X-rays are the first portion of the electromagnetic spectrum to be ionizing, (high frequency ultraviolet rays also have this property) meaning that they have high enough energy to dislodge electrons from the outer shells of atoms, thus converting them into ions.
This part of the electromagnetic spectrum varies in wavelength from .01 nanometers to 10 nanometers, covering three orders of magnitude. X-rays are primarily used for medical purposes, although they can be carcinogens in larger doses. It is estimated that a common dental X-ray does not meaningfully increase cancer risk, but more invasive CAT scans may significantly increase the risk.
The first known instance of a medical X-ray, taken in 1895 by William Roentgen of his wife's hand. The X-rays penetrate skin but not bone, and the lack of X-rays passing through creates the image above. Note the ring present on the ring finger.
Sheets of lead are often used to contain this radiation, as it is the most cost efficient metal for doing so. Varied thicknesses are used, although 10 mm is sufficient for most X-rays.
In astronomy, X-ray telescopes are used to detect sources of radiation throughout the galaxy. The closest source of X-rays is our Sun, (these are often reflected off the moon) but the Sun predominantly emits light at higher wavelengths, and large sources of X-rays are relatively rare. The accretion disc of a black hole emits large amounts of X-rays as it is heated to extreme temperatures. These sources are particularly prominent when the black hole has a large source of material, e.g. a binary star companion.
An artist's conception of a binary star system in which one of the stars has become a black hole. The gravitational pull from the black hole pulls in material from its companion star, and this material rotates the black hole at extreme speeds, (often over a million miles per hour) causing it to emit X-rays. The most well-known example of a binary system containing a black hole is Cygnus X-1, named for being a strong X-ray source in the constellation Cygnus.
X-ray image of Cygnus X-1 (false-color). The blue supergiant companion star is not prominent in this part of the spectrum, and is therefore not visible.
Other astronomical sources of X-rays include very massive stars, and the supernovae they result in, as well as black holes at galaxy centers. In the former case, the massive stars are often very unstable, and shed material in terrific explosions periodically, culminating in a supernova, in which, huge amounts of X-rays are emitted in an extreme explosion. The latter case works on the same principle as stellar black holes: with accreted matter increasing in temperature and energy, and subsequently releasing X-rays before being sucked in to the black hole.
Sources: http://en.wikipedia.org/wiki/X-ray, http://science.hq.nasa.gov/kids/imagers/ems/xrays.html, http://en.wikipedia.org/wiki/Astrophysical_X-ray_source
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