently in terms of A.U. since they are lengths on the Sun's disk as observed from Earth. Suppose an observer (P) sees Venus cross the Sun's disk from A to B while another observer (P') sees the transit happens from A\textquoteright to B'. The two paths are separated by the distance D, whose measurement is crucial for determining the A.U.

From the similar triangles in (b),

\frac{PP'\text{(in km)}}{(1-R_{V})\text{(in A.U.)}}=\frac{D\text{(in A.U.)}}{R_{V}\text{(in A.U.)}}

Here R_{V} is the radius of Venus' orbit around the Sun, which can be found in terms of the A.U. using Kepler's third law. Rearranging the terms, we have

\begin{equation}
1\text{A.U.}=\frac{R_{V}PP'}{(1-R_{V})D}\text{km}
\end{equation}

We can determine D through fundamental geometrical reasoning and accurate measurements of the transit times recorded by the two observers. The transit time as measured by observer P is proportional to the chord AB.

t_{AB}=k\text{AB}=2kR\sin\theta

with k the proportionality constant. We note that the angle subtended by the chord A'B' is slightly smaller than that subtended by the chord AB, by say 2 \delta. We can write the transit time as measured by observer P'.

\begin{align*}
t'_{A'B'} & =2kRsin(\theta-\delta)\\
 & =2kR(\sin\theta\cos\delta-\sin\delta\cos\theta)\\
 & \approx2kR(\sin\theta-\sin\delta\cos\theta)\\
 & =t_{AB}-2kR\sin\delta\cos\theta
\end{align*}

From the above equations we can obtain \sin\delta which will be useful later

\begin{align*}
t_{AB}-t'_{A'B'} & =2kR\sin\delta\cos\theta\\
\sin\delta & =\frac{t_{AB}-t'_{A'B'}}{2kR\cos\theta}\\
 & =\frac{t_{AB}-t'_{A'B'}}{t_{AB}}\frac{\sin\theta}{\cos\theta}
\end{align*}

In the third step, we have used the small angle (\delta\ll1) approximation \cos\delta\approx1 on the first term.

The distance between the two paths D can be written and approximated as

\begin{align*}
D & =h'-h\\
 & =R\cos(\theta-\delta)-R\cos\theta\\
 & =R(\cos\theta\cos\delta+\sin\theta\sin\delta)-R\cos\theta\\
 & =R(\cos\theta+\sin\theta\sin\delta)-Rcos\\
 & =R\sin\theta\sin\delta\\
 & =\frac{R\sin^{2}\theta}{\cos\theta}\frac{t_{AB}-t'_{A'B'}}{t_{AB}}
\end{align*}

Putting this in Eq. (1),

1 \text{A.U.} = \frac{R_{V}PP'}{(1-R_{V})} \frac{\cos(\theta)}{R \sin^2(\theta)} \frac{t_{AB}}{t_{AB}-t'_{A' B'}} \text{km}

The above is a simplified calculation to bring out the essence of the method to determine the astronomical unit. With the use of radar to accurately measure the astronomical unit, this method is now obselete. Nevertheless the transit of Venus is still a spectacular and rare event to observe. The previous two transit happened in 2004 and 2012. The next transit of Venus will happen in 2117!