Photometry

Photometry is the measurement of the flux of an object. Photometry is particularly important for eclipsing binaries since a long series of images can be used to construct a light curve which can be used to detect variations (e.g. eclipses) in the object’s luminosity over time. Flux measurements can be filterless (also known as white light), where all of the light received by the detector is used, or, more commonly, a filter is inserted. A filter only allows light of certain wavelengths through to the detector. For example, the Sloan Digital Sky Survey (SDSS)u-band filter allows the measurement of the flux of an object over the wavelength range of 3000 ˚A₋4000 ˚A. The filter profiles for the high-speed camera ULTRACAM are shown in Figure 2.3. Several calibration steps are required to ensure that the extracted object flux

is accurate. The following sections outline these steps.

Bias subtraction

Since the analog to digital conversion process introduces a small amount of sta- tistical fluctuation in the pixel values, the zero value of the CCD is offset. This avoids potential negative pixel values (which would require a “sign bit” to be used). However, this bias level needs to be subtracted so that errors can be properly deter- mined. This can be estimated by using a bias frame which is a zero exposure-time frame, an example of which is shown in the left hand panel of Figure 2.4. Several bias frames can be combined to reduce the statistical noise and the resultant frame subtracted from the science images.

Bias frames are not possible using infrared detectors since zero second expo- sures are not possible. Therefore, any bias level is subtracted using dark frames.

Dark-current subtraction

Electrons in the detector can occasionally gain enough thermal energy to jump into the conduction band without the influence of a photon. This extra signal is known as the dark current and can be a substantial source of noise, especially for long exposures. The dark current can be reduced by cooling the detector. An estimate of the dark current can be made using dark frames, which are long exposures with the shutter closed. For CCDs the dark current is proportional to the exposure time, therefore the dark frame can be scaled to match the exposure time of the science image. Infrared detector materials have a dark current that is not linear with time and hence dark frames must be taken with the same exposure time as the science frames. Usually several dark frames are combined to reduce the statistical noise and the resultant frame is subtracted from the science frames. For CCDs the bias level is removed before subtraction whilst the dark frame of an infrared detector acts as both a dark and bias frame.

Flat-fielding

Each pixel in a detector array has a slightly different sensitivity to the pixels around it. Flat fields are used to correct for this effect. A flat field frame is an exposure of a uniform source of light. These are usually a series of exposures of the twilight sky (sky flats) taken with the telescope spiraling to ensure that no stars remain on the same part of the chip. Any counts introduced by stars in the field can then be removed by median averaging the set of flat field frames, this is then normalised to

Figure 2.4: left: a bias frame from CCD2 of ULTRACAM. The CCD is split into two channels with different bias levels, here the left hand channel has been multiplied up to match the mean level of the right hand one. The difference between the highest and lowest value is only 3%. Right: a flat field frame from CCD2 (g′

band) of ULTRACAM.

one. All science images are then divided by this frame. Alternately, special flat field screens are provided in many observatories (known as dome flats). An example of a flat field frame is shown in the right hand panel of Figure 2.4.

Aperture photometry

Turning a two dimensional array of numbers into a flux measurement is usually achieved using aperture photometry. An example of aperture photometry is shown in Figure 2.5. It involves picking a small region of pixels (an aperture) that enclose the target. This aperture is usually circular, like those shown in Figure 2.5, to match the star profile. Once we have placed the aperture on the image we simply add up the counts within this window. However, this will also include an offset due to the sky background. The sky background can be estimated by placing a second aperture around the target aperture. This is usually an annulus like those shown in Figure 2.5. The mean of the pixels within this sky aperture is calculated and subtracted from each pixel in the target aperture. A larger sky aperture gives a more reliable estimate of the sky background, but care must be taken that it does not include flux from any other stars, although a mask can be applied to remove any nearby stars.

Figure 2.5: An NTT+ULTRACAM g′

band image of the sky with several stars visible. The CCD was windowed and only half is shown. Three stars have apertures placed over them. For each star, the smallest circle is the object aperture whilst the two larger circles denote the sky annulus.

The usual way of choosing the size (radius) of the apertures is profile fitting. This approach assumes that the profile of the star resembles some known function (e.g. Gaussian or Moffat functions, the latter based on stellar images (Moffat, 1969)) and thus we can fit the profile to measure variables such as peak intensity and Full- Width at Half-Maximum (FWHM). This allows us to get a good idea of the current atmospheric conditions. The size of the apertures is then based on the FWHM multiplied by a suitable scaling factor. Initially a range of scale factors are used (e.g. ranging from 1.5 to 2.0, in steps of 0.1), then the scale factor that produces the highest signal-to-noise ratio is chosen for the final reduction.

Differential photometry

If we perform aperture photometry of a star over multiple images we can build up a light curve. However, variations in atmospheric conditions (e.g. cloud) can affect the shape of the light curve. This can be corrected for by extracting the flux of a nearby star, known as a comparison star. Dividing the object counts by the comparison star’s counts conserves the intrinsic variability of the target but also corrects for variations in conditions. This is known as differential photometry. Care must be taken when choosing a comparison star: it should, if possible, be brighter than the target and have no intrinsic variability itself.

Flux calibration

In some cases it is desirable to turn count values into magnitudes or milli-Janskys (mJy). These values can be compared to external data sets and models. This conversion is achieved using a standard star. Standard stars are relatively bright, non-variable stars with well measured magnitudes in several different bands. The list presented by Smith et al. (2002) is generally used to flux calibrate observations made with the SDSSugriz filters.

The magnitude of the standard star is

mµ=−2.5 log10(Cµ) +xµ−κµX, (2.1)

where µ represents the filter of interest, Cµ is the counts per second, xµ is the

instrumental zero point, κµ is the extinction coefficient in theµband and X is the

airmass.

Since the magnitude of a standard star is known, the only unknowns are the instrumental zero point and the extinction coefficient. A theoretical extinction coefficient can be used but it can also be measured. Using a long observing run (covering a large range in airmass), a linear fit to counts vs. airmass yields the extinction coefficient. Once this is measured the zero point can be obtained and applied to the comparison star and hence the object itself.

To ensure an accurate flux calibration, large apertures must be used to ensure that all the flux is collected. The usual method used to flux calibrate a target’s light curves is to use the longest run on the target to determine the comparison star’s magnitude. Once determined, the comparison star’s magnitude is then fixed and used to flux calibrate all other observations of that target.