1002M. J. Griffin et al.for converting the standard SPIRE extended pipeline surface brightness to the value for a fully extended source of a given spectrum, areplotted in Fig. 14(a) (tabulations of this and the other results shownbelow for SPIRE are given in the SPIRE Observer’s Manual). Theseresults are not sensitive to a small change in the adopted value of γ ,the power-law index of the frequency scaling of the beam profile.For a ν 3 source, using γ 0.75 or 0.95 instead of 0.85 results inKColE (3, , 1, ν 0 ) changing by less than 0.1 per cent for all threebands. However, assuming that the beam profile is independent offrequency (γ 0) results in values which are up to 10 per centdifferent for the three bands.KColE (T, β, , 1, ν 0 ), the correction factor for an assumedmodified black body spectrum, is shown as a function of modifiedblackbody temperature in Fig. 14(b) for β 1.5 and 2. The dependence on β is small, and the dependence on T is also small fortemperatures above 5 K for the 500 μm band and 10 K for the250 μm band.An earlier version of the SPIRE calibration scheme (Swinyardet al. (2010) and SPIRE Observer’s Manual v2.4 and earlier) accounted for the variation of beam size across the band by weightingthe SRF by the square of the wavelength (equivalent to γ 1 andδ 2), producing a larger throughput at the longer wavelength endof the band. Different colour-correction factors were derived andquoted for the case of point and extended sources, and the conversion from flux density to surface brightness was carried out bydividing by the broad-band beam area as measured on Neptune.Compared to the new method presented here, this earlier methodproduced surface brightness values which are higher by approximately (7, 7, 12) per cent at (250, 350, 500) μm for a ν 3 source.We note that these systematic errors were within the 15 per centquoted uncertainties.It is also interesting to compare the results obtained for a fullyextended source using the new method and by simply dividingthe pipeline output by the measured broad-band beam area. For asource of spectral index α, this produces an estimate of the skysurface brightness that departs from the true value by a factor ofG(α) given byG(α) KMonP (α, ν0 ).KUniform (α, ν0 ) Meas(31)The effect is greater for sources with spectra that are more different from that of Neptune. For a ν 2 source, G(2) (0.991, 0.990,0.988) and for ν 3 , G(3) (0.976, 0.976, 0.966). The sky intensity,which depends on the source spectral index, is thus underestimatedby a small extent, and can be up to 3 per cent at 500 μm for a ν 3source.6.2.4 Results for a partially extended Gaussian sourceA partially extended source lies between the point-like and fullyextended sources. The conversion factor from the extended pipelinepeak surface brightness (which applies to a fully extended source,as adopted for the SPIRE extended source pipeline) to the peaksurface brightness for a partially extended source, KColE (α, θ 0 , 1,ν 0 ) is plotted versus the source FWHM in Fig. 15 for the case ofα 3 (a typical value for a cold dust source observed by SPIRE).For large source widths (i.e. θ 0 ), it converges on the colourcorrection parameters plotted in Fig. 14(a), which are close to unity,Figure 14. The colour-correction factor for a fully extended source, assuming (a) a power-law source spectrum and (b) a modified blackbody sourcespectrum with β 2 (lines) and 1.5 (symbols).Figure 15. Conversion factor from the surface brightness produced bythe extended source pipeline to the peak surface brightness of a partiallyextended source, plotted against the source FWHM, for a source withα 3.
Flux calibration of far-IR/sub-mm photometers1003AC K N OW L E D G E M E N T SHerschel is an ESA space observatory with science instrumentsprovided by European-led Principal Investigator consortia and withimportant participation from NASA.SPIRE has been developed by a consortium of institutes ledby Cardiff Univ. (UK) and including: Univ. Lethbridge (Canada);NAOC (China); CEA, LAM (France); IFSI, Univ. Padua (Italy);IAC (Spain); Stockholm Observatory (Sweden); Imperial CollegeLondon, RAL, UCL-MSSL, UKATC, Univ. Sussex (UK); andCaltech, JPL, NHSC, Univ. Colorado (USA). This developmenthas been supported by national funding agencies: CSA (Canada);NAOC (China); CEA, CNES, CNRS (France); ASI (Italy); MCINN(Spain); SNSB (Sweden); STFC, UKSA (UK); and NASA (USA).This research made use of APLPY, an open-source plotting packagefor PYTHON hosted at http://aplpy.github.comFigure 16. Conversion factor from the surface brightness produced by theextended source pipeline to the total flux density of a partially extendedGaussian source, plotted against the source FWHM, for the case of α 3.since the peak value tends towards the surface brightness of a fullyextended source. For small sources, this conversion also includesthe compensation for the fact that the flux is from a smaller solidangle, defined by the combination of the source size and the effectivebeam, and so the peak value of the true source surface brightnessmust increase.The total flux density of a partly extended source can be calculated by multiplying the peak surface brightness by the effectivesource area. The corresponding conversion factor between the peakpipeline surface brightness and the total integrated flux density ofthe source is plotted as a function of source FWHM in Fig. 16. Forvery small sources (i.e. θ 0 0), this converges on a value thatreturns the colour-corrected point source flux. For large sources thetotal flux density increases with the source area.7 CONCLUSIONSWe have developed a methodology for flux calibration of FIR–submillimetre observations made with broad-band instruments, using either antenna-coupled or absorber-coupled detectors. It takesinto account the variation of beam width and aperture efficiencyacross the photometric band. It can accommodate arbitrary sourceSEDs and, in the case of extended emission, arbitrary source surfacebrightness profiles. Accurate knowledge of the instrument properties and of the beam profile is needed to ensure that extendedemission can be calibrated with respect to a point source standard.Application of this scheme to the case of the Herschel-SPIREphotometer produces results which are a few per cent higher thanthose obtained by ignoring the fact that the beam profile variesacross the passband. Although not large, these systematic effects arecomparable to the absolute uncertainties of the primary calibrator,and so need to be understood and eliminated from the overall errorbudget.Similar considerations will apply to other broad-band photometric instruments (space borne in the submillimetre region or groundbased at longer wavelengths). Additional practical aspects of SPIREflux calibration are covered in a companion paper Bendo et al. (inpreparation) and in the SPIRE Observer’s Manual.REFERENCESAde P. A. R. et al., 2010, A&A, 520, A11Armitage-Caplan C. et al., 2011, preprint (arXiv:1102.2181)Beichman C. A., Neugebauer G., Habing H. J., Clegg P. E., Chester T. E.,1988, Infrared Astronomical Satellite (IRAS) Catalogs and Atlases, Vol.1: Explanatory Supplement. GPO, Washington, DCBendo G. et al., 2013, MNRAS, 433, 3062Bernard J.-P. et al., 2010, in 38th COSPAR Scientific Assembly, Vol. 38, p.4075Cesarsky C. J. et al., 1996, A&A, 315, L32Glenn J. et al., 2003, in Phillips T. G., Zmuidzinas J., eds, Proc. SPIE, Vol.4855, Millimeter and Submillimeter Detectors for Astronomy. SPIE,Bellingham, p. 30Griffin M. J., Orton G. S., 1993, Icarus, 105, 537Griffin M. J., Bock J. J., Gear W. K., 2002, App. Opt., 41, 6543Griffin M. J. et al., 2010, A&A, 518, L3Hauser M. G., Kelsall T., Leisawitz D., Weiland J., 1998, COBE DiffuseInfrared Background Experiment (DIRBE) Explanatory Supplement,version 2.3. NASA, GreenbeltHolland W. S. et al., 1999, MNRAS, 303, 659Holland W. et al., 2006, Proc. SPIE, 6275Lemke D. et al., 1996, A&A, 315, L64Moreno R., 1998, PhD thesis, Univ. ParisMoreno R., 2012, Technical Report, Neptune and Uranus Planetary Brightness Temperature Tabulation. ESA Herschel Science Centre, availableat: taryModels/ESA4/Pascale E. et al., 2008, ApJ, 681, 400Poglitsch A. et al., 2010, A&A, 518, L2Reichborn-Kjennerud B. et al., 2010, Proc. SPIE, 7741Ruhl J. et al., 2004, in Bradford C. M. et al., eds, Proc. SPIE, Vol. 5498, Millimeter and Submillimeter Detectors for Astronomy II. SPIE, Bellingham, p. 11Runyan M. C. et al., 2003, ApJS, 149, 265Sibthorpe B., Ferlet M., Bendo G., Papageorgiou A., Technical Report,SPIRE ICC, 2011, Spire Beam Model Release Note, Version 1.1.ESA Herschel Science Centre, available at: eCalibrationWeb/beam release note v1-1.pdfSiringo G. et al., 2009, A&A, 497, 945Siringo G. et al., 2010, The Messenger, 139, 20Stansberry J. A. et al., 2007, PASP, 119, 1038Swinyard B. M. et al., 2010, A&A, 518, L4Ulich B. L., Haas R. W., 1976, ApJS, 30, 247APPENDIX AA list of symbols used in the paper is given in Table A1.
1004M. J. Griffin et al.Table A1. List of symbols.SymbolB(ν, T )B(ν, θ , φ)F(ν)f(ν, ν 0 )g(θ , θ 0 )G(α)IPipE (α 0 , ν 0 )I(ν, θ )I(ν, θ , φ)KBeam (θ P , θ Beam )KColE (f, g, α 0 , ν 0 )KColP (f, α 0 , ν 0 )KMonE (f, g, ν 0 , θ 0 )KMonP (f, ν 0 )KUniform (f, ν 0 )P(ν, θ )P(ν, θ , φ)Pinner (θ ), Pouter (θ )PMeas (θ , α)Pmod (θ , ν, ν eff )PPred (θ , α, ν eff )RS(ν)SCS MeasSPip (α 0 , ν 0 )Ty(ν)y (ν, θ 0 )αα0α Nepβγδη(ν)θθ0θ BeamθPλ, ν λ, νν0ν effφ(ν)eff (f)Meas (α)norm (ν, ν 0 )Pred (α, ν eff )DefinitionPlanck function for temperature, T, and frequency, νThe beam response as a function of position (θ , φ) and frequency, νSpectral response function (SRF) of a photometric band in terms of frequency, νSource spectrum normalized to the flux density at frequency ν 0Radial intensity profile, as a function of radial offset angle θ , of a source with a width characterized by parameter θ 0Factor by which a naı̈ve approach to extended source calibration results is an incorrect estimate of the sky surface brightnessMonochromatic peak surface brightness of a fully extended source produced by the extended source pipeline, assuming a spectral indexα0 .Radial source surface brightness profile as a function of frequency ν and radial offset angle θ .Radial source surface brightness as a function of frequency ν and angular position (θ , φ).Beam-correction factor for a Gaussian main beam coupling to a uniform disc sourceFactor to convert the monochromatic pipeline extended source surface brightness, IPipE (ν 0 ), to the true peak surface brightness of asource with spectrum f(ν, ν 0 ) and spatial variation g(θ , θ 0 ).Colour-correction factor to convert the monochromatic pipeline point source flux density at nominal frequency ν 0 , SPip (ν 0 ), to thatcorresponding to a different assumed source spectrum f(ν, ν 0 ).Factor to convert SRF-weighted flux density, S Meas , to monochromatic surface brightness, I(ν), at frequency ν for an extended sourcewith spectrum f(ν, ν 0 ) and spatial variation g(θ , θ 0 ).Factor to convert SRF-weighted flux density, S Meas to monochromatic flux density, S(ν 0 ), at frequency ν 0 for a point source with aspectrum given by f(ν, ν 0 ).The conversion parameter for a fully extended source, i.e. KMonE (f, g, α 0 , ν 0 ) with θ 0 and therefore g(θ , θ 0 ) 1Normalized, azimuthally averaged beam profile as a function of radial offset angle θ and frequency νNormalized beam response as a function of orthogonal offset angles θ and φ, and frequency νInner and outer portions of the monochromatic beam profile. The inner portion is scaled with frequency, while the outer portion is not.Broad-band beam profile measured on a source of spectral index αModelled monochromatic beam profile at frequency ν, assuming the measured beam is equivalent to the monochromatic beam atfrequency ν effPredicted broad-band beam profile for a source spectral index α and effective frequency ν effPassband width relative to the central frequencySource flux density at frequency νSRF-weighted flux density for a calibration sourceSRF-weighted flux density for an observed sourceMonochromatic flux density at frequency ν 0 produced by a pipeline that assumes point source calibration and a source spectral index ofα0Blackbody or modified blackbody temperatureMonochromatic flux density at frequency ν for an extended source with surface brightness I(ν, θ , ν), integrated over a monochromaticbeam with response B(ν, θ , φ)Normalized monochromatic flux density at frequency ν for an extended source with radial intensity profile g(θ , θ 0 ), integrated over amonochromatic beam with radial profile P(ν, θ )Astronomical source power-law spectral indexNominal source spectral index for which SPIRE flux densities are quotedSpectral index adopted for Neptune when used as a SPIRE photometric calibratorModified blackbody emissivity index such that emissivity ν βPower-law index for adopted variation of main beam FWHM with frequencyPower-law index for adopted variation of beam solid angle with frequencyAperture efficiency (total power coupled to detector from an on-axis source) as a function of frequency, νRadial offset angle from the centre of the beamWidth parameter of a source with a circularly symmetric intensity profileBeam FWHM used for calibrationAngular radius of observed planetary disc used as a calibratorRadiation wavelength and frequencyBandwidth of photometer passband in terms of wavelength and frequencyNominal frequency at which monochromatic source flux density is to be quotedEffective frequency at which the monochromatic beam profile is assumed to be equal to the measured beam profile, and constrainedsuch that Pred is equal to MeasAzimuthal offset angle relative to the beam centre, used in non-circularly symmetric casesMonochromatic beam solid angle at frequency νEffective beam solid angle for observations of a source with spectrum f(ν, ν 0 )Broad-band beam solid angle as measured on a point source of spectral index αMonochromatic beam solid angle at frequency ν normalized to the value at ν 0Predicted beam solid angle for a source with spectral index α, assuming an effective frequency ν effThis paper has been typeset from a TEX/LATEX file prepared by the author.
into account for both antenna-coupled and absorber-coupled focal plane architectures. The scheme covers point source and extended source cases, and also the intermediate case of a semi-extended source proﬁle. We apply the new method to the Spectral and Photometric Imaging Receiver (S