Galaxy Photomty Fo galaxis, w masu a sufac flux, that is, th couts i ach pixl. Though calibatio, this is covtd to flux dsity i Jaskys ( Jy -6 W/m/Hz). Fo a galaxy at som distac, d, a pixl of sid D subtds a agl, α, giv by d D α. d D α Th sufac bightss is th amout of light i that patch of sky dividd by its aa (i asc): I() Jy Flux. α acsc callig th latioship btw flux ad lumiosity, Flux bightss bcoms Lumiosity, th sufac 4 πd L F L d L Watts I() o. 4 πd D 4 πd pc acsc α Which is oft giv i sola lumiositis p pasc. To covt this to magituds, call that th appat magitud is a masu of flux, F m m.5 log. F So th sufac bightss i magituds p asc is F α µ µ.5 log, F α I µ µ.5 log. I EXAMPLE: Fo th Su i th optical V-Bad, takig I L /pc fo µ 6.4 magituds/acsc, yilds µ.5 log ( I ) + 6.4 mag/acsc. Givig th sufac bightss (i th V-bad i this cas), as IV.4(6.4 µv ) L V pc. Fom Amia Hlmi s Kapty cous at www.asto.ug.l/~ahlmi/tachig/gal/llipt.pdf
Galaxy Photomty Udgaduat ALFALFA Tam AOD 6/8/6 Th total (appat) magitud of th galaxy is th sufac bightss itgatd ov th ti galaxy. To do this, w ally d to modl th galaxy sic th sufac bightss is uv ad th dg is vy had to dtmi. So astooms us sufac pofils, paticulaly thos basd o Sésic with dvlopmts by d Vaucoulus ad Ptosia ad th xpotial cas. Th Sésic Pofil Sésic Pofil (Gaham & Div, Whittl 3) dscibs th itsity of a difftial aulus at a distac fom th ct of a galaxy: I() I b ( ) / d wh I is th itsity i flux p acscod (f/ ) at th ffctiv adius closig half th total light. Th lumiosity withi som adius,, is th th itgal of this tims th aa of th difftial aulus (π d) fom zo to : L() πi b ( ) / d () This sults i a xpssio that icluds a icomplt Gamma fuctio, b L ( < ) πi γ (, x ) b Fo >, this has a solutio of b o t t x wh t γ (, x ) t dt () dt γ (,b ) Γ ( ) γ (,b ) γ (,b ) Γ ( ) (3) Th IcompltGamma Fuctio fo 4 Th plot of γ(, b) fo 4 shows that ½Γ(*4) 5 cospods to b 7.67. 5 4 3 8 5 3 b 4 5 Sésic, J., L., 968, Atlas d Galaxias Austals. Obsvatoio Astoomico, Codoba adsabs.havad.du/abs/5pasa...8g ad Pag
Galaxy Photomty Udgaduat ALFALFA Tam AOD 6/8/6 Th d Vaucoulus Pofil placig 4 ad b 7.67 i Equatio () givs th d Vaucoulus I () I dv ( ) /4 7.67 (4) Plottig th total lumiosity as a fuctio of adius, i abitay uits, shows that half th total lumiosity is at ad 88.5% of th total lumiosity is withi 5, as Gaham & Div (5) poit out..885 L Total.75 L Total Lumiosity L Total Total Lumiosity as a Fuctio of adius.5 L Total.5 L Total 5 Th total lumiosity of th galaxy is foud by itgatig to ifiity: Tot Which sults i a omal Gamma Fuctio ( ) / b L πi d b Tot L πi Γ ( ) πi ( ) b b Γ Sic Γ() (-)!, *Γ() ()!. Thus, fo 4 ad b 7.669, b 7.669 L π I Tot 8 ( * 4 )! 7.5 πi. 7.669 (5) Pag 3
Galaxy Photomty Udgaduat ALFALFA Tam AOD 6/8/6 Th Expotial Pofil If is placd i Equatio (3) ( ) Γ( ) γ,b I () I xp th plot of γ(, b ) fo shows that ½Γ(*).5 cospods to b.68. Equatio () th givs th xpotial pofil: ( ).68 (6)..8.6.4. Th IcompltGamma Fuctio fo b 4 6 8 Plottig th total lumiosity as a fuctio of adius, i abitay uits, shows that half th total lumiosity is at ad 84.8% of th total lumiosity is withi. Lumiosity Total Lumiosity as a Fuctio of adius L Total.848 L Total.75 L Total.5 L Total.5 L Total 5 Galactic Scal adius Fo all of ths modls, th adius (g., xpad i SDSS) is th adius at which I I /. Pag 4
Galaxy Photomty Udgaduat ALFALFA Tam AOD 6/8/6 Th Ptosia Pofil Th Ptosia atio usd by SDSS is dfid at a adius fom th ct of a objct. It is calculatd fom th itgal of th atio of th local sufac bightss avag ov a aulus at (SDSS taks it fom.8 to.5) to th total sufac bightss withi. Th avagig is just th total sufac bightss withi th aulus dividd by th aa of th aulus)..8.5 Total sufac bighss withi aulus Aa withi Aulus (.8 -.5) ( ) Total sufac bighss withi Aa withi Thus,.5 π'i ( ')d'.8 ( ) ( ) π.5.8 π'i(')d' Plot fom Michal ichmod s SPSP 4 cous (http://stupdous.it.du/ichmod/) (7) π Ia ( ) I (') d'.38 couts pixl Th adius at which this factio π ( ) is o is quit small, though it is fomally dfid as th Ptosia adius. Fo th plot to th ight, th sufac bightss at pixls is.8 couts/pixl (i th aulus pixl wid) whas th I ( ).8 couts pixl itgatd itsity is about.38 couts/pixl. Goig out to lag adii would dcas both valus, but at som poit, thy would b qual ad that would dfi th Ptosia adius. I pactic, howv, th Ptosia adius tus out to b quit small, so SDSS (ad oths) dfi it as wh th itgatd flux is som factio of th aula flux, η, that SDSS taks as η. I ( P ) η (8) Pag 5
Galaxy Photomty Udgaduat ALFALFA Tam AOD 6/8/6 Th total Ptosia flux is dfid as th sum of all th flux withi k tims th Ptosia adius. Th facto k is usd to impov th sigal-to-light atio by icludig mo of th galaxy s light tha falls withi P. SDSS taks k th dfis th oth Ptosia adii as: P, 5 adius cotaiig half th total Ptosia Flux P, 9 adius cotaiig 9% of th total Ptosia Flux ad th Ptosia coctatio idx is th atio of ths two adii: P, 9 (9) P P, 5 I th 4 UAT Wokshop Scavg Hut 3A, Matha stats: C Whil you might thik that it would b asy to masu all th light, ad hc th magitud, it tus out that it's asi said tha do, bcaus th out gios of th galaxy a fait tha th bightss of th ight sky. So astooms masu magituds i difft ways. Th a whol paps witt o th subjct of how to calculat magituds! W ly o th commdatios of th SDSS staff to us "modl magituds" wh w wat to calculat colos ad "Ptosia magituds" wh w wat to calculat lumiositis o absolut magituds. Ivstigat th diffc btw th two fo th obsvd i bad magituds by plottig th sidual PtoMag_I - ModlMag_I vsus th Ptosia magitud; xamiig siduals is a good way to look fo tds. Hit: Us th aithmtic capability i TOPCAT to gat a w colums as dd. Th plot shows th Magitud siduals a ot zo, which thy would b if th w o diffc. Sic th diffc is mo positiv tha gativ, th Ptosia Magituds a lag tha th Modl Magituds, paticulaly fo dimm galaxis. This mas that th galaxis a bight i th modl magituds tha th Ptosia Magituds. Pag 6