Relative Signal Strengths Using Lunar Retroreflectors

Introduction

Apollo 11 Lunar Retroreflector NASA JSC Scan
Lunar Retroreflectors Desrcription
(click for external web site)
These days, one can run across people who question the reality of the Moon landings. Many ham radio operators received signals not only from the Apollo spacecraft, but from the ALSEP instruments before they were shut down in the late 70's. They could convince reasonable doubters with their compelling evidence. Sadly these hams are getting old, with ever more expiring.

There is equipment still functioning that can convince the more reasonable doubter - namely the retroreflectors. Unusual in their ability to return reflected light back to its origin regardless of incident angle, they are used even today by (among others) the McDonald Observatory in Texas and the Apache Point Observatory Lunar Laser-ranging Operation McDonald Laser Ranging Station Dome
McDonald Laser Ranging Station
(click for external web site)
in New Mexico. These observatories employ earthbound lasers to illuminate the retroreflectors left on the Moon and use the reflected signals to measure the Moon's APOLLO Laser Ranging Telescope
A.P.O.L.L.O.
(click for external web site)
distance with great accuracy.

A question doubters sometimes ask is why a retroreflector is necessary for reception of a return signal. This page is an attempt to answer that question via specification, measurement and calculation.

Two cases are examined here, one being the signal returned by reflection of a laser beam from just the surface of the Moon. This is contrasted with the signal returned by a retroreflector on the lunar surface illuminated by a laser beam of like strength.

Specifications of Actual Equipment Used and Other Real Measurements (MLRS Telescope and Apollo 11 Retroreflector)

  1. Albedo of Moon Surface = 0.073
  2. Distance to Moon = 3.6×108m

  3. Average Laser Output Power = 15W

  4. Diameter of Laser Spot on Moon = 7km
  5. Diameter of Retroreflector Spot on Earth = 20km

  6. Collecting Area of Telescope = 0.42m2
    (0.75m primary obstructed by a 15cm secondary)
  7. Telescope Optical Path Efficiency ≈ 90%

  8. Light Collecting Area of Retroreflector = 0.11m2
  9. Retroreflector Efficiency (when new) ≈ 90%

Calculations

These calculations do not take into account some effects, not the least of which are atmospheric losses. While such can affect the actual photon count, they apply to both cases. Thus they do not interfere with the relative measurement - how much gain one case has over the other.

Without Retroreflector

First we'll look at the case of the laser illuminating a spot on the Moon without a retroreflector. The light reflecting from the illuminated surface loses coherence and is scattered. Also, a spot 7km in diameter on the Moon is small when viewed from the Earth. So it can be treated as a point source of light shining equally in all directions possible from the Moon's surface - a half sphere of illumination.

  1. Given the optical efficiency of the telescope, the power of the beam output by it is:

    15W × 0.9 = 13.5W [from c and g]

  2. Given the albedo of the Moon, the average power of the reflected light is:

    13.5W × 0.073 ≈ 0.99W [from 1 and a]

  3. Using ½(4πr2), the area of the half sphere of illumination reaching the Earth's distance is:

    2π(3.6×108m)2 ≈ 8.1×1017m2 [from b]

  4. The collecting area of the receiving telescope is 0.42m2. So the ratio of collected light to total returned light is:

    0.42m2 / 8.1×1017m2 ≈ 5.2×10-19 [from f and 3]

  5. The average power of the light entering the scope is:

    0.99W × 5.2×10-19 ≈ 5.1×10-19W [from 2 and 4]

  6. Thus, given the telescope's optical efficiency, the average power collected is:

    0.9 × 5.1×10-19W ≈ 4.6×10-19W [from g and 5]

With Retroreflector

Now we'll look at the case where the laser illuminates a lunar retroreflector. For simplicity's sake, we'll ignore the energy returned from the lunar surface proper and consider just the energy returned by the retroreflector. Before continuing, here is a link to a very high resolution NASA JSC scan/image of the Apollo 11 retroreflector on the surface of the Moon (about a 1.5MB download from an external web site - click on the highlighted text to view the image).

  1. Using πr2, the area of the illuminated spot on the Moon is:

    π(3500m)2 ≈ 3.8×107m2 [from d]

  2. The ratio of the retroreflector's light collecting area to total illuminated spot area is:

    0.11m2 / 3.8×107m2 ≈ 2.9×10-9 [from h and 7]

  3. The power of the laser light collected by the retroreflector is:

    13.5W × 2.9×10-9 ≈ 3.9×10-8W [from 1 and 8]

  4. The power returned by the 90% efficient reflector is:

    0.9 × 3.9×10-8W ≈ 3.5×10-8W [from i and 9]

  5. Using πr2, the area of the retroreflected spot on the Earth is:

    π(10000m)2 ≈ 3.1×108m2 [from e]

  6. Again, the collecting area of the receiving telescope is 0.42m2. So the ratio of collected light to total retroreflected light is:

    0.42m2 / 3.1×108m2 ≈ 1.4×10-9 [from f and 11]

  7. The average power of the light entering the telescope is:

    3.5×10-8W × 1.4×10-9 = 4.9×10-17W [from 10 and 12]

  8. Thus, given the scope's efficiency, the average power collected when using a retroreflector is:

    0.9 × 4.9×10-17W ≈ 4.4×10-17W [from g and 13]

Conclusion

Using 10log(Pwr1 / Pwr2), the power dB gain when using the retroreflector is:

10log( 4.4×10-17W / 4.6×10-19W ) ≈ 20dB [from 14 and 6]

NASA LRRR Experiment Logo
NASA LRRR Experiment
(click for external web site)
Use of the retroreflector results in two orders of magnitude more return. This 20dB gain is a very large difference in signal strength, especially when operating at the limits of possible sensitivity. Given how difficult it is to detect the laser's reflected signal when using the retroreflector (44 attoWatts), it would not be practical without it (0.46 attoWatts).

Remember that it is the difference between the retroreflector and non-retroreflector cases being emphasized here. Enviromental factors affecting both absolute measurements cancel as they apply to both MLRS Ranging the Moon
International Laser Ranging Service
(click for external web site)
cases. The relative gain is the important number.

One final note: it doesn't matter how strong or weak the returning signal is. It would not be possible to calculate the Moon's distance to centimeter accuracy were it not for the retroreflectors. Determining why is left as an exercise for the reader. As a hint, examine the problems that bedevil ham radio operators engaging in Moonbounce.

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