Space telescopes such as Hubble can see farther into the cosmos than Earth-based instruments -- but lunar telescopes will be far better.
The Hubble Space Telescope gets its advantage over Earth telescopes by sailing above most of the atmosphere, which distorts the vision of Earth instruments. This results in a 10-fold improvement in resolution.
However, space telescopes have to fight other problems, such as stability. It's hard to make a space telescope perfectly still for long exposures when there's no solid ground to anchor it. And every 45 minutes space telescopes move from the night side to the day side of the Earth, or vice versa. The telescope will suddenly start heating up or cooling down, causing it to flex. And it when it moves from night to day, it gets a small impact shock due to the higher density of the thin atmosphere that still exists at 200 miles, which is higher over the day side than the night side of Earth.
Moon scopes will be sharper
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Large lunar telescopes will improve on Earth resolution by 100,000 times, not the 10 times of the Hubble telescope. Lunar scopes have the advantage of a very steady mounting platform (the Moon) which is far more stable than a satellite. The Moon is even more stable than the Earth -- Moonquakes have much less energy than Earth tremors and are much less frequent.
Each day and night lasts 14.75 days each, so there is ample time for the telescope to settle down in a hot or cold state. What's more, any stars would remain visible for 14 days, allowing enormously long exposures. Earth telescopes can make an exposure for a few hours, and Hubble generally is limited to 40 minutes before the Earth gets in the way.
A large lunar telescope probably would be built as an array of smaller telescopes that are electronically joined. An array of 27 telescopes with four-foot mirrors, laid out in a Y-shape with each arm of the Y 3.7 miles long, would have the resolution of a telescope with a mirror six miles in diameter. Shipping weight would be about seven tons, not a large amount. With such an array, astronomers could watch star-spots and flares on the surface of virtually any star. And out to a distance of 10 to 20 light years, astronomers could see well enough to spot continents on Earth-size planets and analyze the planets' atmospheres.
Even before such a fantastic telescope could be built on the Moon, more modest instruments would still be very valuable. LunaCorp could land a one-meter scope that would be teleoperated from Earth. Because there is no atmosphere, even a one-meter telescope would have a resolution of about 0.1 arc second, better than the largest Earth-based observatories. And because there is no atmosphere, it could photograph in ultraviolet wavelengths that don't reach the Earth. Given a wide field of view, it could complete a map of the entire sky that would show fainter objects and better detail than the Palomar Sky Survey, today's standard reference.
Lunar telescopes could be financed by governments, individuals or foundations. Given the chance to name the first planet found around another star, a wealthy individual might endow a lunar observatory. Foundations which could pay for a lunar telescope include the W.M. Keck Foundation, which gave a $70 million grant to build the Keck Telescope in Hawaii.