Tips for Amateurs Observing Faint Asteroids
B. L. Gary, Hereford Arizona Observatory (G95)
This web page has two purposes: 1) to help amateurs with modest hardware who want to observe faint asteroids (V-mag > 17), and 2) to document for skeptical professionals that amateurs are capable of producing scientifically useful light curves for asteroids as faint as mid-19th magnitude. The methods described are based on 6 months of observing many Near Earth Asteroids with H > 20. The hardware used can be described as modest by amateur standards: a 14" Meade LX200GPS telescope with a SBIG ST-10XME CCD camera in the Cassegrain location. It is shown that rate of motion is an important constraint on the faintness of asteroids for which light curve generation is feasible. I conclude that the faintest asteroids for which scientifically useful rotation light curves can be obtained with an average quality 14" aperture telescope is V-mag ~ 19.6.
Links on this web page:
    Observing Procedure  
    Image Reduction Procedure  
    Excel Spreadsheet Analysis  
    Asteroid Feasibility "Regions"  


This web page was motivated by a perception among professional astronomers that amateurs cannot observe asteroids fainter than ~ 17th magnitude for the purpose of generating scientifically useful light curves. I became aware of this belief when I encountered "raised eyebrows" and polite disbelief in describing to a professional astronomer my measurement of a 17th magnitude phase-folded rotation light curve with a 15-minute period based on observations with my 14" telescope. Since then I have observed almost a couple dozen Near Earth Asteroids (NEAs) fainter than ~ 17th magnitude in support of this astronomer's NASA-funded project, and there is no longer any doubt that amateur hardware is capable of providing useful LCs for asteroids as faint as magnitude 18.5.

I became curious to find out how faint I could go with my modest 14" telescope system, so I began experimenting with various hardware configurations (HyperStar and Cassegrain) and observing and analysis procedures. Part of my motivation was in anticipation of the time when my LCs were to be published and reviewer comments would question the feasibility of such good performance from amateur hardware, operated by an amateur. In order to answer this justified skepticism I realized that I would have to document my procedures and show how such performance can be achieved; that is one of the purposes for this web page, which might eventually take the form of a publication.

After investigating the source of stochastic and systematic uncertainties I have improved my capability sufficiently for the measurement of mid-19th magnitude asteroids and I believe that I have arrived at a fundamental limitation for the modest hardware that I can afford. Any additional improvements will require expensive mounts (e.g., Mathis), better quality optics (e.g., Hyperion) or larger apertures.

Here's an example of a phase-folded rotation light curve for a 19.2 magnitude asteroid, demonstrating that a 14" telescope with a dubious reputation for quality is capable of producing useful observations of asteroids this faint (provided they are "slow movers").

Figure 1a. Phase-folded LC of a Trojan asteroid made with a 14" telescope from typical observing sessions on 8 dates. Each data point is an average of the magnitude readings from 9 images. All images were "star subtracted" (explained in the text). The median SE for these data is 0.057 mag.

Figure 1b. Same data, but averaged in groups of 10. The median SE for these averaged data is 0.018 mag.

Many amateurs use the commercial program Canopus for image processing and light curve (LC) generation. Canopus was developed by Brian Warner, one of the most experienced and prolific asteroid observers who ranks higher than his official status of amateur (he was the first recipient of the Chambliss Award given by the American Astronomical Association in 2006 for his extensive contributions to asteroid observing, characterization, publication and archiving). Canopus is good for many things, especially bright asteroids, and those with well-established orbits, but to make use of some of the techniques that I employ it is necessary to rely upon MaxIm DL for image analysis and photometry readings plus specially designed Excel spreadsheets for calibration and LC generation.


I refer to my observatory as HAO, for Hereford Arizona Observatory. The HAO is in Hereford, AZ, which is 90 miles SSE of Tucson, at a dark site near the border with Mexico, at an altitude of 4656 feet. I has a MPC site code of G95. A 14" Meade LX200GPS telescope (vintage 2006) is located in an 8-foot diameter ExploraDome in my backyard. Everything is controlled from my house via 100-foot cables in buried conduit. A more detailed description of my observatory is given at

Until very recently I autoguided using the 2nd chip in SBIG CCD cameras. The main problem with this arrangement is that the autoguider chip is small, and there are sometimes only faint stars within its small FOV. Autoguiding quality suffers when using faint stars, and poor autoguiding leads to increased point-spread-function (PSF) size and oblong shapes, which translate to reduced limiting magnitude and the inability to observe faint asteroids. The prime focus HyperStar configuration helped in this regard, since the autoguider's FOV was larger, but the penalty was an image scale that caused main chip PSF to be defined by image scale instead of the atmosphere. The HyperStar produces good quality images, but with an image scale of 1.95 "arc/pixel the PSF FWHM was never smaller than ~ 5 "arc. Limiting magnitude is related to PSF size that has large payoffs for small PSFs, as shown by the following equation for limiting magnitude (for a CCD with ~80% QE maximum):

    LM = 15.1 + 2.5 * LOG ((D^2 / PSF) + 1.25 * LOG (g/60s)

where D is telescope effective diameter [inches], PSF is FWHM ["arc], and g = exposure time [sec]. Limiting magnitude, LM, is defined for SNR = 3 when the photometry aperture radius [pixels] is twice the PSF's FWHM [pixels]. The importance of small PSF for limiting magnitude is shown in the next figure.

Figure 2. Limiting magnitude vs.PSF size for a selection of telescope apertures (assumes CCD QE max ~ 80%).

My Meade telescope was PEC trained (in RA) twice, but ~10% of the untrained 8-minute RA variation is still present without autoguiding. This imperfect tracking leads to oblong PSF shapes when exposure times exceed 10 or 15 seconds for the Cassegrain configuration, even with autoguiding using an SBIG 2nd chip. For the HyperStar configuration the imperfect tracking wasn't noticeable until exposure times exceeded 30 seconds, even with autoguiding using the SBIG 2nd chip.

The HyperStar configuration was good for having a large FOV, which was really important for fast-moving NEAs. But the limiting magnitude penalty caused by large PSFs, plus the loss of aperture due to 30% blockage from my large 10-position CFW (color filter wheel), convinced me to switch back to a Cassegrain configuration. This switch motivated me to explore a better way to autoguide. Dean Koenig, owner of Starizona (in Tucson), recommended an off-axis autoguiding system consisting of a 80 mm, f/5 telescope and a Starlight Xpress LodeStar X2 CCD. This is what I implemented in April, 2015.

Figure 1. Autoguiding system mounted on a Meade 14" telescope, consisting of a Orion Short Tube telescope and Starlight Xpress LodeStar X2 CCD. At the Cassegrain focus of the Meade 14" is a focuser, SBIG 10-position filter wheel, and SBIG ST-10XME CCD. The focuser is controlled by a Starizona MicroTouch wireless focuser.

The autoguiding system has an image scale of 4.3"arc/pixel and a field-of-view (FOV) of 53x42 'arc. Typical exposure times of 1 second produce autoguiding scatter of ~ 1.0 & 0.7 "arc in the RA/DE axes (Fig. 2). For the main chip of my SBIG ST-10XME CCD, a typical PSF is ~ 3.2"arc FWHM (it can be as small as 2.4 "arc), so the atmosphere dominates the PSF size.

Figure 2. Tracking performance on the first night of operation of the Orion Short Tube telescope and Starlight Xpress LodeStar X2 autoguiding system.

The SBIG CCD has an image scale of 0.73 "arc, unbinned, and a FOV = 27 x 18 'arc. I use 2x2 binning, partly because the atmospheric PSF does not require 0.73" arc resolution but also to reduce read noise and download time (from 10 seconds to ~ 3 seconds). This increases duty cycle, which improves "information rate" (the bottom line for evaluating any observing system). Note that even when atmospheric seeing improves to ~ 2.0" arc, as it does occasionally, there are ~ 3 pixels per FWHM, and this assures accurate photometry (even for high precision, bright stars).

For my Meade "mirror flop" is actually "mirror creep" as the telescope tracks through a large range of hour angles and gravity loading shifts. With an off-axis autoguiding system this mirror creep causes the star field to move slowly with respect to the pixel field. This movement has been measured to be ~ 0.4 'arc/hour (in RA only). This is small compared with my FOV = 27 x 18 'arc, and since FOV centering is performed a few times per observing session, typically, mirror creep is not a problem.

A wireless surveillance camera provides auditory information about dome and telescope movements, as well as wind sounds.

Observing Procedure  

MaxIm DL v5.24 is used for control of the telescope, CCD, 10-position CFW, focuser and dome. A computer is dedicated to control of the observatory and is not used for any other purpose in order to not diminish computer resources for this task. The only exception is a UT/LST clock and TheSkyX for keeping track of asteroid location.

Exposure times are set to no more than the time required for the asteroid to move across a PSF's FWHM. For slow moving asteroids exposure time is limited to 2 minutes. A clear filter is used for all asteroid observing, except when the asteroid's spectral slope is desired (when I use g'r'i'z' SDSS filters).

 An observing session begins with a manual focus measurement. The temperature coefficient for focusing is known, so focus checks are usually not needed for the rest of the night since either automatic focus changing is turned-on or manual changes are made. Occasionally a 2nd (or 3rd) focus measurement is made after cooling stability has been achieved.

Spot checks are made of a chosen star's magnitude to be sure the dome is synchronized with the telescope pointing, and is not blocking telescope aperture. Occasional trips to the backyard dome are also made to assure dome azimuth synchronization.

An observing log is maintained, noting when target observations begin, filter used, exposure times, focus measurements, dome changes, visual sightings of clouds, etc.

Approximately once per week I produce a new master flat for any filter bands that are expected to be used during the next week, and a new master dark and master bias are produced. My flats are made shortly after sunset, with a two T-shirt diffuser covering the aperture. A dark is automatically obtained for each light, and exposure times are increased to maintain ~ 45,000 counts for the brightest FOV region.

A computer printout of the asteroid's RA/DE etc for hourly intervals, obtained from the JPL Horizons web site, is available for double-checking FOV location. When the asteroid is not easily seen in any of the images I will copy image files to a flash card and load them into my main computer for calibration and viewing to be sure the asteroid is present within the FOV. 

Slow moving asteroids will stay within a FOV during an entire night's observing session. Fast movers may require FOV changes at 1.5-hour intervals. I try to not observe asteroids than move faster than ~ 900 "arc/hour because they require FOV changes at intervals shorter than 1.5 hours. If frequent FOV changes are needed I may go to bed at midnight and set an alarm for whenever a FOV change is needed. But if FOV changes aren't needed, then I'll go to bed typically at 1 or 2 AM. I then set an alarm for sometime before sunrise to manually stop imaging. (I know, I should use CCD Commander or CCD Autopilot to do this, and one of these days I will!).

Image Reduction Procedure   

Processing images for each FOV requires about 1.5 hours. My main computer's RAM (8 GB) will allow ~ 500 images to be loaded by MaxIm DL without invoking virtual RAM from the hard disk (with a serious penalty for processing time). This is one reason to use 2x2 binned images, because 1x1 images are 4 times larger and I can load only ~ 150 of these images at a time for processing. MaxIm DL (MDL) version 6.x is supposed to overcome this limitation, but it has some user-unfriendly "upgrades" that I dislike.

All images that have been loaded into MDL are calibrated using master dark, bias and flat. If necessary, all images are subjected to a hot pixel removal (typically 30%). Then all images are star-aligned, usually using just one star near the FOV center. Automatic star alignment sometimes is inadequate, especially when a saturated star is in any of the images. My polar axis is aligned very accurately, so image rotation is never noticeable.

Poor quality images are deleted. The star-aligned images are saved to a folder. All images are then median combined to produce a FOV master images. This image is solved using MDL's PinPoint and saved. It is then subtracted from all individual images to produce what I call "star subtracted" images. These are saved to another folder.

The entire process of star subtraction adds an extra 4 minutes to each FOV's processing time. For faint asteroids, or any asteroids in a crowded star field, the 4 minutes of extra work is very worthwhile. In almost every case the benefits for star subtraction are evident, and this is one reason I can achieve useable LCs for faint asteroids. I therefore highly recommend use of star subtraction. (More info on my star subtraction process is given at

I add an "artificial star" to all of the star subtracted images (using a DLL created for me by Ajai Sehgal, to whom I shall be forever indebted). This star is located in the upper-left corner of each image, and it has the same total flux for every image. This provides a way to keep track of atmospheric extinction losses, due to cirrus clouds for example. It is not needed when using MDL v6, but is is needed for all earlier versions of MDL because they record photometry files consisting of only magnitudes (not fluxes, or "intensities" - as MDL refers to them). (More info on my use of the artificial star is given at

The MDL photometry tool is then invoked. Recall, we are still working with the star subtracted images. Set the photometry signal aperture to ~ 1.5 times the FWHM [pixel value], and set the gap and sky background annulus to 12 pixels each. Assuming the asteroid can be seen in at least one of the early images, and one of the late images, it can be specified as a "moving object." (If the asteroid can't be seen in some images, don't use the "snap to centroid" feature; just be careful with positioning the photometry circle for the early and late images where the asteroid can be seen). When the asteroid can't be seen in any images, other tricks are possible, but they won't be described here. For 19.2 magnitude, slow-moving asteroids, the asteroid can be easily seen in every image (with SNR => 5). Specify the artificial star to be the Reference star. Save the photometry results CSV-file for later use by Excel.

Load the un-star-subtracted images, add the artificial star, and invoke the photometry tool again. Specify the "moving object" again, using the same precautions mentioned above. Specify the artificial star to be the Reference star. Now, we have lots of stars in all images, and can choose a couple or 3 dozen of them to be "Check Stars." Check Star photometry readings have zero effect on the real-time LC MDL displays; their mag's are just included in the CSV-file for later use (by Excel). After specifying 20 or 30 stars as "Check" save the CSV-file.

Excel Spreadsheet Analysis 

Over the years I've developed an Excel file that I use as a template for all LC analyses. The first "page" is a list of targets (RA/DE, mags and transit parameters for exoplanet stars). A cell is used to indicate which target is to be processed. The 2nd page is where I import CSV-files. One area is for importing the CSV-files that include many check stars, and another section is where I import CSV-files that have only the star-subtraction readings (their rows are synchronized).

The 3rd page is where I copy mag data from the first page with the intent that all subsequent pages use those data. So I copy the mag's with all the check star mag's (from page 2) to this page. Then I copy the column of star-subtracted mag's to this page, replacing those that came from the image set with all stars present. Invariably, a plot shows that the target mag's from the star-subtracted photometry have "better behavior" than the target mag's where background stars could cause negative and positive artifacts to the asteroid's mag readings. Columns of this page calculate air mass, and also converts each check star's mag to flux, and creates a column for total check star flux and corresponding mag. (Of course, this page has the observer's latitude and longitude).

The next page uses mag's from the previous page 3 user adjusted cells for achievinge an extinction "fit" to the total flux mag's from the previous page. A graph of total mag versus air airmass for guidance.

The next page is complicated because it identifies which check stars are well behaved and therefore recommended for inclusion for use in correcting for extinction variations vs. UT. It has a row of numbers with associated slide bars for incrementing or decrementing the numbers; when the number for a specific check star is odd it is included in a total extinction correction; when it is even it is excluded. The user has feedback cells showing how the LC fit is affected (and yes, I'm getting ahead of things here).

The next page is where extinction variations from the previous page are applied to the target to show what the target's mag would be if it was above the atmosphere (air mass zero) and there were no extinction variations. RMS noise is calculated here using neighbor differences. Many other quality check parameters are displayed. For example, the user is invited to specify whether data is accepted or not based on total extinction departure from the model fit (i.e., "extra losses," due, for example, to cirrus clouds). The user may also specify a criterion for when data is too noisy for inclusion.

The next page is unimportant, as it is used to review the LC for all check stars to look for EBs, transiting exoplanets, or misbehaving variations among the check stars).

The next page shows the target's LC (e.g., Fig. 1). A model fit can be determined by adjusting a dozen or more parameters. When the target is an exoplanet, for example, the user can play with transit depth, ingress and egress times, transit shape and many more parameters. When the target is an asteroid the user may specify a sinusoidal variation superimposed on an average level (where period, amplitude and phase can be specified). Provision is made for a slope (linear change of target magnitude with UT) and "air mass curvature" (caused by the target having an unusual color, differennt from the check stars chosen to be used as reference stars). Much of this is described in greater detail in my book Exoplanet Observing for Amateurslink

The next page is where chi-square is calculated for the model specified in the previous page. The chi-square result can be used in the previous page for automatic parameter solution solving (using Excel's Solver tool).

The next page is where mag calibration occurs. (It is actually dealt with before the previous two Excel pages.) Every check star is a potential calibrator if it has APASS mag's (in the UCAC4 catalog, which contains DR6 mags). I go offline (shell out of Excel) to run the program C2A: link. C2A is a wonderful planetarium program, free, for getting APASS mags and creating a CSV-file of them for import to Excel. TheSkyX can also show APASS mag's, but I don't think you can export them to a CVS-file. With C2A you center the FOV and set the scale to show all of the FOV, and then export all APASS mag's to a CVS-file. This file is imported to the Excel calibration page. Columns are reserved for the check stars, and the user has to enter a star ID number in a cell. A search of the APASS mag section then fills cells below the ID number with BVg'r'i' mag's. Tentative mag's from several pages back are compared with the APASS mags, and an offset and slope fit can be determined.  Figure 3 shows a plot of "APASS r' minus instrumental mag" vs. star color (g'-r' - 0.45). 

Figure 3. Star color sensitivity plot for one observing session, using 15 stars with APASS g'r' magnitudes used for achieving a "CCD transformation" calibration, leading to r' magnitudes even though a clear filter was used.

In this plot a sloped line (with an optional quadratic term) is fit to the measure mag differences. In this case the slope is -90 mmag/mag. This corresponds to one of the "transformation coefficients" in the horribly ill-conceived "CCD transformation equations" that old fashioned astronomers probably still use. For more information than any sane person wants about the derivation of CCD Equations, I refer you to: link. On that page I explain why CCD Equations were OK when everyone used log tables instead of calculators, or Marchand calculators before computers, and why they are totally cumbersome and unnecessary in today's age of computers with spreadsheets. Having a display like this is useful in identifying "outlier" stars. These are stars that vary slowly and weren't identified by the APASS project as variable, so their mag's at the time of a few measurements were included in the APASS catalog. But this is years later, and some of those stars in the APASS catalog have mag's that are totally different. (Incidentally, CCD Transformation equations don't allow for this!) I estimate that about 3 to 5% of all stars are variable at the 20 mmag level (and vary slow enough to have not been identified as variable by APASS), and therefore can't be used when 10 mmag accuracy is needed (which is the case for asteroid work).

Inspection of the above figure shows one star that appears to be an outlier (upper-left). Indeed, when I use an equation that compares it's departure from the fitted line with the SE of all data it is shown to have a <5% probability of "belonging." An additional check can be made of the 15 APASS stars using a color/color scatter plot (next figure).

Figure 4. Star color/color scatter diagram showing the 15 APASS candidate reference stars (open red squares) and a set of ~100 well-calibrated stars (SMith et al, 2002).

One of the stars in this scatter plot appears to be an outlier, and indeed it is the same one identified in the previous figure as an outlier. I therefore reject it from use (by changing a cell for it from 1 to 0), and this leads to the following plot.

Figure 5. Revised star color sensitivity plot with an outlier candidate reference star removed.

In Fig. 5 it can be seen that the check star data exhibit a scatter about the model fit that is 0.032 mag. There are 14 check stars with useable APASS mag's, so the ensemble calibration SE is 0.009 mag. There is rarely an occasion when the ensemble SE exceeds 0.015 mag.

One Fig. 5 subtlety should be mentioned. In this plot I've assumed the target asteroid has a g'-r'-0.45 color of zero, which is the sun's color (note: the constant -0.45 was chosen so that the x-axis is star color difference with respect to the sun). When there's no information about the asteroid I adopt g'-r'-0.45 = 0.15, which is typical for asteroids. In this case I know that the asteroid's color is the same as the sun (described later), so I set the target color to zero (using the above definition for color).

You might wonder why so much attention is given to achieving an accurate calibration since we just want to know how the asteroid varies in brightness as it rotates. The reason we need a good mag calibration is because almost all asteroids will require observations with different FOV placements, or on different dates, and LC segments have to be compared with correct offsets in order to produce a multi-FOV rotation LC.

Figure 6. Example light curve for one observing session (details in the text).

Figure 6 shows a LC segment for one FOV for the 19.4 mag Trojan asteroid 65000 (depicted in Fig. 1), made on one of the 8 dates on which it has been observed. The information box in the upper-left corner shows the JD range of this FOV's observations (with 2450000 subtracted).  It shows that 26 check stars were chosen for use as "reference stars" and 13 stars were used for calibration (similar to Fig 3). The term "reference star" needs explanation. A few percent of stars with APASS mags are slow variables, and they can't be used for calibration but they can be used to monitor variations due to extinction and cirrus cloud "extra losses" (because during an observing session they don't vary). Among those 13 stars used for calibration their RMS about a star-color-sensitivity slope fit was 13 mmag, and the ensemble calibration is expected to be accurate to 4 mmag (assuming APASS mag's are perfect). A comment is given providing a subjective assessment of how much benefit was afforded by the use of star-subtraction. Faint asteroids typically benefit the most from star-subtraction, and asteroids near the Milky Way always benefit from star-subtraction.

Figure 6 also shows information (lower-left box) about the model fit. I like using 10-minute RMS as a measure for describing precision, which is based on the fact that for exoplanet transit LCs this is an acceptable averaging interval for evaluating transit depth and shape. For this LC the 10-minute RMS was 77 mmag (including data for air mass < 2.5). Exposure time for the individual images was 120 seconds (the small crosses correspond to individual images). The MDL photometry aperture parameters were 4, 12 and 12 pixels (signal aperture radius = 4 pixels, gap = 12 pixels, sky background annulus = 12 pixels). All images were taken with 2x2 binning (hence "b2"). 94% of the data were included (some outliers were rejected from use). The filled circles are averages of 5 individual image magnitudes. A model extinction of 154 mmag/air mass was used, with a temporal trend of +2 mmag/hour. No slope was used in the model fitting, and no air mass curvature was used. The reduced chi-square for the model fit is 1.09. A sinusoidal variation was used for estimating rgw 10-minute RMS performance, and it had an adopted period of 8.0 hours and semi-amplitude of 127 mmag. This sinusoidal model is not used for any subsequent analysis.

The lower panel shows air mass vs. UT (red trace) and "extra losses" (blue trace) that departed from the extinction model (that incorporated a linear trend). Extra losses are usually due to cirrus clouds, but may occur if the dome is not synchronized with the telescope. The "extra losses" change starting at ~ 10.7 UT, when air mass ~ 4 , or EL ~ 14.5 degrees, may be due to the dome beginning to obstruct the telescope aperture.

Returning to the 10-minute RMS value of 77 mmag, noted in the lower-left information box, we may convert this to the RMS for individual images by noting that the exposure time is 2 minutes. Considering that the download time is 5 seconds, and another 5 seconds is needed for synchronizing with the autoguider, the cadence is one image per 2.17 minutes. Therefore, within 10 minutes there are 4.6 images, so each image must exhibit an RMS scatter of 165 mmag (i.e., 0.165 mag). It is commonly thought that LC quality is limited by the uncertatinty of each image's magnitude uncertainty; this is true only because the SE per image determines how averaging of images reduces SE per average of images. For example, in Fig. 1b the median SE for averages of 90 images is 0.018 mag (this is to be expected since 0.165/sqrt(90) = 0.017).

This completes analysis for an individual FOV. Next we consider how to combine LC segments from many FOVs, taken at different times of an observing session, or on different dates. The first matter to consider is that an asteroid is continually changing brightness even if it doesn't rotate, due to changing distances (r and d) and changing viewing geometry (phase angle). I use the JPL Horizons ephemeris as a guide in correcting measured magnitude for a given JD to what it is expected to have at a standard JD (usually chosen to be the first JD of the first FOV for that asteroid). The next figure shows JPL Horizons predicted ephemeris magnitude vs. time for the Trojan 65000 observations during the 4 weeks of observations.

Figure 7. Ephemeris magnitude vs. observing date and 3rd-order model fit. Solid blue diamonds are the ephemeris values, circles are the fitted values and small green symbols represent values for offsetting observations.

The ephemeris data were fitted by a polynomial which was then used for offsetting measured magnitudes to a standard date 0 hours) before phase-folding analysis is performed.

Figure 8. Phase-folded LC for observations on  8 dates during a 4 week period (same as Fig. 1). Each data point is an average of magnitude readings from 9 images. Using chi-square > 10 as a guide outlier data were identified (8 out of 186 images) and excluded from the fitting process.

The phase-folded LC, above, is based on 286 images, all subjected to the star-subtraction process, with a 96% acceptance using chi-square > 10 as a rejection criterion. A 3-term harmonic sinusoidal fit was performed using a fundamental frequency (corresponding to 1/2 the rotation period) and two others (corresponding to the rotation period and 1/4th the rotation period). Free parameter status was given to the amplitude and phase for each sinusoid, as well as the period of the fundamental and an offset.

After a solution was found, free parameter status was given to an offset for each observing session, normalized by 0.020 mag, an estimate of maximum "observing session offset" due to calibration uncertainties. This was done because such offsets are expected to exist.due to an imperfect calibration using a finite number of APASS stars for each observing session (as well as imperfect ephemeris magnitude changes with date due to the use of a default phase effect, using G = 0.15). The "observing offsets" increased with phase angle in a way that implies the need for a larger G (i.e., G = 0.30). The RMS difference between "observing session offset" and a model with a better G value is 0.023 mag. This is considered acceptable. After the "observing session offsets" were solved for another iteration of the phase-folded magnitudes is performed.

The information box in Fig. 8 states that "Red'd Chi-Sqr = 1.43" which means that either another component of variation is present in the data that has not been modeled or my estimated SE values are slightly optimistic (by 20%). If this asteroid was a binary then the smaller component could produce another variation with a period corresponding to that component's rotation period, which isn't necessarily the same as the primary's rotation period. It is therefore worth checking to see if the "residuals" off the above phase-folded LC exhibit a variation that can be fit by another sinusoidal model.

Figure 9. Phase-fold fitting of residuals off the model in Fig. 6 showing that there is no significant variation in addition to the one in the Fig. 6 model.

As this graph shows the Fig. 8 residuals do not exhibit a statistically significant variation. Therefore, we have no evidence for a second component contributing to brightness variability for this asteroid. We therefore should investigate the possibility that a higher order sinusoidal harmonic fit is required.

Figure 8. This is a 4th-order fit (4th harmonic of the fundamental), providing a slightly better fit to the data.

This slightly better fit now includes 98% of data as acceptable (4 rejected out of 186), using a chi-square criterion of 10. However, it still has a reduced chi-square greater than one (1.34), and I don't think the data deserves any more higher order terms for the model. What's normally done in this situation is to arbitrarily increase the SE estimates (all of them) by the suggested 16% and repeat the chi-square fitting. This isn't necessary in this situation because we would just arrive at the same solution and the only difference would be a slightly greater SE for the model parameters that are solved for. There is negligible effect on the period; the solution SE for P has an uncertainty that is still 0.007 hrs.

Asteroid Feasibility Regions   

The "exercise" above illustrates the feasibility of observing a 19.3 magnitude asteroid with a 14" telescope, provided the asteroid moves slow enough that long exposure times are possible. The observations in the example were made with 2-minute exposure times, and the the PSF FWHM was typically 3.2 "arc. In order for the crossing time for this PSF to be less than 2 minutes the asteroid motion must be < 100 "arc/hour. This should include all Trojan asteroids, for example, and most main belt asteroids. The fast movers will be NEAs. The following diagram is a plot of magnitude versus rate of motion showing "observability" for various telescope systems.

Figure 11. "Feasibility" diagram for asteroids, showing area within which observations are feasible for various telescopes (when the sky is dark). The red dot corresponds to Trojan 65000 (on 2015 May 11), a slow mover. The green square is typical of a Near Earth Asteroid (NEA), with a typical rate of motion of typically ~850 "arc/hour. The % numbers at the bottom show what percentage of NEAs are moving slower than their rate of motion location. More details are described in the text.

This graph is useful in quickly determining if an asteroid can be observed with a specific telescope system. For example, the Trojan 65000 used for this case study was moving at ~ 20 "arc/hour and was at V-mag ~ 19.4 on the last day I observed it. This is shown by the red dot, and it is "within" the feasibility region for the telesccope system labeled "HAO 14", Cass." A typical NEA might be at the location indicated by the green square, at 850 "arc/hour and V-mag = 17.7. It too would be observable by my 14" telescope. But a NEA moving at the same rate that is fainter than ~ 18.4 would be too faint for my telescope. If 20th magnitude is desired then a Hyperion 16" would be acceptable, for example.

This graph assumes that the asteroid must register in most images with a SNR > 4.5 in order to be capable of producing a useable LC. With my 14" I can "see" 20.0 stars when the sky is dark, images are sharp and exposure times are 2 minutes; so that's a limiting magnitude for 2-minute exposures (for the standard 60 second exposures subtract 0.4 mag, yielding a standard limiting magnitude of 19.6 for my telescope).

Moonlight can reduce limiting magnitude by about 1 magnitude (full moon, 45 degrees away), so that should be taken into account when using the feasibility graph.

CCD QE is also a factor in limiting magnitude. The above feasibility graph assumes 80% QE, which corresponds to my SBIG ST-10XME (KAF-3200E chip).

    HAO description
    Master list of B.Gary web pages
    B.Gary resume


This site opened:  2015.04.20.  by Bruce L. Gary (B L G A R Y at u m i c h dot e d u).  Last updated: 2016.04.06