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.
Received Photon Count
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.