The Relationship Between Flux Density and Brightness Temperature

Transcript

1 The Relationship Between Flux Density and Brightness Temperature Jeff Mangum (NRAO) June 3, 015 Contents 1 The Answer 1 Introduction 1 3 Elliptical Gaussian Source 3 Uniform Disk Source 5 1 The Answer In this document I derive the general relationship between the flux density of a source and its image brightness temperature for an antenna beam measuring two different source geometries, uniform disk and gaussian. For the impatient I give the answer here. For those interested in the details, subsequent sections provide those details. For a point source, measured with a gaussian antenna beam from an antenna with main reflector diameter D, the relationship between the source s flux density (Sν point (Jy)) and its Rayleigh-Jeans equivalent brightness temperature ( ) is given by: Introduction S point ν (Jy) πk (ln()d) (1) D (m) In the following I derive the relationship between the flux density of a source and its brightness temperature for an antenna beam measuring two different source geometries, 1

2 uniform disk and gaussian. The general relation between the flux density of a source and its brightness temperature is S ν k T λ B (Ω)dΩ () Note that Equation assumes that the Rayleigh-Jeans approximation (hν kt ) applies. In a way, then, this is a fictitious temperature which is useful for describing the output power from a radio source using a relation that is linearly proportional to temperature. For a source with brightness temperature distribution ψ(ξ, η), we can define the source solid angle as follows Ω s ψ(ξ, η)dξdη. (3) The solid angle of the source normalized by the primary beam pattern of the antenna (f(ξ, η)), sometimes called the effective source solid angle, is given by f(ξ, η)ψ(ξ, η)dξdη, () while the main beam solid angle is given by Ω m mb b f(ξ, η)dξdη, (5) and the full beam solid angle is given by Ω A f(ξ, η)dξdη. (6) As the main beam full-width at half-maximum (FWHM) beam width θ B is defined as follows (see Baars 007, The Paraboloidal Reflector Antenna in Radio Astronomy and Communication, Chapter ): θ B bλ D, (7) it is convenient in the following to parameterize the main beam solid angle in terms of the illumination taper employed by the measurements: Ω m π ( ) bλ D πb θ Ba θ Bb, (8) where b is the illumination taper factor. For a Gaussian beam b 1/ ln() 1.. With measurements of the solid angle of the source normalized by the primary beam pattern of the antenna (), the primary (main) beam (Ω m ), and the main beam efficiency ( ), we can write the relation between the measured antenna temperature ( ) and the source brightness temperature (T B ) as follows Ω s Ω m T B (9)

3 Substituting for the quantity T B dω s T B Ω s in Equation results in the following general relation between flux density and measured antenna temperature S ν k λ T Ω s AΩ A k Ω s Ω λ m where we have defined the main beam efficiency as follows: (10) Ω m Ω A (11) Note that the term Ωs defines the coupling of the measured source to the beam of the antenna. In the following I calculate Ωs for two standard source distributions. For these calculations I use the simplification that the antenna beam is circular (θ Ba θ Bb θ B ). This is a common situation in practice, and avoids the need to include a position angle term in our beam and source major (θ Ba and θ Sa ) and minor (θ Bb and θ Sb ) axis terms 1. 3 Elliptical Gaussian Source For an elliptical Gaussian source with dimensions θ Sa θ Sb and an elliptical beam with FWHM θ B, Ω S ξ exp ln() + η dξdη θsa θsb πθ Saθ Sb ln() (1) and Ω m ( exp b ξ θ Ba )] + η θbb πb θb. (13) With Equations 1 and 13 the solid angle of the source multiplied by the primary beam pattern of the antenna becomes ( )] Ω ξ S exp ln() + η exp b ξ + η dξdη θsa θsb θb πbθ B ln() ( θ Sa θ Sa + θ B ) 1/ ( θ Sb θ Sb + θ B ) 1/, (1) 1 This entails including the scaling factor cos P A in all equations that include major and minor axis variables. 3

4 where I have used the fact that 0 exp( r x )dx π r. (15) We can now write the source correction factor defining the coupling between a Gaussian source and a Gaussian beam as follows Ω s Inserting this value for Ωs π (θ ln() Saθ Sb ) ( ) πbθb θ 1/ ( Sa θ Sb ln() θsa +θ B ( 1 θ b Sa + θb ln() θ B θ Sb +θ B ) 1/ ) 1/ ( ) θ Sb + θb 1/. (16) θ B and using Ω m πb θ B in Equation 10 yields ( ) S ν (Jy) kν πb θb θ Sa + θ 1/ ( ) B θ Sb + θ 1/ B c θb θb ] k (π)(10 9 ) πb ν (GHz)θ c 103 B (arcsec) ( ) θ Sa + θb 1/ ( ) θ Sb + θb 1/ θ B θ B b ν (GHz)θB(arcsec) ( ) θ Sa + θb 1/ ( ) θ Sb + θ 1/ B θb θb ( ) bν(ghz) Sν gauss θb (Jy) (arcsec) ( ) θ Sa + θb 1/ ( ) θ Sb + θb 1/. (17) θ B Note that for a point source θ Sa, θ Sb θ B, and Equation 17 becomes: S ν (Jy) kν c k c 103 S point ν (Jy) πb θb (π)(10 9 ) θ B ] πb ν (GHz)θ B (arcsec) b ν (GHz)θB(arcsec) ( ) bν(ghz) θb (arcsec) (18)

5 Still assuming a point source, If you prefer to use the reflector diameter (D) rather than the beam size (θ B ), Equation 18 can be written as follows Uniform Disk Source ( ) Sν point (Jy) kν πb cb TA c νd πkb D 10 3 πkb D (m) (100) b D (m) (19) For a uniform disk source, such as a planet or asteroid, with equatorial and poloidal angular sizes θ eq and θ pol, and whose area, for a source radius R s, is given by R s πrdr: 0 Rs Ω S πr exp r dr 0 b θb πb θ B where I have used the fact that d exp r dr b θb θ eqθ pol b θ B 1 exp θ eqθ pol b θ B, (0) 8r exp r. (1) b θb b θb Since the integral over the source is given by Ω s R s πr πr 0 s, we can write the source correction factor defining the coupling between a disk source and a Gaussian beam as follows Ω s πr s 1 exp θ 1 eqθ pol Ω πb θb s b θb 1 1 exp. () θ eqθ pol b θ B 5

6 Inserting this value for Ωs and using Ω m πb θ B in Equation 10 yields S ν (Jy) kν πb θb 1 θ eq θ pol c b θb ( )] 1 TA 1 exp θ eqθ pol b θb ] k (π)(10 9 ) πν (GHz)θ eq (arcsec)θ pol (arcsec) c exp θ 1 eqθ pol TA b θb ν (GHz)θ eq (arcsec)θ pol (arcsec) ( )] 1 TA 1 exp θ eqθ pol b θb ( ) ν(ghz) θ eq (arcsec)θ pol (arcsec) ν (Jy) ( S disk 1 exp θ eqθ pol b θ B )] 1 TA. (3) Note that for a point source θ eq, θ pol θ B, and Equation 3 becomes: S ν (Jy) kν c k c 103 S point ν (Jy) πb θb (π)(10 9 ) ] πb ν (GHz)θ B (arcsec) b ν (GHz)θB(arcsec) ( ) bν(ghz) θb (arcsec) , () which is exactly the same relation derived from the assumption of an elliptical gaussian source and beam in Equation 18 (as it should be). 6