Lab 3

CCD Characterization

Sean Lockwood1

November 19, 2007

ABSTRACT

In this lab, we characterize many of the properties of the 14” reflecting telescope

and CCD located at Boston University. We determined a gain of 2.50 ± 0.12 e-/ADU,

a read noise of 22.8 ± 1.0 e-, a mean dark rate of 0.00427 ± 0.00087 ADU/pixel/s, a

lower-bound on the nonlinear regime of 53,000 counts/pixel, and a shutter time of 0.126

s. We then discuss image reduction and flat-fielding techniques. Then, we measure a

local sky brightness of 15.5 mag/arcsec in the B-band and 15.0 mag/arcsec in the V-

band. Next, we determined that the plate scale is 0.4726 arcsec/pixel and a total CCD

field of view of 12.05′ wide by 8.03′ tall. Then we discuss the development of aperture

photometry software and its application to the Pleiades Cluster to measure extinction

and color transformation coefficients.

1. Observations

We used the 14” reflecting telescope located on the roof of Boston University’s CAS build-

ing. Through the course of our observations, we took data in the B-, V-, & R-filters, as well as

observations without any filter when applicable. We observed for various durations on 9/19/2007,

10/16/2007, 10/29/2007, 11/1/2007, & 11/2/2007.

2. Gain and Read Noise

According to Howell, p. 73:

Gain =(F̄1 + F̄2) − (B̄1 + B̄2)

σ2F1−F2

− σ2B1−B2

(1)

Read noise =Gain · σB1−B2√

2(2)

The gain was found by studying 15 pairs of dome-flats of three different exposure lengths.

According to eqn. 1, the gain is the inverse of the slope in fig. 1, or 2.50±0.12 e-/ADU. Furthermore,

the read noise can be determined from eqn. 2 to give 22.8 ± 1.0 e-.

1email: lockwood@bu.edu

– 2 –

Fig. 1.— Gain and read noise were found by plotting the variance of the difference of two flat-field

images versus the mean of the sum of the flat-field images, corrected for bias (see eqn. 1). The

read noise was determined from this value and the FWHM of the difference of two bias images (see

eqn. 2).

3. Dark Rate

Random thermal fluctuations in the CCD cause a random uncertainty called dark noise. In

order to correct for this effect, we are unable to subtract synthesized random counts, because this

would simply add to the uncertainty in each pixel. However, we can subtract off a constant that

has been appropriately scaled for exposure duration in order to account for differences in total

measured flux. In order to determine this value, we observed 28 dark exposures of various exposure

durations (see fig. 2). We found a mean dark rate of 0.00427 ± 0.00087 ADU/pixel/s. If we had

not corrected for this effect, it would have resulted in a minor error in measured fluxes.

Of course, this value is dependent upon the temperature of the CCD. To characterize this

dependence and find an optimal operating CCD temperature, we should find dark rate as a function

of temperature.

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Fig. 2.— Dark frames of various exposure time durations were taken. The slope of the linear fit

(red line) is the CCD’s dark rate at −20o C of 0.00427 ± 0.00087 ADU/pixel/s. Ideally, we should

have taken more exposures with intermediate durations.

4. Saturation and Nonlinearity

In order to quantify the linear regime of the CCD, we observed a set of 75 dome flats of various

exposure duration. Each exposure time was observed in a set of 5 observations, which were median

stacked in order to remove cosmic rays and read noise. The average pixel value from a 501 x 501

pixel box centered in the image was plotted against exposure time (see fig. 3).

We determined that our measured linear regime ran from 0.15 s to 0.7 s for this set of observa-

tions. Upon trying linear fits both with and without the point at 0.7 s, we found little difference in

the residual distribution about other points. Thus, we had no justification for excluding it from our

linear regime. Hence, we can place a lower limit on the nonlinear threshold of 53,195 counts/pixel.

Our observations at 1 s saturated the CCD, leaving us with 65,535 counts, as expected from the

upper limit of the analog-to-digital converter. Further observations between 53,195 counts/pixel

and saturation are needed to narrow in on the linear limit and calibrate counts within the nonlinear

regime. Furthermore, we expect appropriate nonlinear calibrations to be pixel-dependent as is the

case with linear flat-fielding.

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Fig. 3.— The mean number of counts in a 501 x 501 pixel box centered in median-stacked dome

flats of various exposure duration. The red line is a fit to the points in the linear regime. The

lower green line represents the minimum exposure time capable due to finite shutter speed. The

upper green line through 65,535 counts represents the maximum value that the CCD’s ADC can

represent with 16 bits. Note that most of the scatter about the line in the linear regime is probably

due to slight variations in diffuser position between exposures due to movement.

The flat detector response at lower exposure times was due to the shutter taking a finite amount

of time to operate. By determining the intercept of this flat region with a fit to the linear regime,

we are able to determine that the minimum exposure time achievable by our instrument is 0.126 s.

5. Image Reduction

5.1. Bias

Bias images instantaneous dark images taken without opening the camera’s shutter. These

are used to compensate for the CCD’s positive voltage offset that would otherwise result in an

artificially high count rate in all images. Since this rate can vary by location on the CCD, we

need to correct each pixel independently, rather than using a constant offset value across the CCD.

– 5 –

We median-stacked a number of bias images in order to reduce the read noise and cosmic ray

contamination. Since bias images take a very short time to obtain, there is little overhead to

observing a large enough number of them with each observation run.

5.2. Flats

We experimented with three different types of flat-field images: twilight flats, dark-sky flats,

and dome flats. Twilight flats were sensitive to changes in the illumination of the sky. This makes

it difficult to median-stack them to remove read noise and cosmic rays. However, they have the

advantage of following the same beam path as our science images, making twilight flats potentially

more applicable. (See fig. 4.)

Fig. 4.— While the twilight flat has a better signal-to-noise ratio than a comparable dark-sky flat,

it fails to meet the standard of a dome-flat.

We attempted to create a sky flat by spatially dithering some of our science images. By moving

the telescope between similar observations, the hope was to remove stellar sources from our sky flat

via median stacking (see fig. 5). Although we were removed the brightest of the unwanted stellar

sources, many blotches still remained. Also, the sky flux was much lower than needed to achieve

an acceptable signal-to-noise ratio. Many of the features we wish to remove are not distinguishable

– 6 –

(i.e. dust aberrations, hot pixels). Furthermore, moving the telescope proved to be overly time

consuming.

Fig. 5.— A dark-sky flat. Note that background sources are not completely removed and that the

noise is quite high, as compared to other types of flat-field images.

Finally, dome flats were taken by pointing the telescope at a diffuser located within the tele-

scope’s dome that was illuminated by a light bulb. Thanks to their artificial source, they maintained

a more consistent illumination over time. However, as the target was not located at infinity, the

light traveled down a slightly different beam path through the telescope. Despite this drawback

our dome flats proved applicable in the reduction of the data. They had a high enough flux in

all filters to achieve a much higher signal-to-noise ratio than sky flats. Notice that we are able to

see torus-shaped dust patterns in the dome flats due to out of focus dust, whereas these were not

visible in the sky flats (see fig. 6). Also, some features visible in the dome flats were not visible in

the twilight flats. Having worked with all three types of flat images, it is apparent that dome flats

are the easiest and probably most accurate choice.

– 7 –

Fig. 6.— A dome-flat. Like the other flats, this image has been median-stacked to reduce read

noise and cosmic ray contamination.

5.3. Total Correction

The “pixel math” formula to reduce a raw data frame, Imageo, is as follows:

Image =Image

o− Bias − dark rate · t

(Flat − Bias)/median(Flat − Bias)

where dark rate is the constant determined in sec. 3 and t is the exposure duration.

5.4. Local Sky Brightness

We estimated the local sky brightness by measuring the counts in a selected box without any

stars (using the plate scale to scale to 1 arcsec2—found below) and comparing the flux to a star of

known magnitude in the same image. We obtained a value of 15.5 mag/arcsec in the B-band and

15.0 mag/arcsec in the V-band.

– 8 –

6. Plate Scale

Rather than calculating the plate scale of the telescope/CCD system from known dimensions

and assumed optical specifications, we chose to measure it directly by observing a cluster of stars

with known angular separation. By plotting all permutations of pixel distances between six stars

in the Pleiades Cluster against their known angular separation (see fig. 7, we obtained a plate scale

of 0.4726 arcsec/pixel.) This gives a total CCD field of view of 12.05′ wide by 8.03′ tall.

Fig. 7.— The plate scale of 0.4726 arcsec/pixel was determined by comparing all permutations of the

distances between six stars in the Pleiades Cluster with distances calculated from the coordinates

given by SIMBAD sources. Spherical distances were used. As spherical distance depends on the

absolute declination, the pixel distances were found by iterating the routine to converge on the

appropriate solution.

7. Aperture Photometry

It was necessary to develop an aperture photometry routine that would return the number of

counts received from a target star. In order to do this, we needed to tackle a number of problems:

finding the center of target stars, determining an appropriate aperture over which to sum flux, and

– 9 –

the subtraction of local sky.

7.1. Star Centering

For the Pleiades data, the stars did not move relative to one-another over time. So, it was

necessary to find the pixel locations of each target star only once and shift this set of coordinates

an appropriate amount for each exposure (see fig. 8). The initial locations were found by using

daophot’s find procedure. The brightest (non-saturated) target stars were manually selected

from the returned matches.

In order to determine each image’s offset from the reference locations, we used find to make

a guess for the brightest source and then used the correl images function from The IDL As-

tronomer’s Library to cross-correlate the images in two dimensions.

Fig. 8.— A typical Pleiades observation, taken in the V-band. The red X’s mark the locations

of star targets for our photometry. The brightest star was not chosen because we intentionally

saturated it in order to achieve better signal with the dimmer stars. (Some of the stars may be

hard to see in this scale.)

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7.2. Aperture & Sky Subtraction

We used a circular aperture with a radius determined by the flux contained in each star.

Daophot’s user manual recommends using an aperture radius approximately the size of a star’s

FWHM. To determine this value (or at least a consistent value), we generated a curve-of-growth

for each star (see fig. 9). This plot shows the number of counts contained within a circle of growing

radius (see fig. 10). To account for background sky flux, we subtracted the mean value of an

annulus centered on each star, weighted appropriately for area.

Fig. 9.— A typical curve-of-growth of one of the Pleiades. The radius is set to the point that has

an average count of 14 the baseline. Note that the chosen radius will change, depending on the star’s

growth profile. Also, there is a centering error ∼4 pixels, as is seen in the location of the peak.

In order to isolate pixels within our aperture, we multiplied each reduced science image by

an array that was 1 inside the aperture and 0 outside. In order to reduce quantization errors, we

weighted pixels on the edge of the aperture by the fraction of their area within the circle using the

pixwt function from The IDL Astronomer’s Library.

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Fig. 10.— The Pleiades star corresponding to fig. 9 with a selected aperture radius of 17 pixels.

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8. Atmospheric Extinction Coefficients

As our target stars travel through various airmasses (see fig. 11), they change in brightness

as a function of frequency. We shall make the assumption that frequencies can be adequately

approximated by bandpasses and solve the standard photometric transformation equations for the

B- and V-bandpasses.

Fig. 11.— We observed the Pleiades Cluster over a variety of airmasses.

We used our aperture photometry techniques to solve for raw instrumental magnitudes of two

stars in the Pleiades cluster (see figs. 12 & 13). These stars (HD 23608 & HD 23607) both have

known B and V absolute magnitudes in the Simbad database. By observing these stars at various

airmasses (a minimum of two points), we are able to solve for the photometric extinction and color

transformation coefficients (see tables 1 & 2, respectively).

Errors in our absolute instrumental magnitudes, m, may be due to variations through the night

(i.e. passing clouds) and errors in the photometric reduction (i.e. centering errors, bad aperture

radii). We really did not have the weather we needed to obtain highly accurate extinction and

transformation coefficients, but we shall go through the exercise anyway.

v = V + kvX + Cv(B − V ) + k′X(B − V ) + zv (3)

– 13 –

Fig. 12.— Magnitude of HD 23608 (#2) in the Pleiades Cluster versus airmass for three filters:

Red, Blue, & Visual (green). Simbad gives: B=9.15, V=8.72

b = B + kbX + Cb(B − V ) + k′X(B − V ) + zb

where X is the airmass; V & B are known absolute magnitudes; v & b are measured instrumental

magnitudes; kv & kb are broad-band extinction coefficients; and k′, Cv, & Cb are broadband

transformation coefficients.

First, we assume X = 0 (i.e. no atmospheric extinction). Now, let’s work with the V-band.

We find:

(v − V )∗ = Cv(B − V )∗ + zv (4)

where variables indexed with a ∗ refer to the two known stars.

Now, we simply solve two linear equations for Cv and zv:

Cv =(v − V )1 − (v − V )2(B − V )1 − (B − V )2

= 1.056

zv = (v − V )1 − Cv(B − V )1 = −23.7 mag

– 14 –

Fig. 13.— Magnitude of HD 23607 (#3) in the Pleiades Cluster versus airmass for three filters:

Red, Blue, & Visual (green). Simbad gives: B=8.51, V=8.26

Similarly, we can work in the B-band to solve for Cb and zb when X = 0:

(b − B)∗ = Cb(B − V )∗ + zb (5)

Cb =(b − B)1 − (b − B)2

(B − V )1 − (B − V )2= −1.167

zb = (b − B)1 − Cb(B − V )1 = −22.3 mag

Finally, plug back into eqn. 3 to find k′:

k′ =

v−V −kvX−Cv(B−V )−zv

X(B−V )

b−B−kbX−Cb(B−V )−zb

X(B−V )

=v − V − kvX − Cv(B − V ) − zv

b − B − kbX − Cb(B − V ) − zb

(6)

Eqn. 6 can be solved by plugging in two independent values into each of v(X), b(X), & X.

However, this is a second–order correction and is prone to large errors given that only two points

were used to solve for Cm and zm. Thus, we shall not solve for k′ here.

– 15 –

Star Band M (Simbad) m (Instrumental) km (mag/AM)

HD 23608 (mag 2) B 9.15 -13.61 −0.059*

V 8.72 -14.53 1.11

R - -15.52 2.16

HD 23607 (mag 3) B 8.51 -14.04 0.051

V 8.26 -15.18 1.29

R - -16.23 2.26

Table 1: Stellar statistics.

*Negative value is within error limits of positive regime.

Band km (mag/AM) Cm zm

average

B -0.004 -1.167 -22.3

V 1.20 1.056 -23.7

R 2.21 - -

Table 2: Bandpass statistics.

References

Howell, Steve B. Handbook of CCD Astronomy, 2nd edition (2006).

Simbad Astronomical Database, <http://simbad.u-strasbg.fr/simbad/> .

Taylor, John R. An Introduction to Error Analysis: The Study of Uncertainties in Physical Mea-

surements, 2nd edition (1997).