We consider below two circumstances. The first case deals with sources smaller than the main beam, i.e., where the source size can range from as small as a point to as large as the distance between the first “nulls” of the main beam. Most observations of spectral lines from external galaxies fall into this category. Therefore, this category is the principal subject of this chapter. The second case deals with objects of an angular size larger than the main beam, such as observations of galactic molecular clouds with millimeter wave telescopes of intermediate to large diameters (≥10 m).

### F.2.1 Sources Smaller Than the Beam Size

In the commonly used “on–off” observing technique, we measure the direct product

^{2} of the telescope response and the source distribution over each point, (θ, φ), within the solid angle of the source, Ω

_{ S }. In this case, we require Ω

_{ S } to be smaller than the solid angle of the main beam, Ω

_{ B }. For simplification, we omit the “subscript” ν in

*T* and in

*S* although most quantities are functions of ν. Here, the measured antenna temperature

*T*_{ A } of a source with a brightness temperature distribution

*T*_{ R }(θ,φ) observed with an antenna with a normalized beam

*f*(θ,φ) is (see, e.g., Baars (1973))

$$T_A = \frac{\eta_R}{\Omega_A} \int_{\rm source} T_R(\theta,\phi) f(\theta,\phi) d\Omega,$$

(F.5)

where η

_{ R } is the radiation efficiency of the antenna accounting for ohmic losses, and

\(\Omega_A = \int_{4\pi}f(\theta,\phi) d\Omega\) is the solid angle of the antenna pattern. Normally, η

_{ R } is close to 1 for a well‐designed telescope surface. Using the relationship

$$\frac{\eta_R}{\Omega_A} = \frac{G}{4\pi} = \frac{A}{\lambda^2},$$

(F.6)

where

*G* is the antenna gain, λ is the wavelength of the observations, and

*A* ≡ η

_{ A } π (

*D*/2)

^{2} is the effective area of the antenna of diameter

*D* with an

*aperture efficiency*, η

_{ A }, we obtain

$$T_A = \frac{A}{\lambda^2} T_R \int_{\rm source} \psi(\theta,\phi) f(\theta,\phi) d\Omega,$$

(F.7)

where we have introduced the normalized source brightness distribution function, ψ(θ,φ). The parameter

*T*_{ R } is the source brightness temperature at the position (θ,φ) = (0,0).

The substitution of (

F.2) into (

F.7) yields (Baars, 1973)

$$T_A = \frac{SA}{2k} \frac{1}{\Omega_S} \int_{\rm source} \psi(\theta,\phi) f(\theta,\phi) d\Omega$$

(F.8)

$$= \frac{SA}{2k} \frac{\Omega_\Sigma}{\Omega_S},$$

(F.9)

where we have defined the source solid angle

$$\Omega_S \equiv \int_{\rm source}\psi(\theta,\phi) d\Omega$$

(F.10)

and the beam‐weighted source solid angle

$$\Omega_\Sigma \equiv \int_{\rm source}\psi(\theta,\phi)f(\theta,\phi) d\Omega.$$

(F.11)

The factor

*K* ≡ Ω

_{ S }/Ω

_{Σ} corrects the measured antenna temperature for the weighting of the source distribution by the large antenna beam. Therefore, (

F.9) gives the spectral flux density of a source smaller than the beam as

$$S = \frac{2k}{A} K T_A,$$

(F.12)

and the flux density received in a spectral line, given by (

F.3), becomes

$$F = \frac{8k}{\pi D^2} \frac{K}{\eta_A} \int_{{\rm line}} T_A d\nu, \quad \Omega_B > \Omega_S,$$

(F.13)

in terms of observational units.

For sources with Gaussian or disk distributions, the correction factor

*K* can be written explicitly as

$$K = \begin{cases} 1+x^2 & {\rm Gaussian \ source} \\ \frac{x^2}{1-\exp(-x^2)} & {\rm disk \ source} \quad x \leq 1, \end{cases}$$

(F.14)

where the quantity

*x* is defined by

$$x = \begin{cases} \theta_S/\theta_B & {\rm Gaussian \ source} \\ \sqrt{\ln 2} \theta_D/\theta_B & {\rm disk \ source} \end{cases}$$

(F.15)

where θ

_{ S } and θ

_{ B } are the widths of the source and beam at half‐intensity, respectively, and θ

_{ D } is the angular diameter of the disk source. Table

F.1 tabulates

*K* as a function of source size for both a Gaussian and disk source.

Table F.1 Correction factor *K*

Note that the basic characteristic of the antenna required to evaluate (

F.13) is the aperture efficiency η

_{ A } that can normally be accurately determined

^{3} from the observation of a point source (

*K* = 1) or a small source of known size and brightness distribution such as a planet where

*K* may be determined from Table

F.1. As long as the source is smaller than the beam, there is no need to invoke the

*beam efficiency*, defined as

$$\eta_B \equiv \frac{1}{\Omega_A} \int_{\rm mainbeam} f(\theta,\phi) d\Omega = \Omega_B/\Omega_A,$$

(F.16)

which is more difficult to determine since the entire main beam shape must be measured.

From (

F.6) and (

F.16), we obtain

$$\eta_B = \frac{\pi \eta_A D^2 \Omega_B}{4 \eta_R \lambda^2}.$$

(F.17)

Combining (

F.12) and (

F.17) and putting η

_{ R } = 1, we find

$$S = \frac{2 k}{\lambda^2} \frac{1}{\eta_B} T_A K \Omega_B,$$

(F.18)

which for a uniformly bright source that just fills the main beam (Ω

_{Σ} = Ω

_{ B }) is reduced to the well‐known relationship

$$S = \frac{1}{\eta_B} \frac{2k}{\lambda^2} T_A \Omega_S.$$

(F.19)

If the beam shape is known, one can convert the antenna efficiency η

_{ A } into beam efficiency η

_{ B } using (

F.17). For example, the solid angle of a symmetrical Gaussian beam is

$$\Omega_B = 1.133 \theta_B^2,$$

(F.20)

where the full width at half‐flux density is given by

$$\theta_B = F_t\frac{\lambda}{D}.$$

(F.21)

For a quadratic illumination function, Table

F.2 gives the taper factor

*F*_{ t } as a function of the edge taper.

Table F.2 Taper factor *F*_{ t }

Assuming a feed with a 12‐db taper – a common illumination for parabolic reflectors used in radio astronomy – and substituting (

F.20) and (

F.21) into (

F.17), we find

$$\eta_B = 1.2 \frac{\eta_A}{\eta_R},$$

(F.22)

where, in many cases, the radiation efficiency η

_{ R } ≈ 1.

We now arrive at a telescope‐independent expression for the line flux density

*F* of a spectral line observed from a source of angular size less than the beam. Evaluating the factors in (

F.13), we find

$$\begin{array}{rcl} \frac{F}{[{\rm W \ m}^{-2}]} & = & 3.515 \times 10^{-23} \left(\frac{D}{[{\rm m}]}\right)^{-2} \frac{K}{\eta_A} \times \\ & & \int_{\rm line}\frac{T_A}{[{\rm K}]} \frac{d\nu}{[{\rm Hz}]}, \quad \Omega_B > \Omega_S. \end{array}$$

(F.23)

To convert *F* to (ergs s^{−1} cm^{−2}) or to (Jy Hz), multiply by 10^{3} or 10^{26}, respectively.

Note that *F* is independent of the telescope size. (F.9) shows the measured *T*_{ A } to be proportional to η_{ A }*D*^{2}/*K*, thus precisely canceling out the factor *K*/(η_{ A }*D*^{2}) in (F.23).

Sanders et al. (1991) have also considered this problem, but their (A6) and (A8) do not quite follow standard antenna theory. Furthermore, the denominator of (A11) is missing a factor of 2 ln 2 ≈ 1.4. Therefore, the numerical results in their resulting equations (A12)–(A15) that relate the observations to astrophysical quantities need to be multiplied by this factor.

### F.2.2 Sources Larger Than the Beam Size

This case is more difficult. The random imperfections of most telescopes give rise to an error beam that is many times wider than the main diffraction beam of the telescope. Even weak radiation entering the error beam can contribute significantly to the resulting spectrum because of the large solid angle of the error beam. The coupling of the overall beam to the source region is often too complex to be corrected by a simple mathematical procedure. This situation is encountered when observing giant molecular clouds in our galaxy with a large millimeter wave antenna.

To obtain accurate measurements of the line flux density *F* from such extended sources, one needs a detailed knowledge of the antenna pattern out to an angle at least as large as that of the source *and* a detailed knowledge of the source brightness distribution so as to calculate the coupling of the antenna and source. The large‐scale antenna pattern can be difficult to measure and the source distribution is, of course, usually unknown.

As a practical approach, we suggest the use of a quantity that we call the

*effective beam efficiency*,

\(\eta^\prime_B\),

$$\eta^\prime_B(\Theta) \equiv \int_\Theta f(\theta,\phi) d\Omega / \Omega_A,$$

(F.24)

in which the integration is extended over a solid angle Θ equal to that of the source. So, if the source size is known,

\(\eta^\prime_B\) will be the best representation of the coupling of the beam to the source. If we also assume that the source is uniformly bright over its angle Θ, then the measured antenna temperature relates to the effective brightness temperature of the source by

$$T_A = \eta^\prime_B T_R.$$

(F.25)

Alternatively, (F.19) is valid for this case if \(\eta^\prime_B\) replaces η_{ B }.

Most observers follow procedures described by Kutner and Ulich (1981), which describe observations in terms of the parameter

\(T^\ast_R\) that corrects observations for all telescope‐dependent parameters

*except* the coupling of the antenna to the source brightness distribution. In terms used by Kutner and Ulich (1981), and under the assumption of a uniformly bright source,

$$\eta^\prime_B = \eta_c \eta_s,$$

(F.26)

where η

_{ c } and η

_{ s } are their “coupling” and “extended source” efficiencies, respectively. Unfortunately, η

_{ c } generally cannot be measured and can be calculated only with simplifying assumptions.

In our approach, it is possible to estimate \(\eta^\prime_B\) by observing a series of sources of different sizes using the planets (a few arcseconds to an arcminute) and the moon (≈ 30′). Interpolation between 1′ and 30′ could result in large errors. Extrapolation of \(\eta^\prime_B\) beyond 30′ could be determined using the complete forward beam efficiency (over 2π sr), which may be obtained from the standard “sky tips” used to measure the atmospheric extinction. Note that the contribution from the main beam, sidelobes, and error pattern are all present in \(\eta^\prime_B\). Thus, a direct measurement of \(\eta^\prime_B\) is more accurate than any theoretical calculation. We repeat that such measurements require that the source size is known and that the brightness distribution is constant over the source.

If a reasonable estimate of \(\eta^\prime_B(\Theta)\) over a source size Θ is available, one could correct the measured antenna temperature at each point into a “main beam” value by multiplying by \(\eta_B/\eta^\prime_B\). The mapped source could then be processed as if it had been observed with a “clean beam” of efficiency η_{ B }.

For extragalactic sources that are only a few times larger than the main beam of a well‐behaved antenna, we recommend that observers map the source at θ_{ B }/2 (Nyquist sampling) intervals with respect to θ_{ B } given by (F.21). Each measurement of line flux density can be corrected to telescope‐independent quantities using expressions given in this chapter. The total line flux density for the source would then be the sum of these measurements. Although significant errors due to beam imperfections would still exist in the sum owing to radiation entering the sidelobes and error beam, the restriction to sources only a few times larger than the beam would minimize these contributions. Although the resulting total *F* would still overestimate the actual line flux density, we do not know of a better, alternative procedure.

### F.2.3 Antenna Temperature Scale

Filled‐aperture, centimeter wave telescopes calibrated by hot and cold loads placed in front of the receiver produce spectra in units of *T*_{ A } described in this chapter. We assume that such observations have been corrected for atmospheric extinction, if present.

Unlike centimeter wave telescopes, millimeter wave telescopes use the atmosphere in calibration procedures involving choppers or vanes and produce spectra in intensity units of

\(T^\ast_A\),

\(T^\ast_R\), or derivatives thereof (see especially Kutner and Ulich (1981), Guilloteau (1988), and Downes (1989)). In effect, these units presume angular sizes for the emitting region. Using the definitions given by Kutner and Ulich (1981),

^{4} we find

$$T_A = T^\ast_A \eta_l$$

(F.27)

and

$$T_A = T^\ast_R \eta_s,$$

(F.28)

to relate millimeter wave intensity units to our unit

*T*_{ A }. Here, η

_{ l } is the “forward beam efficiency” and η

_{ s } is the “extended source efficiency” defined to be η

_{ l } η

_{ fss }, where η

_{ fss } is called the “forward spillover and scattering efficiency.” The efficiency η

_{ l } results from a sky tip by extrapolation of the measured antenna temperature as a function of air mass to the point where the air mass is zero. The determination of η

_{ s } is less straightforward, because it involves a choice for the size of the “diffraction” beam as described by Kutner and Ulich (1981). Usually, η

_{ s } is measured by observations of the moon.

The NRAO 12‐m telescope produces spectra in intensity units of \(T^\ast_R\). The temperature scale of its spectra can be converted into our units by using an efficiency η_{ s } of ≈ 0.64 for observations from 70 to 310 GHz and ≈ 0.59 for observations from 330 to 360 GHz (Jewell, 1990).

The IRAM 30‐m telescope produces spectral intensities in a variety of units depending upon what the observer enters in the command SET EFFICIENCY of the observing program OBS. Entering the “forward efficiency” (η_{ l }) produces spectra in units of \(T^\ast_A\); entering the “extended efficiency” (η_{ s }), \(T^\ast_R\); and entering the “main beam efficiency,” *T*_{ mb }.

Table

F.3 lists efficiencies that obtain for the IRAM 30‐m telescope at this writing that have been taken from Thum (1986), Mauersberger et al. (1989), Baars et al. (1989), and Greve (1992). Depending upon which efficiency was entered into OBS, either (

F.27) or (

F.28) may be used to convert spectral intensities taken with the IRAM 30‐m telescope into the general units of

*T*_{ A } used in this chapter. In addition, if spectra are reported in units of main beam brightness temperature, one should multiply these intensities by η

_{ mb } to convert them into our traditional units of

*T*_{ A }.

Table F.3 Efficiencies for the IRAM 30‐m telescope

Similar procedures should apply to the temperature scales used at other millimeter wave telescopes.