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Aperture (Stars) Measure inside aperture of given size !, which contains "#\$% pixels.

FWHM

& = ( )*#+, − )./0 ×"#\$%

2 = −2.5 log & + (calibration terms)

Optimal size of aperture does not encompass all the 0.4” pixels, 1.2” FWHM seeing, Howell et al 1989 . Need to employ aperture corrections. Measuring Signal-to-Noise

Consider measuring the flux from a star in an aperture that contains !"#\$ pixels.

Signal: • %∗, the total number of photons from the star.

Noise: These N’s all refer to • Total Poisson noise from the star: ' = %∗ photons or electrons, • Per-pixel Poisson noise from the sky: ' = %) not counts!

• Per-pixel Poisson noise from dark current: ' = %* • Per-pixel CCD read noise: ' = %+ (note: not squared!)

, % = ∗ % / %∗ + !"#\$(%) + %* + %+ )

“The CCD Equation” see Howell, Chapter 4.4 + , Example: Schmidt Telescope + CCD = ∗ • gain = 2.5 e-/ADU , , + ! (, + , + ,() • read noise = 3.6 e- ∗ "#\$ B C D • ND = 0 ADU

In a 60s in the M filter, we get • Sky = 80 ADU = 200 photons (±14) per pixel • A star with a peak of 20,000 ADU has 136,000 ADU (340,000 photons) inside ( a circular aperture of r=5 pixels (so !"#\$ ≈ π5 ≈ 80)

+ 340,000 = = 570 , 340,000 + 80×(200 + 0 + 3.6()

For a star that peaks at 100 ADU, the same calculation gives S/N = 12.

Error in Magnitude: ?@ Very important: This refers to random error in the • 9: = 1.0857 9=⁄> ≈ (+⁄,) ; calibration uncertainties set a • Star 1: S/N = 570, so 9: ≈ 0.002 mag floor to the final photometric uncertainty. • Star 2: S/N = 12, so 9: ≈ 0.09 mag S/N scaling with exposure time

\$ ! = ∗ ! 3 !∗ + .*/)(!1 + !2 + !+ )

Case 1: “Source limited”

!∗ dominates the counts, so \$⁄! ≈ !∗⁄ !∗ ≈ !∗ Since !∗ scales with exposure time, \$⁄! ∝ '()*

Case 2: “Detector limited”

!+ dominates the counts, so \$⁄! ≈ !∗⁄!+ Since !+ is independent of exposure time, \$⁄! ∝ !∗ ∝ '()* PSF Fitting Photometry (Stars)

Aperture photometry good here.

Aperture photometry not so good here! PSF Fitting Photometry (Stars)

Model the 2D point spread function (PSF) based on bright (not saturated!) stars.

Fit the model to individual stars, varying of model to minimize residuals.

1D analogy: fitting two Gaussians to a curve, easy to fit even with overlap

• Good for crowded fields. • Often gives best magnitudes, as long as PSF model is well-determined • But beware PSF variations (frame-to-frame, across the field of view, etc.) Surface Photometry M101 Surface Photometry M101

Galaxy annuli and Sky annuli – accurate sky estimate is critical!

Beware: foreground stars, background galaxies, artifacts must all be masked! Surface Photometry Spiral galaxies: exponential disks HJ⁄K \$ E = \$FG Then (conceptually): 2.5 D E = DF + E 1. Sky corrected flux in annulus: ℎ ln 10 M101

! = # \$%&& − \$()* ×,-./

2. Total magnitude in annulus:

0 = −2.5 log ! + (calibration terms)

3. Surface brightness in annulus:

D = 0 + 2.5 log(area)

O − P = DQ − DR Surface Photometry M49 Surface Photometry M49 Surface Photometry Elliptical galaxies: not exponential, classically r1/4 or, now, Sersic profiles:

,⁄( &'( *⁄*+ &- ! " = !\$%

2.53 . " = . + 4 "⁄" -⁄4 − 1 \$ ln 10 \$ M49