# Measuring the Distance to the Moon and Photon Counting

## Introduction

While looking up some information on the Moon, I ran into an interesting set of web pages that describes an experiment to measure the distance to the Moon with centimeter-level accuracy. This experiment sends a stream of laser pulses and determines the delay between when the pulses were sent and when a tiny fraction of them return to Earth. The project is called APOLLO, which stands for Apache Point Observatory Lunar Laser-ranging Operation.

On one of their pages, I encountered the following statement:

... the APD [Avalanche Photo-Diode] array enables simultaneous measurement of multiple photons returning from the moon. Throughput estimates for APOLLO predict a mean photon return rate as high as 5 photons per pulse.

The photons are reflecting off of a retro-reflector left on the Moon by the Apollo space project. I started to become curious about the small number of photons that were being returned with each pulse. I wondered if I could understand that number. This is another Fermi-type calculation. Let's dig in ...

## Background

The APOLLO folks are trying to measure the distance between the Earth and Moon VERY accurately. Here is their approach:

• Send a stream of pulses toward one of the various reflectors left on the moon by robots or people
• Detect the returning photons and determine the time delay between the transmission and reception of the pulses.

While this sounds simple, achieving the required level of accuracy requires a tremendous effort. They are trying to model all sorts of subtle effects, like:

My analysis will simply look at the their photon budget to see if I understand (1) how many photons are being launched, and (2) where their photons are being lost in the measurement process.

## Analysis

### Number of Photons Transmitted Per Pulse

Figure 1 shows my calculations for the number of photons per pulse being launched toward the Moon.

Figure 1: Number of Photons Launched Toward the Moon.

As I thought, an enormous number of photons are being launched toward the Moon.

### Reflected Photon Count

Figure 2 shows my calculations for the number of photons per pulse that will be reflected back toward Earth. I guessed at the percentage of photons that were lost because of absorption when the transmit pulse passed out of the Earth's atmosphere.

Figure 2: Count of the Photons Reflected from The Lunar Reflector.

Only a very tiny fraction of the photons reflect back toward the Earth.

Figure 3 shows my calculations for the number of photons received at the detector. I had to make some guesses here for parameters like:

• The percentage of the photons absorbed by passing through the atmosphere again (kAT= 48%)
• The percentage of photons lost because of imperfect sensor alignment (kA=20%)
• The percentage of photons lost because photons get lost in the receiver (kDE= 40%)

These are all guesses. However, my experience says that they are not unreasonable.

Figure 3: Calculation of the Number of Photons Received Per Pulse Transmitted.

Given all the losses, just a handful of photons are available for measuring the time delay.

## Conclusion

Making reasonable assumptions, I got a number very close to 5 photons received per pulse. I think I understand what they are doing.

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