Difference imaging photometry of blended gravitational microlensing events with a numerical kernel
Identifieur interne : 001E88 ( Istex/Corpus ); précédent : 001E87; suivant : 001E89Difference imaging photometry of blended gravitational microlensing events with a numerical kernel
Auteurs : M. D. Albrow ; K. Horne ; D. M. Bramich ; P. Fouqué ; V. R. Miller ; J.-P. Beaulieu ; C. Coutures ; J. Menzies ; A. Williams ; V. Batista ; D. P. Bennett ; S. Brillant ; A. Cassan ; S. Dieters ; D. Dominis Prester ; J. Donatowicz ; J. Greenhill ; N. Kains ; S. R. Kane ; D. Kubas ; J. B. Marquette ; K. R. Pollard ; K. C. Sahu ; Y. Tsapras ; J. Wambsganss ; M. ZubSource :
- Monthly Notices of the Royal Astronomical Society [ 0035-8711 ] ; 2009-08-21.
Abstract
The numerical kernel approach to difference imaging has been implemented and applied to gravitational microlensing events observed by the PLANET collaboration. The effect of an error in the source-star coordinates is explored and a new algorithm is presented for determining the precise coordinates of the microlens in blended events, essential for accurate photometry of difference images. It is shown how the photometric reference flux need not be measured directly from the reference image but can be obtained from measurements of the difference images combined with the knowledge of the statistical flux uncertainties. The improved performance of the new algorithm, relative to isis2, is demonstrated.
Url:
DOI: 10.1111/j.1365-2966.2009.15098.x
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Cassan</name><affiliation><mods:affiliation>Astronomisches Rechen-Institut (ARI), Zentrum fur Astronomie (ZAH), Heidelberg University, Monchhofstrasse 12-14, 69120 Heidelberg, Germany</mods:affiliation></affiliation></author><author><name sortKey="Dieters, S" sort="Dieters, S" uniqKey="Dieters S" first="S." last="Dieters">S. Dieters</name><affiliation><mods:affiliation>Institut dAstrophysique de Paris, CNRS - UMR 7095, Universite Pierre & Marie Curie, 98bis Bd Arago, 75014 Paris, France</mods:affiliation></affiliation></author><author><name sortKey="Prester, D Dominis" sort="Prester, D Dominis" uniqKey="Prester D" first="D. Dominis" last="Prester">D. Dominis Prester</name><affiliation><mods:affiliation>Department of Physics, University of Rijeka, Omladinska 14, 51000 Rijeka, Croatia</mods:affiliation></affiliation></author><author><name sortKey="Donatowicz, J" sort="Donatowicz, J" uniqKey="Donatowicz J" first="J." last="Donatowicz">J. Donatowicz</name><affiliation><mods:affiliation>Technical University of Vienna, Department of Computing, Wiedner Hauptstrasse 10, Vienna, Austria</mods:affiliation></affiliation></author><author><name sortKey="Greenhill, J" sort="Greenhill, J" uniqKey="Greenhill J" first="J." last="Greenhill">J. Greenhill</name><affiliation><mods:affiliation>University of Tasmania, School of Maths and Physics, Private bag 37, GPO Hobart, Tasmania 7001, Australia</mods:affiliation></affiliation></author><author><name sortKey="Kains, N" sort="Kains, N" uniqKey="Kains N" first="N." last="Kains">N. Kains</name><affiliation><mods:affiliation>SUPA, School of Physics and Astronomy, University of St. Andrews, North Haugh, St Andrews, KY16 9SS</mods:affiliation></affiliation></author><author><name sortKey="Kane, S R" sort="Kane, S R" uniqKey="Kane S" first="S. R." last="Kane">S. R. Kane</name><affiliation><mods:affiliation>NASA Exoplanet Science Institute, Caltech, MS 100-22, 770 South Wilson Avenue Pasadena, CA 91125, USA</mods:affiliation></affiliation></author><author><name sortKey="Kubas, D" sort="Kubas, D" uniqKey="Kubas D" first="D." last="Kubas">D. Kubas</name><affiliation><mods:affiliation>European Southern Observatory, Casilla 19001, Vitacura 19, Santiago, Chile</mods:affiliation></affiliation></author><author><name sortKey="Marquette, J B" sort="Marquette, J B" uniqKey="Marquette J" first="J. B." last="Marquette">J. B. Marquette</name><affiliation><mods:affiliation>Institut dAstrophysique de Paris, CNRS - UMR 7095, Universite Pierre & Marie Curie, 98bis Bd Arago, 75014 Paris, France</mods:affiliation></affiliation></author><author><name sortKey="Pollard, K R" sort="Pollard, K R" uniqKey="Pollard K" first="K. R." last="Pollard">K. R. Pollard</name><affiliation><mods:affiliation>Beatrice Tinsley Institute, Department of Physics and Astronomy, University of Canterbury, Private Bag 4800, Christchurch, New Zealand</mods:affiliation></affiliation></author><author><name sortKey="Sahu, K C" sort="Sahu, K C" uniqKey="Sahu K" first="K. C." last="Sahu">K. C. Sahu</name><affiliation><mods:affiliation>Space Telescope Science Institute, 3700 San Martin Drive, Baltimore, MD 21218, USA</mods:affiliation></affiliation></author><author><name sortKey="Tsapras, Y" sort="Tsapras, Y" uniqKey="Tsapras Y" first="Y." last="Tsapras">Y. Tsapras</name><affiliation><mods:affiliation>Las Cumbres Observatory Global Telescope Network, 6740 Cortona Drive, Suite 102, Goleta, CA 93117, USA</mods:affiliation></affiliation><affiliation><mods:affiliation>Astrophysics Research Institute, Liverpool John Moores University, Liverpool CH41 1LD</mods:affiliation></affiliation></author><author><name sortKey="Wambsganss, J" sort="Wambsganss, J" uniqKey="Wambsganss J" first="J." last="Wambsganss">J. Wambsganss</name><affiliation><mods:affiliation>Astronomisches Rechen-Institut (ARI), Zentrum fur Astronomie (ZAH), Heidelberg University, Monchhofstrasse 12-14, 69120 Heidelberg, Germany</mods:affiliation></affiliation></author><author><name sortKey="Zub, M" sort="Zub, M" uniqKey="Zub M" first="M." last="Zub">M. Zub</name><affiliation><mods:affiliation>Astronomisches Rechen-Institut (ARI), Zentrum fur Astronomie (ZAH), Heidelberg University, Monchhofstrasse 12-14, 69120 Heidelberg, Germany</mods:affiliation></affiliation></author></titleStmt><publicationStmt><idno type="wicri:source">ISTEX</idno><idno type="RBID">ISTEX:A3134BD63E717DD5413C4892BAB348A34A5FB296</idno><date when="2009" year="2009">2009</date><idno type="doi">10.1111/j.1365-2966.2009.15098.x</idno><idno type="url">https://api.istex.fr/document/A3134BD63E717DD5413C4892BAB348A34A5FB296/fulltext/pdf</idno><idno type="wicri:Area/Istex/Corpus">001E88</idno><idno type="wicri:explorRef" wicri:stream="Istex" wicri:step="Corpus" wicri:corpus="ISTEX">001E88</idno></publicationStmt><sourceDesc><biblStruct><analytic><title level="a">Difference imaging photometry of blended gravitational microlensing events with a numerical kernel</title><author><name sortKey="Albrow, M D" sort="Albrow, M D" uniqKey="Albrow M" first="M. D." last="Albrow">M. D. Albrow</name><affiliation><mods:affiliation>Beatrice Tinsley Institute, Department of Physics and Astronomy, University of Canterbury, Private Bag 4800, Christchurch, New Zealand</mods:affiliation></affiliation><affiliation><mods:affiliation>E-mail: michael.albrow@canterbury.ac.nz</mods:affiliation></affiliation><affiliation><mods:affiliation></mods:affiliation></affiliation><affiliation><mods:affiliation>E-mail: michael.albrow@canterbury.ac.nz</mods:affiliation></affiliation></author><author><name sortKey="Horne, K" sort="Horne, K" uniqKey="Horne K" first="K." last="Horne">K. Horne</name><affiliation><mods:affiliation>SUPA, School of Physics and Astronomy, University of St. Andrews, North Haugh, St Andrews, KY16 9SS</mods:affiliation></affiliation></author><author><name sortKey="Bramich, D M" sort="Bramich, D M" uniqKey="Bramich D" first="D. M." last="Bramich">D. M. Bramich</name><affiliation><mods:affiliation>Isaac Newton Group of Telescopes, Apartado de Correos 321, E-38700 Santa Cruz de la Palma, Canary Islands, Spain</mods:affiliation></affiliation></author><author><name sortKey="Fouque, P" sort="Fouque, P" uniqKey="Fouque P" first="P." last="Fouqué">P. Fouqué</name><affiliation><mods:affiliation>LATT, Universite de Toulouse, CNRS, 14 avenue Edouard Belin, F-31400 Toulouse, France</mods:affiliation></affiliation></author><author><name sortKey="Miller, V R" sort="Miller, V R" uniqKey="Miller V" first="V. R." last="Miller">V. R. Miller</name><affiliation><mods:affiliation>Beatrice Tinsley Institute, Department of Physics and Astronomy, University of Canterbury, Private Bag 4800, Christchurch, New Zealand</mods:affiliation></affiliation></author><author><name sortKey="Beaulieu, J P" sort="Beaulieu, J P" uniqKey="Beaulieu J" first="J.-P." last="Beaulieu">J.-P. Beaulieu</name><affiliation><mods:affiliation>Institut dAstrophysique de Paris, CNRS - UMR 7095, Universite Pierre & Marie Curie, 98bis Bd Arago, 75014 Paris, France</mods:affiliation></affiliation></author><author><name sortKey="Coutures, C" sort="Coutures, C" uniqKey="Coutures C" first="C." last="Coutures">C. Coutures</name><affiliation><mods:affiliation>Institut dAstrophysique de Paris, CNRS - UMR 7095, Universite Pierre & Marie Curie, 98bis Bd Arago, 75014 Paris, France</mods:affiliation></affiliation><affiliation><mods:affiliation>CEA, DSM, DAPNIA, Centre d'Études de Saclay, 91191 Gif-sur-Yvette Cedex, France</mods:affiliation></affiliation></author><author><name sortKey="Menzies, J" sort="Menzies, J" uniqKey="Menzies J" first="J." last="Menzies">J. Menzies</name><affiliation><mods:affiliation>South African Astronomical Observatory, PO Box 9, 7935 Observatory, South Africa</mods:affiliation></affiliation></author><author><name sortKey="Williams, A" sort="Williams, A" uniqKey="Williams A" first="A." last="Williams">A. Williams</name><affiliation><mods:affiliation>Perth Observatory, Walnut Road, Bickley, Perth 6076, Australia</mods:affiliation></affiliation></author><author><name sortKey="Batista, V" sort="Batista, V" uniqKey="Batista V" first="V." last="Batista">V. Batista</name><affiliation><mods:affiliation>Institut dAstrophysique de Paris, CNRS - UMR 7095, Universite Pierre & Marie Curie, 98bis Bd Arago, 75014 Paris, France</mods:affiliation></affiliation></author><author><name sortKey="Bennett, D P" sort="Bennett, D P" uniqKey="Bennett D" first="D. P." last="Bennett">D. P. Bennett</name><affiliation><mods:affiliation>University of Notre Dame, Department of Physics, 225 Nieuwland Science Hall, Notre Dame, IN 46556, USA</mods:affiliation></affiliation></author><author><name sortKey="Brillant, S" sort="Brillant, S" uniqKey="Brillant S" first="S." last="Brillant">S. Brillant</name><affiliation><mods:affiliation>European Southern Observatory, Casilla 19001, Vitacura 19, Santiago, Chile</mods:affiliation></affiliation></author><author><name sortKey="Cassan, A" sort="Cassan, A" uniqKey="Cassan A" first="A." last="Cassan">A. Cassan</name><affiliation><mods:affiliation>Astronomisches Rechen-Institut (ARI), Zentrum fur Astronomie (ZAH), Heidelberg University, Monchhofstrasse 12-14, 69120 Heidelberg, Germany</mods:affiliation></affiliation></author><author><name sortKey="Dieters, S" sort="Dieters, S" uniqKey="Dieters S" first="S." last="Dieters">S. Dieters</name><affiliation><mods:affiliation>Institut dAstrophysique de Paris, CNRS - UMR 7095, Universite Pierre & Marie Curie, 98bis Bd Arago, 75014 Paris, France</mods:affiliation></affiliation></author><author><name sortKey="Prester, D Dominis" sort="Prester, D Dominis" uniqKey="Prester D" first="D. Dominis" last="Prester">D. Dominis Prester</name><affiliation><mods:affiliation>Department of Physics, University of Rijeka, Omladinska 14, 51000 Rijeka, Croatia</mods:affiliation></affiliation></author><author><name sortKey="Donatowicz, J" sort="Donatowicz, J" uniqKey="Donatowicz J" first="J." last="Donatowicz">J. Donatowicz</name><affiliation><mods:affiliation>Technical University of Vienna, Department of Computing, Wiedner Hauptstrasse 10, Vienna, Austria</mods:affiliation></affiliation></author><author><name sortKey="Greenhill, J" sort="Greenhill, J" uniqKey="Greenhill J" first="J." last="Greenhill">J. Greenhill</name><affiliation><mods:affiliation>University of Tasmania, School of Maths and Physics, Private bag 37, GPO Hobart, Tasmania 7001, Australia</mods:affiliation></affiliation></author><author><name sortKey="Kains, N" sort="Kains, N" uniqKey="Kains N" first="N." last="Kains">N. Kains</name><affiliation><mods:affiliation>SUPA, School of Physics and Astronomy, University of St. Andrews, North Haugh, St Andrews, KY16 9SS</mods:affiliation></affiliation></author><author><name sortKey="Kane, S R" sort="Kane, S R" uniqKey="Kane S" first="S. R." last="Kane">S. R. Kane</name><affiliation><mods:affiliation>NASA Exoplanet Science Institute, Caltech, MS 100-22, 770 South Wilson Avenue Pasadena, CA 91125, USA</mods:affiliation></affiliation></author><author><name sortKey="Kubas, D" sort="Kubas, D" uniqKey="Kubas D" first="D." last="Kubas">D. Kubas</name><affiliation><mods:affiliation>European Southern Observatory, Casilla 19001, Vitacura 19, Santiago, Chile</mods:affiliation></affiliation></author><author><name sortKey="Marquette, J B" sort="Marquette, J B" uniqKey="Marquette J" first="J. B." last="Marquette">J. B. Marquette</name><affiliation><mods:affiliation>Institut dAstrophysique de Paris, CNRS - UMR 7095, Universite Pierre & Marie Curie, 98bis Bd Arago, 75014 Paris, France</mods:affiliation></affiliation></author><author><name sortKey="Pollard, K R" sort="Pollard, K R" uniqKey="Pollard K" first="K. R." last="Pollard">K. R. Pollard</name><affiliation><mods:affiliation>Beatrice Tinsley Institute, Department of Physics and Astronomy, University of Canterbury, Private Bag 4800, Christchurch, New Zealand</mods:affiliation></affiliation></author><author><name sortKey="Sahu, K C" sort="Sahu, K C" uniqKey="Sahu K" first="K. C." last="Sahu">K. C. Sahu</name><affiliation><mods:affiliation>Space Telescope Science Institute, 3700 San Martin Drive, Baltimore, MD 21218, USA</mods:affiliation></affiliation></author><author><name sortKey="Tsapras, Y" sort="Tsapras, Y" uniqKey="Tsapras Y" first="Y." last="Tsapras">Y. Tsapras</name><affiliation><mods:affiliation>Las Cumbres Observatory Global Telescope Network, 6740 Cortona Drive, Suite 102, Goleta, CA 93117, USA</mods:affiliation></affiliation><affiliation><mods:affiliation>Astrophysics Research Institute, Liverpool John Moores University, Liverpool CH41 1LD</mods:affiliation></affiliation></author><author><name sortKey="Wambsganss, J" sort="Wambsganss, J" uniqKey="Wambsganss J" first="J." last="Wambsganss">J. Wambsganss</name><affiliation><mods:affiliation>Astronomisches Rechen-Institut (ARI), Zentrum fur Astronomie (ZAH), Heidelberg University, Monchhofstrasse 12-14, 69120 Heidelberg, Germany</mods:affiliation></affiliation></author><author><name sortKey="Zub, M" sort="Zub, M" uniqKey="Zub M" first="M." last="Zub">M. Zub</name><affiliation><mods:affiliation>Astronomisches Rechen-Institut (ARI), Zentrum fur Astronomie (ZAH), Heidelberg University, Monchhofstrasse 12-14, 69120 Heidelberg, Germany</mods:affiliation></affiliation></author></analytic><monogr></monogr><series><title level="j">Monthly Notices of the Royal Astronomical Society</title><title level="j" type="abbrev">Monthly Notices of the Royal Astronomical Society</title><idno type="ISSN">0035-8711</idno><idno type="eISSN">1365-2966</idno><imprint><publisher>Blackwell Publishing Ltd</publisher><pubPlace>Oxford, UK</pubPlace><date type="published" when="2009-08-21">2009-08-21</date><biblScope unit="volume">397</biblScope><biblScope unit="issue">4</biblScope><biblScope unit="page" from="2099">2099</biblScope><biblScope unit="page" to="2105">2105</biblScope></imprint><idno type="ISSN">0035-8711</idno></series></biblStruct></sourceDesc><seriesStmt><idno type="ISSN">0035-8711</idno></seriesStmt></fileDesc><profileDesc><textClass></textClass></profileDesc></teiHeader><front><div type="abstract">The numerical kernel approach to difference imaging has been implemented and applied to gravitational microlensing events observed by the PLANET collaboration. The effect of an error in the source-star coordinates is explored and a new algorithm is presented for determining the precise coordinates of the microlens in blended events, essential for accurate photometry of difference images. It is shown how the photometric reference flux need not be measured directly from the reference image but can be obtained from measurements of the difference images combined with the knowledge of the statistical flux uncertainties. The improved performance of the new algorithm, relative to isis2, is demonstrated.</div></front></TEI><istex><corpusName>oup</corpusName><author><json:item><name>M. D. Albrow</name><affiliations><json:string>Beatrice Tinsley Institute, Department of Physics and Astronomy, University of Canterbury, Private Bag 4800, Christchurch, New Zealand</json:string><json:string>E-mail: michael.albrow@canterbury.ac.nz</json:string><json:null></json:null><json:string>E-mail: michael.albrow@canterbury.ac.nz</json:string></affiliations></json:item><json:item><name>K. Horne</name><affiliations><json:string>SUPA, School of Physics and Astronomy, University of St. Andrews, North Haugh, St Andrews, KY16 9SS</json:string></affiliations></json:item><json:item><name>D. M. Bramich</name><affiliations><json:string>Isaac Newton Group of Telescopes, Apartado de Correos 321, E-38700 Santa Cruz de la Palma, Canary Islands, Spain</json:string></affiliations></json:item><json:item><name>P. Fouqué</name><affiliations><json:string>LATT, Universite de Toulouse, CNRS, 14 avenue Edouard Belin, F-31400 Toulouse, France</json:string></affiliations></json:item><json:item><name>V. R. Miller</name><affiliations><json:string>Beatrice Tinsley Institute, Department of Physics and Astronomy, University of Canterbury, Private Bag 4800, Christchurch, New Zealand</json:string></affiliations></json:item><json:item><name>J.-P. Beaulieu</name><affiliations><json:string>Institut dAstrophysique de Paris, CNRS - UMR 7095, Universite Pierre & Marie Curie, 98bis Bd Arago, 75014 Paris, France</json:string></affiliations></json:item><json:item><name>C. Coutures</name><affiliations><json:string>Institut dAstrophysique de Paris, CNRS - UMR 7095, Universite Pierre & Marie Curie, 98bis Bd Arago, 75014 Paris, France</json:string><json:string>CEA, DSM, DAPNIA, Centre d'Études de Saclay, 91191 Gif-sur-Yvette Cedex, France</json:string></affiliations></json:item><json:item><name>J. Menzies</name><affiliations><json:string>South African Astronomical Observatory, PO Box 9, 7935 Observatory, South Africa</json:string></affiliations></json:item><json:item><name>A. Williams</name><affiliations><json:string>Perth Observatory, Walnut Road, Bickley, Perth 6076, Australia</json:string></affiliations></json:item><json:item><name>V. Batista</name><affiliations><json:string>Institut dAstrophysique de Paris, CNRS - UMR 7095, Universite Pierre & Marie Curie, 98bis Bd Arago, 75014 Paris, France</json:string></affiliations></json:item><json:item><name>D. P. Bennett</name><affiliations><json:string>University of Notre Dame, Department of Physics, 225 Nieuwland Science Hall, Notre Dame, IN 46556, USA</json:string></affiliations></json:item><json:item><name>S. Brillant</name><affiliations><json:string>European Southern Observatory, Casilla 19001, Vitacura 19, Santiago, Chile</json:string></affiliations></json:item><json:item><name>A. Cassan</name><affiliations><json:string>Astronomisches Rechen-Institut (ARI), Zentrum fur Astronomie (ZAH), Heidelberg University, Monchhofstrasse 12-14, 69120 Heidelberg, Germany</json:string></affiliations></json:item><json:item><name>S. Dieters</name><affiliations><json:string>Institut dAstrophysique de Paris, CNRS - UMR 7095, Universite Pierre & Marie Curie, 98bis Bd Arago, 75014 Paris, France</json:string></affiliations></json:item><json:item><name>D. Dominis Prester</name><affiliations><json:string>Department of Physics, University of Rijeka, Omladinska 14, 51000 Rijeka, Croatia</json:string></affiliations></json:item><json:item><name>J. Donatowicz</name><affiliations><json:string>Technical University of Vienna, Department of Computing, Wiedner Hauptstrasse 10, Vienna, Austria</json:string></affiliations></json:item><json:item><name>J. Greenhill</name><affiliations><json:string>University of Tasmania, School of Maths and Physics, Private bag 37, GPO Hobart, Tasmania 7001, Australia</json:string></affiliations></json:item><json:item><name>N. Kains</name><affiliations><json:string>SUPA, School of Physics and Astronomy, University of St. Andrews, North Haugh, St Andrews, KY16 9SS</json:string></affiliations></json:item><json:item><name>S. R. Kane</name><affiliations><json:string>NASA Exoplanet Science Institute, Caltech, MS 100-22, 770 South Wilson Avenue Pasadena, CA 91125, USA</json:string></affiliations></json:item><json:item><name>D. Kubas</name><affiliations><json:string>European Southern Observatory, Casilla 19001, Vitacura 19, Santiago, Chile</json:string></affiliations></json:item><json:item><name>J. B. 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The effect of an error in the source-star coordinates is explored and a new algorithm is presented for determining the precise coordinates of the microlens in blended events, essential for accurate photometry of difference images. It is shown how the photometric reference flux need not be measured directly from the reference image but can be obtained from measurements of the difference images combined with the knowledge of the statistical flux uncertainties. 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D. Albrow et al.</json:string><json:string>Schechter, Mateo & Saha 1993</json:string><json:string>Gaudi et al. 2008</json:string><json:string>Alard & Lupton 1998</json:string><json:string>Aubourg et al. 1993</json:string><json:string>Kains et al. 2008</json:string><json:string>Gaudi et al. 2002</json:string><json:string>Gilliland et al. 2000</json:string><json:string>Albrow et al. 2001</json:string><json:string>Dong et al. 2008</json:string><json:string>Albrow et al. 1998</json:string><json:string>Cassan et al.</json:string><json:string>Udalski et al. 1994</json:string><json:string>Alcock et al. 1993</json:string><json:string>Bond et al. 2004</json:string><json:string>Arellano Ferro et al. 2008</json:string><json:string>Udalski 2003</json:string><json:string>Bramich (2008)</json:string><json:string>Udalski et al. 2005</json:string><json:string>Blu, Thevenaz & Unser 2001</json:string><json:string>Einstein 1936</json:string></ref_bibl><bibl></bibl></unitex></namedEntities><ark><json:string>ark:/67375/HXZ-L43CXH2Z-T</json:string></ark><categories><wos><json:string>science</json:string><json:string>astronomy & astrophysics</json:string></wos><scienceMetrix><json:string>natural sciences</json:string><json:string>physics & astronomy</json:string><json:string>astronomy & astrophysics</json:string></scienceMetrix><scopus><json:string>1 - Physical Sciences</json:string><json:string>2 - Earth and Planetary Sciences</json:string><json:string>3 - Space and Planetary Science</json:string><json:string>1 - Physical Sciences</json:string><json:string>2 - Physics and Astronomy</json:string><json:string>3 - Astronomy and Astrophysics</json:string></scopus><inist><json:string>sciences appliquees, technologies et medecines</json:string><json:string>sciences exactes et technologie</json:string><json:string>terre, ocean, espace</json:string><json:string>geophysique externe</json:string></inist></categories><publicationDate>2009</publicationDate><copyrightDate>2009</copyrightDate><doi><json:string>10.1111/j.1365-2966.2009.15098.x</json:string></doi><id>A3134BD63E717DD5413C4892BAB348A34A5FB296</id><score>1</score><fulltext><json:item><extension>pdf</extension><original>true</original><mimetype>application/pdf</mimetype><uri>https://api.istex.fr/document/A3134BD63E717DD5413C4892BAB348A34A5FB296/fulltext/pdf</uri></json:item><json:item><extension>zip</extension><original>false</original><mimetype>application/zip</mimetype><uri>https://api.istex.fr/document/A3134BD63E717DD5413C4892BAB348A34A5FB296/fulltext/zip</uri></json:item><istex:fulltextTEI uri="https://api.istex.fr/document/A3134BD63E717DD5413C4892BAB348A34A5FB296/fulltext/tei"><teiHeader><fileDesc><titleStmt><title level="a">Difference imaging photometry of blended gravitational microlensing events with a numerical kernel</title></titleStmt><publicationStmt><authority>ISTEX</authority><publisher scheme="https://publisher-list.data.istex.fr">Blackwell Publishing Ltd</publisher><pubPlace>Oxford, UK</pubPlace><availability><licence><p>© 2009 The Authors. Journal compilation © 2009 RAS</p></licence><p scheme="https://loaded-corpus.data.istex.fr/ark:/67375/XBH-GTWS0RDP-M">oup</p></availability><date>2009-08-11</date></publicationStmt><notesStmt><note type="research-article" scheme="https://content-type.data.istex.fr/ark:/67375/XTP-1JC4F85T-7">research-article</note><note type="journal" scheme="https://publication-type.data.istex.fr/ark:/67375/JMC-0GLKJH51-B">journal</note><note>http://www.beatricetinsleyinstitute.org/</note></notesStmt><sourceDesc><biblStruct type="inbook"><analytic><title level="a">Difference imaging photometry of blended gravitational microlensing events with a numerical kernel</title><author xml:id="author-0000"><persName><forename type="first">M. D.</forename><surname>Albrow</surname></persName><email>michael.albrow@canterbury.ac.nz</email><email>michael.albrow@canterbury.ac.nz</email><affiliation>Beatrice Tinsley Institute, Department of Physics and Astronomy, University of Canterbury, Private Bag 4800, Christchurch, New Zealand</affiliation><affiliation></affiliation></author><author xml:id="author-0001"><persName><forename type="first">K.</forename><surname>Horne</surname></persName><affiliation>SUPA, School of Physics and Astronomy, University of St. Andrews, North Haugh, St Andrews, KY16 9SS</affiliation></author><author xml:id="author-0002"><persName><forename type="first">D. M.</forename><surname>Bramich</surname></persName><affiliation>Isaac Newton Group of Telescopes, Apartado de Correos 321, E-38700 Santa Cruz de la Palma, Canary Islands, Spain</affiliation></author><author xml:id="author-0003"><persName><forename type="first">P.</forename><surname>Fouqué</surname></persName><affiliation>LATT, Universite de Toulouse, CNRS, 14 avenue Edouard Belin, F-31400 Toulouse, France</affiliation></author><author xml:id="author-0004"><persName><forename type="first">V. R.</forename><surname>Miller</surname></persName><affiliation>Beatrice Tinsley Institute, Department of Physics and Astronomy, University of Canterbury, Private Bag 4800, Christchurch, New Zealand</affiliation></author><author xml:id="author-0005"><persName><forename type="first">J.-P.</forename><surname>Beaulieu</surname></persName><affiliation>Institut dAstrophysique de Paris, CNRS - UMR 7095, Universite Pierre & Marie Curie, 98bis Bd Arago, 75014 Paris, France</affiliation></author><author xml:id="author-0006"><persName><forename type="first">C.</forename><surname>Coutures</surname></persName><affiliation>Institut dAstrophysique de Paris, CNRS - UMR 7095, Universite Pierre & Marie Curie, 98bis Bd Arago, 75014 Paris, France</affiliation><affiliation>CEA, DSM, DAPNIA, Centre d'Études de Saclay, 91191 Gif-sur-Yvette Cedex, France</affiliation></author><author xml:id="author-0007"><persName><forename type="first">J.</forename><surname>Menzies</surname></persName><affiliation>South African Astronomical Observatory, PO Box 9, 7935 Observatory, South Africa</affiliation></author><author xml:id="author-0008"><persName><forename type="first">A.</forename><surname>Williams</surname></persName><affiliation>Perth Observatory, Walnut Road, Bickley, Perth 6076, Australia</affiliation></author><author xml:id="author-0009"><persName><forename type="first">V.</forename><surname>Batista</surname></persName><affiliation>Institut dAstrophysique de Paris, CNRS - UMR 7095, Universite Pierre & Marie Curie, 98bis Bd Arago, 75014 Paris, France</affiliation></author><author xml:id="author-0010"><persName><forename type="first">D. 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Dominis</forename><surname>Prester</surname></persName><affiliation>Department of Physics, University of Rijeka, Omladinska 14, 51000 Rijeka, Croatia</affiliation></author><author xml:id="author-0015"><persName><forename type="first">J.</forename><surname>Donatowicz</surname></persName><affiliation>Technical University of Vienna, Department of Computing, Wiedner Hauptstrasse 10, Vienna, Austria</affiliation></author><author xml:id="author-0016"><persName><forename type="first">J.</forename><surname>Greenhill</surname></persName><affiliation>University of Tasmania, School of Maths and Physics, Private bag 37, GPO Hobart, Tasmania 7001, Australia</affiliation></author><author xml:id="author-0017"><persName><forename type="first">N.</forename><surname>Kains</surname></persName><affiliation>SUPA, School of Physics and Astronomy, University of St. Andrews, North Haugh, St Andrews, KY16 9SS</affiliation></author><author xml:id="author-0018"><persName><forename type="first">S. R.</forename><surname>Kane</surname></persName><affiliation>NASA Exoplanet Science Institute, Caltech, MS 100-22, 770 South Wilson Avenue Pasadena, CA 91125, USA</affiliation></author><author xml:id="author-0019"><persName><forename type="first">D.</forename><surname>Kubas</surname></persName><affiliation>European Southern Observatory, Casilla 19001, Vitacura 19, Santiago, Chile</affiliation></author><author xml:id="author-0020"><persName><forename type="first">J. B.</forename><surname>Marquette</surname></persName><affiliation>Institut dAstrophysique de Paris, CNRS - UMR 7095, Universite Pierre & Marie Curie, 98bis Bd Arago, 75014 Paris, France</affiliation></author><author xml:id="author-0021"><persName><forename type="first">K. 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<name>
<surname>Albrow</surname>
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<surname>Bramich</surname>
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<surname>Cassan</surname>
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<surname>Dieters</surname>
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<given-names>D. Dominis</given-names>
</name>
<xref ref-type="aff" rid="a12">12</xref>
</contrib>
<contrib contrib-type="author">
<name>
<surname>Donatowicz</surname>
<given-names>J.</given-names>
</name>
<xref ref-type="aff" rid="a13">13</xref>
</contrib>
<contrib contrib-type="author">
<name>
<surname>Greenhill</surname>
<given-names>J.</given-names>
</name>
<xref ref-type="aff" rid="a14">14</xref>
</contrib>
<contrib contrib-type="author">
<name>
<surname>Kains</surname>
<given-names>N.</given-names>
</name>
<xref ref-type="aff" rid="a2">2</xref>
</contrib>
<contrib contrib-type="author">
<name>
<surname>Kane</surname>
<given-names>S. R.</given-names>
</name>
<xref ref-type="aff" rid="a15">15</xref>
</contrib>
<contrib contrib-type="author">
<name>
<surname>Kubas</surname>
<given-names>D.</given-names>
</name>
<xref ref-type="aff" rid="a10">10</xref>
</contrib>
<contrib contrib-type="author">
<name>
<surname>Marquette</surname>
<given-names>J. B.</given-names>
</name>
<xref ref-type="aff" rid="a5">5</xref>
</contrib>
<contrib contrib-type="author">
<name>
<surname>Pollard</surname>
<given-names>K. R.</given-names>
</name>
<xref ref-type="aff" rid="a1">1</xref>
</contrib>
<contrib contrib-type="author">
<name>
<surname>Sahu</surname>
<given-names>K. C.</given-names>
</name>
<xref ref-type="aff" rid="a16">16</xref>
</contrib>
<contrib contrib-type="author">
<name>
<surname>Tsapras</surname>
<given-names>Y.</given-names>
</name>
<xref ref-type="aff" rid="a17">17</xref><xref ref-type="aff" rid="a18">18</xref>
</contrib>
<contrib contrib-type="author">
<name>
<surname>Wambsganss</surname>
<given-names>J.</given-names>
</name>
<xref ref-type="aff" rid="a11">11</xref>
</contrib>
<contrib contrib-type="author">
<name>
<surname>Zub</surname>
<given-names>M.</given-names>
</name>
<xref ref-type="aff" rid="a11">11</xref>
</contrib>
</contrib-group>
<aff id="a1"><label>1</label>Beatrice Tinsley Institute,<xref ref-type="fn" rid="fn3">3</xref> Department of Physics and Astronomy, University of Canterbury, Private Bag 4800, Christchurch, New Zealand</aff>
<aff id="a2"><label>2</label>SUPA, School of Physics and Astronomy, University of St. Andrews, North Haugh, St Andrews, KY16 9SS</aff>
<aff id="a3"><label>3</label>Isaac Newton Group of Telescopes, Apartado de Correos 321, E-38700 Santa Cruz de la Palma, Canary Islands, Spain</aff>
<aff id="a4"><label>4</label>LATT, Universite de Toulouse, CNRS, 14 avenue Edouard Belin, F-31400 Toulouse, France</aff>
<aff id="a5"><label>5</label>Institut dAstrophysique de Paris, CNRS - UMR 7095, Universite Pierre & Marie Curie, 98bis Bd Arago, 75014 Paris, France</aff>
<aff id="a6"><label>6</label>CEA, DSM, DAPNIA, Centre d'Études de Saclay, 91191 Gif-sur-Yvette Cedex, France</aff>
<aff id="a7"><label>7</label>South African Astronomical Observatory, PO Box 9, 7935 Observatory, South Africa</aff>
<aff id="a8"><label>8</label>Perth Observatory, Walnut Road, Bickley, Perth 6076, Australia</aff>
<aff id="a9"><label>9</label>University of Notre Dame, Department of Physics, 225 Nieuwland Science Hall, Notre Dame, IN 46556, USA</aff>
<aff id="a10"><label>10</label>European Southern Observatory, Casilla 19001, Vitacura 19, Santiago, Chile</aff>
<aff id="a11"><label>11</label>Astronomisches Rechen-Institut (ARI), Zentrum fur Astronomie (ZAH), Heidelberg University, Monchhofstrasse 12-14, 69120 Heidelberg, Germany</aff>
<aff id="a12"><label>12</label>Department of Physics, University of Rijeka, Omladinska 14, 51000 Rijeka, Croatia</aff>
<aff id="a13"><label>13</label>Technical University of Vienna, Department of Computing, Wiedner Hauptstrasse 10, Vienna, Austria</aff>
<aff id="a14"><label>14</label>University of Tasmania, School of Maths and Physics, Private bag 37, GPO Hobart, Tasmania 7001, Australia</aff>
<aff id="a15"><label>15</label>NASA Exoplanet Science Institute, Caltech, MS 100-22, 770 South Wilson Avenue Pasadena, CA 91125, USA</aff>
<aff id="a16"><label>16</label>Space Telescope Science Institute, 3700 San Martin Drive, Baltimore, MD 21218, USA</aff>
<aff id="a17"><label>17</label>Las Cumbres Observatory Global Telescope Network, 6740 Cortona Drive, Suite 102, Goleta, CA 93117, USA</aff>
<aff id="a18"><label>18</label>Astrophysics Research Institute, Liverpool John Moores University, Liverpool CH41 1LD</aff>
<author-notes>
<corresp id="c1"><label>*</label>E-mail: <email>michael.albrow@canterbury.ac.nz</email></corresp>
<fn id="fn3">
<label>†</label><p><ext-link ext-link-type="uri" xlink:href="http://www.beatricetinsleyinstitute.org/">http://www.beatricetinsleyinstitute.org/</ext-link></p>
</fn>
</author-notes>
<pub-date pub-type="ppub">
<day>21</day>
<month>08</month>
<year>2009</year>
</pub-date>
<pub-date pub-type="epub">
<day>11</day>
<month>08</month>
<year>2009</year>
</pub-date>
<volume>397</volume>
<issue>4</issue>
<fpage>2099</fpage>
<lpage>2105</lpage>
<history>
<date date-type="accepted"><day>18</day><month>5</month><year>2009</year></date>
<date date-type="received"><day>17</day><month>5</month><year>2009</year></date>
</history>
<copyright-statement>© 2009 The Authors. Journal compilation © 2009 RAS</copyright-statement>
<copyright-year>2009</copyright-year>
<abstract>
<p>The numerical kernel approach to difference imaging has been implemented and applied to gravitational microlensing events observed by the PLANET collaboration. The effect of an error in the source-star coordinates is explored and a new algorithm is presented for determining the precise coordinates of the microlens in blended events, essential for accurate photometry of difference images. It is shown how the photometric reference flux need not be measured directly from the reference image but can be obtained from measurements of the difference images combined with the knowledge of the statistical flux uncertainties. The improved performance of the new algorithm, relative to <sc>isis2</sc>, is demonstrated.</p>
</abstract>
<kwd-group>
<kwd>methods: statistical</kwd>
<kwd>techniques: image processing</kwd>
<kwd>techniques: photometric</kwd>
</kwd-group>
</article-meta>
</front>
<body>
<sec id="ss1">
<title>1 INTRODUCTION</title>
<p>Over the last 15 years, gravitational microlensing (<xref ref-type="bibr" rid="b20">Einstein 1936</xref>) has been observed routinely and used in the study of dark baryonic matter (<xref ref-type="bibr" rid="b10">Alcock et al. 1993</xref>; <xref ref-type="bibr" rid="b12">Aubourg et al. 1993</xref>) and stellar atmospheres (<xref ref-type="bibr" rid="b4">Albrow et al. 1999</xref>, <xref ref-type="bibr" rid="b6">2001a</xref>,<xref ref-type="bibr" rid="b7">b</xref>; <xref ref-type="bibr" rid="b21">Fields et al. 2003</xref>; <xref ref-type="bibr" rid="b18">Cassan et al. 2004</xref>), and in the search for extrasolar planets (<xref ref-type="bibr" rid="b5">Albrow et al. 2000</xref>, <xref ref-type="bibr" rid="b8">2001c</xref>; <xref ref-type="bibr" rid="b22">Gaudi et al. 2002</xref>; <xref ref-type="bibr" rid="b16">Bond et al. 2004</xref>; <xref ref-type="bibr" rid="b29">Udalski et al. 2005</xref>; <xref ref-type="bibr" rid="b13">Beaulieu et al. 2006</xref>; <xref ref-type="bibr" rid="b19">Dong et al. 2008</xref>; <xref ref-type="bibr" rid="b23">Gaudi et al. 2008</xref>). The PLANET collaboration (<xref ref-type="bibr" rid="b3">Albrow et al. 1998</xref>) operates a number of 1-m class telescopes distributed around the southern hemisphere and performs round-the-clock CCD photometry of microlensing events that have been discovered and alerted in real time by the OGLE (<xref ref-type="bibr" rid="b28">Udalski et al. 1994</xref>; <xref ref-type="bibr" rid="b27">Udalski 2003</xref>) and MOA (<xref ref-type="bibr" rid="b15">Bond et al. 2002</xref>) microlensing surveys.</p>
<p>In this paper, we discuss recent advances in the PLANET difference imaging reduction pipeline, focusing on several subtleties inherent in the reduction of blended microlensing events. In what follows, we use the convention
<disp-formula id="m1">
<label>1</label>
<graphic xlink:href="mnras0397-2099-m1.gif"></graphic>
</disp-formula>
for difference image <italic>D</italic>, reference image <italic>R</italic>, target image <italic>T</italic> and convolution kernel <italic>K</italic>. That is, a difference image is defined as the convolved reference minus the target, normalized so that it is on the effective exposure scale of the reference.</p>
<p>We illustrate the methods using a sample data set of images of microlensing event OGLE 2008-BLG-229 that were taken using the Elizabeth 1.0-m telescope at the South African Astronomical Observatory (SAAO) during PLANET operations in 2008. The microlensing event was alerted by OGLE on 2008 May 3 and initially was predicted to have low magnification. Subsequent observations revealed a blended moderate magnification event, peaking with magnification <italic>A</italic><sub>0</sub>= 7.24 on 2008 July 18. OGLE data and parameters for the event can be obtained from the OGLE Early Warning System website.<xref ref-type="fn" rid="fn1">1</xref> The SAAO observations consist of 84 images, spanning the time from 7 days before maximum until 23 days after maximum.</p>
</sec>
<sec id="ss2">
<title>2 THE REDUCTION PIPELINE</title>
<sec id="ss2-1">
<title>2.1 Introduction</title>
<p>In our first years of operation, PLANET photometry was performed both in real time at the telescopes and offline using the <sc>dophot</sc> point spread function (PSF)-fitting code (<xref ref-type="bibr" rid="b26">Schechter, Mateo & Saha 1993</xref>) under a reduction pipeline written mainly by JPB. Following the development of the <sc>isis</sc> code (<xref ref-type="bibr" rid="b2">Alard & Lupton 1998</xref>), we adopted the difference imaging method for obtaining our best photometry offline, while still employing the <sc>dophot</sc> pipeline at the telescope sites. In the 2006 season, we began using a difference imaging photometric pipeline at the telescope sites and for our final offline photometry. The offline version, known as <sc>pysis2</sc>, was developed by MDA, and is based on the <sc>isis2</sc> code of <xref ref-type="bibr" rid="b1">Alard (2000)</xref>. An adaptation of <sc>pysis</sc> by CC, known as <sc>wisis</sc>, is used for most of the real-time at-telescope reductions, while the <sc>pysis2</sc> code is used at the Perth Observatory. The pipelines allow single images to be reduced immediately after observation using an existing reference template. As better quality images are acquired, the reference template can be updated and the previously observed images rereduced.</p>
</sec>
<sec id="ss2-2">
<title>2.2 Image registration</title>
<p>The <sc>isis</sc> code requires that all images be fully registered to an astrometric reference. Bright stars are located on all frames and cross-correlation of their positions followed by an iterative rejection scheme is used to define an astrometric transformation for each target image.</p>
<p>An innovation introduced to our offline pipeline in 2007 was the removal of the requirement to fully register images. Instead, we register the images only by a shift in X and Y to the nearest pixel, thus avoiding the need for interpolation and resampling. Resampling is generally undesirable since it introduces correlations between adjacent pixels, meaning that their flux uncertainties are no longer described by Poisson statistics. In the case of images that are close to or below critical spatial sampling, resampling introduces an artefact where stellar PSFs are not constant or slowly varying across an image, but depend on the subpixel location of their centroids. Such images usually do not subtract cleanly. Integer-pixel registration was handled in our modified version of <sc>isis</sc> by offsetting the kernel centroid by the subpixel registration residual. We note that this approach is somewhat less flexible than the standard <sc>isis</sc> code, in that it cannot work with sets of images with rotations relative to each other.</p>
</sec>
<sec id="ss2-3">
<title>2.3 Difference imaging with a numerical kernel</title>
<p>In 2008, we have developed a new version of the code, <sc>pysis3</sc>, that is no longer based on <sc>isis</sc> image subtraction. Instead, for the difference-imaging step, we have implemented the algorithm of <xref ref-type="bibr" rid="b17">Bramich (2008)</xref>. In this method, the kernel is represented as a numerical pixel array, rather than the decomposition of Gaussians multiplied by polynomials used in <sc>isis</sc>. The numerical kernel is able to accommodate images with irregular PSFs, for instance trailed images, that <sc>isis</sc> cannot cope with. An implicit feature of the method is that complete registration is not required and the kernel naturally incorporates subpixel offsets. Image registration in <sc>pysis3</sc> is hence restricted to integer pixel shifts. <xref ref-type="bibr" rid="b17">Bramich (2008)</xref> shows examples of how the new algorithm outperforms <sc>isis</sc>. DMB's code has been used successfully to discover new variable stars in the globular cluster NGC 6366 (<xref ref-type="bibr" rid="b11">Arellano Ferro et al. 2008</xref>).</p>
<p>Our implementation has been used for the analysis of several microlensing events, appearing in forthcoming papers on MOA 2007-BLG-197 (Cassan et al., in preparation), OGLE 2004-BLG-482 (Zub et al., in preparation) and OGLE 2007-BLG-472 (<xref ref-type="bibr" rid="b25">Kains et al. 2008</xref>), and for a transit search (Miller & Albrow, in preparation). There are several specific details of our implementation that we note here.</p>
<p>First, the algorithm is generally more computationally intensive than <sc>isis</sc>, and the computation time scales strongly with the number of pixels required for the kernel array. For microlensing events, we generally reduce only a subsection of the images, typically 250 × 250 pixels centred on the microlens. Our plate scales are typically around 0.3 arcsec pixel<sup>−1</sup>.</p>
<p>Secondly, results depend on the size of the pixel array chosen for the kernel. The kernel needs to be large enough to encapsulate the transformation between reference and target, but not so large that it introduces noise into the convolution. We have found the best results by employing a circular kernel with a radius (in pixels) given by
<disp-formula id="m2">
<label>2</label>
<graphic xlink:href="mnras0397-2099-m2.gif"></graphic>
</disp-formula>
where FWHM<sub>target</sub> and FWHM<sub>reference</sub> are the full width at half-maximum (in pixel units) of the microlens on the target and reference images. Regions of the kernel that are located more that 7 pixels from its centre are represented by 3 × 3 binned pixels in order to reduce noise. The values for the binned kernel pixels are computed from the equations in <xref ref-type="bibr" rid="b17">Bramich (2008)</xref>, but using a 3 × 3 boxcar-smoothed version of the reference image.</p>
<p>Thirdly, to prevent saturated stars that have irregular PSFs from entering the kernel determination, we mask a circular area of radius 15 pixels around all pixels that are saturated in either the reference or target image as well as masking the microlens itself. The default behaviour is to use all the remaining image pixels to determine the kernel using the <xref ref-type="bibr" rid="b17">Bramich (2008)</xref> algorithm. In cases where images are contaminated by artefacts, such as diffraction spikes, that are not easily masked, we have found that using ‘stamps’ around bright unsaturated stars rather than the entire unmasked image renders a kernel that is less prone to contamination.</p>
<p>Fourthly, the photometric scaling factor,
<disp-formula id="m3">
<label>3</label>
<graphic xlink:href="mnras0397-2099-m3.gif"></graphic>
</disp-formula>
where <italic>K</italic> is the convolution kernel, represents the relative difference in <italic>effective</italic> exposure time between the reference and target images, i.e. it accounts for differences in both exposure time and atmospheric transparency. For each target, if <italic>s</italic> is significantly different from the ratio of true exposure times, this usually indicates a poorly subtracted target or one affected by cloud.</p>
<p>Fifthly, for images that have poor spatial sampling – either close to or even below critical sampling – we use the following technique. The registered images are oversampled by a factor of 2 in each direction using cubic O-MOMS interpolation (<xref ref-type="bibr" rid="b14">Blu, Thevenaz & Unser 2001</xref>). This type of resampling does not transfer flux across original-pixel boundaries. A stack of typically 10 of the best of these images are then mapped on to the best-seeing image and combined to make a reference image. Since our individual images usually have random subpixel dithers, this process generally results in an oversampled reference so long as the initial undersampling is not too severe. This approach is similar to that employed in R. Gilliland's code for difference imaging of undersampled <italic>HST</italic> WFPC2 images (<xref ref-type="bibr" rid="b24">Gilliland et al. 2000</xref>; <xref ref-type="bibr" rid="b9">Albrow et al. 2001d</xref>). We note that this approach is still under development and is not used for the sample data set of images for OGLE 2008-BLG-229 in this paper, which are not undersampled.</p>
</sec>
<sec id="ss2-4">
<title>2.4 Photometry</title>
<p>To extract photometric measurements from our difference images, we first use the <sc>bphot</sc> program from <sc>isis</sc> to compute the PSF of the reference image. This PSF is then convolved with the previously computed kernel to produce a PSF for each target image. The PSF is then normalized and resampled at the subpixel lens coordinates using cubic O-MOMS interpolation. Any residual background is removed from the difference image using a low-order polynomial model. Finally, the PSF is fitted to the difference image using optimal extraction, i.e. each pixel weighted by the inverse of its flux variance.</p>
<sec id="ss2-4-1">
<title>2.4.1 Flux errors from imprecise coordinates</title>
<p>All of the microlensing events towards the Galactic bulge are blended to some extent in regular ground-based imaging, i.e. the PSF of the microlensed source star overlaps with nearby stars. One implication of this is that the coordinates of the true source star are often displaced by perhaps several tenths of an arcsec from the centroid of the PSF.</p>
<p>When performing photometry on a set of difference images by PSF fitting, small errors in target location lead to systematic underestimates of the flux. In <xref ref-type="fig" rid="f1">Fig. 1</xref>, we show how the flux measurement depends on coordinate displacement, Δ<italic>R</italic> (measured in units of FWHM), under a Gaussian PSF assumption. The two limiting cases, shown as lines, are (i) optimal PSF fitting in the zero-background limit and (ii) unweighted PSF fitting, essentially the background-limited case. In both limits, the flux error scales as the square of the ratio of the coordinate error to the FWHM of the PSF. This means that a flux error due to an incorrectly positioned PSF is more serious for images with better seeing and that such a coordinate error introduces scatter into a light curve derived from a set of variable-seeing (or even variable-background) images.</p>
<fig position="float" id="f1">
<label>Figure 1</label>
<caption>
<p>Flux determined at offset coordinates relative to the flux at the correct coordinates as a function of coordinate offset, Δ<italic>R</italic> in units of PSF FWHM. Solid line for unweighted PSF fitting, dashed line for optimal PSF fitting with zero background, both under the assumption of a Gaussian PSF. The three distinct groups of data points are computed from the SAAO difference images for OGLE 2008-BLG-229, with the PSF shifted in <italic>x</italic> by 0.2 (green), 0.5 (red) and 1.0 pixels (blue) (0.06, 0.16, 0.31 arcsec, respectively) from their correct value.</p>
</caption>
<graphic xlink:href="mnras0397-2099-f1.gif"></graphic>
</fig>
<p>We can demonstrate this effect using our test case of SAAO OGLE 2008-BLG-229. We have created difference images using as a reference template a single good-seeing image, number 18, acquired close to the peak of the microlensing event. We have performed optimal photometry on the set of difference images at the correct coordinates (i.e. zero-offset) and again with the coordinates shifted in X by 0.2, 0.5 and 1.0 pixels (0.06, 0.16, 0.31 arcsec) from their correct position. The three displayed groups of data points in <xref ref-type="fig" rid="f1">Fig. 1</xref> show the flux measured at the offset coordinates relative to the flux at the correct position for the three sets of displaced-coordinate measurements. The figure shows that, as predicted, the measured difference flux decreases and its dispersion increases with coordinate offset.</p>
<p>It is important to note that, for difference-image photometry, the above effect applies to difference fluxes, Δ<italic>F<sub>i</sub></italic>. The total flux is given by <italic>F<sub>i</sub></italic>=<italic>F</italic><sub>0</sub>−Δ<italic>F<sub>i</sub></italic>, where <italic>F</italic><sub>0</sub> is the flux on the reference image, <italic>R</italic> in <xref ref-type="disp-formula" rid="m1">equation (1)</xref>. This means that the place in the light curve where coordinate errors are manifested most strongly depends on the choice of reference image. The flux error is largest for regions of the light curve where the magnification is most different from that of the reference image. Consequently, such errors can be minimized for a given part of a light curve (for instance, some part of a light curve suspected to display an anomaly) by choosing a reference image where the source star has a similar magnification.</p>
</sec>
<sec id="ss2-4-2">
<title>2.4.2 Precise coordinates for blended events</title>
<p>For high-magnification events, the true source location may be discerned from images taken near peak magnification, when the flux from the true microlensed source star dominates that from nearby blended stars. For data sets comprised of images that have precise registration, a sum of the absolute values of the difference images can be used successfully to refine the coordinates from their initial estimate, even for relatively low-magnification events. This method was used in <sc>pysis2</sc>.</p>
<p>In our current circumstance, we have sets of images that are registered only to the nearest pixel. The method of stacking difference images could, in principle, be applied to these data, provided each of the difference images is first shifted by the subpixel registration residual. However, such a shift requires accurate knowledge of the subpixel residuals and involves interpolation and resampling, a process that is inaccurate for images with near- or below-critical spatial sampling.</p>
<p>A better way, that retains the original sampling, is to use the residuals from PSF fits to the difference images. In <xref ref-type="sec" rid="ss6">Appendix A</xref>, we introduce a new algorithm to refine the source-star coordinates by minimizing these residuals over all images. In our photometric code, the algorithm operates as an integral part of the measurement process.</p>
<p>In <xref ref-type="fig" rid="f2">Fig. 2</xref>, we show direct, difference and residual images for two sample observations, numbered 78 (good seeing) and 83 (poor seeing) in the SAAO data set for OGLE 2008-BLG-229. Both images were taken during the final days of data, when the source was at a magnification, <italic>A</italic>≈ 2.5. The reference image was again image number 18, taken near peak magnification, <italic>A</italic>≈ 7.</p>
<fig position="float" id="f2">
<label>Figure 2</label>
<caption>
<p>Sample images 78 (left; good seeing) and 83 (right; poor seeing) from the SAAO observations of OGLE 2008-BLG-229. Top row: direct images registered to the nearest pixel, linear grey-scale encompassing 95 per cent of pixel values. Second row: difference images, linear grey-scale range −400 to +400. Third row: difference images after subtraction of PSF fitted at correct coordinates. Bottom row: difference images after subtraction of PSF fitted at coordinates shifted by +1 pixel in X.</p>
</caption>
<graphic xlink:href="mnras0397-2099-f2.gif"></graphic>
</fig>
<p>Our coordinate algorithm resulted in a change of 0.54 pixels (0.17 arcsec) in the location of the target star relative to the position found from our best-seeing image (which was adopted as the astrometric reference). Using the correct coordinates, the residual images (difference images after subtraction of the fitted PSF) are very clean.</p>
<p>The effect of a coordinate offset of 1 pixel (0.3 arcsec) is to produce a residual image with large positive and negative flux features remaining. The effect of such an offset on the light curve can be tested. We compare difference-flux light curves in a blend-free way by mapping them to a point-source point-mass lens model as follows. At time <italic>t<sub>i</sub></italic>, the unblended flux from the source star is given by
<disp-formula id="m4">
<label>4</label>
<graphic xlink:href="mnras0397-2099-m4.gif"></graphic>
</disp-formula>
where <italic>A<sub>i</sub></italic> is the magnification that we constrain to be defined by the OGLE geometric parameters for the event (<italic>u</italic><sub>0</sub>= 0.139, <italic>t</italic><sub>E</sub>= 53.994 d, <italic>t</italic><sub>0</sub>= JD2454665.780), <italic>F</italic><sub>0</sub> is the unblended source flux on the reference image and <italic>F</italic><sub>base</sub> is the unblended baseline source flux. For each light curve, we solve for <italic>F</italic><sub>0</sub> and <italic>F</italic><sub>base</sub> by minimizing
<disp-formula id="m5">
<label>5</label>
<graphic xlink:href="mnras0397-2099-m5.gif"></graphic>
</disp-formula></p>
<p>Light curves for the whole data set are shown in <xref ref-type="fig" rid="f3">Fig. 3</xref> for the correct coordinates and for those with a 1-pixel offset. An increased scatter is visible in the light curve corresponding to the offset coordinates. In the same figure, we also show the best light curves we have derived using a template created from the 10 best-seeing images using <sc>pysis3</sc> and <sc>isis2</sc>. <xref ref-type="fig" rid="f4">Fig. 4</xref> shows the corresponding residuals. The effect of a coordinate error can be seen through comparison of the upper two panels, particularly during the last 10 days' data points when the magnification is most different from that of the reference image. The superior performance of the new numerical-kernel algorithm can be discerned through comparison of the two lower panels, where the <sc>isis2</sc> light curve residuals (panel d) have a 53 per cent greater rms scatter than the <sc>pysis3</sc> residuals (panel c). We note that there is an apparent systematic residual in all the displayed light curves, where the earliest data points lie below the magnification curve. This is likely due to the fact that we have constrained, rather than fitted, the underlying geometric model.</p>
<fig position="float" id="f3">
<label>Figure 3</label>
<caption>
<p>Light curves for OGLE 2008-BLG-229: (a) using single template image from near peak, (b) as for (a) but with a 1 pixel coordinate offset, (c) combination of 10 images for reference template, (d) best light curve obtained using <sc>isis2</sc>.</p>
</caption>
<graphic xlink:href="mnras0397-2099-f3.gif"></graphic>
</fig>
<fig position="float" id="f4">
<label>Figure 4</label>
<caption>
<p>Light curve residuals for OGLE 2008-BLG-229 corresponding to <xref ref-type="fig" rid="f3">Fig. 3</xref> after subtraction of a best-fitting PSPL model with the geometric parameters set to be those found from OGLE. Error bars are suppressed for clarity.</p>
</caption>
<graphic xlink:href="mnras0397-2099-f4.gif"></graphic>
</fig>
</sec>
<sec id="ss2-4-3">
<title>2.4.3 Reference flux</title>
<p>The output of our difference-image photometry is the difference in flux, Δ<italic>F<sub>i</sub></italic>, between each target image, <italic>i</italic>, and a photometric reference image. The reference image may be a single observation, as in the preceding sections, but more commonly is created from a combination of images with the best seeing and lowest sky background. In order to interpret our observations, we require the flux, <italic>F<sub>i</sub></italic>=<italic>F</italic><sub>0</sub>−Δ<italic>F<sub>i</sub></italic>, where <italic>F</italic><sub>0</sub> is the flux on the reference image. In a model-dependent sense, a deblended <italic>F</italic><sub>0</sub> can be derived as a fitting parameter as done above. More usually, particularly during the time when observations are being acquired, <italic>F</italic><sub>0</sub> is measured from a PSF-fit at the lens coordinates on the reference image. Often this estimation is in error due to the crowded nature of the Galactic bulge fields in which we observe.</p>
<p>An incorrect value for <italic>F</italic><sub>0</sub> can, to some extent, be compensated for in the microlens blend fraction,
<disp-formula id="m6">
<label>6</label>
<graphic xlink:href="mnras0397-2099-m6.gif"></graphic>
</disp-formula>
where for magnification <italic>A</italic>, the baseline flux (<italic>F<sub>s</sub></italic>+<italic>F<sub>b</sub></italic>) increases to
<disp-formula id="m7">
<label>7</label>
<graphic xlink:href="mnras0397-2099-m7.gif"></graphic>
</disp-formula> and the blend fraction is derived from light curve fitting. However, blending parameters so derived are inconsistent between different data sets for the same event, may take on non-physical values, and certainly no longer have the correct physical interpretation as the fraction of light contributing to the stellar PSF at baseline from the microlensing source star.</p>
<p>A more successful approach that we have developed is to choose <italic>F</italic><sub>0</sub> so that the photometric uncertainties in the set of Δ<italic>F<sub>i</sub></italic> measurements are consistent with Poisson noise for fluxes <italic>F<sub>i</sub></italic>=<italic>F</italic><sub>0</sub>−Δ<italic>F<sub>i</sub></italic>. The algorithm for this determination, referred to as the <italic>Poisson reference flux</italic>, is detailed in <xref ref-type="sec" rid="ss7">Appendix B</xref>. The method uses information from all suitable images and the variance in the Poisson reference flux scales roughly with the inverse of the number of images. This variance is generally smaller than the variance for a direct flux measurement from the reference image, which scales with the inverse of the number of individual images incorporated into the reference. For our sample light curve, the reference flux is measured directly as <italic>F</italic><sub>0</sub>= 101, 820 ± 90 ADU (likely to include extra blended light), while the Poisson method yields <italic>F</italic><sub>0</sub>= 93, 570 ± 40 ADU.</p>
</sec>
</sec>
</sec>
<sec id="ss3">
<title>3 SUMMARY</title>
<p>Difference imaging has proved to be a powerful technique in the measurement of gravitational microlensing flux variations and for variable stars and transiting extrasolar planets. The numerical kernel method introduced by <xref ref-type="bibr" rid="b17">Bramich (2008)</xref> represents a significant advance over the analytic kernel of <xref ref-type="bibr" rid="b2">Alard & Lupton (1998)</xref>.</p>
<p>In this paper, we have shown that, for blended microlensing events, precise determination of the coordinates of the microlens is necessary to obtain accurate photometry. We have presented a new algorithm, based on photometric residuals, to measure such coordinates to high precision.</p>
<p>Additionally, we have introduced a new method to measure the reference flux in a manner that does not depend on an (often inaccurate) analysis of the photometric reference image. The <italic>Poisson reference flux</italic> method produces a more accurate and precise determination of the unmagnified source flux than a direct measurement from the reference image.</p>
<p>A comparison has been made between photometry from our new code and photometry using <sc>isis</sc> for SAAO images of microlensing event OGLE 2008-BLG-229. The new code produces measurements that display significantly less scatter about a point-source point-mass-lens light curve based on the OGLE-determined geometric parameters for the event.</p>
</sec>
</body>
<back>
<fn-group>
<fn id="fn1">
<label>1</label><p><ext-link ext-link-type="uri" xlink:href="http://ogle.astrouw.edu.pl/ogle3/ews/ews.html">http://ogle.astrouw.edu.pl/ogle3/ews/ews.html</ext-link></p>
</fn>
<fn id="fn2">
<label>2</label><p>Note that Δ<italic>x</italic> > 0 shifts the PSF peak to smaller <italic>x</italic>, and similarly for <italic>y</italic>. We adopt this sign convention to simplify the equations.</p>
</fn>
</fn-group>
<ack>
<p>This work was supported by the Marsden Fund of New Zealand under contract UOC302 and by ANR grant HOLMES during MDA's visit to the Observatoire Midi-Pyrénées, Toulouse, in 2008 May–June. We thank the referee, Scott Gaudi, for his thorough report.</p>
</ack>
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<app-group id="s3">
<title>Appendices</title>
<app id="ss6">
<title>APPENDIX A: REFINING THE LENS POSITION</title>
<sec id="ss6-1">
<title>A1 Target position from a single difference image</title>
<p>We wish to find subpixel offsets, Δ<italic>x</italic> and Δ<italic>y</italic>, that create the best match of a PSF model <italic><bold>P</bold></italic> to a difference image <italic><bold>D</bold></italic>. Expand the PSF model to first order in Δ<italic>x</italic> and Δ<italic>y</italic>,
<disp-formula id="m8">
<label>(A1)</label>
<graphic xlink:href="mnras0397-2099-m8.gif"></graphic>
</disp-formula>
and thereby write the residual image<xref ref-type="fn" rid="fn2">2</xref> as
<disp-formula id="m9">
<label>(A2)</label>
<graphic xlink:href="mnras0397-2099-m9.gif"></graphic>
</disp-formula>
where <bold><italic>P</italic></bold><sub>0</sub> is the unshifted PSF, Δ<italic>F</italic> is the difference flux, and the <italic>x</italic> and <italic>y</italic> gradients of the unshifted PSF are
<disp-formula id="m10">
<label>(A3)</label>
<graphic xlink:href="mnras0397-2099-m10.gif"></graphic>
</disp-formula>
The PSF is normalized to
<disp-formula id="m11">
<label>(A4)</label>
<graphic xlink:href="mnras0397-2099-m11.gif"></graphic>
</disp-formula>
when summed over pixels <italic>i</italic>. With σ<sup>2</sup><sub><italic>i</italic></sub>= Var[<italic>D<sub>i</sub></italic>] the variance of the data <italic>D<sub>i</sub></italic> in pixel <italic>i</italic>, the χ<sup>2</sup> statistic measuring the ‘badness-of-fit’ is
<disp-formula id="m12">
<label>(A5)</label>
<graphic xlink:href="mnras0397-2099-m12.gif"></graphic>
</disp-formula>
Here we adopt the convenient notation
<disp-formula id="m13">
<label>(A6)</label>
<graphic xlink:href="mnras0397-2099-m13.gif"></graphic>
</disp-formula>
for the inverse-variance weighted ‘dot product’ of ‘image vectors’<italic><bold>A</bold></italic> and <italic><bold>B</bold></italic>, and note that χ<sup>2</sup> is the squared norm of the residual image <italic><bold>R</bold></italic>.</p>
<p>Starting with Δ<italic>F</italic>=Δ<italic>x</italic>=Δ<italic>y</italic>= 0, we calculate the difference flux Δ<italic>F</italic>, for fixed Δ<italic>x</italic> and Δ<italic>y</italic>, by optimally scaling the shifted PSF <italic><bold>P</bold></italic> to fit the difference image <italic><bold>D</bold></italic>, giving
<disp-formula id="m14">
<label>(A7)</label>
<graphic xlink:href="mnras0397-2099-m14.gif"></graphic>
</disp-formula>
and the corresponding variance
<disp-formula id="m15">
<label>(A8)</label>
<graphic xlink:href="mnras0397-2099-m15.gif"></graphic>
</disp-formula>
Next, we update the image position, for fixed Δ<italic>F</italic>, by scaling the PSF gradient images to fit the residuals,
<disp-formula id="m16">
<label>(A9)</label>
<graphic xlink:href="mnras0397-2099-m16.gif"></graphic>
</disp-formula>
<disp-formula id="m17">
<label>(A10)</label>
<graphic xlink:href="mnras0397-2099-m17.gif"></graphic>
</disp-formula>
with variances
<disp-formula id="m18">
<label>(A11)</label>
<graphic xlink:href="mnras0397-2099-m18.gif"></graphic>
</disp-formula>
<disp-formula id="m19">
<label>(A12)</label>
<graphic xlink:href="mnras0397-2099-m19.gif"></graphic>
</disp-formula></p>
<p>The above results minimize χ<sup>2</sup>= |<bold><italic>R</italic></bold>|<sup>2</sup> if Δ<italic>F</italic>, Δ<italic>x</italic> and Δ<italic>y</italic> are independent, and if σ<sub><italic>i</italic></sub> are fixed. As these assumptions are only approximately true, iteration is required. We find that the iteration is faster and more stable if we take account of <italic><bold>R</bold></italic> being a linear function of Δ<italic>x</italic> and Δ<italic>y</italic>. We then have two coupled equations,
<disp-formula id="m20">
<label>(A13)</label>
<graphic xlink:href="mnras0397-2099-m20.gif"></graphic>
</disp-formula>
<disp-formula id="m21">
<label>(A14)</label>
<graphic xlink:href="mnras0397-2099-m21.gif"></graphic>
</disp-formula>
Write these in matrix form as
<disp-formula id="m22">
<label>(A15)</label>
<graphic xlink:href="mnras0397-2099-m22.gif"></graphic>
</disp-formula>
with the Hessian matrix
<disp-formula id="m23">
<label>(A16)</label>
<graphic xlink:href="mnras0397-2099-m23.gif"></graphic>
</disp-formula>
The solution is
<disp-formula id="m24">
<label>(A17)</label>
<graphic xlink:href="mnras0397-2099-m24.gif"></graphic>
</disp-formula>
where the inverse of the Hessian matrix is
<disp-formula id="m25">
<label>(A18)</label>
<graphic xlink:href="mnras0397-2099-m25.gif"></graphic>
</disp-formula>
with the Hessian determinant
<disp-formula id="m26">
<label>(A19)</label>
<graphic xlink:href="mnras0397-2099-m26.gif"></graphic>
</disp-formula>
The subpixel shift is then
<disp-formula id="m27">
<label>(A20)</label>
<graphic xlink:href="mnras0397-2099-m27.gif"></graphic>
</disp-formula>
<disp-formula id="m28">
<label>(A21)</label>
<graphic xlink:href="mnras0397-2099-m28.gif"></graphic>
</disp-formula></p>
<p>Since <italic>H</italic><sup>−1</sup> is the parameter covariance matrix, the diagonal elements give the variances
<disp-formula id="m29">
<label>(A22)</label>
<graphic xlink:href="mnras0397-2099-m29.gif"></graphic>
</disp-formula>
<disp-formula id="m30">
<label>(A23)</label>
<graphic xlink:href="mnras0397-2099-m30.gif"></graphic>
</disp-formula>
and the off-diagonal element gives the covariance
<disp-formula id="m31">
<label>(A24)</label>
<graphic xlink:href="mnras0397-2099-m31.gif"></graphic>
</disp-formula></p>
<p>Note that with Δ<italic>F</italic> in the denominator, these expressions become problematic when Δ<italic>F</italic>≈ 0. Such images carry very little information about the target location. Fortunately, when we optimally average over several images, the inverse-variance weights shift the Δ<italic>F</italic> factors to the numerator, so that these images receive low weight.</p>
</sec>
<sec id="ss6-2">
<title>A2 Lens position from many images</title>
<p>In fitting a microlensing data set, we have many difference images <bold><italic>D</italic></bold><sub><italic>j</italic></sub>, and the corresponding PSFs <bold><italic>P</italic></bold><sub><italic>j</italic></sub>. The above analysis provides estimates (with error bars) that we correspondingly label Δ<italic>F<sub>j</sub></italic> for the difference fluxes, Δ<italic>x<sub>j</sub></italic> and Δ<italic>y<sub>j</sub></italic> for the subpixel offsets.</p>
<p>The difference fluxes are different for each image, but the subpixel shift establishing the lens position on the reference image should be the same for all images. The optimal average of the estimates Δ<italic>x<sub>j</sub></italic> from individual images is
<disp-formula id="m32">
<label>(A25)</label>
<graphic xlink:href="mnras0397-2099-m32.gif"></graphic>
</disp-formula>
with inverse-variance weights <italic>w<sub>j</sub></italic>= 1/Var[Δ<italic>x<sub>j</sub></italic>]. Explicit evaluation using <xref ref-type="disp-formula" rid="m27">(A.20)</xref> and <xref ref-type="disp-formula" rid="m29">(A.22)</xref> gives
<disp-formula id="m33">
<label>(A26)</label>
<graphic xlink:href="mnras0397-2099-m33.gif"></graphic>
</disp-formula>
<disp-formula id="m34">
<label>(A27)</label>
<graphic xlink:href="mnras0397-2099-m34.gif"></graphic>
</disp-formula>
The corresponding expressions for Δ<italic>y</italic> and Var [Δ<italic>y</italic>] are found by reversing <italic>x</italic> and <italic>y</italic>. For clarity, we omit the index <italic>j</italic> that labels every term in the sums over images <italic>j</italic>.</p>
<p>Note that the Δ<italic>F</italic> factors appear in the numerator only, so that difference images with Δ<italic>F<sub>j</sub></italic>≈ 0 are included in the sums but with appropriately low weight. In our implementation, the algorithm typically converges to ∼ 10<sup>−3</sup> pixels in ∼4 iterations.</p>
</sec>
</app>
<app id="ss7">
<title>APPENDIX B: COMPUTING THE POISSON REFERENCE FLUX</title>
<p>We retain here the convention (<xref ref-type="disp-formula" rid="m1">equation 1</xref>) where a difference image is on the same effective exposure scale as the reference image and a negative difference flux results when the target star is brighter than it is on the reference. The expected pixel-integrated star flux on a target image is then
<disp-formula id="m35">
<label>(B1)</label>
<graphic xlink:href="mnras0397-2099-m35.gif"></graphic>
</disp-formula>
where <italic>F</italic><sub>0</sub> is the pixel-integrated flux of the lens star in the reference image (ADU), Δ<italic>F</italic> is the pixel-integrated differential flux of the lens star in the difference image (ADU) and <italic>s</italic> is the exposure scale factor between the reference image and the target image (<xref ref-type="disp-formula" rid="m3">equation 3</xref>). Assuming a noiseless reference image, the variance in flux of the lens on a single difference image is given approximately by
<disp-formula id="m36">
<label>(B2)</label>
<graphic xlink:href="mnras0397-2099-m36.gif"></graphic>
</disp-formula>
where σ<sup>2</sup><sub>0</sub> is the readout noise variance (<italic>e</italic><sup>−</sup>/pix)<sup>2</sup>, <italic>g</italic> is the gain (<italic>e</italic><sup>−</sup>/ADU), <italic>F</italic><sub>sky</sub> is the background flux (ADU/pixel) on the target image and <italic>N</italic><sub>pix</sub> is the effective number of pixels in the photometric aperture. The different denominators for each of the three terms in <xref ref-type="disp-formula" rid="m36">equation (B2)</xref> are due to the readout noise being measured in units of electrons, <italic>F</italic><sub>sky</sub> in units of target-frame ADU and <italic>F</italic><sub>0</sub>−Δ<italic>F</italic> in units of reference-frame ADU. An estimate of the reference flux from this single difference image is therefore
<disp-formula id="m37">
<label>(B3)</label>
<graphic xlink:href="mnras0397-2099-m37.gif"></graphic>
</disp-formula>
with variance
<disp-formula id="m38">
<label>(B4)</label>
<graphic xlink:href="mnras0397-2099-m38.gif"></graphic>
</disp-formula>
An optimal estimate for <italic>F</italic><sub>0</sub> is obtained by combining such measurements from all target images,
<disp-formula id="m39">
<label>(B5)</label>
<graphic xlink:href="mnras0397-2099-m39.gif"></graphic>
</disp-formula>
with associated variance,
<disp-formula id="m40">
<label>(B6)</label>
<graphic xlink:href="mnras0397-2099-m40.gif"></graphic>
</disp-formula>
For aperture photometry measurements, <italic>N</italic><sub>pix</sub> is the number of pixels in the photometric aperture. In the case of optimal PSF-fitting photometry, <italic>N</italic><sub>pix</sub> is the effective number of sky pixels, which we define to be equal to the variance in the flux measurement that is due to the background divided by the background variance per pixel. To estimate <italic>N</italic><sub>pix</sub>, consider the background-limited case, where the noise variance is the same on each pixel, <italic>V</italic><sub>0</sub>, and is the dominant contributor to the variance in the flux measurement. If our optimal extraction is confined to some aperture, then the variance in the measured flux is
<disp-formula id="m41">
<label>(B7)</label>
<graphic xlink:href="mnras0397-2099-m41.gif"></graphic>
</disp-formula>
where <italic>P</italic>(<italic>x</italic>, <italic>y</italic>) is the PSF and hence
<disp-formula id="m42">
<label>(B8)</label>
<graphic xlink:href="mnras0397-2099-m42.gif"></graphic>
</disp-formula>
Under a Gaussian PSF,
<disp-formula id="m43">
<label>(B9)</label>
<graphic xlink:href="mnras0397-2099-m43.gif"></graphic>
</disp-formula>
where Δ is the FWHM,
<disp-formula id="m44">
<label>(B10)</label>
<graphic xlink:href="mnras0397-2099-m44.gif"></graphic>
</disp-formula>
equivalent to an aperture of radius 2σ. For a Gaussian truncated at <italic>r</italic>=<italic>R</italic>, this equates to
<disp-formula id="m45">
<label>(B11)</label>
<graphic xlink:href="mnras0397-2099-m45.gif"></graphic>
</disp-formula></p>
</app>
</app-group>
</back>
</article></istex:document></istex:metadataXml><mods version="3.6"><titleInfo><title>Difference imaging photometry of blended gravitational microlensing events with a numerical kernel</title></titleInfo><titleInfo type="alternative" contentType="CDATA"><title>Difference imaging photometry of blended gravitational microlensing events with a numerical kernel</title></titleInfo><name type="personal"><namePart type="given">M. D.</namePart><namePart type="family">Albrow</namePart><affiliation>Beatrice Tinsley Institute, Department of Physics and Astronomy, University of Canterbury, Private Bag 4800, Christchurch, New Zealand</affiliation><affiliation>E-mail: michael.albrow@canterbury.ac.nz</affiliation><affiliation></affiliation><affiliation>E-mail: michael.albrow@canterbury.ac.nz</affiliation><role><roleTerm type="text">author</roleTerm></role></name><name type="personal"><namePart type="given">K.</namePart><namePart type="family">Horne</namePart><affiliation>SUPA, School of Physics and Astronomy, University of St. Andrews, North Haugh, St Andrews, KY16 9SS</affiliation><role><roleTerm type="text">author</roleTerm></role></name><name type="personal"><namePart type="given">D. M.</namePart><namePart type="family">Bramich</namePart><affiliation>Isaac Newton Group of Telescopes, Apartado de Correos 321, E-38700 Santa Cruz de la Palma, Canary Islands, Spain</affiliation><role><roleTerm type="text">author</roleTerm></role></name><name type="personal"><namePart type="given">P.</namePart><namePart type="family">Fouqué</namePart><affiliation>LATT, Universite de Toulouse, CNRS, 14 avenue Edouard Belin, F-31400 Toulouse, France</affiliation><role><roleTerm type="text">author</roleTerm></role></name><name type="personal"><namePart type="given">V. R.</namePart><namePart type="family">Miller</namePart><affiliation>Beatrice Tinsley Institute, Department of Physics and Astronomy, University of Canterbury, Private Bag 4800, Christchurch, New Zealand</affiliation><role><roleTerm type="text">author</roleTerm></role></name><name type="personal"><namePart type="given">J.-P.</namePart><namePart type="family">Beaulieu</namePart><affiliation>Institut dAstrophysique de Paris, CNRS - UMR 7095, Universite Pierre & Marie Curie, 98bis Bd Arago, 75014 Paris, France</affiliation><role><roleTerm type="text">author</roleTerm></role></name><name type="personal"><namePart type="given">C.</namePart><namePart type="family">Coutures</namePart><affiliation>Institut dAstrophysique de Paris, CNRS - UMR 7095, Universite Pierre & Marie Curie, 98bis Bd Arago, 75014 Paris, France</affiliation><affiliation>CEA, DSM, DAPNIA, Centre d'Études de Saclay, 91191 Gif-sur-Yvette Cedex, France</affiliation><role><roleTerm type="text">author</roleTerm></role></name><name type="personal"><namePart type="given">J.</namePart><namePart type="family">Menzies</namePart><affiliation>South African Astronomical Observatory, PO Box 9, 7935 Observatory, South Africa</affiliation><role><roleTerm type="text">author</roleTerm></role></name><name type="personal"><namePart type="given">A.</namePart><namePart type="family">Williams</namePart><affiliation>Perth Observatory, Walnut Road, Bickley, Perth 6076, Australia</affiliation><role><roleTerm type="text">author</roleTerm></role></name><name type="personal"><namePart type="given">V.</namePart><namePart type="family">Batista</namePart><affiliation>Institut dAstrophysique de Paris, CNRS - UMR 7095, Universite Pierre & Marie Curie, 98bis Bd Arago, 75014 Paris, France</affiliation><role><roleTerm type="text">author</roleTerm></role></name><name type="personal"><namePart type="given">D. P.</namePart><namePart type="family">Bennett</namePart><affiliation>University of Notre Dame, Department of Physics, 225 Nieuwland Science Hall, Notre Dame, IN 46556, USA</affiliation><role><roleTerm type="text">author</roleTerm></role></name><name type="personal"><namePart type="given">S.</namePart><namePart type="family">Brillant</namePart><affiliation>European Southern Observatory, Casilla 19001, Vitacura 19, Santiago, Chile</affiliation><role><roleTerm type="text">author</roleTerm></role></name><name type="personal"><namePart type="given">A.</namePart><namePart type="family">Cassan</namePart><affiliation>Astronomisches Rechen-Institut (ARI), Zentrum fur Astronomie (ZAH), Heidelberg University, Monchhofstrasse 12-14, 69120 Heidelberg, Germany</affiliation><role><roleTerm type="text">author</roleTerm></role></name><name type="personal"><namePart type="given">S.</namePart><namePart type="family">Dieters</namePart><affiliation>Institut dAstrophysique de Paris, CNRS - UMR 7095, Universite Pierre & Marie Curie, 98bis Bd Arago, 75014 Paris, France</affiliation><role><roleTerm type="text">author</roleTerm></role></name><name type="personal"><namePart type="given">D. Dominis</namePart><namePart type="family">Prester</namePart><affiliation>Department of Physics, University of Rijeka, Omladinska 14, 51000 Rijeka, Croatia</affiliation><role><roleTerm type="text">author</roleTerm></role></name><name type="personal"><namePart type="given">J.</namePart><namePart type="family">Donatowicz</namePart><affiliation>Technical University of Vienna, Department of Computing, Wiedner Hauptstrasse 10, Vienna, Austria</affiliation><role><roleTerm type="text">author</roleTerm></role></name><name type="personal"><namePart type="given">J.</namePart><namePart type="family">Greenhill</namePart><affiliation>University of Tasmania, School of Maths and Physics, Private bag 37, GPO Hobart, Tasmania 7001, Australia</affiliation><role><roleTerm type="text">author</roleTerm></role></name><name type="personal"><namePart type="given">N.</namePart><namePart type="family">Kains</namePart><affiliation>SUPA, School of Physics and Astronomy, University of St. Andrews, North Haugh, St Andrews, KY16 9SS</affiliation><role><roleTerm type="text">author</roleTerm></role></name><name type="personal"><namePart type="given">S. R.</namePart><namePart type="family">Kane</namePart><affiliation>NASA Exoplanet Science Institute, Caltech, MS 100-22, 770 South Wilson Avenue Pasadena, CA 91125, USA</affiliation><role><roleTerm type="text">author</roleTerm></role></name><name type="personal"><namePart type="given">D.</namePart><namePart type="family">Kubas</namePart><affiliation>European Southern Observatory, Casilla 19001, Vitacura 19, Santiago, Chile</affiliation><role><roleTerm type="text">author</roleTerm></role></name><name type="personal"><namePart type="given">J. B.</namePart><namePart type="family">Marquette</namePart><affiliation>Institut dAstrophysique de Paris, CNRS - UMR 7095, Universite Pierre & Marie Curie, 98bis Bd Arago, 75014 Paris, France</affiliation><role><roleTerm type="text">author</roleTerm></role></name><name type="personal"><namePart type="given">K. R.</namePart><namePart type="family">Pollard</namePart><affiliation>Beatrice Tinsley Institute, Department of Physics and Astronomy, University of Canterbury, Private Bag 4800, Christchurch, New Zealand</affiliation><role><roleTerm type="text">author</roleTerm></role></name><name type="personal"><namePart type="given">K. C.</namePart><namePart type="family">Sahu</namePart><affiliation>Space Telescope Science Institute, 3700 San Martin Drive, Baltimore, MD 21218, USA</affiliation><role><roleTerm type="text">author</roleTerm></role></name><name type="personal"><namePart type="given">Y.</namePart><namePart type="family">Tsapras</namePart><affiliation>Las Cumbres Observatory Global Telescope Network, 6740 Cortona Drive, Suite 102, Goleta, CA 93117, USA</affiliation><affiliation>Astrophysics Research Institute, Liverpool John Moores University, Liverpool CH41 1LD</affiliation><role><roleTerm type="text">author</roleTerm></role></name><name type="personal"><namePart type="given">J.</namePart><namePart type="family">Wambsganss</namePart><affiliation>Astronomisches Rechen-Institut (ARI), Zentrum fur Astronomie (ZAH), Heidelberg University, Monchhofstrasse 12-14, 69120 Heidelberg, Germany</affiliation><role><roleTerm type="text">author</roleTerm></role></name><name type="personal"><namePart type="given">M.</namePart><namePart type="family">Zub</namePart><affiliation>Astronomisches Rechen-Institut (ARI), Zentrum fur Astronomie (ZAH), Heidelberg University, Monchhofstrasse 12-14, 69120 Heidelberg, Germany</affiliation><role><roleTerm type="text">author</roleTerm></role></name><typeOfResource>text</typeOfResource><genre type="research-article" displayLabel="research-article" authority="ISTEX" authorityURI="https://content-type.data.istex.fr" valueURI="https://content-type.data.istex.fr/ark:/67375/XTP-1JC4F85T-7">research-article</genre><originInfo><publisher>Blackwell Publishing Ltd</publisher><place><placeTerm type="text">Oxford, UK</placeTerm></place><dateIssued encoding="w3cdtf">2009-08-21</dateIssued><dateCreated encoding="w3cdtf">2009-08-11</dateCreated><copyrightDate encoding="w3cdtf">2009</copyrightDate></originInfo><abstract>The numerical kernel approach to difference imaging has been implemented and applied to gravitational microlensing events observed by the PLANET collaboration. The effect of an error in the source-star coordinates is explored and a new algorithm is presented for determining the precise coordinates of the microlens in blended events, essential for accurate photometry of difference images. It is shown how the photometric reference flux need not be measured directly from the reference image but can be obtained from measurements of the difference images combined with the knowledge of the statistical flux uncertainties. The improved performance of the new algorithm, relative to isis2, is demonstrated.</abstract><note type="footnotes">http://www.beatricetinsleyinstitute.org/</note><subject><genre>keywords</genre><topic>methods: statistical</topic><topic>techniques: image processing</topic><topic>techniques: photometric</topic></subject><relatedItem type="host"><titleInfo><title>Monthly Notices of the Royal Astronomical Society</title></titleInfo><titleInfo type="abbreviated"><title>Monthly Notices of the Royal Astronomical Society</title></titleInfo><genre type="journal" authority="ISTEX" authorityURI="https://publication-type.data.istex.fr" valueURI="https://publication-type.data.istex.fr/ark:/67375/JMC-0GLKJH51-B">journal</genre><subject><topic>Papers</topic></subject><identifier type="ISSN">0035-8711</identifier><identifier type="eISSN">1365-2966</identifier><identifier type="PublisherID">mnras</identifier><identifier type="PublisherID-hwp">mnras</identifier><part><date>2009</date><detail type="volume"><caption>vol.</caption><number>397</number></detail><detail type="issue"><caption>no.</caption><number>4</number></detail><extent unit="pages"><start>2099</start><end>2105</end></extent></part></relatedItem><identifier type="istex">A3134BD63E717DD5413C4892BAB348A34A5FB296</identifier><identifier type="DOI">10.1111/j.1365-2966.2009.15098.x</identifier><accessCondition type="use and reproduction" contentType="copyright">© 2009 The Authors. 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