Information content of radiation

All astronomical telescope and detector systems have the same purpose, namely, the study of incoming photons with the maximum possible sensitivity, and with the optimum frequency, timing, and angular resolution. One can not always attain the best possible performance in all these aspects at the same time.

A stellar object at a great ("infinite") distance appears to us as a "point" source; its angular size is smaller than is resolvable by our eye or instrument. Light rays may diverge isotropically from it, but at the great distance of our telescope, the small subset of rays impinging on it is effectively parallel. The beam of photons arriving at the earth is thus like rain falling everywhere parallel to itself. In terms of waves, the wavefronts are everywhere normal to the propagation direction. This signal from a point-like source is called a plane wave.

Telescopes capture the portion of the incoming energy that impinges on the telescope aperture. A larger telescope can collect more energy (rain) each second. The instruments on the telescope are used to determine the properties of the collected electromagnetic radiation (or incoming photons). The properties that can be measured are limited in number. They are:

(i) The rate (number per unit time) of arriving photons. This rate follows from the total power radiated by the source of the radiation, the average energy of the individual photons, and the distance to the source. (At radio frequencies one measures the amplitude of the electromagnetic wave in lieu of counting photons.) This rate can vary with time, for example from variable stars and pulsars. The former variations arise from periodic changes in radius and brightness of a star, while the latter arise from the rotation of a neutron star.

(ii) The arrival directions of the photons, or equivalently, the regions of the sky from which they originate. This allows one to describe the angular shape of the source on the sky. The photon numbers and energies from different directions determine, for example, the brightness distribution of a diffuse nebula.

(iii) The photon energy hv (or equivalently the frequency or wavelength) of the radiation. This allows one to determine how the incoming radiation is distributed in frequency (the spectrum). For example, a concentration of energy at one frequency

(a spectral line) would indicate the existence of a particular atom, such as hydrogen, undergoing a specific atomic transition. The existence of such transitions gives information about the temperatures and densities in the atmospheres of stars as well as their speeds from Doppler shifts of the frequency. (iv) The polarization, i.e., the directions of the transverse electric vector E of the incoming electromagnetic wave. A predominance of vectors in one direction is indicative of polarized light. This can indicate, for example, that the emitting particles (electrons) are significantly influenced by magnetic fields in the emitting region or that the light has been scattered by dust grains in the interstellar medium.

The sensitivity or precision with which a given telescope-detector system is able to measure these quantities is crucial to understanding the data obtained with it. For example, the angular resolution is the capability of the system to distinguish (or resolve) two adjacent objects, expressed as the minimum separation angle. This is typically 1'' for a ground-based optical telescope, about 0.05'' for a large space-borne optical telescope (e.g., the Hubble Space Telescope), and better than 0.001'' for several radio telescopes operating together from locations on different continents.

Similarly, frequency resolution is the ability to distinguish two spectral lines at closely adjacent frequencies. High-dispersion, echelle-grating spectrometers used on optical telescopes attain resolutions in wavelength of AX < 0.10 nm. Since the wavelength of optical radiation is ~500 nm, the resolution, defined as X/AX, is ~5000. Timing resolution is the ability of the instrument to distinguish the arrival times of single photons (or groups of photons) that arrive at closely spaced times. Pulses of radio emission arrive from spinning neutron stars with separations as small as 1.6 ms.

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