4. ANGULAR RESOLUTION:
This term refers to sharpness of image. With normal "20/20" vision, you can read letters 1 cm high at a distance of 10 meters. But with the Hubble Space Telescope, you could read letters 1 cm high at a distance of 12 km -- 1200 times farther! We characterize the angular resolution by the smallest angle, q, that can be discerned in an image. For an eye with 20/20 vision, the angular resolution is q = 1 arcminute (1/60th of a degree). The HST has an angular resolution q = 0.05 arcseconds, 1200 times sharper.
There are four main factors that might limit the sharpness of a telescope image. The first is the accuracy of the mirrors, both in the smoothness of the surface and in the shape of the mirror. Every astronomer knows what is necessary: the primary mirror surface must have shape defined by a mathematical curve called a parabola, and the mirror surface must not deviate from this shape by more than about 1/10 the wavelength of the light -- less than 1000 atoms thick and a tiny amount compared to the size of the mirror. Otherwise the image will be blurred. But achieving this goal may be very difficult in practice.
The primary mirrors
on the large ground-based telescopes are so big that they flex by much more
than this amount under their own weight. To compensate for this flexing, the
mirror must be supported on its backside by hundreds of actuators that can be
continually adjusted to remove these distortions.
Left: Backside of the primary mirror of the 3.5 meter WIYN telescope, showing the 66 actuators that control the shape of the mirror. Source.
Since X-rays have wavelengths comparable to the size of a single atom, it is not possible (yet!) to polish an X-ray telescope mirror smooth enough that it doesn't blur the image somewhat. In fact, the smoothest astronomical mirror ever made is the one on the Chandra X-ray telescope, and the angular resolution of this telescope (about 1/3 arcseconds) is limited by the tiny distortions of the mirror surface.
Moreover, without extreme care it is possible to grind the mirror in the wrong shape. That happened with the Hubble Space Telescope. After the HST was launched, astronomers discovered that they couldn't focus the telescope. Soon after, they traced the problem to an error in manufacturing the primary mirror. They were able to fix the problem by sending up astronauts to install in small secondary mirrors ground specially to correct for the error in the primary mirror. But it was a very costly mistake. We had a bad telescope for two years and the repair cost more than $ 0.5 Billion.
The second limiting factor is called the diffraction limit. Even with perfect optics, there is a fundamental limit to the sharpness of the image. The angular resolution is proportional to the ratio of the wavelength, l, of the radiation divided by the telescope diameter: q = l/D. If you have 20/20 vision, your visual acuity is limited by the diameter of the pupil of your eye, and glasses cannot further improve that vision. A camera can take a picture that is sharper than you can see because the lens aperture is bigger than the pupil of your eye. Now that the Hubble Telescope is repaired, its angular resolution is limited only by its aperture (mirror diameter). The only way to get a sharper image than the HST is to build a new space telescope with a bigger primary mirror. However, note that the HST images are sharper at ultraviolet (UV) wavelengths and less sharp at infrared wavelengths, because UV radiation has a smaller wavelength than infrared, so the ratio of the wavelength/(mirror diameter) is smaller for UV.
Third, the detector resolution can also be a limiting factor. If the pixels are too large, the sharpness of the image will be limited by the pixel spacing, even though the optics and diffraction limit of the telescope deliver a sharper image to the detector. Today, the quality of pictures made with a digital camera is limited by the pixel size of the CCD detector. Pictures from a digital camera are not as good as they would be on the same camera using film. Generally, telescopes are designed with sufficient magnification so that CCD pixel size is not a problem.
Fourth, the angular resolution of a large ground-based telescope is limited by the distortion of light waves by the Earth's atmosphere. When you come to Sommers-Bausch Observatory, you will have a chance to see the image of a bright star through a 40 cm telescope. It shimmers and dances around like a little flame. This phenomenon is called seeing. Boulder is not a good site for astronomy because even on a clear night, the air is usually turbulent due to high altitude winds. Typically, the image of a star through one of the Sommers-Bausch telescopes is blurred to about 5 arcsec. Astronomers call this "bad seeing." We can improve the seeing by locating the telescope at a site where the high altitude winds are very smooth. The best places in the world for seeing are on high mountains on islands -- specifically, the top of the Mauna Kea volcano in Hawaii and the top of Tenerife mountain on the Canary islands. There the seeing is commonly 1 arcsec, and often as good at 0.5 arcsec. That means that the images will be 5 - 10 times as sharp as the ones you can see through the Sommers-Bausch telescopes. Seeing is also very good on mountains in California, Arizona, and Chile, and that is why some of the world's greatest telescopes are located there.
But even at the best sites, seeing limits the angular resolution of ground-based telescopes to far less than their optical design and diffraction limit would permit. Without this distortion, the 10-meter Keck telescopes would be able to produce images 4 times sharper than the 2.4-meter Hubble Space Telescope.
ADAPTIVE OPTICS: Since ground-based telescopes are far less expensive than telescopes in space, astronomers are devoting great efforts to finding ways to correct the telescope optics for atmospheric distortion. This is a big technological challenge, because the atmospheric distortion is very unsteady. A correction that works at one instant will be no good 0.01 second later, because the atmosphere has changed. Thus, to correct a ground-based telescope for seeing, one must build a system that will sense the distortion of the light waves, calculate the necessary correction shape, and deform a mirror to this shape, and do this at a rate greater than 100 times a second. It is now possible, but very difficult, to build such a system, called adaptive optics. The correction is done with a very thin flat mirror that is deformed by dozens of actuators on its backside. The actuators are controlled by a computer, which calculates the necessary deformation from the blurred image. This mirror is placed in the light beam outside the telescope, after the primary and secondary mirrors.
To accurately sense the distortion of the light waves, the adaptive optics system must be able to see a fairly bright star in nearly the same direction as the object of interest. But many important astronomical objects are not near a star that is bright enough. To solve this problem, astronomers are building systems to project "laser guide stars" in the same direction as the object they want to observe. Laser guide stars are not stars -- they are bright spots in the sky that appear when the astronomers project a very bright laser beam in the direction of the source they are observing. The spots are caused by reflection of the laser beam by atoms in the upper layers of the Earth's atmosphere.
Right now, it is possible to make adaptive optics systems that work well at infrared wavelengths, and such systems are working or under development on many major telescopes. For example, have a look at the description of adaptive optics on the Canada-France-Hawaii Telescope (CFHT Adaptive Optics Home Page), which also points you to all the major adaptive optics efforts in the world.
Infrared image (1.6 micron) of a bright star without (left) and with (right) adaptive optics. With the adaptive optics system turned off, the star image on the left has an angular diameter of about 0.6 arcsec. With the adaptive optics system on, the star image on the right has an angular diameter of about 0.044 arcsec. Source: Keck Observatory.
The distortion of optical light waves by the Earth's atmosphere is much greater than the distortion of infrared waves. To work at optical wavelengths, an adaptive optics system may require many thousands of actuators that can readjust at rates of thousands of times per second (compared to dozens of actuators working at hundreds of times per second for infrared). This very difficult technical problem has not been solved yet.
INTERFEROMETRY: To obtain the ultimate in angular resolution, astronomers try to combine the light from different telescopes separated by relatively large distances. If they can control the optical path lengths of the light waves from the two different telescopes to an accuracy of a small fraction of a wavelength, the angular resolution is limited, not by the diameter of either telescope, but by the separation between the telescopes. Astronomers are trying hard to employ the technique of interferometry at optical and infrared wavelengths. If they succeed, they will be able to see cosmic sources with angular resolution hundreds of times better than the Hubble Space Telescope. The technical challenges of optical and infrared interferometry are enormous. For telescopes on the ground, the bending of light waves by the Earth's atmosphere compounds the problem. You can read here about the many efforts to make optical interferometry work on ground-based telescopes.
In space, the technical problem is easier in principle because the light waves are not affected by the Earth's atomosphere. NASA is now building the Space Interferometry Mission (SIM). SIM is designed to do optical interferometry with 3small telescopes on a 10-meter beam. SIM will be able to locate stars with exquisite precision: about 10-5 arcseconds, about 100 times sharper than has been accomplished up to now. The planned launch date for SIM is 2009.
As we shall learn later in this course, one of the main reasons that astronomers want to develop the technique of interferometry is to find planets orbiting stars other than the Sun. They have already found Jupiter-like massive planets around dozens of nearby stars. Finding planets like Earth (500 times less massive than Jupiter) is a much greater technical challenge. But NASA is going to try. NASA scientists are now designing a mission called Terrestrial Planet Finder. If Congress funds its development, perhaps we will be observing planets like Earth in other solar systems, some 15 - 20 years from now.
Even higher angular resolution would be possible if we could build an interferometer that works at X-ray wavelengths. Recently, Professor Webster Cash of the CU Department of Astrophysical and Planetary Sciences succeeded in demonstrating the technique of X-ray interferometry in his laboratory, and now NASA is developing plans for a mission called MAXIM that is intended to image cosmic X-ray sources with this technique. It will probably take more than 20 years before MAXIM is ready to launch into space, but when it does, it will have incredible angular resolution: better than 10-6 arcseconds. To get an idea of how sharp that is, imagine that this computer screen was on the Moon and you could read it from Earth!
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Last modified January 29, 2002
Copyright by Richard McCray