Why Can't We See The Lunar Landers From The Earth? (I Do The Math For You)

In the summer of 1969 Neil Armstrong and Buzz Aldrin landed on the Moon in the Sea of Tranquillity. Their mooncraft consisted of a lander plus an ascent module, which took the astronauts back up to the orbiting command module, leaving the lander part behind.

The Apollo Lunar Module was an artifact that 9.4 metres in width on each side and it is just sitting there on the surface for all to see. So why can't our Earth bound telescopes actually see it?

The Wave Nature Of Light

It all boils down to the wave nature of light. Have you ever watched waves on a lake or ocean approach a rock or a pier or a dock? Have you noticed how the waves tend to 'wrap around' that object. This phenomenon is called diffraction and it not only occurs for water waves but for light waves as well.

Angular Resolution of A Telescope

Building on that knowledge we turn our attention now to the operation of a telescope.

The light wave will enter the aperture of the telescope which is finite in size and the edges of the aperture will act as a type of obstruction for the light wave. The light which was essentially a perfectly parallel series of wavefronts before it hit the telescope's aperture will be warped somewhat by the edges of this aperture. This will distort the wavefront from being a perfectly parallel series of wavefronts to one that will now be slightly curved and distorted from this pattern.

This will essentially to make it harder for the telescope to tell the difference between a wavefront coming in from one angle and another wavefront coming in from a very slightly different angle (as you would get from two closely spaced objects at a large distance).

The overall effect is to smear out the details of objects that are very close to one another. In more technical terms it means that there will be a practical limit to the telescopes resolution.

Rayleigh Criterion and Dawes Limit

This has all been worked out mathematically and it has been found that the angular resolution of a telescope depends on the wavelength of the light and the diameter of a telescope's aperture is is given by:

Essentially this equation is saying that the shorter the wavelength of the light the better the resolution that can be achieved. It is also saying that the larger the diameter of the telescope the better the resolution that can be achieved.

W.R. Dawes developed out a more practical working formula for this:

where D is in centimetres, R is in arc-seconds and the equation is valid for light in the visible range.

The Angular Size of the Apollo Moon Lander

As we mentioned above the moon lander is 9.4 metres in width. It is on the moon which means that it is 362,600 kilometres away (362,600,000 metres).

This means that the angular size of the moon lander is 9.4 m / 362,600,000 m = 2.6 x 10^-8 radians. This is a very small angular size.

I am going to be lazy and use Google to convert radians to arc-seconds:

So the angular size of the moon lander on the Moon is 0.00535 arc-seconds.

We can re-arrange Dawe's equation above to figure how large a telescope would have to be to see something this small:

Plugging 0.00535 arc-seconds in for R we get:

D = 2168 centimetres = 21.7 metres

This means that an Earth bound telescope would need to be 21.7 metres in diameter to have a chance to see the lunar landers. This would only happen under ideal atmospheric conditions so in reality an Earth telescope would need to be even larger than this.

Looking up the list of largest telescopes in the world we see the current record holder is the Gran Telescopio Canarias at 10.4 metres in diameter. This is about one-half of the diameter needed.

The Large Binocular Telescope which is two 10 m telescopes mounted side-by-side is claimed to have the effective resolution of a 22.8 metre wide telescope. I have found no reports of it ever being trained on the Moon. I have a suspicion that the Moon is too bright an object and you would not want to train a 10 m telescope on the Moon and gather in that much light (you would likely burn out some optics).

Closing Words

Light is a wave and it diffracts. Diffraction of light waves caused by the finite aperture size of a telescope fuzzes out images slightly and places real world limits on the maximum angular resolution that can be achieved.

The lunar landers are small (9.4 m) and they are very far away (362,600 km). This means that their angular sizes are very very small. In order to resolve an object that small would require a very large telescope (>>21 m) operating under perfect atmospheric conditions.

We don't have telescopes this large and we also don't have perfect atmospheric conditions so this all adds up to answer: you cannot see the lunar landers from Earth (yet).

Thank you for reading my post.

Post Sources

Equations generated using the Codecogs LaTeX equation editor.

[1] Telescope Resolving power.
[2] Rayleigh limit and Dawe's Limit.
[3] Angular resolution.
[4] Rayligh Limit.
[5] Dawe's Limit.
[6] Apollo Lunar Module.
[7] Apollo 11.
[8] List of largest optical telescopes.
[9] Gran Telescopio Canarias.