Experiment 13: Distance to the Moon

Objective :

To measure the distance to the moon using parallax method

Introduction :

The method of parallax is a technique used to measure the distance to an object by taking two observations of it from two separate locations located in space. Here in this experiment we shall use this technique to find the distance to the moon by observing it from two locations on the same line of longitude. Observations shall be taken at the time of meridian transit. The observations shall be used to calculate the horizontal parallax of the Moon, which will give us the distance to the moon.

Theory:

Let us first understand what parallax is. Consider a certain object O which is being observed from two different locations, $P_1$ and $P_2$ (See Fig 1). The parallactic angle is defined as p = ${\angle P_1OP_2}$. Note that this is equal to angle ${\angle OP_2 Q}$. Thus, if we are observing O with respect to a distant background, then the parallactic angle is the change in the apparent position of the object w.r.t. the fixed background. We say that $p$ is the parallactic angle of O between $P_1$ and $P_2$. The line $P_1P_2$ is called the baseline.

We define the geocentric parallax, $p$, to be the parallactic angle for any object, between the actual observer, and a hypothetical observer located at the center of the earth. (See Fig 2). We define the horizontal parallax, $P$, to be the geocentric parallax when the object appears on the horizon for the observer.

First we shall derive a relation between $P$, and $p$.
We notice that \begin{eqnarray} &&\frac{\sin p}{R_E} &=& \frac{\sin(180^\circ -z_0)}{d}\\ &\Rightarrow& \sin p &=& \frac{R_E}{d} \sin(z_0) \end{eqnarray} Where $R_E$ is the radius of the earth, $d$ is the distance of the object from the centre of the earth, $z_0$ is the zenith distance for the real observer, and $z$ is the zenith distance for the hypothetical observer at the centre of the earth who has the same zenith as the actual observer ( the geocentric zenith distance). Also, we see that ${\frac{R_E}{d} = \sin P}$. Thus
\begin{eqnarray} \sin p &=& \sin P \times \sin(z_0) \end{eqnarray}
Also, since both $P$, and $p$ are small angles,
\begin{eqnarray} p &=& P \sin(z_0) \end{eqnarray}
Note that $\sin P$ gives the distance to the object measured in Earth Radii ($R_E$).

Let us now get back to using these newly derived equations for measuring the distance to the moon. Consider two observatories $O_1$ and $O_2$ located on the same longitude on the surface of the earth. Both of them shall be used to measure the zenithal distance of the moon during meridinal transit (The moons azimuthal distance is ${0^\circ}$ or ${180^\circ}$). Let the observatories have latitude ${\phi_{1}}N$ and ${\phi_{2}}S$.

Let $z_{10}$ and $z_{20}$ be the observed zenith distance from the two observatories; and $z_1$ and $z_2$ be the corresponding geocentric zenith distance. Let $p_1$ and $p_2$ be the two parallactic angles for the two observatories.

NOTE: We take all angles as positive counterclockwise and negative clockwise.Further latitude in northern hemisphere is taken positive and in southern hemisphere is negative. Here in Fig 3, ${z_{1}}$ and ${\phi_{1}}$ are positive and ${z_{2}}$ and ${\phi_{2}}$ are negative.

Now, we see that \begin{eqnarray} \phi_{1} - \phi_{2} &=& z_{1} - z_{2} &&\hspace{2cm} \mbox{Eqn. 2}\\ z_{1} &=& z_{10} - p_{1} && \hspace{2cm} \mbox{Eqn. 3a}\\ z_{2} &=& z_{20} - p_{2} && \hspace{2cm} \mbox{Eqn. 3b}\\ p_{1} &=& P \sin(z_{10}) &&\hspace{2cm} \mbox{Eqn. 4a}\\ p_{2} &=& P \sin(z_{20}) &&\hspace{2cm} \mbox{Eqn. 4b}\\ \end{eqnarray} Using equation 4 \begin{eqnarray} P &=& (p_{1} - p_{2})/(\sin(z_{10})-\sin(z_{20})) \hspace{2cm} \mbox{Eqn. 5} \end{eqnarray} We now use Eqn. 3 to eliminate ${p_{1} - p_{2}}$ \begin{eqnarray} p_{1} - p_{2} &=& (z_{10} - z_{20}) - (z_{1} - z_{2}) \end{eqnarray} We now use Eqn 2 to eliminate ${z_{1} - z_{2}}$, and use this in Eqn. 5 \begin{eqnarray} P &=& (z_{10} - z_{20} - \phi_{1} + \phi_{2})/(\sin(z_{10})-\sin(z_{20})) \hspace{2cm} \mbox{Eqn. 6} \end{eqnarray}
Here, we can measure all the values in the RHS. Thus we can calculate $P$. Then, ${\sin P}$ will give us the distance to the moon in units of the earths Radii.

Procedure:

• Step 1: Note down the current latitude(${\phi_{1}}$) and longitude of the current location. (F6)
  
• Step 2: Find the Moon. (Use Ctrl+F for search)
  
• Step 3: Advance or retreat the time using keys J and L, until the azimuth of the Moon is either ${0^\circ}$ or ${180^\circ}$
  
• Step 4: Pause the time. (Press K)
  
• Step 5: Note down the altitude. This angle subtracted from ${90^\circ}$ will give us the observed zenithal distance ${z_{10}}$
  
• Step 6: Change the location to another place, maintaining the same longitude. Note down the new latitude ${\phi_{2}}$ (F6)
  
• Step 7: Notice that the moon is already on the meridian. Note down the altitude of the moon. This angle subtracted from ${90^\circ}$ will give us the observed zenithal distance ${z_{20}}$
  
• Step 8: Use Eqn. 6 to find angle $P$
  
• Step 9: The Inverse of sine of P should give the distance to the moon in earth radii.
  

CAUTION: Please note that: We take all angles as positive counterclockwise and negative clockwise.Further latitude in northern hemisphere is taken positive and in southern hemisphere is negative. Here in Fig 3, ${z_{1}}$ and ${\phi_{1}}$ are positive and ${z_{2}}$ and ${\phi_{2}}$ are negative.