Astronomers have proposed using interferometry to make an extremely high- resolution telescope. This proposal involves placing a number of infrared telescopes in space, separating them by thousands of kilometers, and combining the light from the individual telescopes. One design of this kind has an effective diameter of \(6000 \mathrm{~km}\) and uses infrared radiation with a wavelength of \(10 \mathrm{~mm}\). If it is used to observe an Earthlike planet orbiting the star Epsilon Eridani, \(3.22\) parsecs (10.5 light-years) from Earth, what is the size of the smallest detail that this system will be able to resolve on the face of that planet? Give your answer in kilometers.

When we talk about the clarity with which a telescope can distinguish small details, we're referring to its 'angular resolution'. Think of this as the telescope's ability to take sharp photos rather than blurry ones, even from great distances. The angular resolution is measured in radians and signifies the smallest angle in which we can tell apart two points of light or objects in space.

Using this concept, astronomers can calculate how fine the details they can see with their telescopes are. They use the formula \theta = 1.22 \frac{\lambda}{D}\, where \theta\ is the angular resolution, \lambda\ is the wavelength of the light being observed, and \(D\) is the diameter of the telescope's aperture. The '1.22' comes from a calculation involving the diffraction pattern of a circular aperture.

Therefore, astronomers use this formula to evaluate the capability of a telescope's design, like the one proposed using interferometry, which allows for the creation of an 'effective' aperture much larger than what a single telescope can provide.

Infrared telescopes are the special 'glasses' that astronomers use to see a different 'color' of the universe. While our eyes can see visible light, infrared telescopes detect infrared radiation - a form of light that has longer wavelengths than what we can see. These telescopes can observe celestial objects that are too cool or too far away to emit much visible light, but which glow warmly in the infrared.

Unlike visible light, infrared can pass through dust clouds in space, allowing us to peek into regions where stars are being born. This capability makes infrared telescopes like the ones in our evoked astronomical interferometry system invaluable for understanding the more secretive corners of the cosmos. They are also placed in space to avoid Earth's atmosphere, which absorbs many of the infrared wavelengths.

A parsec is like a galactic ruler. It's a unit of measurement that astronomers use to gauge the staggering distances between stars and galaxies. One parsec is equivalent to 3.086 x \(10^{16}\) meters. To convert parsecs into meters, we multiply the number of parsecs by this conversion factor, giving us a distance in meters - something a bit more familiar to our everyday ways of measuring.

When a star, like Epsilon Eridani, is 3.22 parsecs away, we can convert this into meters by multiplying 3.22 by 3.086 x \(10^{16}\) meters. This sort of conversion allows astronomers to use Earth-based mathematical principles and physics to understand the properties and behaviors of celestial bodies at these immense distances.

Scientific notation is like shorthand writing for numbers that are too big or too small to be conveniently written in decimal form. In astronomy, where we deal with huge distances and timescales, it's essential for communication and calculation.

For example, the distance to a nearby star might be 40,000,000,000,000 kilometers, which is cumbersome to write and read. In scientific notation, this number becomes simply 4 x \(10^{13}\) kilometers. It's shorthand that makes big or tiny numbers more manageable by using powers of ten. This method reduces errors and streamlines calculations, such as those required when dealing with astronomical distances and the technical specs of equipment like telescopes.