S16_MoonDistance

The distance to the moon at any time is given by as shown in figure 1 and the distance of an observer from the center of the Earth to the surface is then . We measure the distance to the Moon as the distance from the Earth's center to the Moon’s center. If we assume the Earth is spherical then we can equate where RE is simply the radius of the Earth. . From this we can then determine the distance from the observer to the Moon as and using the Law of Cosines we get the following.

z is then the geocentric zenith angle of the Moon. However z is not an observable quantity and so we need to replace it with a topocentric zenith angle to the moon z’. This is done using trigonometric identities to obtain

Here and x is the ratio of the Earth radius and the distance to the moon, R_E/r. We know that the distance to the moon is much larger than the radius of the Earth and so x will be much less than 1, using the expansion and we can then ignore terms of order x² or higher.

Looking at figure 1 we can see that γ is also on the order of and therefore sin γ is at most of the same order as x. This makes the second term in Eq. (3) of the order x² which we can safely ignore. Due to the small angles we can approximate cos γ ≈1 and then disregarding terms of order x², we obtain the following to first order in R_E/r.

The moon however is not a static object and so the distance will vary in two ways. First the distance r, varies with a period of 27 days due to the orbital motion of the moon. Second r’ varies due to the rotation of the Earth over the course of one day. Our experiment will only last on the time scale of a few hours so we only care of the changing distance due to the rotation of the Earth. The variation in r can be approximated with a linear function.

In equation (5) r₀ is the distance to the moon at a time t_0­ and t is simply the time since that initial time. We also need to know the angular radius of the disk of the moon at any time and this is given by

Where R_M is the radius of the moon. We can use the linear approximation of the sin function because α ≈ 0.5 degrees. By using our observable α, we can then rewrite Eq. 5 to obtain

This gives us the variation in the angular radius as a function of time as measured by the observer on Earth’s surface. The second term gives us the change in distance due to the orbital motion of the moon and from the third term gives us the change in distance due to the Earth’s rotation. On the time scale of a few hours the second term is linear as we showed in Eq. 5 however on the same time scale the third term is not linear. In our experiment we are seeking to determine the dependence of α on z' and from this we can determine r₀, the distance to the Moon. At all times except when the Moon is at apogee or perigee v_r does not equal 0 and so we must take into account the second term in Eq. 7. To deal with the orbital term we will measure the angular size at two different times with the same zenith angle, one on the ascent and one on the descent. As both have the same zenith angle the terms describing the Earth’s rotation will be the same. We apply this correction and substitute it into the previous equation to obtain our final equation.

Results:

In order to see the consistent trend of the varying angular size of the moon over the course of a night, I will use a raw plot of angular size(in arc seconds) versus a dimensionless x-axis. The x axis has no dimension because in this plot we are only observing the change in angular size, not the rate at which the angular size of the moon changes as to get a picture of the basis of our measurements. This plot is shown below in figure 2:

Figure 2: A plot of the Angular size of the moon in arc seconds over the span of six hours.

The orbital speed of the moon needs to be corrected for using a plot of equation eight. The slope of this plot yields the desired correction factor of v_r/R_M. Since our data was taken the day after the moon was at apogee^[1], this correction term is extremely small since the velocity of the moon was almost entirely tangential. From the plot of equation eight, we obtained a correction term value of v_r/R_m=(2 ± 5.8) X 10^-9. This uncertainty is so large because the moon was almost at apogee so the radial velocity was almost zero. Using this correction term, the value of β^-1 can be determined as shown in equation 9, thus allowing a determination of the distance to the moon. In equation nine the (t-t₀) term will be replaced with (t_d-t_a) as in equation eight. With this correction term known, a final plot of our experiment can be completed yielding the distance to the moon. This plot is shown below in figure 3:

Figure 3: a final plot of our determination of the instantaneous distance to the moon.

Using the slope of this plot, the radius of the moon can be determined with the radius of the Earth as a given of 6371 km^[1]. Our radius of the moon is R_M=(604.2 ± 1.533) km. Using this result, the instantaneous radial velocity of the moon can be determined with the correction factor calculated earlier. We obtained a result of v_r=(1.21 ± 8.89) X 10^-6 km/s. From our intercept to slope ratio of equation ten, we obtained a final result for the distance to the moon of r₀=(1.40 ± 0.71) X 10⁵km. There are a few reasons why the uncertainty in this experiment is so large, and for the large deviation of our results to literature values. First, on the night we took data, the range in zenith angles of the moon was only 15 degrees, when it is normally around 40 degrees^[3].This small angle range makes the uncertainty in the measurements much larger since there is less data to choose from with the small range. Also, since the moon stayed at a low altitude during our data taking (17 to 32 degrees), atmospheric seeing played an effect in our results. For how low the moon was in the sky, this seeing could have distorted our angular radius measurements by up to three arc seconds^[4]. This effect nearly doubles our angular radius uncertainty, which is why the error bars in figures 2 and three are so large. Finally, the camera that was used did not have the precision needed to complete the experiment. In order to obtain accurate results, a precision of up to one arc second was needed. This was not the case as discussed in the experimental setup section using the edge finder feature in ImageJ.

If data had been taken during a full moon with a large altitude range, greater than 40 degrees, then we could have either eliminated the effect of seeing, or greatly increased our data range. Both would cut down on the uncertainty of the experiment and yield results closer to literature values. Unfortunately for the time frame of this experiment, this was not possible.