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Let us assume we have some radiation passing through a surface element dA (Fig. 4.1). Some of the radiation will leave d A within a solid angle dw; the angle between dw and the normal to the surface is denoted by 0. The amount of energy with frequency in the range [v, v + d v] entering this solid angle in time dt is d Ev = Iv cos 0 d A dv dw dt. (4.1)

Here, the coefficient Iv is the specific intensity of the radiation at the frequency v in the direction of the solid angle dw. Its dimension is Wm-2 Hz-1 sterad-1.

The projection of the surface element dA as seen from the direction 0 is d An = d A cos 0, which explains the factor cos 0. If the intensity does not depend on direction, the energy d Ev is directly proportional to the surface element perpendicular to the direction of the radiation.

The intensity including all possible frequencies is called the total intensity I, and is obtained by integrating Iv over all frequencies:

More important quantities from the observational point of view are the energy flux (L v, L) or, briefly, the flux and the flux density (Fv, F). The flux density gives the power of radiation per unit area; hence its dimension is W m-2 Hz-1 or W m-2, depending on whether we are talking about the flux density at a certain frequency or about the total flux density.

Observed flux densities are usually rather small, and Wm-2 would be an inconveniently large unit. Therefore, especially in radio astronomy, flux densities are often expressed in Janskys; one Jansky (Jy) equals 10-26Wm-2 Hz-1.

When we are observing a radiation source, we in fact measure the energy collected by the detector during some period of time, which equals the flux density integrated over the radiation-collecting area of the instrument and the time interval.

The flux density Fv at a frequency v can be expressed in terms of the intensity as

where the integration is extended over all possible directions. Analogously, the total flux density is

Hannu Karttunen et al. (Eds.), Photometric Concepts and Magnitudes.

In: Hannu Karttunen et al. (Eds.), Fundamental Astronomy, 5th Edition. pp. 83-93 (2007)

DOI: 11685739_4 © Springer-Verlag Berlin Heidelberg 2007

For example, if the radiation is isotropic, i. e. if I is independent of the direction, we get

The solid angle element Am is equal to a surface element on a unit sphere. In spherical coordinates it is (Fig. 4.2; also c. f. Appendix A.5):

Substitution into (4.3) gives

so there is no net flux of radiation. This means that there are equal amounts of radiation entering and leaving the surface. If we want to know the amount of radiation passing through the surface, we can find, for example, the radiation leaving the surface. For isotropic radiation this is n/2 2n

In the astronomical literature, terms such as intensity and brightness are used rather vaguely. Flux density is hardly ever called flux density but intensity or (with luck) flux. Therefore the reader should always carefully check the meaning of these terms.

Flux means the power going through some surface, expressed in watts. The flux emitted by a star into a solid angle m is L = mr2 F, where F is the flux density observed at a distance r. Total flux is the flux passing through a closed surface encompassing the source. Astronomers usually call the total flux of a star the luminosity L. We can also talk about the luminosity L v at a frequency v ([Lv] = WHz-1). (This must not be confused with the luminous flux used in physics; the latter takes into account the sensitivity of the eye.)

If the source (like a typical star) radiates isotropically, its radiation at a distance r is distributed evenly on a spherical surface whose area is 4nr2 (Fig. 4.3). If the flux density of the radiation passing through this surface is F, the total flux is

Fig. 4.4. An observer sees radiation coming from a constant solid angle w. The area giving off radiation into this solid angle increases when the source moves further away (A a r2). Therefore the surface brightness or the observed flux density per unit solid angle remains constant

Fig. 4.4. An observer sees radiation coming from a constant solid angle w. The area giving off radiation into this solid angle increases when the source moves further away (A a r2). Therefore the surface brightness or the observed flux density per unit solid angle remains constant

If we are outside the source, where radiation is not created or destroyed, the luminosity does not depend on distance. The flux density, on the other hand, falls off proportional to 1/r2.

For extended objects (as opposed to objects such as stars visible only as points) we can define the surface brightness as the flux density per unit solid angle (Fig. 4.4). Now the observer is at the apex of the solid angle. The surface brightness is independent of distance, which can be understood in the following way. The flux density arriving from an area A is inversely proportional to the distance squared. But also the solid angle subtended by the area A is proportional to 1/r2 (w = A/r2). Thus the surface brightness B = F/w remains constant.

The energy density u of radiation is the amount of energy per unit volume (Jm-3):

This can be seen as follows. Suppose we have radiation with intensity I arriving from a solid angle dw perpendicular to the surface dA (Fig. 4.5). In the time dt, the radiation travels a distance c dt and fills a volume dV = c dt dA. Thus the energy in the volume dV is (now cos 0 = 1)

Hence the energy density du of the radiation arriving from the solid angle dw is dE 1 du = — = -1 dw, dV c and the total energy density is obtained by integrating this over all directions. For isotropic radiation we get

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