Testing the Limits of Optical Telescopes

As I continue to answer questions from my earlier solicitation, I am going to skip ahead to the question:

"How large would a cherry clafouti near the Moon's equator have to be to be easily identifiable as a cherry clafouti, assuming clear conditions of observation?"

At first glance, this appears to be an absurd question, but a cherry clafouti is one of my favorite desserts. It is also a good lead-in to a discussion of what we can actually see from Earth. Whether it be a delicious dessert, a meteor crater, or a distant galaxy, it is very difficult to build instruments that are capable of resolving distant objects.

1). First of all, just how distant is this caflouti?

The average distance from the center of the Earth to the center of the moon is about 385,000 km, or 38,500,000,000 cm. As I did some research online for this post, I read someone's comment that this is about 10 round-trip flights to Australia from SFO.

2). Second of all, how big is a clafouti?

This of course depends on the baker, but for now, I declare that the clafouti is baked in a 9-inch pie pan. Considering that there are 2.54 cm in one inch, this equates to 9 in * 2.54 cm/1 in = 22.86 cm. So why do I care about these measurements? Well, to predict what can be seen from earth, we need to know at how large an angle the object appears. For small objects, this angle is simply the height divided by the distance. In the example above, the angle (in units of radians) is the diameter of the clafouti divided by distance to the clafouti, or 22.86 cm/38,500,000,000 cm = 5.94 * 10^-10 radians. The angular resolution of a telescope depends on the wavelength of the light being observed and the diameter of the telescope. This angle is called the diffraction limit. Skipping the complicated math, the diffraction limit of a telescope is 1.22 times the wavelength divided by the diameter. We'll assume that the telescope and camera are sensitive to light that is visible to the eye. The wavelength of light in this range is about 400 nm to 700 nm, ranging from blue to red. We'll take green light for our calculations, in the middle around 500 nm, or 500 nm * 1 m/1,000,000,000 nm = 5 * 10^-7 m. Looking through the 1-meter telescope at Yerkes Observatory that Michael mentioned, we can resolve objects that appear at angles larger than 1.22 * 5 * 10-7 m/1 m, or 6.1 * 10^-7 radians. To see down to clafouti scales as small as 5.94 * 10-10 radians, we would need a telescope ~1000 meters in diameter. The biggest optical telescope in the world at the Keck Observatory, and is only 10 meters in diameter.

In theory, this telescope can resolve objects on the moon as small as 100 clafoutis across, or 75 feet across, about the size of a baseball diamond.

However, there is one obstacle to obtaining this resolution from the ground, any ideas what that can be?

Kyle S. Dawson is engaged in post-doctorate studies of distant supernovae and
development of a proposed space-based telescope at Lawrence Berkeley National Laboratory.