Ronald J Maddalena

September 9, 2014

September 9, 2014

BeamShapes.tclsh -- Allows one to explore how various parameters influence the

theoretical beam shapes of telescopes.

SYNOPSIS

BeamShapes.tclsh [-h | -help] [-Te t] [-DishDiam d] [-lambda w]

[-srcType Disk | Gaussian] [-srcDiam x] [-fracCntr f]

[-minDB m] [-phiMax p] [-phiStep s] [-nSteps n]

DESCRIPTION

Produces up to 4 tables that illustrate how feed taper and source size

will alter the measured antenna temperature and a telescope's beam shape.

The algorithms used, which can be found in various texts and papers on Radio

Telescopes, are based on using Bessel functions or Gaussians.

For this application, the results and explanations are based

on the nomenclature used by Baars (1973, IEEE Trans on Ant and Prop,

Vol AP-21, No. 4, p. 461) and Goldsmith (2002, in Single-Dish Radio

Astronomy: Techniques and Applications, ed. Stanimorivic, Altschuler,

Goldsmith, and Salter (ASP, Vol. 278), p. 45.). See these papers and

references therein. Note that I have used the assumption that etaR, the

radiation efficiency, is 1, as is true with most telescopes.

The following tables are produced:

Table 1 : Normalized Beam Shape for Point Sources

Table 2 : Corrections Factors for Extended Sources of Various Diameters

Table 3 : General Values

Table 4 : (Optional) Normalized Beam Shape for Given Source Diameter

See the Results section below for details about these tables.

OPTIONS

The user can supply the following optional arguments:

-h or -help Brings up a help page. If specified, all other

options and operands are ignored

-Te
t Feed taper in
dB. Algorithms assume a Gaussian taper with

user-specified edge illumination. (See, for example,

eq. 31-35 of Goldsmith). The sign of the taper is ignored.

Default: 13 dB

-DishDiam d Dish diameter in the same units as lambda (wavelength).

Default: 10000.

-lambda w Wavelength in the same units as the dish diameter, D.

Default: 10.

-srcType str Either Disk or Gaussian. Whether the source brightness

distribution is to be modeled as a uniform Disk (default)

of Gaussian. Used in determining values in Tables 2-3 or

the optional Table 4.

-srcDiam x For disk source, the source diameter in degrees. For

Gaussian source, FWHM source diameter. Only used in

generating table 4. If zero, the default, Table 4 is not

produced.

-fracCntr f The fraction of the aperture's diameter that is covered by

a central obstruction (Secondary reflector, receiver).

Default: 0 (a clear aperture).

-minDB m Sets a limit (in dB), along with the limit set by -phiMax,

for the extent of the beam profile, that will be rendered

in Table 1 and 2. Default: 99 (the sign is ignored).

-phiMax p The extent of the bean profile in degrees, along with the

limit set by -minDB, for the extent of the beam profile,

that will be rendered in Table 1 and 2. If not specified,

Table 1 and 2 will extend to the beam's first null.

-phiStep s The distance in degrees between rows in Tables 1, 2, and 4.

If not specified, a phiStep will be chosen that is

approximately 1/20 the FWHM of the beam profile.

-nStep n
The number of steps to use when evaluating the integral for

determining the normalized beam shape (eq. 61 of Goldsmith).

Also used in determining the location of the beam's first

null (See Results for Table 3). Default: 100

RESULTS

The program produces the following tables:

Table 1 : Normalized Beam Shape for Point Sources:

--------------------------------------------------

Generates a table with rows that give the normalized power pattern as a

function of bore-sight angle. Compares the results of Eq 61 of Goldsmith

(Bessel function approximation with a feed taper and optional central

obstruction) to the Gaussian approximation (independent of taper and no

central obstruction. The main dependencies are on -Te and -fracCntr while

-D and -lambda mainly set the angular scaling.

Columns:

Phi: Bore-sight angle in either degrees, arc minutes, or arc seconds

Phi/FWHM: Bore-sight angle in units of the FWHM beam width.

Intensity: The power at the given phi for the Bessel approximation

in both linear units and in dB.

Gaussian: The power at the given phi for the Gaussian approximation

in both linear units and in dB.

dBDiff: The difference in dB between the Bessel and Gaussian

approximation

Table 2 : Corrections Factors for Extended Sources of Various Diameters

-----------------------------------------------------------------------

Correction factors that assist one in converting intensities in units of

antenna temperature, flux density, or brightness temperature for extended

sources. Follows the approach and nomenclature of section V.A of Baars.

The main dependencies are on -Te, -srcType, and -fracCntr while -D and

-lambda mainly set the angular scaling.

Columns:

D: Source diameter (in whatever units were used for Phi in Table 1)

D/FWHM: Ratio of source diameter to FWHM beam width. See -srcDiam

for
the definition of source diameter for the kinds of -srcType's.

X: The value of X defined right after Eq. 12 of Baars. Definition

depends upon -srcType.

OmegaSig: Beam-weighted source solid angle, as defined by Baars,

section V.A., in the designated units.

OmegaSrc: Source Solid angle in the designated units.

L = Ta/Tb/etaA: Source coupling efficiency used for converting between

antenna temperature (Ta) and brightness temperature (Tb) for an

extended source whose size given in the first column and the

specified -srcType. etaA = aperture efficiency. Ignores

atmospheric attenuation. Also, L = OmegaSig/OmegaA (See Table 3).

LBaars: The equivalent of Ta/Tb/etaA for the Gaussian beam

approximation of the definition of OmegaSigma using the notation

of
Baars. From Baars, derived by combining eq. 10, 11, and the

definitions of OmegaSig, OmegaMB, and OmegaA (see Table 3).

%Diff: The percentage difference between L for the Bessel and

Gaussian approximation

K: Ratio of OmegaSrc/OmegaSigma and used in converting between flux

density (S) and Ta for an extended source: S = 2k Ta K /(etaA*Area)

where Area is the area of the dish, k = Boltzman's constant, and

etaA = aperture efficiency.

KBaars: The value of K using the Gaussian approximation as in Baars.

%Diff: The percentage difference between K for the Bessel and

Gaussian approximation

Table 3 : General Values

------------------------

Some general values that can be derived from the results of the last

two tables or from first principles:

EtaA*OmegaA: The product of the antenna pattern solid angle and the

aperture efficiency. From the antenna theorem (= lambda^2/Dish Area)

EfA(GS): Estimate of the aperture efficiency from the fractional central

obstruction and the feed taper. Product of eq. 36, 38, and 41 of

Goldsmith. Always seems to be an overestimate.

FWHM(GS): Estimate of the FWHM beam size using eq. 67 of Goldsmith.

OmegaMB(GS): Estimate of the main-beam solid angle using Gaussians and

given by eq. 75 of Goldsmith.

EfMB(GS)/EfA: Estimate of the ratio of main beam to aperture efficiency

for Gaussian beams and following eq. 73, 75, 67, and 86 of Goldsmith.

FWHM(J0): FWHM beam size as found by searching for the half-power point

for the Bessel approximation of the bean shape. Uses a 'regula falsi'

zero-searching algorithm.

1stNull(J0): Bore-sight angle of the beam's first null for the Bessel

approximation to the beam shape. Uses a brute-force search that is good

to FWHM/nSteps.

omegaMBJ0: Estimate of the main-beam solid angle using Table 2 results

for the Bessel approximation to beam shape.

EfMB(J0)/EfA: Estimate of the ratio of main beam to aperture efficiency

for the Bessel approximation to the beam shape.

Table 4 : Normalized Beam Shape for Given Source Diameter

---------------------------------------------------------

Gives the antenna convolution integral for a source (normalized) whose size

is given by -srcDiam and with brightness distribution given by -srcType.

Also depends upon -Te and -fracCntr while -D and -lambda mainly set the

angular scaling. Table 4 will not be generated if -srcDiam is zero, the

default. The angular extent of the table is chosen to be significantly

larger than twice the location of the 1st null of the point-source beam or

twice the source size, whichever is larger. Depending upon -nStep,

-phiStep, -phiMax, and -srcDiam, Table 4 may take some time to produce.

Columns:

Phi: Bore-sight angle in either degrees, arc minutes, or arc seconds

Phi/FWHM: Bore-sight angle in units of the FWHM beam width.

Intensity: The power at the given phi for the Bessel approximation

in both linear units and in dB.