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Analysis Details of SPIRE Photmeter Beam Profiles( Bernhard Schulz, September 2012)  
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Initial Beam ProfileThe beam profile was reconstructed from four fine scan maps with obsids: 1342186522, 1342186523, 1342186524, 1342186525 Pointing for Neptune proper motion was corrected. 
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Analysis Details of SPIRE Photmeter Beam Profiles( Bernhard Schulz, September 2012)  
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As the beam profiles and the solid angles were derived from a map of Neptune, there is a discrepancy to the n*Fnu = const. spectrum that all photometry is color corrected for.  
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< <  To determine the magnitude of this variation, a radial beam profile model was derived from the final background subtracted 2D beam profiles, following the idea in North, Griffin, AmaralRogers 2011 (v2.6). The radial beam profiles are split up into a constant and a variable core part whose radial scale varies with nu^gamma as mentioned above. The solid angle can be found by integrating the product of beam profile, relative spectral response function (RSRF) and source spectrum over radius and wavelength and dividing by the integral over RSRF and source spectrum. The source spectrum for Neptune was approximated by a power law with exponent 1.39. The ratio of the solid angles found for the Neptune power law and a power law with exponent 1 was used to correct the Neptune solid angles to the standard SPIRE reference spectrum.  
> >  To determine the magnitude of this variation, a radial beam profile model was derived from the final background subtracted 2D beam profiles, following the idea in Griffin et al. 2013. The radial beam profiles (download page) are split up into a constant and a variable core part whose radial scale varies with nu^gamma as mentioned above. The solid angle can be found by integrating the product of beam profile, relative spectral response function (RSRF) and source spectrum over radius and wavelength and dividing by the integral over RSRF and source spectrum. The source spectrum for Neptune was approximated by a power law with exponent 1.39. The ratio of the solid angles found for the Neptune power law and a power law with exponent 1 was used to correct the Neptune solid angles to the standard SPIRE reference spectrum.  

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Analysis Details of SPIRE Photmeter Beam Profiles( Bernhard Schulz, September 2012)  
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This map shows green rings at 600, 650, and 1000 arcsec radius. The maps become less reliable outside of about 700 arcsec due to decreasing coverage and S/N. The background seems to rise again in the plots of average annular signal vs. radius, which is likely to be due to the lower coverage.
Results 
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Analysis Details of SPIRE Photmeter Beam Profiles( Bernhard Schulz, September 2012)
Initial Beam ProfileThe beam profile was reconstructed from four fine scan maps with obsids: 1342186522, 1342186523, 1342186524, 1342186525 Pointing for Neptune proper motion was corrected. HIPE 9.0.1782 and SPIA 1.8 were used. The map reconstruction was performed after median subtraction excluding a radius of 4 arcmin around the source. The maps have 1 arcsec sky bin size.The resulting original files are:
Complications that arise when using these maps directly for derivation of beam profiles and solid angles were discussed in a presentation and led to a few corrections as follows.
Background fit and removalThe background was not flat at the level 0.1e5 from the peak, most likely due to Zodiacal Light background variation, however galactic Cirrus could also be the reason. A two dimensional polynomial was fitted and removed. The removed background can be examined in the following files:
Overlaying the ecliptic coordinate system over the background fit (contours) shows that a an interpretation by a Zodiacal light gradient is at least qualitatively consistent. Using IRSKY to obtain a rough estimate through extrapolation of its Zodiacal model to SPIRE wavelengths at the position of the observation and one degree towards the ecliptic pole, the following gradients were obtained.
Background source removalA large number of extragalactic sources contaminated the maps. These were removed in HIPE using Sussextractor.
Solid AnglesAfter background tilt subtraction and background source removal the signal was averaged over concentric annuli around the source and this radialized beam profile was plotted against radius (Plot 1). To ensure symmetry, e.g. flatness of the background, the same was also plotted for three sectors (Plot 2 and 3). From the radialized beam profile a new background was determined and subtracted based on the necessity to have only positive flux within the errors (Plot 4). The average background outside a given radius was plotted for all radii (Plot 5). The integrated solid angle within a given radius was plotted for all radii (Plot 6). All the aforementioned diagrams are seen for the three detector arrays in files:
This map shows green rings at 600, 650, and 1000 arcsec radius. The maps become less reliable outside of about 700 arcsec due to decreasing coverage and S/N. The background seems to rise again in the plots of average annular signal vs. radius, which is likely to be due to the lower coverage.
ResultsThe diagrams showing solid angle vs. radius show a plateau around 600 arcsec which may signal the end of significant contribution from the beam profile. At different integration radii, the following values in arcsec^2 result:
These data are between 3.6 and 6.7% different from the North&Griffin numbers for 1000 arcsec integration radius and between 2.0 and 3.9% for 600 arcsec integration radius. SPIRE plans to conduct shadow observations of the same region on the sky without Neptune in it in the fall of 2012. These will be used to determine whether the apparent rise outside 700 arcsec is real and they will improve the accuracy of the background level determination substantially, especially for the longer wavelength observations. A presentation of this data was given to the HCalSG, the Hfi/Spire Cross Calibration Group and the SDAG.
ErrorsAnother presentation was given at the SDAG, which discussed various sources of uncertainties and concluded that an error of +/ 4% is a conservative estimate at this time.
Correction between Isophotal and Reference WavelengthIt is important to point out that solid angles are color dependent. The FWHM of the beam varies with frequency proportional to nu^gamma. The most recent estimate from Griffin (priv. comm) for gamma is 0.78, 0.85, 0.85, respectively for PSW, PMW and PLW. As the beam profiles and the solid angles were derived from a map of Neptune, there is a discrepancy to the n*Fnu = const. spectrum that all photometry is color corrected for. To determine the magnitude of this variation, a radial beam profile model was derived from the final background subtracted 2D beam profiles, following the idea in North, Griffin, AmaralRogers 2011 (v2.6). The radial beam profiles are split up into a constant and a variable core part whose radial scale varies with nu^gamma as mentioned above. The solid angle can be found by integrating the product of beam profile, relative spectral response function (RSRF) and source spectrum over radius and wavelength and dividing by the integral over RSRF and source spectrum. The source spectrum for Neptune was approximated by a power law with exponent 1.39. The ratio of the solid angles found for the Neptune power law and a power law with exponent 1 was used to correct the Neptune solid angles to the standard SPIRE reference spectrum.
Summary of derived solid angles for Neptune spectrum and a standard reference spectum. Different numbers apply for other colors.
This diagram shows the frequency dependency of the solid angles of PSW, PMW, and PLW in comparison to conveniently scaled power law curves with different exponents.
Normalized Beam Profile Products
 BernhardSchulz  19 Oct 2012 