Photometry

Astronomers refer to the determination of a celestial object's brightness as photometry. Even though a star image may sprawl over many pixels, because CCDs are linear, they accurately record the total amount of light in each pixel, and therefore, also record the total light in any feature. Star images present something of a problem, however, because the starlight is mixed with light from the background sky. Extracting an accurate measurement of the total light in a star image is tricky, but it is one of the most important "instruments" in the digital astronomer's tool kit. Chapter 10 covers photometry in detail.

7.9.1 Defining the Star Image

Photometry involves calculating the statistics of small regions of interest; specifically, the region encompassing a star image and the dark-sky region immediately surrounding it. However, the two regions of interest must be defined in terms of their dimensions in the focal plane of the telescope rather than the pixel space, somewhat complicating the analysis for CCDs with pixel widths and heights that are not the same.

Star images are comprised of three regions:

1. a core consisting of the diffraction disk and the inner diffraction rings, containing perhaps 90% of the star's total light;

2. an aureole several times larger than the core containing light that, to the eye, is either hidden by the dazzle of the core or is too faint to see, containing perhaps 90% of the remaining light, and;

3. the surrounding sky, containing not only sky light, but also a tiny fraction of light from the star itself, parasitic light such as field flooding, and possibly a few faint background stars.

Both the core and the aureole also contain background sky light, and they also quite possibly contain unwanted background stars.

The basic task of stellar photometry is to determine the total amount of light contained in the image of the star. To accomplish this, not only must the starlight

Min: 141.9209 Profile Range [in ADUs] -pS/WhB C Auto Min/Mex

Figure 7.7 Although features can fool the human eye, they do not fool the image profile.

The profile reveals that spiral arms, which dominate the visual impression in this image, are really rather minor ripples in the overall radial decline in brightness outward from this galaxiy's nucleus.

Min: 141.9209 Profile Range [in ADUs] -pS/WhB C Auto Min/Mex

Figure 7.7 Although features can fool the human eye, they do not fool the image profile.

The profile reveals that spiral arms, which dominate the visual impression in this image, are really rather minor ripples in the overall radial decline in brightness outward from this galaxiy's nucleus.

be totaled, but the light from the sky must also be measured and subtracted from the starlight. On the image, therefore, the procedure must measure two regions of interest, one containing the star (the aperture), and a second consisting of a representative sample of sky (the annulus). The aperture contains the star plus the sky background, and the annulus contains only the sky. Subtracting the area-weighted annulus from the aperture should yield the total starlight.

1. From an initial rough estimate of the star's location, determine the cen-troid of the star image;

2. determine which pixels belong to a region of interest (i.e., the star aperture) centered on the centroid of the star image, containing both its core and aureole;

3. count the number of pixels and add up the total pixel value contained inside the star aperture;

4. determine which pixels belong to a ring-shaped region of interest (i.e., the sky annulus) surrounding the aperture; or alternatively, locate and determine which pixels belong to a nearby region of interest (i.e., a sky aperture) containing only sky pixels;

5. make a meaningful determination of the sky pixel value in the annulus or sky aperture;

6. from the total pixel value in the aperture, subtract the product of sky pixel value times the number of pixels in the aperture. The result is the total pixel value of the star.

The radius of the star aperture should be four to five times the full-width half-maximum of the star image on the CCD, or between 5 and 20 arcseconds— depending on the seeing quality, focal length of the telescope, tracking quality, and size of the pixels on the CCD. For typical amateur instruments and CCD cameras, the optimum radius usually lies between 4 and 10 pixels, so that the region of interest contains between 60 and 360 pixels.

The most difficult part of measuring stellar brightness is determining a meaningful pixel value of the sky. For an aperture containing 100 pixels, an error of one ADU in the sky brightness generates an error of 100 ADUs in the total pixel value of the star. Thus, a key requirement in digital photometry is an accurate determination of the sky brightness.

The radius of the annulus is typically twice the radius of the aperture, and contains approximately three times as many pixels. The larger the number of sky pixels, the greater the statistical accuracy of the sky background measurement, but the greater the risk of encountering contaminating background stars.

The average brightness of the annulus is one obvious possibility, but the average is sensitive to contamination from background stars. Their light adds to the sky brightness and drops the measured star brightness. The median pixel value of the sky background is another option, good because the scattering of high pixel values from faint background stars will have a negligible affect on the median. The median value of a set of pixels, however, is quantized into steps of one ADU, so that there is an automatic error of Vi ADU built into the photometry.

Similar to the median, but free of quantization, is an average built from the middle of the distribution of sky pixel values. Excluding the top and bottom 20% of sky pixel values and taking the average of the middle 60%, this "mean of the median half" method avoids both contamination from stars in the annulus and quantization effects. Repeated measurements on images with uniform sky backgrounds suggest that the "mean of middle" method of measuring the sky background is consistent to better than 0.1 ADU.

• Tip: AIP4Win's Star Image Tool extracts the basic measurements used in measuring the properties of star images. For stellar photometry, the suite of photometry tools is designed to carry out high-quality measurements of star brightness.

7.9.2 Photometric Image Profile

To measure the total light from a star with any accuracy, the aperture must contain

Figure 7.8 The profile of a star image shows a smooth decline in brightness from its centroid. From the profile, you can estimate the radius (or "half-width") of the star image at half maximum as about 1.1 pixels. The curve of growth shows how much of the star's light falls inside a circle with a given radius.

Figure 7.8 The profile of a star image shows a smooth decline in brightness from its centroid. From the profile, you can estimate the radius (or "half-width") of the star image at half maximum as about 1.1 pixels. The curve of growth shows how much of the star's light falls inside a circle with a given radius.

as much of its light as possible. Superficially, a star in a CCD image looks like a blocky circle with a distinct edge; but in reality, the image is bright in the center (the core) and fades smoothly into the background sky (the aureole). Good photometry requires measuring not only the light in the core but also the ligjit in the aureole.

To form a realistic estimate of the extent of the star image, you can examine its profile. The profile is simply a graph of pixel values around a star image—the pixel value versus the radial distance of the pixel from the centroid of the star image. With the exception of bright stars near the saturation level of the CCD, all star images in a given CCD image show exactly the same profile independent of the star's brightness. A typical profile falls rapidly from a maximum value, then more gradually approaches the sky value until the contribution of the starlight is lost in the noise of the sky background. This profile closely resembles the Gaussian, or normal, curve of distribution.

From the star profile plot, you can easily estimate the radius at which the star's light falls to half the peak. This radius is called the "half-width at half-maximum," or HWHM; the diameter of the image is its full-width at half maximum, or FWHM. If the profile were truly Gaussian, then the radius containing 99% of the star's light is 2.2 x HWHM; and the radius containing 99.9% of it is 2.8 x HWHM. To be sure of including all the starlight and minimizing the star's contribution to the sky background, many observers prefer to set the radius of the aperture to at least 4 x HWHM.

• Tip: The Star Image Tool in AIP4Win displays a photometric profile of a star image.

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