Index: trunk/doc/release.2015/ps1.analysis/analysis.tex =================================================================== --- trunk/doc/release.2015/ps1.analysis/analysis.tex (revision 40687) +++ trunk/doc/release.2015/ps1.analysis/analysis.tex (revision 40689) @@ -2676,4 +2676,163 @@ models for each galaxy model measured for the stack image. +\subsection{Galaxy Lensing Parameters} + +\begin{verbatim} +* background : KSB, related (mention Deacon et al here or at the end?) +* second moments are discussed above (same values, including window function as given) +* write out the KSB formalism +* stellar parameters using PSF stars +* output parameters +* this is only done on warp -- move to ForceWarp section? +\end {verbatim} + +Weak-lensing studies frequently use non-parametric measurements of the +ellipticities of galaxies to quantify the strength of gravitational +lensing, and thus directly measure mass distributions in the Universe. +The classic approach was originally described by +\cite{1995ApJ...449..460K} and applied to a set of deep HST +observations. The details of the technique were further refined by +\cite{1998ApJ...504..636H}; in the discussion below we primarily use +their notation, though we explicitly cast their integrals as sums over +discrete pixels. + +The KSB-style analysis of object ellipticities has also been used by +several authors to search for margially-resolved binary stars +in wide-field imaging data. The use of the lensing statistics for +this application was described by \cite{2005ApJ...626.1070H} in the +context of vetting planet transit events in data from the Optical +Gravitational Lensing Experiment (OGLE). \cite{2013ApJS..206...18T} +applied the techinique to PTF data to search for binary stars and +\cite{2017MNRAS.468.3499D} used the same technique to search for +binary companions to known ultracool dwarfs using Pan-STARRS $\3pi$ +data. The work by \cite{2017MNRAS.468.3499D} used images and their +own analysis of the pixels with the program Sextractor +\citep{Bertin.ref}. + +For the Pan-STARRS $\3pi$ PV3 analysis, we have measured the full set +of KSB lensing parameters for \note{which subset?} of the data to +enable both lensing studies and binary / multiple star searches. Here +we describe the measurements as performed within \ippprog{psphot}, +reviewing the mathematical framework as described by +\cite{1995ApJ...449..460K} and \cite{1998ApJ...504..636H}. + +The goal of the KSB technique is to measure the intrinsic ellipticity +of objects (i.e., galaxies, in the case of weak lensing studies) as +would be observed sky on the without instrumental effects. The +analysis starts with the observed ellipticity of the object as +represented by the two polarization components derived from the second +moments (see Section~\ref{sec:moments}): +\begin{eqnarray} + e_1 = \frac{M_{xx} - M_{yy}}{M_{xx} + M_{yy}} \\ + e_2 = \frac{2 M_{xy}}{M_{xx} + M_{yy}}. \\ +\end{eqnarray} +These two quantities have values which range from -1 to +1, with a +circularly-symmetric object having $(e_1,e_2) = 0.0,0.0$. The two +polarization components vary sinusoidally with twice the position +angle of the object: an elongated object aligned with the $x$-pixel +axis will have positive values of $e_1$ and $e_2$ values near zero, +while the same object aligned with the $y$-pixel axis will negative +$e_1$ values. An object with a position angle on the 45\degree\ lines +between the pixel axes will have large positive or negative values of +$e_2$ and low absoluate values of $e_1$. + +\note{need for the window function}. + +The observed ellipticity of an object observed in a real instrument +will be affected by the instrumental signature of the instrument. To +first order, the effect on the polarization components can be +described as a combination of ``smear'', in which the observed shape +is more circularized (driving $e_1,e_2$ to low absolute values) and +``shear'', in which the observed shape is stretched in one direction +relative to the others (driving $e_1,e_2$ to larger absolute values). +With sufficient understanding of the image PSF, both shear and smear +terms can be corrected. + + + +@ARTICLE{2017MNRAS.468.3499D, + author = {{Deacon}, N.~R. and {Magnier}, E.~A. and {Best}, W.~M.~J. and + {Liu}, M.~C. and {Dupuy}, T.~J. and {Chambers}, K.~C. and {Draper}, P.~W. and + {Flewelling}, H. and {Metcalfe}, N. and {Tonry}, J.~L. and {Wainscoat}, R.~J. and + {Waters}, C.}, + title = "{Identification of partially resolved binaries in Pan-STARRS 1 data}", + journal = {\mnras}, +archivePrefix = "arXiv", + eprint = {1702.05491}, + primaryClass = "astro-ph.SR", + keywords = {binaries: visual, brown dwarfs}, + year = 2017, + month = jul, + volume = 468, + pages = {3499-3515}, + doi = {10.1093/mnras/stx440}, + adsurl = {http://adsabs.harvard.edu/abs/2017MNRAS.468.3499D}, + adsnote = {Provided by the SAO/NASA Astrophysics Data System} +} + +@ARTICLE{1995ApJ...449..460K, + author = {{Kaiser}, N. and {Squires}, G. and {Broadhurst}, T.}, + title = "{A Method for Weak Lensing Observations}", + journal = {\apj}, + eprint = {astro-ph/9411005}, + keywords = {COSMOLOGY: OBSERVATIONS, COSMOLOGY: DARK MATTER, GALAXIES: FORMATION, COSMOLOGY: GRAVITATIONAL LENSING, COSMOLOGY: LARGE-SCALE STRUCTURE OF UNIVERSE}, + year = 1995, + month = aug, + volume = 449, + pages = {460}, + doi = {10.1086/176071}, + adsurl = {http://adsabs.harvard.edu/abs/1995ApJ...449..460K}, + adsnote = {Provided by the SAO/NASA Astrophysics Data System} +} +@ARTICLE{1998ApJ...504..636H, + author = {{Hoekstra}, H. and {Franx}, M. and {Kuijken}, K. and {Squires}, G. + }, + title = "{Weak Lensing Analysis of CL 1358+62 Using Hubble Space Telescope Observations}", + journal = {\apj}, + keywords = {GALAXIES: CLUSTERS: INDIVIDUAL ALPHANUMERIC: CL 1358+62, GALAXIES: FUNDAMENTAL PARAMETERS, COSMOLOGY: GRAVITATIONAL LENSING, galaxies: clusters: individual (Cl 1358 + 62), Galaxies: Fundamental Parameters, Cosmology: Gravitational Lensing}, + year = 1998, + month = sep, + volume = 504, + pages = {636-660}, + doi = {10.1086/306102}, + adsurl = {http://adsabs.harvard.edu/abs/1998ApJ...504..636H}, + adsnote = {Provided by the SAO/NASA Astrophysics Data System} +} +@ARTICLE{2005ApJ...626.1070H, + author = {{Hoekstra}, H. and {Wu}, Y. and {Udalski}, A.}, + title = "{An Algorithm to Detect Blends with Eclipsing Binaries in Planet Transit Searches}", + journal = {\apj}, + eprint = {astro-ph/0501353}, + keywords = {Stars: Binaries: Eclipsing, Stars: Planetary Systems}, + year = 2005, + month = jun, + volume = 626, + pages = {1070-1078}, + doi = {10.1086/430299}, + adsurl = {http://adsabs.harvard.edu/abs/2005ApJ...626.1070H}, + adsnote = {Provided by the SAO/NASA Astrophysics Data System} +} +@ARTICLE{2013ApJS..206...18T, + author = {{Terziev}, E. and {Law}, N.~M. and {Arcavi}, I. and {Baranec}, C. and + {Bloom}, J.~S. and {Bui}, K. and {Burse}, M.~P. and {Chorida}, P. and + {Das}, H.~K. and {Dekany}, R.~G. and {Kraus}, A.~L. and {Kulkarni}, S.~R. and + {Nugent}, P. and {Ofek}, E.~O. and {Punnadi}, S. and {Ramaprakash}, A.~N. and + {Riddle}, R. and {Sullivan}, M. and {Tendulkar}, S.~P.}, + title = "{Millions of Multiples: Detecting and Characterizing Close-separation Binary Systems in Synoptic Sky Surveys}", + journal = {\apjs}, +archivePrefix = "arXiv", + eprint = {1210.4550}, + primaryClass = "astro-ph.SR", + keywords = {binaries: close, methods: data analysis, stars: statistics, surveys, techniques: image processing }, + year = 2013, + month = jun, + volume = 206, + eid = {18}, + pages = {18}, + doi = {10.1088/0067-0049/206/2/18}, + adsurl = {http://adsabs.harvard.edu/abs/2013ApJS..206...18T}, + adsnote = {Provided by the SAO/NASA Astrophysics Data System} +} + \section{Difference Image Photometry}