Asteroid Hunting
Bruce L. Gary, Hereford Arizona Observatory (G95)

Overview

Asteroid hunting, or trying to discover an asteroid that's not already cataloged, is a big challenge for anyone using a telescope with an aperture smaller than 20 inches. This generally acknowledged wisdom, while true, does not mean that someone with only a 14-inch telescope can't discover an asteroid. The smaller telescope just means that the search will be "labor intensive" in both observing and image analysis compared with using a larger telescope for which observing and image analysis are relatively stratightforward. This web page explains what must be dealt with by the small telescope user wishing to try for an asteroid discovery.

I have two purposes in creating this web page: 1) explaining a thing helps me understand it better myself, and 2) having spent a lot of effort to derive an insight into relevant factors that should influence an observing and data analysis strategy, I want to share these insights with other amateurs who might be floundering the way I have with the same problem.

The Goal

The Minor Planet Center, MPC, requires that for two dates, preferably one or two days apart, positions of a candidate new asteroid be submitted that can "link" together with ressiduals of no greater than 1.5 "arc for all positions. Although magnitude estimates are not required it will be helpful for you as an observer to know how you're performing (SNR and residuals) for the magnitude range where you should be searching. As explained in the next section, you need to be capable of measuring the positions of asteroids fainter than 20th magnitude since almost all brighter asteroids have been cataloged.

Undiscovered Asteroid Statistics

There are more than a million asteroids already cataloged with known orbits and approximate brightnesses. A model for asteroid size distrribution and orbit sizes has been used to create a model for the total number of asteroids (discovered plus undiscovered) that are present in the night sky. This is shown in the next figure. Figure 1. Total number of asteroids versus brightness at the standard distance of 1 a.u. (for opposition illumination and viewing).
Numbers at the top are approximate magnitude at opposition (assuming a typical main belt orbit). Data from Bidstrup
et al (2004).

By comparing statistics for the already discovered asteroids with what would be expected using the model just referred to it is possible to estimate how many asteroids of any given size (brightness) remain undiscovered. This is shown in the next figure. Figure 2. Total number of "undiscovered asteroids" versus brightness at opposition (assuming a typical main belt orbit).

The number of undiscovered asteroids brighter than V-mag = 20 is less than one per 10 square degrees, and the number becomes vanishingly small for brighter asteroids. This is why the goal for discovery is to be able to detect and measure the position of asteroids fainter than 20th magnitude.

Notice that there's an approximate 10-fold increase in the number of undiscovered asteroids in going from V-mag = 20.0 to 21.0. The rewards for "going deep" are great. But so are the rewards for having a large field-of-view (FOV). As will be see below, striving for one payoff causes a loss in the other.

Limiting Magnitude

Limiting magnitude is usually defined as the V-magnitude of a point-source that produces a signal-to-noise ratio (SNR) of 3.0. As a practical matter when someone is asked to visually inspect an image and indicate an example of the faintest star in the image it will turn out toi have SNR of about 6, not 3.  The following figure shows limiting magnitude, ML, for a 14-inch Celestron. Figure 3. Limiting magnitude (unfiltered) versus total exposure time and FWHM ("atmospheric seeing"), derived empirically using an SBIG CCD and a clear filter. It is assumed that moonlight is not present (and there is minimal light pollution from nearby cities).

Notice that there's strong dependence of ML upon the half-power diameter of stars in the image (FWHM). Usually, FWHM is determined by "atmospheric seeing" but for a prime focus configuration, or a Cassegrain configuration with 2x2 or 3x3 binning, the value for FWHM may be so large that it is relatively unaffected by atmospheric seeing. For example, my Celestron at prime focus (using a Starizona HyperStar field flattening lens) produces star images that have FWHM ~9 "arc (7.2 "arc for the best seeing and 10 or 11 "arc for the worst seeing). At prime focus the image scale is 2.8 "arc/pixel, so even with perfect optics and no atmospheric seeing degradation the best possible FWHM would be ~5 "arc. Cassegrain configurations can employ a focal reducer lens for enlarging the FOV. With a 3.3x Celestron focal reducer attached to the back of the telescope the FWHM is typically ~3 "arc, whereas without the focal reducer it is typically 2.5 "arc. (My site is at 4650 feet altitude in Southern Arizona.)

In the figure the term "Exposure TIme" refers to the total exposure time of all images used to produce the image whose ML is being described. Every time an image is "read out" from a CCD "read noise" isadded to the noise that existed before read-out. Therefore, a one hour exposure will have a deeper ML than the combination of 60 one-minute images. Combining images can be done by averaging or median combining, and this choice will affect the final image's ML. These are some of the reasons that the slope of the lines in this figure do not agree with the theoretical slope that would exist for making just one exposure for the indicated time; instead, the slope is based on empirical data, using typical observing and image combining procedures.

It is important to understand why FWHM is so important in setting ML. I am assuming the user employs aperture photometry with the signal circle diameter equal to ~ 2.0 times FWHM. This is larger than the theoretically best SNR choice of 1.5 times FWHM, and it is prudent to not use anything smaller than ~2.0 so that small changes in a star's point-spread-function (PSF) will have negligible effect on the measured star flux. Note that PSF shape and size is likely to vary across an image, especially if the f-ratio is small (for Prime focus my f-ratio is 1.86). The PSF will also vary from images ot image due to seeing changes and focus setting errors. When FWHM is small, the number of pixels containing the star's signal will be small, and this allows the user to set the photometry signal circle to a smaller number of pixels. Since each pixel has "noise" (attributable to thermal effects in the instrument as well as temporal fluctuations of the sky background level) the smaller the number of pixels within the signal aperture the greater the ratio of star flux to noise. When FWHM is doubled, the area of the signal aperture must be quadrupled. This will lead to a doubling of noise due to stocahstic processes (such as thermal noise and sky background noise). Therefore, a doubling of FWHM causes SNR to be reduced by a factor of two. This, in turn, is equivalent to a 0.75 change in magnitude.That's what can be seen in the graph, above.

Asteroid Motion and Exposure Times

At opposition an asteroid is moving westward, or retrograde, due to the Earth's greater orbital velocity. There are locations along the ecliptic where the asteroid's motion is small, as it switches from retrograde to prograde, and this is typically at 70 degrees from the anti-solar point, as shown by the following figure. Figure 4. Rate of motion of an asteroid in an uninclined circular orbit versus angular distance from the sun along the ecliptic.

If an asteroid at opposition moves at 1 "arc/minute and if FWHM = 3 "arc, an exposure of 3 minutes would produce an oval shaped asteroid with dimensions ~4.5 x 3.0 "arc. A longer exposure would yield ever decreasing improvements in SNR, and ever-increasing problems of interpretation as well as a greater probability that the asteroid would be close to a background star that would bias the measurement of its position and magnitude. My rule-of-thumb is to keep the exposure shorter than 2/3 FWHM. In this case, the maximum exposure would be 2 minutes.

If the same asteroid were observed when it was ~70 degrees from the anti-solar point, however, its motion would considerably less. This would allow longer exposures, but two factors compensate for this gain of SNR due to a longer exposure: 1) the asteroid is farther from the Earth, and hence fainter, and 2) it is being viewed at less than full illumination (i.e., gibbous phase) without the benefit of the opposition brightening effect, which also makes it fainter. For example, asteroid 2005 US157 is predicted to be 1.3 magnitudes fainter when it's at 70 degrees from the anti-solar point than when it's at opposition. Equivalent exposure times for the two magnitudes (19.8 and 21.2) are 2 and 20 minutes (for SNR = 1 with a 14-inch telescope and 3.0 "arc FWHM). In other words, a 2-minute exposure at opposition, producing a 2 "arc smearing of the asteroid image in its direction of motion, will have the same SNR = 1 as a 20-minute exposure when it's at a location where it's moving much slower. For this specific asteroid case, chosen at random, it is better to observe it at opposition and stack 2-minute images in order to produce an image with sufficient SNR for the accurate measurement of its coordinates. The relationship between position accuracy and SNR is dealt with in the next section.

Position Accuracy versus SNR

When using PinPoint to plate solve (leading to coefficients that convert pixel coordinates to RA and Dec coordinates) it is common to achieve average residuals of 0.1 "arc for a Cassegrain configuration and 0.2 "arc for a prime focus configuration. These residuals are for the stars used in the plate solving procedure, which I am assuming are from the UCAC 2.0 star catalog. When I started observing asteroids I naievley assumed this average residual would also apply to any asteroid I had in the image. For bright asteroids this should be correct, but for faint ones the expected coordiante accuracy must degrade as SNR decreases. The way to understand this is to imagine the background level of the image to be a topographic surface with hills and valleys. The topography bumpiness is caused by sky background noise, CCD readout noise, thermal noise of the CCD electronics, imperfections of the dark frame used, and imperfections of the flat field frame used. The fainter the asteroid the greater these topographic features shift the apparent centrroid position ffect of the asteroid feature. I have empirically derived the following equation for for position uncertainty versus SNR:

Position Uncertainty  = Star Fit Residuals + 2.2 * FWHM ["arc] / SNR

For example, when SNR is large, Position Uncertainty = Star Fit Residuals (the thing PinPoint reports as average residual). This makes sense. But when SNR is small the second term dominates. For example, when FWHM = 9 "arc (forced high due to the prime focus image scale of 2.8 "arc) and SNR = 15, the average residual is ~1.5 "arc. This "expected average residual" is the same as the MPC requirement of 1.5 "arc, and is therefore unacceptable since about half the position errors will exceed the MPC threshold. A reasonable goal is to achieve an expected average residual of 1.0 "arc. Figure 5. Average residual position uncertainty versus SNR for three image scale values, based on Declination measurements of a faint asteroid on 4 dates.

If small SNR is to be useful it must be accompanied by a small FWHM and this favors Cassegrain configurations over a prime focus one. For example, when FWHM = 3 "arc (typical for one of my Cassegrain configurations) the desired average residual of ~1.0 "arc can be achieved with SNR = 7.

Image analysis Strategies

I have already described how each individual image should not be exposed for longer than the time it takes for the asteroid to move ~2/3 of a FWHM distance against the background stars. Recall that we are searching for an asteroid that is probably too faint to be seen in any of the individual images. How, then are we to align the images for stacking (using either averaging or median combining)? If we use "star alignment" to register the images the resultant stacked im age will further smear the asteroid. Although we can't use the asteroid for alignment we can make use of our knowledge of a typical motion for asteroids in the part of the sky we're observing. One way to get such a motion vector is to use the MPC's web page for producing a list of asteroids in the part of the sky we're observiong. Another way is to quickly process the bright asteroids in our FOV (in a prime focus FOV I typically see about a dozen asteroids). The MPC web page http://scully.harvard.edu/%7Ecgi/CheckMP allows the user to enterRA/Dec coordinates and retrieve information about asteroids in that area, including rates of motion in both RA and Dec.

Once a typical pair of motion rates have been established it is possible to stack images so that most asteroids in the FOV are approximately aligned. This can be accomplished by pixel editing an "offset alignment dot" (OAD) in each image and then combining the images using a "single star alignment" method where the OAD is the specified "star." In this way the SNR for most asteroids in the FOV will increase to the level where they can be "seen." The best way to "see" the asteroids after this SNR enhancement is to blink two or three such images (using star alignment, so that the asteroids move). The procedure for adding OADs is summarized here:

1) Measure the x,y coordinates of a star to be used as an offset reference
2) Note the UT for that image
3) Using a spreadsheet (or calculator) calculate how much movement a typical asteroid is expected to move in both x and y
4) Add an offset to the above delta-x and delta-y values (so that the OAD will end up in a star free region)
5) Add the results of steps 1 and 4; this is the coordinate for the OAD
6) Pixel edit a "bright" dot at this location in the iamge
7) Repeat the above steps for other images; do this for 8 images, for example
8) Median combine (or average) the first 4 images using the OAD for alignment; repeat for the second 4 images
9) Blink the two new images with enhanced SNR and search for asteroids

Note that if groups of 4 images are combined for the SNR enhancement the asteroid's SNR will approximately double. This is comparable to being able to expose 4 times longer without asteroid smear, or observe for the same time with a telescope having an aperture root-two larger (i.e, a 14-inch telescope will perform as if it's a 20-inch telescope).

Determining RA and Dec coordinates for an asteroid that's seen only in a SNR-enhanced image can be difficult. If the stars aren't smeared too much a PinPoint plate solving should be acceptable. But if the stars are smeared a lot the asteroid coordinated may be determined by noting the asteroid's x,y location and then going back to the first image in the SNR-enhanced set (of 4 images) and reading the RA/Dec coordiantes for that x,y location. (This works with MaxIm DL; I don't kow about other programs.)

FOV and Deepness Tradeoffs

In this section I bring together all the considerations of the previous sections to demonstrate a trade-off analysis meant to choose the best telescope/CCD configuration. I'll use my telescope to illustrate specific configuration options. My hardware consists of a Celestron CGE-1400 (14-inch aperture Schmidt-Cassegrain), a SBIG ST-8XE CCD, a SBIG CFW-8 color filter wheel, a Celestron 3x focal reducer lens, a SBIG AO-7 tip/tilt image stabilizer and a Starizona HyperStar prime focus field flattening lens. Various combinations are possible, leading to large, medium and small FOVs, as summarized here (using abbreviations for some of this hardware):

Config A:  Prime focus (HyperStar, CFW, CCD), FOV = 72 x 48 'arc, image scale 2.8 "arc/pixel, FWHM typically 9 "arc
Config B:  Cassegrain large FOV (focal reducer, AO, CFW, CCD), FOV =  24.5 x 16.3 'arc, image scale = 0.96 "arc/pixel, FWHM typically 4 "arc
Config C:  Cassegrain small FOV (CFW, CCD), FOV = 12.7 x 8.5 'arc, image scale = 0.50 "arc/pixel, FWHM typically 3 "arc

The longest exposure for individual images (assuming an asteroid motion of 0.8 "arc/minute, and adopting my 2/3 of FWHM rule) are 7.5 minutes, 3.3 minutes and 2.5 minutes for the three configurations. Note that it is possible to expose for 1/3 as long and median combine 3 images (to be rid of cosmic ray defects, or imperfect dark frame subtraction). Doing this, however, adds "CCD read noise" to each image, and median combining will add an additional 15% to the background noise level.

Let's now assume that we want to submit 3 positions to MPC, with a UT midpoint spacing of at least one hour. Let's devote an hour of observation to each position (i.e., calling for a 3-hour observing session, following set-up time). We shall now pose the question: What is the probability of discovering an asteroid with this 3-hour observing session corresponding to each of the three configurations?

Let's consider Config A. Referring to Fig. 5 we note that for "Config A" we need SNR >26 to assure position accuracies of <1.0 "arc (for each coordinate). From Fig 3 we learn that a 7.5-minute image with FWHM = 9 "arc produces SNR = 3 for CV = 19.3. Combining 4 such images (using the OAD procedure) yields SNR = 6. This set of 4 images requires 30 minutes of observing, so in one hour we could combine another set of 4 images to achieve SNR = 8.5. We need SNR = 26 for position accuracy purposes, but CV = 19.3 produces only SNR = 8.5 in one hour. To overcome an SNR ratio = 26 / 8.5 = 3.1 we must resort to asteroids that are 3.1 times brighter (1.2 magnitude). Thus, after one hour of observing we will have an SNR sufficient to assure 1.0 "arc positional accuracy for asteroids with CV brighter than 18.1. According to Fig. 2 there are no undiscovered asteroids this bright. Thus, this configuration appears to be unuseable for discovering asteroids.

Let's consider Config B. Referring to Fig. 5 we note that for "Config B" we need SNR >10 to assure position accuracies of <1.0 "arc (for each coordinate). From Fig 3 we learn that a 3.3-minute image with FWHM = 4 "arc produces SNR = 3 for CV = 19.7. Combining 4 such images (using the OAD procedure) yields SNR = 6. This set of 4 images requires ~14 minutes of observing, so in one hour we could combine a total of 4 sets of 4 images to achieve SNR = 12. We need SNR = 10 for position accuracy purposes, and CV = 19.7 produces SNR = 12 in one hour. We have more SNR than necessary for this magnitude, so we can convert the extra SNR ratio = 12 / 10 = 1.2 to an asteroids that is 1.2 times fainter (0.2 magnitude). Thus, after one hour of observing we will have an SNR sufficient to assure 1.0 "arc positional accuracy for asteroids with CV brighter than 19.9. According to Fig. 2 there are ~0.13 undiscovered asteroids per square degree that are brighter than this. Our FOV area is 0.11 square degree, so for this one observing session we have a probability of discovering an asteroid of ~0.11 * 0.13 = 1.4%.

Let's consider Config C. Referring to Fig. 5 we note that for "Config C" we need SNR >8 to assure position accuracies of <1.0 "arc (for each coordinate). From Fig 3 we learn that a 2.5-minute image with FWHM = 3 "arc produces SNR = 3 for CV = 19.7. Combining 4 such images (using the OAD procedure) yields SNR = 6. This set of 4 images requires ~10 minutes of observing, so in one hour we could combine a total of 6 sets of 4 images to achieve SNR = 14.7. We need SNR = 8 for position accuracy purposes, and CV = 19.7 produces SNR = 14.7 in one hour. We have more SNR than necessary for this magnitude, so we can convert the extra SNR ratio = 14.7 / 8 = 1.8 to an asteroids that is 1.8 times fainter (0.6 magnitude). Thus, after one hour of observing we will have an SNR sufficient to assure 1.0 "arc positional accuracy for asteroids with CV brighter than 20.3. According to Fig. 2 there are ~0.4 undiscovered asteroids per square degree that are brighter than this. Our FOV area is 0.030 square degree, so for this one observing session we have a probability of discovering an asteroid of ~0.030 * 0.4 = 1.2%.

This is discouraging. However, the result is sensitive to the model for residual uncertainty versus SNR and FWHM, as the next section demonstrates.

Sensitivty of Result on Position Uncertainty versus SNR Model

Notice that Fig. 5 is based on Declination measurements of a faint asteroid on 4 dates. The following figure shows observed residuals for both declination and right ascension. Figure 6. Observed residuals for declination (top pasnel) and right ascension (bottom panel) for a faint asteroid (CV = 20.3) using a 14-inch telescope in a prime focus configuration (FWHM = 9 "arc). The grey dashed trace is for Residual = 0.2 + 2.2 * FWHM / SNR, whereas the green trace is for Residual = 0.1 + 0.7 * FWHM / SNR.

Notice that the top panel has 4 "outliers" (above the grey dashed trace) out of a total 16 data points. These may have been produced by faint stars too close to the asteroid when the image was made that the declination position was "pulled" away from the asteroid's declination. If this set of data suffered from "bad luck" then the lower traces should be adopted. If the four outliers can be identified as outliers by the observer before they're submitted for MPC to evaluate, such as by their unusual brightness compared to other readings, then they can be rejected by the observer before submission. Since I do not yet know if these are outliers that can be identified by the observer before submission to MPC it is prudent to consider the more optimistic green trace for evaluating the feasiblity of discovering asteroids with a small telescope. It may be worth the effort of employing a technique of "image subtraction" to remove interfering stars (Gary and Healy, 2005). On the assumption that the outliers can eventually be dealt with I will repeat the analysis of the previous section using the green trace model, as shown for 3 different FWHM values in the next figure. Figure 7. Revised (optimistic) relationship between average residual position uncertainty and SNR for FWHM values 9, 4 and 3 "arc.

These traces can be described by the following equation:

Average Residual Position Uncertainty  = Star Fit Residuals + 0.73 * FWHM ["arc] / SNR

Config A: Referring to Fig. 7 we note that for "Config A" (FWHM = 9 "arc) we need SNR >9 to assure position accuracies of <1.0 "arc (for each coordinate). From Fig 3 we learn that a 7.5-minute image with FWHM = 9 "arc produces SNR = 3 for CV = 19.3. Combining 4 such images (using the OAD procedure) yields SNR = 6. This set of 4 images requires 30 minutes of observing, so in one hour we could combine another set of 4 images to achieve SNR = 8.5. We need SNR = 9 for position accuracy purposes, but CV = 19.3 produces only SNR = 8.5 in one hour. To overcome an SNR ratio = 9 / 8.5 = 1.1 we must resort to asteroids that are 1.1 times brighter (0.1 magnitude). Thus, after one hour of observing we will have an SNR sufficient to assure 1.0 "arc positional accuracy for asteroids with CV brighter than 19.6. According to Fig. 2 there are might be 0.02 undiscovered asteroids per square degree that are brighter than this. Our FOV area is 0.96 square degree, so for this one observing session we have a probability of discovering an asteroid of ~0.96 * 0.02 = 2%. After observing 50 such sky areas there will be an approximate 50% probability of discovering an asteroid.

Config B: Referring to Fig. 7 we note that for "Config B" (FWHM = 4 "arc) we need SNR >3.3 to assure position accuracies of <1.0 "arc (for each coordinate). From Fig 3 we learn that a 3.3-minute image with FWHM = 4 "arc produces SNR = 3 for CV = 19.7. Combining 4 such images (using the OAD procedure) yields SNR = 6. This set of 4 images requires ~14 minutes of observing, so in one hour we could combine a total of 4 sets of 4 images to achieve SNR = 12. We need SNR = 3.3 for position accuracy purposes, and CV = 19.7 produces SNR = 12 in one hour. We have more SNR than necessary for this magnitude, so we can convert the extra SNR ratio = 12 / 3.3 = 3.6 to an asteroids that is 3.6 times fainter (1.4 magnitude). Thus, after one hour of observing we will have an SNR sufficient to assure 1.0 "arc positional accuracy for asteroids with CV brighter than 21.1. According to Fig. 2 there are ~2.5 undiscovered asteroids per square degree that are brighter than this. Our FOV area is 0.11 square degree, so for this one observing session we have a probability of discovering an asteroid of ~0.11 * 2.5 = 27%. Two observing sessions should produce an approximate 50% probability for success in discovering an asteroid.

Config C: Referring to Fig. 7 we note that for "Config C" we need SNR >2.4 to assure position accuracies of <1.0 "arc (for each coordinate). From Fig 3 we learn that a 2.5-minute image with FWHM = 3 "arc produces SNR = 3 for CV = 19.7. Combining 4 such images (using the OAD procedure) yields SNR = 6. This set of 4 images requires ~10 minutes of observing, so in one hour we could combine a total of 6 sets of 4 images to achieve SNR = 14.7. We need SNR = 2.4 for position accuracy purposes, and CV = 19.7 produces SNR = 14.7 in one hour. We have more SNR than necessary for this magnitude, so we can convert the extra SNR ratio = 14.7 / 2.4 = 6.1 to an asteroids that is 6.1 times fainter (2.0 magnitude). Thus, after one hour of observing we will have an SNR sufficient to assure 1.0 "arc positional accuracy for asteroids with CV brighter than 21.7. According to Fig. 2 there are about 5 undiscovered asteroids per square degree that are brighter than this. Our FOV area is 0.030 square degree, so for this one observing session we have a probability of discovering an asteroid of ~0.030 * 5 = 15%. After 4 observing sessions of different star fields near the ecliptic there should be an approximate 50% probability of discovering an asteroid.

The "winner configuration" is "B":
Cassegrain large FOV (focal reducer, AO, CFW, CCD), FOV =  24.5 x 16.3 'arc, image scale = 0.96 "arc/pixel, FWHM typically 4 "arc. This was also the winning configuration under the more pessimistic residual uncertainty model. But there's a large difference in feasibility between the two models.

Tips for Going Deep

Moon: Moonlight can reduce limiting magnitude by 1 or 2 magnitudes, so serious asteroid hunting should be restricted to times when the moon is below the horizon. If you really want to try to observe in the presence of moonlight then use an R-band filter, since the degradation is significantly reduced. There's a SNR penalty for using an R-band filter instead of a clear filter, and for my system it varies from 0.36 at low air mass (m=1.2) to 0.39 at high air mass (m=3). I-band is not worth it although it's relatively unaffected by moonlight; the SNR penalties are ~0.24 for all air masses.

Darks: Calibrating with a dark frame (or master dark frame) adds noise to the light frame image. Therefore, lots of darks are a must when trying to "go deep." One or two dozen should be enough for the master dark frame to not add significantly to the background noise level during the calibration process. I prefer to devote at least a half-hour to dark frames on every night that I'm doing serious observing, and I make sure the darks are all at the same CCD temperature as the asteroid "light" frames. It is well known by observationalists that for the case of no change in star field pixel location you should spend as much time taking dark frames as light frames; the only reason to take fewer darks than lights is that the star field moves with respect to the field of pixels between exposures.

Flats: Flat frame calibration adds noise, so there are only two reasons for doing it: 1) when there are dust donuts that could interfere with the measurement of an asteroid's position or flux, and 2) when you want to achieve an accurate brightness measurement, or variation of brightness with time during an observing session (as in "rotation light curve"). For asteroid work, where brightness accuracy is not important (i.e., when it's OK for accuracy to be no better than 5%, or 0.05 magnitude), it is not important to have an accurate flat frame but it is important to have a precise one. OK, that needs explanation. Making a flat frame that is precise is easy: you just set the focus to what it should be and average, or median combine, lots of images of a uniformly illuminated white screen (or the zenith sky at sunset with a couple T-shirts over the aperture). A master flat made from averaging 16 flats has 4 times less noise than an individual flat frame. The goal is to minimize additional background noise during the flat frame calibration process.

CCD Temperature: Cooling the CCD reduces thermal noise. For every 6 degrees C of additional cooling the CCD thermal noise ("dark current") can be reduced by a factor of two. For example, going from an uncooled +10 C to a cooled -20 C leads to a 30-fold reduction in the CCD's thermal noise level. However, the sky background contribution of noise will be present so there will be diminishing returns by cooling below a temperature where the two noise levels are comparable. Hence, when moonlight is present there is less to be gained by cooling than when the sky is dark. Also, cooling the CCD will have greater payoffs near zenith than close to the horizon. Professional observatories are located at dark sky locations, and they can benefit by additional cooling. That's why they use liquid nitrogen (at -173 C) to cool their CCD to a constant value near -100 C.

Seeing and Focus: The better the seeing, the deeper you can go! This point is abundantly clear from Fig. 3. Of course, to take advantage of good seeing you need to stay well focused. The two factors work together; for example, good seeing with poor focus is equivalent to poor seeing with good focus. I use a graph with lines showing how focus setting changes with temperature. I have different colored lines for each filter (note: filter parfocality is worse for "fast" f-ratios). My focus setting can be read as a number (between 0 and 9999) using a wireless focuser (Starizona's MicroTouch). After the telescope has equilibrated to the ambient air temperature (an hour or two after sunset) this graph can be a useful guide in maintaining an approximately correct focus. Every couple hours I interrupt observing to perform a manual focus check (plotting FWHM versus focus setting, and plotting by hand). A too-casual attitude about maintaining the best focus is equivalent to not caring about the cost of buying whatever aperture telescope and support equipment you have.

Photometry Aperture Size: The best SNR is achieved when the diameter of the photometry aperture is 1.5 x FWHM. Smaller apertures not only suffer from a larger SNR, they introduce systematic errors when attempting to measure brightness (doing "photometry"). To avoid the introduction of systematic errors in any photometry that might be attempted it is prudent to use an aperture diameter of about 2 x FWHM. For faint objects there's another reason for using a larger aperture than for bright objects, and that has to do with the influence of noise in the background on the measured flux and location of the faint object. Using a larger signal aperture reduces this noise biasing. The lower the SNR the more influence background noise will have on the pixel placement corresponding to a maximum flux reading. When SNR <~10 I choose a pixel location corresponding to the "centroid" shown in MaxIm DL's real-time display; the aperture's pixel location is always close to the one giving a maximum flux reading (called "intensity" by MaxIm DL), but it is sometimes not the same and it is my impression that the centroid location agrees with my visual impression of the asteroid's location. I can't give quantitative descriptions of this because I haven't studied it yet and I haven't read about it.

Aperture Recovery Fraction: If astrometry is the only goal then this paragraph can be ignored. This tip is meant for those who will want to do photometry of the faint object. Since the object is faint, you will be using a small signal aperture diameter, ~3 x FWHM. If the PSF is a perfect Gaussian use of this aperture size will lead to a reading of ~99% of the flux that's registered in the image. I'll refer to this fraction as "F" and express it as a %. The PSF is never a perfect Gaussian, and this means that the aperture circle will recover an even smaller percentage of what's in the image (the central obstruction produces a non-Gaussian PSF; image movement during the exposure adds to this). When diameter ~3 x FWHM it is common for F ~95 to 98%; using 2 x FWHM can prodcue F = 90%. This last F value would lead to a magnitude error of 0.1 if it is not dealt with properly. I evaluate fr using a bright star with no others nearby (in the region of the image close to the asteroid). The asteroid's measured flux can then be converted to a magnitude using the F value, as illustrated by the following equation:

CV = 21.35 - 2.5 * LOG ( S / (g * F) ) - 0.15 * m + 0.67 * C

where CV is a clear-filter V-magnitude equivalent, S is measured star flux ("or Intensity"), g is exposure time, F =s recovery fraction, m is air mass and C is star color (defined as V-R-0.31, or 0.57 *(B-V)-0.30, whichever is more convenient). The constants 21.35 and +0.67 have been determined empirically for my telescope and CCD system (using Landolt standard stars). The coefficient 0.15 is a zenith extinction [magnitudes per air mass] for my site, which varies a small amount with season and has been determined using many flux versus air mass plots. This "Simplified Magnitude Equation" (SME)is for converting clear filter observations to a V-magnitude equivalent, and it assumes that the star's "color" is known. The color parameter C has been defined so that when the star's color is unknown a good first assumption is that C = 0 (since a typical star has C = 0). For asteroids the most likely value for C = +0.19. More information on this method for converting measured flux to magnitude can be found at http://brucegary.net/photometry/x.htm

Averaging versus Median Combining: Median combining (MC) is meant to reduce the influence of cosmic ray artifacts and imperfect dark subtraction defects (caused by using a dark frame with a different CCD temperature or exposure time). MC does a good job of this, but you pay a SNR price of about 15% to 20%. Because of this SNR penalty it is not advisable to MC images that have already been produced by an earlier stage of MC. You need a minimum of 3 images for the MC to work (I prefer to use 4 when they're available and when asteroid motion smearing allows it). If SNR is not a concern, and when asteroid motion smearing is not an issue, then it's OK to MC a large number of images. But for maximum SNR it is better to average images. When averaging instead of MC'g it is prudent to visually inspect each image and reject those with artifacts near the asteroid. When you want to determine the asteroid's brightness it is much safer to use averaging instead of MC'g (and never use the hot pixel removal tool).

Progress Report

So far I've only observed one star field in search of an uncataloged asteroid (using "Config A"). I "detected" an uncataloged asteroid but my residuals exceeded MPC's acceptance threshold of 1.5 "arc on the first 3 dates of observation. The 4th date was excellent, having residuals of ~0.4 "arc in both coordinates, but by then another observatory (Catalina Sky Survey) had observed the same asteroid with acceptably small residuals (using my good observing date and their good observations to establish an orbit). Technically, the discovery of "2005 US157" goes to the Catalina Sky Survey, and that's what happened. But this was a close call, and my luck was probably affected by the asteroid being too close to interfereing stars on the first three dates (producing the outliers discussed in the previous section).

Based on this analysis I'm going to try "Cofig B" a couple times and try to verify the residuals model. Future results will be added to this web page.

References

Bidstrup, Philip R., Rene Michelsen, Anja Andersen and Henning Haack, Astron. and Astrophys., August 10, 2004.

Gary, Bruce L. and David Healy, The Minor Planet Bulletin, now published. (An extended treatment of the same matter can be found at http://brucegary.net/Ast46053/x.htm

-----------------------  SOME OF MY OTHER ASTRONOMY WEB SITES ----------------------

Tutorials

Photometry for Dummies  For several quick/sloppy procedures

Photometry for Smarties  Novel new method (intuitive) for high accuracy

CCD Transformation Equations Explained  And derived (link used by AAAVSO)

Atmospheric Seeing Degradation  Atmospheric theory & movie demo

Exoplanet Observing Strategies  "How to" suggestions (link used by AAVSO)

CCD Imaging Tips  "How to" suggestions

Photometry Error Estimation  Stochastic, systematic and total SE (link used by AAVSO)

Miscellaneous

Hardware "Hereford Arizona Observatory" (G95)

Amateur Counterpart of HST 3.5-day exposure (mag 22.8 vs 30.7)

The Big Picture  Overview of immense scale of time and space of the universe

AstroPhotos Pretty pictures (plus links to many other pages)

Professional and Personal  (everything branches off from this page)

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First created: 2005.11.08   Last updated: 2006.03.30