Observational Astronomy 2018 Part 9 Prof. S.C. Trager Photometry

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Photometry

Observational Astronomy 2018 Part 9

Prof. S.C. Trager

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Photometry is the measurement of magnitudes from images

technically, it’s the measurement of light, but

astronomers use the above definition these days

This sounds simple, but there are several steps required!

We’ll focus here on photometry of objects from two- dimensional images (detectors)

similar (if not all) steps are needed when using single- pixel detectors like photomultiplier tubes

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1. Find the location(s) of the object(s) of interest (perhaps all)

2. Determine the background level(s) B of the object(s) (per pixel)

3. Calculate the integrated source intensity I for each

object. To do this, sum S counts from N pixels. Then

The process of photometry

I = S N ⇥ B

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The process of photometry

4. Determine the magnitude:

where C is the photometric zeropoint

5. Determine the photometric zeropoint(s), airmass correction(s), and color term(s) (if needed) using photometry of standard objects (almost always these are standard stars)

m = 2.5 log₁₀ I + C

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Image centroiding

In a digital (or digitized) image, the image of an object is spread over some number of pixels (hopefully!)

To measure its flux, we want to know where its center is

There are a few ways to do this...

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Image centroiding

The easiest way to centroid an image is to use marginal distributions

Imagine an image of a source with intensity Iij in ADUs in pixel (i,j)

Then extract a 2L+1×2L+1 subarray containing – hopefully! – only the source

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Image centroiding

Then

and the mean intensities in each direction are I_i =

j=LX

j= L

I_ij and J_j =

i=LX

i= L

I_ij

I =¯ 1

2L + 1

i=LX

i= L

I_i and ¯J = 1

2L + 1

j=LX

j= L

J_j

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Image centroiding

Then the intensity-weighted centroids are

and

x_c =

P_i=L

i= L(I_i I)x¯ _i P_i=L

i= L(I_i I)¯ for I_i I > 0¯

y_c =

P_j=L

j= L(J_j J)y¯ _j P_j=L

j= L(J_j J)¯ for J_j J > 0¯

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Image centroiding

This method can locate the peak (center) of an object to a fraction (one-fifth to one-tenth) of a pixel

Use this center to then fit, e.g., a two-dimensional Gaussian (for a star) to get a more precise center

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Background estimation

It is always preferable to find the background as close to the object as possible

When measuring

magnitudes of compact objects (stars, small

galaxies), it is common to determine the

background in an annulus around the object

object

sky annulus

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Background estimation

the annulus is often

circular or elliptical, but can be a more

complicated shape not done when doing surface photometry

object

sky annulus

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Background estimation

Once the counts in the sky annulus have been measured, compute a

histogram of these values then take the mode of the distribution (the

most probable value) to determine the

background level per pixel

Counts

#of pixels

Mode (peak of this histogram)

Arithmetic mean Median (1/2 above, 1/2 below)

Because essentially all deviations from the sky are positive counts (stars and galaxies), the mode is the best approximation to the sky.

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Background estimation

For surface photometry, you should fit a constant value or a plane or a surface to “blank” regions of the image and subtract this from the entire image

Note: surface photometry is the measurement of magnitudes per unit area (on the sky)

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Determining source intensity

There are two approaches to this step, depending on need and source type:

aperture photometry PSF (fitting) photometry

...and there’s also surface photometry, which we’ll describe briefly below

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Aperture photometry

In this case, we count the flux from the object and the sky within some aperture

typically we use circular apertures for stellar

photometry, but the apertures can be arbitrarily shaped for, say, complex galaxies

Then the source intensity is

I = X

aperture

I_ij B ⇥ N^aperture

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Aperture photometry

Problems...

How big should the apertures be?

Ideally, you’d like to get all of the light from your object...

but even stars have very extended

images

Profile of a stellar image on a photographic plate (King 1971)

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Aperture photometry

Note that a Gaussian profile contains 99% of its light within 10σ≈4 FWHM (because FWHM=2.355σ)

But! As the aperture grows, increases, but so does Naperture×B (because N gets bigger)

therefore the noise increases, because N∝r², where r is the radius (size) of the aperture

therefore the maximum S/N occurs at some intermediate radius, depending on FWHM

S = P

aperture I_ij

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Aperture photometry

If we restrict size to maximize S/N, we’re not measuring all of the flux

Either measure and compare all objects through the same aperture ...or...

Use the fact that the profile is the same for all stars (hopefully!) and measure a bright, well-exposed,

unsaturated, isolated star out to 4 FWHM. Then use the magnitude difference between this aperture and your smaller aperture to correct all the photometry

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Aperture Radius

!mag[aper(n+1) - aper(n)]

5 10 15 20 25 30 -0.15

-0.10 -0.05

0 0.05

Curve-of-growth analysis

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Aperture Radius

!mag[aper(n+1) - aper(n)]

5 10 15 20 25 30 -0.15

-0.10 -0.05

0 0.05

Curve-of-growth analysis

Sky over-subtracted

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Aperture photometry

PSF (fitting) photometry

To get around the problem of contamination, or more generically, for crowded fields, use the expectation that all stars have the same shape and vary only in

brightness

This means all the stars have the same point- spread function (PSF)

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PSF (fitting) photometry

For ground-based observations, the PSF is usually a Gaussian

or a Moﬀat profile

I_xy = I₀e ^r²^/2 ² , where r = p

(x x₀)² + (y y₀)²

I_xy = I₀ 2 1

↵²



1 + ⇣ r

↵

⌘2

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PSF (fitting) photometry

By assuming that the width of the stars (σ or α) is

constant (or at least smoothly, slowly varying), we can fit Gaussians or Moffat profiles to every star, just

varying I0

Then the magnitude difference between any two stars in the frame is

and so

m₁ m₂ = 2.5 log₁₀

✓ I_0,1 I_0,2

◆

m = 2.5 log₁₀ I₀

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PSF (fitting) photometry

Advantages

more accurate than aperture photometry

robust against cosmic rays, bad pixels, neighboring objects

can measure crowded fields because overlaps can be controlled for or, better, simultaneously fit

Implemented in DAOPHOT (I+II), DoPHOT, ROMAPHOT, HST/DolPHOT (and others...)

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PSF (fitting) photometry

Disadvantages

Much more computationally expensive than aperture photometry

Still need to perform aperture photometry on some stars to correct for light missed by PSF template

how good is the template?

how big is the template?

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Surface photometry

Like aperture photometry on a large scale

Determines intensity per unit (angular) area on the sky:

Often use elliptical apertures for this

Surface photometry gives light profile and shape information (using parameters of aperture fits)

SB = X

aperture

I_ij B ⇥ N^aperture

!

/area

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Surface brightness profiles

The surface brightness of a galaxy I(x) is the amount of light contained in some small area at a particular point x in an image

Consider a square area with a side of length D of a galaxy at distance d. This length will subtend an

angle

If the total luminosity of the galaxy in that area is L, then the received flux is

= D/d

F = L/(4 d²)

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So the surface brightness is

which is independent of distance

note that this is not true at cosmological distances!

The units of I(x) are usually given in

I(x) = F/ ² = L/(4⇥d²)(d/D)² = L/(4⇥D²)

L pc ²

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Often the magnitude per square arcsecond is quoted as the surface brightness:

In the B-band, the constant is 27 mag/arcsec², which corresponds to

Thus

1 L pc ²

µ (x) = 2.5 log I (x) + constant

I_B = 10 ^0.4(µ^B ²⁷⁾ L _,Bpc ²

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Contours of constant

surface brightness in an image are called

isophotes

NGC 7331

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If we plot the surface brightness profile of NGC 7331 in magnitudes/arcsec² as a function of radius, we find a straight line far away from the center

This implies that its disk has an exponential profile:

I(R) = I₀ exp( R/h_R)

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Elliptical galaxies have very smooth profiles over 2 orders of magnitude in radius,

usually falling off as R^1/4 In NGC 1700, surface brightness falls by 9

magnitudes/arcsec² — 4000x! — over 100x in radius

Light in ellipticals is highly concentrated

Surface brightness profiles

Surface brightness profile for giant elliptical galaxy: as f(R) and f(R^1/4).

Very good fit over 2 decades in radius

The surface brightness falls 9 magnitudes from centre to outskirts:

10⁹ fall-off in projected luminosity!

The light in elliptical galaxies is quite centrally concentrated

NGC 1700

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Photometric calibration

To determine our photometric zeropoints,

we need to observe objects — usually stars — of

known magnitudes and colors at many different hour angles

m = 2.5 log₁₀ I + ZP

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Photometric calibration

We need to correct three major effects:

1. overall magnitude offset: what magnitude corresponds to, say, one e^–/s at X=1?

2. color shifts between your filters and the standard stars’

filters

3. atmospheric extinction

note that there will also be different kλ for different-colored stars, due to the width of the broadband filter compared to the slope of kλ and the shape of the stars’ spectra

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Color terms

• The color terms come about through mismatches between

the effective bandpasses of your filter system and those of

the standard system. Objects with different spectral shapes

have different offsets.

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Photometric calibration

Combining 1) and 2), we have

where c is the color of your object

And 3) means

where c is the color, k is the extinction coefficient, k′ is the differential color–extinction coefficient, and mX is the

magnitude observed at airmass X

V=v₁+a₀

RMS=0.055

just magnitude zeropoint

m_true = m_0,inst + b₀ + b₁c + b₂c² + · · ·

m_0,inst = m_X kX + k⁰cX

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Photometric calibration

And 3) means

V=v_inst+ c₀+ c₁X

RMS=0.032

zeropoint and airmass

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Photometric calibration

And 3) means

V=v_inst+c₀+c₁X+c₂(B-V)

RMS=0.021

zeropoint, airmass, and color

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Photometric calibration

Thus, for a star of known mtrue and c observed at X,

Since each star satisfies this equation, a system of

linear polynomial equations exist, and we can invert this system to get our necessary coefficients ai

Lists of standard stars can be found in papers by

Landolt, Graham, and Stetson, and should be available at any observatory!

m_true = m_X + a₀ + a₁X + a₂c + a₃cX + a₄c² + · · ·

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Photometric calibration of the Gratama Telescope

Using data from 2015 September 9, I found

Figure 1: The photometric solutions.

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I have used the data from 09.09.2015 to determine the photometric calibration for the BV R filters of the Gratama Telescope at the Blaauw Observatory.

I used images of the fields SA 110, SA 35, and SA 38. The beginning of this night appears to have been reasonably photometric, although the scatter in the photometry is occasionally larger than the photon noise would suggest. I had to remove one R-band image of SA 38 whose photometry had a significant o↵set from the other two R-band images.

I have fit functions of the form

mag_obs mag_true = a₀+ a₁c + a₂X + a₃cX,

where magobs is the “instrumental magnitude” (the magnitude produced by phot in IRAF, with a zeropoint of 25.0 corresponding to 1 e s ¹), mag_true is the true magnitude (from Landolt 2009, 2013), c is the true color of the star, and X is the airmass at which the image of the star was taken. I performed this step for all three filters BV R and all three colors (B V ), (B R), and (V R). The results of these fits are show in the figure on the next page.

I then inverted the equations to determine the true magnitudes from the observed magnitudes and colors. In the following equations, a capital letter (like “B”) represents the true magnitude (here in the B filter) and a small letter (like “b”) represents the observed magnitude (here again in the B filter).

B = b (4.501 + 0.618X) (0.060 0.233X)( 0.048 + 0.607X)

1.032 0.261X (b r)

R = r (4.549 + 0.011X) (0.028 + 0.028X)( 0.048 + 0.607X)

1.032 0.261X (b r)

B = b (4.505 + 0.617X) (0.017 0.132X)( 0.132 + 0.436X)

0.942 0.114X (b v)

V = v (4.637 + 0.181X) (0.074 0.018X)( 0.132 + 0.436X)

0.942 0.114X (b v)

V = v (4.644 + 0.173X) (0.150 0.040X)(0.086 + 0.171X)

1.055 0.106X (v r)

R = r (4.558 + 0.002X) (0.095 + 0.066X)(0.086 + 0.171X)

1.055 0.106X (v r)