# Detecting and measuring faint point sources with a CCD

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1 Detecting and measuring faint point sources with a CCD Herbert Raab a,b a Astronomical ociety of Linz, ternwarteweg 5, A-400 Linz, Austria b Herbert Raab, chönbergstr. 3/1, A-400 Linz, Austria; tars, Asteroids, and even the (pseudo-)nuclei of comets, are point-sources of light. In recent times, most observers use CCDs to observe these objects, so it might be worthwhile to think about some details of detecting and measuring point sources with a CCD. First, this paper discusses the properties of point-sources, and how they can describe them with a small set of numerical values, using a Point pread Function (PF). Then, the sources of noise in CCD imaging systems are identified. By estimating the signal to noise ratio (NR) of a faint point source for some examples, it is possible to investigate how various parameters (like exposure time, telescope aperture, or pixel size) affect the detection of point sources. Finally, the photometric and astrometric precision expected when measuring faint point sources is estimated. Introduction Modern CCD technology has enabled amateur astronomers to succeed in observations that were reserved to professional telescopes under dark skies only a few years ago. For example, a 0.3m telescope in a backyard observatory, equipped with a CCD, can detect stars of 0 mag. However, many instrumental and environmental parameters have to be considered when observing faint targets. Properties of Point ources In long exposures, point sources of light will be smeared by the effects of the atmosphere, the telescope optics, vibrations of the telescope, and so forth. Assuming that the optics are free of aberrations over the field of the CCD, this characteristic distribution of light, called the Point pread Function (PF) is the same for all point sources in the image. Usually, the PF can be described by a symmetric Gaussian (bell-shaped) distribution (figure 1) [1]: 0 0 ( x x ) + ( y y ) σ I ( x, y) = H e + B (1) I (x,y) is the intensity at the coordinates (x,y), which can be measured form the image. By fitting the PF to the pixel values that make up the image of the object (figure 1), the quantities x 0, y 0, I, σ and B can be found, which characterize the point source as follows: Width In equation 1, the width of the Gaussian PF is characterized by the quantity σ. In astronomy, the width of the PF is frequently specified by the socalled Full Width Half Maximum (FWHM). As the name implies, this is the width of the curve at half its height. The FWHM corresponds to approximately.355 σ. Although a number of factors control the FWHM (like focusing, telescope optics, and vibrations), it is usually dominated by the seeing. The FWHM is the same for all point-sources in the image (if optical aberrations can be neglected). Most notably, it is independent of the brightness of the object. Bright stars appear larger on the image only because the faint outer extensions of the PF are visible. For faint stars, these parts drown in the noise and are therefore not visible. Background During the exposure, the CCD not only collects signal from the object, but also light from the sky background and the thermal signal generated within the detector. These signals result in a pedestal (B) on which the PF is based. Ideally, the background signal is the same over the whole field for calibrated images. In practice, however, it will vary somewhat over the field. Position The position of the object in the CCD frame can be expressed in rectangular coordinates (x 0, y 0 ), usually along the rows and columns of the CCD. Fitting a PF to the image will allow to calculate the position of the object to a fraction of the pixel size. Intensity The height of the PF (H) is proportional to the magnitude of the object. The total flux of the objects corresponds to the integrated volume of the PF, less the background signal (see below). Figure 1: Image of a star on a CCD (left), and the Gaussian PF fitted to the image data (right).

4 in limiting magnitude is about 1. mag, as compared to example 1, or 1.4 mag as compared to example 3. Pixel ize and ampling Apparently, the relative size of the pixel to the FWHM of the stellar images is an important factor in obtaining the highest possible NR. By performing calculations similar to the NR estimates in the previous chapters, it can be shown that the highest Peak NR is obtained when the pixels are about 1. FWHM in size (figure 4). Figure 5: Undersampled (left), critically sampled (center) and oversampled (right) stellar images (top row), and the PF fitted to the image data (bottom row). Error Estimates Figure 4: Variation of peak NR for various pixel scales. The pixel size is measured in units of FWHM. With such large pixels, most of the photons are collected by the single pixel on which the PF of the stellar image is centred, whilst only the fainter, noisy wings of the PF fall on the neighbouring pixels, resulting in a high NR. However, with almost all the light concentrated in a single pixel, it would be very difficult to distinguish real objects from image artefacts (like hot pixels or cosmic ray strikes), and it is impossible to calculate the precise position of the object to sub-pixel accuracy. To retain the information of the objects on the CCD image, the scale must be chosen so that the FWHM of stellar sources spans at least 1.5 to pixels [4]. This scale is called critical sampling, as it preserves just enough information that the original PF can be restored by some software analysing the image. With even larger pixels (i.e., less than 1.5 pixels per FWHM), the PF can not be restored with sufficient precision, and astrometric or photometric data reduction is inaccurate, or not possible at all. This situation is called undersampling. In the other extreme ( oversampling ) the light of the object is spread over many pixels: Although the PF of stellar objects can be restored with high precision in this case, the NR is decreased (figure 5). Critically sampled images will give the highest NR and deepest limiting magnitude possible with a given equipment in a certain exposure time, without loosing important information contained in the image. For applications that demand the highest possible astrometric or photometric precision, one might consider some oversampling. The same is true for pretty pictures, as stars on critically sampled images look rather blocky. Fitting a PF profile to a faint, noisy detection is naturally less precise than for bright stellar images with a high NR (figure 6). Position and brightness calculated for faint detections are therefore expected to be less precise than for bright objects. Figure 6: Gaussian PF fitted to a faint (Peak NR ~4) and a bright (Peak NR ~100) stellar image. The fractional uncertainty of the total flux is simply the reciprocal value of the ignal to Noise Ratio, 1 NR (sometimes also called the Noise to ignal Ration). By converting this uncertainty to magnitudes, we get: PHOT 1 Log(1 + ) = NR Log(.5) σ (6) Here, σ PHOT is the one-sigma random error estimated for the magnitude measured, and NR is the total NR of all pixels involved (e.g., within a synthetic aperture centred on the object). By modifying equation 5, we can find this value from: NR = (7) + n ( B + T +σ ) In this formula, is the total integrated signal from the object in the measurement (i.e., within the aperture), and n is the number of pixels within the aperture. The other quantities are identical to equation 3. It should be noted that, as both and n will change with the diameter of the R

5 aperture, the total NR varies with the diameter of the photometric aperture, so photometry can be optimised by choosing the appropriate aperture [5]. Figure 7: The photometric error (in stellar magnitudes) expected for point-sources up to a NR of 50. Figure 7 shows the expected uncertainty in the magnitude for point sources up to NR 50, as calculated from equation 6. Equation 6 only estimates the random error in photometry due to image noise. It does not account for any systematic errors (like differences in spectral sensitivity of the CCD and the colour band used in the star catalogue) that might affect absolute photometric results. Provided that the stellar images are properly sampled, the astrometric error can be estimated using this equation [6]: σ PF σ AT = (8) NR Here, σ AT is the estimated one-sigma error of the position of the object, σ PF the Gaussian sigma of the PF (as in equation 1), and NR is the Peak ignal to Noise Ratio of the object. Note that σ AT will be expressed in the same units as σ PF (usually arc seconds), and that σ PF can be calculated from FWHM.355. calculated from equation 8. Again, equation 8 only estimates the random error in the stellar centroid due to image noise. It does not account for any systematic errors (introduced by the astrometric reference star catalogue, for example) that might affect absolute astrometric results. Returning to example 1, the astrometric one-sigma error expected for the point source with a Peak ignal to Noise Ratio of 4.1 and a FWHM of 4 can now be estimated to ~0.4, using equation 8. Adopting a photometric aperture with a diameter of 3 FWHM (covering 18 pixels), a total ignal to Noise Ratio of about 3.3 is found by using equation 7. From equation 6, the photometric error is estimated to ~0.3 mag. An astrometric error of ~1 is acceptable, particularly if it is a observation of a minor planet with a uncertain orbital solution or a large sky-plane uncertainty (for example, as in the case of late follow-up or recovery observations). Observations of the light curve of a minor planet usually require a precision of 0.05 mag or better, corresponding to a NR of 0 or higher. Obviously, photometric observations are much more demanding than astrometry. ummary and Conclusions This paper first described the characteristics of a Gaussian Point pread Function, and the sources of noise in the imaging system. A few examples, estimating the ignal to Noise Ratio obtained for faint point sources with various telescope setups, highlighted that environmental conditions, telescope equipment, and CCD detector must harmonise to operate at peak performance. Finally, the astrometric and photometric error expected when measuring faint point sources was estimated. Useful astrometric results can be obtained even for very faint targets at the limit of detection, particularly if the skyplane uncertainty for the object under observation is large. For photometric studies, a higher NR is desirable. References 1. Auer, L. H.; van Altena, W. F.: Digital image centering II, Astronomical Journal, 83, (1978). cientific Imaging Technologes, Inc: ITe cientific-grade CCD 3. Berry, R; Burnell, J.: The Handbook of Astronomical Image Processing, Willmann-Bell, Inc. (000) 4. Howell,. B.; Koehn, B; Bowell, E.; Hoffman, M.: Detection and Measurement of poorly sampled Point ources images with -D Arrays, The Astronomical Journal, 11, (1996) 5. Naylor, T.: An optimal extraction algorithm for imaging photometry, Monthly Notices of the Royal Astronomical ociety, 96, (1998) Figure 8: The astrometric error (in units of the FWHM) expected for point-sources up to a Peak NR of 50. Figure 8 shows the expected uncertainty in the position (in units of FWHM) for point sources up to NR 50, as 6. Neuschaefer, L. W.; Windhorst, R.A.: Observation and Reduction Methods of deep Palomar 00 inch 4-hooter Mosaics, The Astrophysical Journal upplement eries, 96, (1995)

6 Appendix ignal to Noise Ratio Estimation Example 1 Example Example 3 Example 4 Example 5 Telescope Mirror Diameter 0.60 m 0.60 m 1.50 m 1.50 m 1.50 m Obstruction 0.0 m 0.0 m 0.50 m 0.50 m 0.50 m Light Collecting Area 0.5 m² 0.5 m² 1.57 m² 1.57 m² 1.57 m² Local Length 1.98 m 1.98 m 7.00 m 7.00 m 7.00 m Focal Ratio Detector Pixel ize 4 µm 4 µm 4 µm 48 µm 4 µm Pixel cale.50 /Pixel.50 /Pixel 0.71 /Pixel 1.4 /Pixel 0.71 /Pixel Dark Current 1 e /s/pixel 1 e /s/pixel 1 e /s/pixel 4 e /s/pixel 1 e /s/pixel Readout Noise 10 e 10 e 10 e 0 e 10 e Quantum Efficiency 70 % 70 % 70 % 70 % 70 % Integration Time 100 s 600 s 100 s 100 s 100 s Object and ky Object Magnitude 0 mag 0 mag 0 mag 0 mag 0 mag ky Background 18 mag / 18 mag / 18 mag / 19 mag / 19 mag / FWHM NR Calculation Object Flux 11'000 γ γ γ γ γ Object ignal e e e e e hare for central Pixel Object Flux in centr. Pixel γ γ γ γ γ Object ignal in centr. Pixel 33 e e e 5 09 e 5 09 e Object Noise in centr. Pixel 47 e 116 e 36 e 71 e 71 e Background Flux γ/pixel ' γ/pixel γ/pixel γ/pixel γ/pixel Background ignal e /pixel e /pixel e /pixel e /pixel e /pixel Background Noise 548 e /pixel 134 e /pixel 390 e /pixel 780 e /pixel 390 e /pixel Dark Current 100 e /pixel 600 e /pixel 100 e /pixel 400 e /pixel 100 e /pixel Dark Noise 10 e /pixel 5 e /pixel 10 e /pixel 0 e /pixel 10 e /pixel Noise in centr. Pixel 550 e 134 e 39 e 783 e 397 e Peak NR Table 1: ummary of the NR calculations mentioned in the text. Example 1 is described in some detail in the paper. Note that, for example 4, the pixel size listed in the table is not the physical size, but the site of the binned pixel, and all other data refer to the binned pixel.

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