## Specific intensity

Here we introduce the quantity that takes into account the spread of directions from which the radiation arrives from an extended source. It could be called directional spectral energy flux density, but we will use the simpler, but less informative, name specific intensity.

### Concept of specific intensity

A hypothetical antenna with a narrow beam and narrow bandwidth could survey the entire sky and produce a map of the brightness of the sky. Your eye does this when it surveys the sky and so does the radio antenna that produces radio sky maps. Since different antennae have different areas, beam sizes, and bandwidths, it is again convenient to normalize the measured power to unit area, etc. Here we have the additional factor of beam size. Thus, we divide the power by the area, bandwidth and solid angle of the antenna beam. Recall that the unit of solid angle is the steradian (sr), and that in spherical coordinates the element of solid angle is dQ = sin 9 d9 d0; see Fig. 3.7.

This normalization defines the specific intensity I(9,0,v) with dimensions Wm-2 Hz-1 sr-1,

I(9,0,v) = Specific intensity (W m-2 Hz-1 sr-1) (8.26)

This quantity is a function of the angular location (9,0) on the sky and of the frequency v of the radiation. If one desires the actual power detected by a given telescope, one simply multiplies I(9,0, v) by the actual area, bandwidth and beam solid angle of the telescope. If I is not constant over angle or frequency, this may require integration as demonstrated below.

The specific intensity function can thus be thought of as the energy received from the angular position (9t,0t) by the antenna per second, per square meter of antenna aperture, per unit bandwidth at frequency v, and per unit solid angle at 9t ,0t. That is, it represents the energy per second that would be received each second by a hypothetical antenna with area 1 m2 facing each portion of the source, with a bandwidth of 1 Hz, and with a beam of solid angle 1 sr, if the source fills the entire beam with constant specific intensity. In fact, the specific intensity is, in almost all instances, not constant over such large angles (57° x 57°), and most telescopes have much smaller beams than 1 sr. Thus one must divide the measured power by the actual solid angle of the beam just as one must divide by the actual area and bandwidth of the telescope to estimate I .

One can develop this concept differently. Take an ideal telescope or antenna of area A and point it to some celestial position, 9t and 0t (Fig. 3) where there is a uniform background radiation. Take the solid angle of the beam AQ and the antenna passband Av to be small with the former centered on the direction (9t, 0t) and the latter centered at frequency v. The power, A P (in watts) received by the antenna must be proportional to the magnitude of each of the quantities A, Av, and AQ. (We assume here the efficiency e does not vary with frequency or with angular position over the small beam and passband.) That is, if the bandwidth Av of the antenna is doubled, the power received is doubled; similarly doubling the

Figure 8.3. Antenna beam observing a diffuse source in the sky of which one element P is shown. The celestial coordinates of the antenna pointing direction are 0t,$t, the coordinates of P are 0P,$P. The coordinates of P relative to the telescope axis and to some azimuthal reference fixed to the telescope are &P,0P.

Figure 8.3. Antenna beam observing a diffuse source in the sky of which one element P is shown. The celestial coordinates of the antenna pointing direction are 0t,$t, the coordinates of P are 0P,$P. The coordinates of P relative to the telescope axis and to some azimuthal reference fixed to the telescope are &P,0P.

area A or the solid angle AQ doubles the power received. Thus AP a AAvAQ, or in differential form (e. g., dQ <AQbeam), dP a A dv dQ (W) (8.27)

where the element of solid angle may also be written as (3.14), dQ = sin 0td0 d0 (8.28)

The missing proportionality constant in (27) is simply the specific intensity I(9t,0t,v) already defined, dP = I(0t,0t,v) A dv dQ (W; Power; defines I) (8.29)

In fact, this expression serves to define the specific intensity. The units of I (W m-2 Hz-1 sr-1), as previously specified, balance the equation.

The specific intensity is properly a vector, I (9,<,v) because the flow of energy at any one position in space has a well defined direction, as do the spectral flux density S and the flux density F. We will typically use its magnitude, I = |I |, which is a scalar.

Power received by antenna If the specific intensity of the sky is not uniform over the received band of frequencies and beam angles, one must sum the contributions by each element of the sky to obtain the received power. In other words, one must integrate the specific intensity I(9,<,v) over the observed angles and frequencies, where 9,< are coordinates on the celestial sphere.

In addition, one takes into account the variation with angle of the efficiency, or response function, of the antenna, e (9,<,v). This is again a dimensionless function with value ranging from 0.0 to 1.0. It will usually be at maximum on the antenna axis, and will be near zero outside the beam. It includes the effect of reduced off-axis projected area and the frequency-dependent diffraction limit on the beam size.

An antenna pointed at a fixed position on the sky will thus receive the following power, where the integration may now be taken over the entire sky because the efficiency function insures that the contribution from each part of the sky is properly weighted. The solid angle is written in terms of 9 and <, from (28), and the units of the several terms are shown.

One must be careful with the meaning of area A of the telescope in (30). The reflecting surfaces will not be perfectly efficient, transmission lines will exhibit ohmic losses, and detector efficiencies will not be perfect. Thus the total power detected, even on the beam axis, is less than that impinging on the geometric collecting area of the telescope. One often uses a hypothetical lesser effective area, A = Aeff < Ageom, which collects all the energy impinging on it along the telescope axis. In this case, the term A(v) carries these efficiencies and e = 1 for radiation impinging along the telescope axis. If one uses A = Ageom, then e would carry the on axis efficiency; in this case, we would have e < 1.0 for radiation impinging on axis.

The antenna efficiency e(9,<) is expressed here in terms of the angles of a sky coordinate system 9,<. Alternatively, it could be defined as a function of the angles in a coordinate system centered on the antenna itself (9',<') as illustrated in

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