Bruce L. Gary, Hereford, AZ


Amateurs with a small telescope (10-inch aperture) and a good CCD are capable of monitoring a 14th magnitude blazar with a nightly precision of ~0.010 magnitude (V-filter) and a 30-minute precision of ~0.007 magnitude.  It is therefore possible for amateurs to search for, and monitor, blazar brightness variations on timescales of hours, days and longer.

This web page illustrates an exploratory set of observations which were used to establish the above precisions.  Blazar "W Com" was observed for a 15-day period in June, 2003, and it was found to fade smoothly over a 0.42 magnitude range until an abrupt rise on June 19.  Day-to-day departures from a simple polynomial fit to the daily data during the fading period is only slightly greater than predicted using check stars, and it is concluded that the blazar may exhibit day-to-day departures from smooth fading and rising periods that are at the ~0.010 magnitude level.  Hourly variations were not detected at the level of ~0.005 magnitude.  Although hardware requirements for this observing capability are modest, data analysis requirements are considerable, which means that only dedicated amateurs will be able to achieve these levels of precise blazar monitoring.


Imagine a distant galaxy with an active nucleus powered by a black hole (consuming nearby gas, dust and stars), with one of its polar jets pointed at us.  The radiation from the jet is so strong that it outshines the galaxy, giving it the appearance of a star, a quasi-stellar object (QSO).  Since the jet is variable, and dominates the object's total output, the QSO is an erratic variable.  The name for this category of QSO is "blazar."

Blazar brightness variations can be 1.7 magnitudes over time scales of 2 months, and possibly ~5 magnitudes over decades.  Brightness changes on hourly time scales are alleged to occur, and more observations are welcome to better characterize them.  My summary of information on what the professionals have determined about W Com variability can be found at Tosti.

The AAVSO (American Association of Variable Star Observers) is beginning a program of blazar monitoring (AAVSO Blazars).  Since blazars tend to be faint, they are well suited to observation by sensitive CCDs.  Although large, professional telescopes can easily monitor blazar variations, competition for telescope usage essentially rules out long hours and many nights for the purpose of monitoring a blazar's variations.  This task is a much better match for advanced amateurs, using telesopes with apertures of at least 8-inches and good CCD cameras.  Larger telescopes aren't necessary, except for the possible study of minute time scale variations (which probably don't exist).

This web page's purpose is to deomstrate that the telescopes and detection systems used by many amateurs in the AAVSO are capable of performing useful monitoring observations on all time scales longer than approximately an hour. I'll illustrate both short time scale and long time scale monitoring using a Meade 10-inch LX200 telescope. (My Celestron 14-inch telescope is back at Celestron for repair and won't be in use until perhaps July).  I'll also illustrate the different advantage of filtered versus unfiltered observing.  Whereas unfiltered observations might be superior to filtered observing for monitoring short term changes (during a night's observing session), filtered observing is best for monitoring night-to-night and longer variations.

Here's a preview of the results I obtained in my attempt to monitor day-to-day changes in a blazar's brightness.

Example of daily mionitoring of blazar W Com by an amateur with a 10-inch aperture (described below).

The remainder of this web page consists of the following sections:

Blazar W Com Location
Unfiltered Short-Term Stability
Filtered Short Term Stability
Filtered Long-Term Stability
Other links to detailed descriptions appear throughout.  They're meant for the observer who has other ideas on how to do things, who might benefit by seeing the flounderings that didn't pay off for me.

1. Detection System (CCD, etc)

My CCD system consists of a JMI focuser, a SBIG AO-7 "adaptive optics" tilt mirror unit, a True Tech color filter wheel and a SBIG ST-8XE CCD camera.  The CCD communicates with a computer using a USB interface (allowing for 5-second full-resolution image downloads).  The CCD camera is a 2-chip CCD.  The main chip is a 1530x1020 KAF1602E, with 9 micron pixels (measuring 13.8x9.2 mm).  The autoguider is a TC237 chip with 657x495 7.4 micron pixels (measuring 4.9x3.7 mm).

The AO7 adaptive optics isn't necessary, of course, but it does permit long exposures with autoguiding quality that is far superior to the standard method of using an autoguiding chip to control the telescope drive motors.  I use MaxIm DL/CCD for control of the telescope, focuser, AO7, CCD and color filter wheel.  The AO7 can adjust the mirror tilt as often as 50 Hz, although for telescopes as small as my 10-inch 2 to 10 Hz is typical.  MaxIm DL/CCD's control of the AO7 senses when the mirror tilt range limits are approached, and the program proceeds to nudge the telescope drive motors to keep the mirror tilt movements within range.  Since the mirror adjusts faster than the telescope can move with a nudge, there's no loss of tracking quality during this procedure,  The AO7 can autoguide the entire observing session, which I find useful because it guarantees that the star field doesn't move with respect to the pixel field (i.e., placing fewer demands on "flat field" quality). Use of the AO7 for long exposures has a smaller resolution penalty than the alternative of using a perfect mount, since a perfect mount will not correct for atmophere-induced image wander.

The AO7 is not a requirement for blazar monitoring.  In fact, a year ago, before acquiring the AO7, I achieved ~0.005 magnitude precision during a 10-hour observation period (detecting the exo-planet transit of HD209458).  Amateurs without the AO-7 should not be discouraged about their prospects for blazar monitoring at comparable levels of precision.

2. Blazar "W Com" Location

The blazar "W Com" is in Coma Berenices, which is a Spring season object.  It's location is RA = 12:21:32 +28:14.0 (epoch 2000.0)  The visual magnitude brightness is stated to vary between 13.4 and 15.4.  AAVSO finding charts can be found at chart.  The AAVSO object designation is:  1216+28

The following image shows the neighborhood of W Com (1216+28).

Figure 1.  The field of view is 24 'arc wide, with north at the top and east to the left. The blazar is circled and AAVSO suggested reference stars are shown as numbers (10 times their visual magnitude; for example, the star to the lower-left of the blazar has Mv = 12.1). All magnitudes are from Arne Henden's photometric measurements, appearing on the AAVSO chart dated "0200301". The star "168" is not a member of Arne's sequence, but is a very unusual (red) object (discussed below). [Meade LX200 10-inch SCT, V-filter, AO-7 autoguider, SBIG ST-8XE CCD, 233 minute total exposure (19x5-minute & 69x2-minutes, median combined & averaged), 2003.06.11 and 12Z]

The next image is a 2-times zoom centered on the blazar, unfiltered and showing fainter stars.

Figure 2.  The field of view of this image is 7.5 x 5.9 'arc (north up, east left).  The 12.1 magnitude star is the brightest one in this image, 51 "arc from the blazar.  The faintest stars have an approximate magnitude of 20.3. The unsaturated stars have FWHM = 4.0 "arc. [80 minute total exposure (16 5-minute exposures median combined), unfiltered, 10-inch Meade LX200, 2003.06.07Z]

Notice that a 19th magnitude star is located 10"arc south of the blazar.  Since it is 4.6 magnitudes fainter than the blazar (1.4% as bright), it's contribution to the blazar's aperture photometry analysis is a "consideration."  An observer might want to use the same photometry aperture/annulus choices for all observations if variations below the 1.4% level are to be studied.

Because W Com transited before sunset when the observations described on this web page were taken, it could be observed for less than 4 hours.  The best time to observe W Com is ~April 2, when it transits at midnight, allowing 10 hours of monitoring (above 25 degrees elevation).  Thus, the short timescale results of this demonstration study are poorly suited to the sarch for hourly variations.

The following sections summarize my observations of W Com for the purpose of developing an observing strategy and analysis procedure suitable for monitoring blazar variability.

3. Unfiltered Observations of Short-Term Stability

Question:  Does "W Com" vary on minute and hourly time scales?

The short answer is that on one date, consisting of a 1.7-hour observing run, the blazar did not vary on a 5-minute time scale by more than about 0.003 magnitudes.

Unfiltered observations have a higher signal-to-noise (S/N) than filtered ones, but they are subject to the shortcoming that as air mass changes, a changing system spectral response will cause the ratio of the blazar to a reference star to change (assuming the reference star has a different color than the blazar).

The following graph shows the magnitude of the blazar, in relation to 3 photometric reference stars, during a 1.7-hour observing period.

Figure 3.  Blazar "W Com" unfiltered magnitude versus time for a 1.7-hour observing period.  Each datum is from a 5-minute exposure. The V-magnitudes for the reference stars 121, 131 and 148 were used to establish a magnitude scale for the blazar, and since these observations are unfiltered the blazar magnitudes plotted here are subject to an offset uncertainty of probably 0.25 magnitude.

The trend of blazar magnitude with time during this 1.7 hour period can be discounted since it is most likely due to changes in air mass (from 1.08 to 1.41) and the use of reference stars having a different color from the blazar.  Nevertheless, departures of measurments from the fitted straight line can be used to estimate how precisely very short time scale changes can be monitored.  Each datum comes from a 5-minute exposure, and the population SE difference from the straight line fit is ~0.0068 magnitude (corresponding to 0.63%).

A "check star" in the same images was subjected to the same analysis procedure. It's magnitude trend was the opposite of the blazars, which is consistent with my assertion that unfiltered observations can't be used for trend studies (unless more exotic analysis procedures than I've performed are employed).  The check star exhibited a 5-minute population SE of 0.0051 magnitude (0.47%).  The check star is 1.5 times brighter than the blazar, so it should have a smaller population SE with respect to a fitted trend line.  The check star's population SE can be used to predict the blazar's population SE.  Using "square-root of 1.5" as a scaling factor, the check star predicts 0.0062 magnitude for the blazar's population SE.  Orthogonal subtraction of 0.0062 from 0.0068 leaves 0.003 magnitudes unaccounted for in the blazar's population SE.  This result is not meant to be taken seriously, but presented merely to illustrate a possible method for estimating a blazar's variability.

With more data it would be possible to perform "structure function" analyses to quantify variability versus time offset.  But the first goal is to determine how variable the blazar must be to be detectable with a given telescope and detection system.  As I conduct more unfiltered measurements of this blazar I'll add them to this web page.

Details of the analysis summarized in this section (for the serious observer intent on starting a monitoring program) can be found at 2003.06.07 Unfiltered.

4. Filtered Obserations of Short-Term Stability

For my observing site the V-filter extinction is typically 0.27 Nepers/air mass (0.28 magnitudes per air mass, or 22% loss per air mass).  B-filter absorptions are ~50% greater, and I-filter absorption is ~50% less.  This is illustrated in the next figure.

Figure 4Spectrum of typical atmospheric absorption.  The four black horizontal bars correspond to the passbands of the filter sequence B, V, R and I. The traces are for a site in Pasadena, where I used a "spectral hygrometer" with several interference filters to characterize atmospheric absorption versus wavelength (before retiring to AZ).

This figure is for "clear skies."  Thin cirrus consists of particles larger than light wavelength, so their Mie scattering will affect all wavelengths the same (all traces in the figure would go down by the same percentage). High altitude sites have higher transparencies, especially for the water vapor complex of resonant absorptions (because of water vapor mixing ratio having a scale height of ~2 km instead of the non-resonant molecular scale height of 9 km).

Star ratios from the same image, for a series of images, is the best way to minimize extinction effects for searches of small blazar brightness changes during a night's observing. Although unfiltered observations may have extremely good signal-to-noise, even star ratios within an image are subject to atmospheric extinction effects that will vary as air mass changes.  Filtered observations will greatly reduce extinction-related effects upon star ratios.

But even over the V-filter's bandpass, extinction varies (~7%, typically), which means that star color will still matter for very precise measurements.  Therefore, we should be prepared to encounter star brightness ratios that depend somewhat on air mass even when using a filter.  Yes, this complicates analysis, but that's what makes milli-magnitude monitoring so much fun!  (From now on, I'll use the abbreviation "mMag" for "milli-magnitude" and "milli-magnitudes.")

Before describing an observing and analysis procedure that seems to work, I want to review some lessons learned from my previous work.  For the serious observer who wants to know more about the floundering observing and analysis exercises that led to the list of "lessons learned" I present a lengthy account at floundering.  Trust me, it's not worth reading.  Spare yourself the trouble by just adhering to the following precautions:

1)  Very good flat fields are essential!  The sky is best light source, shortly after sunset, looking ~20 east of zenith (where sky brightness varies the least over the CCD's FOV).  I took many flats with exposures that increased as the sky brightness decreased, keeping the maximum data number (DN) at ~50% of full scale (i.e., ~30,000 counts).  This assures that my non-anti-blooing CCD was operating in the linear range.  I inspected all images for cosmic ray defects, then summed them, and rescaled the sum image to keep all DN within the 16-bit FITS image range.  Never use median combine with sky flats, because average brightness levels differ for each sky flat by amounts that are large compared to the noise (a median combine with such a series will return the middle image).  Another precaution about taking flat fields:  don't do them when there's a chance that the CCD hasn't completed its cooldown (which might be 30 minutes for my CCD).  I typically do my flat fields before turning on the cooler. And another precaution for taking flat fields: be sure to have a new dark frame for each flat frame light exposure.  If a dark frame is used for all light frames, and the dark frame has a cosmic ray defect, all dark frame calibrated light frames will be defective.  This can be disastrous for a flat field when precision photometry is being done.  Also, if you're taking flat frames of the sky when it might be so dark that you're using long exposures (more than a second), stars may register, and ruin the flat frame.  It's good to nudge the telsecope between flat frame exposures (or command the telescope to move while taking the exposure) to minimize this problem.  An entire web page could be devoted to the proper creation of flat frames.
2)  It's important to keep the star field in the same pixel locations on the CCD.  This helps minimize errors in the flat field calibration.  (In theory, you could do without flat fields if the stars in all images were at the exact same locations for all images, provided star ratio changes was the sole objective for the night's observations - as might be the case for exo-planet transit measurements).  With the AO7 this requirement will be difficult to achieve if it is not allowed to track for the entire night's observing, since the AO7 will produce an unpredictable shift of the image as it first acquires the specified guide star.  When I stopped the AO7 to perform a focus check, then re-started the AO7 for observations, a scatter of ~15 pixels was observed (out of a 1530x1020 pixel field) on successive AO7 acquires.  However, if the AO7 is allowed to "run" during the entire blazar observing session, even during focus tests, the pixel location of the star field will remain fixed.

3)  When a bright reference star is near the object to be monitored, as is the case here with star "121" being <1'arc from the blazar, it is better to use just this one reference star for ratios than to use a large number of reference stars for establishing the magnitude scale (permitted by MaxIm DL's photometry feature).  The reason for this is that reference stars near the CCD's FOV edge are sensitive flat field errors, whereas a reference star near the image center is less sensitive to flat field errors, and reference stars near the object of interest are likely to share the same errors with the object, cancelling out flat field errors when ratios are taken.

4)  Use several aperture/annulus choices consistent with the point spread function of the brightest star to be used.  Think carefully about how this is to be done.  The purpose for this is not only to improve S/N, but to later search for incorrect aperture/annulus choices (in a spreadsheet) and reject outliers from subsequent analysis. A cosmic ray blemish, or even an unsuspected faint star near any of the stars used, can produce bias errors in aperture photometry.  Since such errors will depend upon the radius values for the aperture circle, gap and reference annulus, later (spreadsheet) analyses can be used to detect when this situation exists.

5)  Use many check stars (with the same reference star that's used by the blazar).  With MaxIm DL this is easy; just specify one reference star and many objects (including the blazar as an object).  Record the text files (CSV-files, in MaxIm) that contain the magnitude solutions for that aperture/gap/annulus radii choices so that they can be imported to a spreadsheet for analysis.

6)  The spreadhseet analysis should include "looking for trouble"!  Look for patterns that would invalidate a specific aperture/gap/annulus choice.  Look for outlier data.  Then perform the averages of the aperture/gap/annulus choices for each image, and plot them versus air mass (not time).  Perform least-squares fits to the magnitude versus air mass for each object.

7)  The other "lessons learned" are described in the remainder of this section, using data from one night's observing to illustrate the concepts.

V-filter observations of this blazar were made during a 3.5-hour observing run on June 11, 2003 (UT).  The same 10-inch Meade LX200 telescope and SBIG AO-7 and ST-8XE CCD were used.  All exposures were 5 minutes long, and employed AO-7 guiding.  For this date my observing cycle consisted of 4 "light" exposures and 2 "dark" exposures (I'd recommend something like 5 to 1, in retrospect).  Note: S/N theory says to spend 25% of your time on dark frames, unless you're willing to trust your CCD to not change pixel properties, etc.  A pointing check was performed between most cycles.

The following figure shows blazar magnitude, based the ratio of the blazar's intensity to that of nearby reference star "121", as a function of air mass.

Figure 5Blazar magnitude based on its intensity ratio to Reference Star 121, versus air mass, with a LS fitted line.  The population SE for image-averaged magnitudes is 12 mMags.

If the blazar were redder than the reference star it would have an intensity ratio that would exhibit a positive slope when plotted versus air mass (like the one in this figure).  This is because more of the blazar's photons are arriving through the red side of the filter, compared to the reference star, and the red side is less affected by atmospheric absorption. Note that the simple formulation of CCD transformation equations would not "solve" this problem, it would merely shift the data set vertically, and the slope would remain.  The only analysis procedure that would fix the slope problem is to sue a second order extinction term.  This won't be necessary for present purposes, as will become clear.

Let's look at another star's ratio versus air mass.

Figure 6Reference star "148" ratio to reference star "121" verus air mass.

Refence star "148" (same approximate brightness as the blazar) shows the opposite dependence on air mass.  Apparently it is a bluer star than "121."  All reference stars were processed the same way, and the signs of the slope solutions were about evenly divided.  Therefore, there is nothing unusual about the blazar having the slight positive slope for its intensity ratio versus air mass.

Notice that in both of the above figures the ratios are either above or below the fitted lines at the same air mass values.  In other words, each image has its charactersitc bias.  Perhaps refernce star "121" had a hot pixel buried within the bright region that I didn't notice before performing the aperture photometry.  Most of the "check stars" exhibit this correlated behavior.  Therefore, whatever its origin, these image-to-image biases should be allowed for before searching for variability in the blazar data.

The following figure shows the average of all check star variations versus time for this night's observations.

Figure 7Image biases of only the "check" stars (i.e., reference stars other than the "121" primary reference), after removal of the specific extinction trend for each reference star.

The "image biases" are on the order of 5 mMags, and they are established from individual refernce star deviations from extinction plots having a population SE of ~11 mMags.

When the image biases are removed from the blazar measurements for June 16, the following graph is obtained.

Figure 8Unaccounted for Blazar brightness differences versus time, after removing extinction trends and image biases.  Population SE = 4.7 mMags.

This graph can be used to rule-out blazar brightness changes exceeding ~5 mMag during the 3.5-hour observing period (please control your urge to fit a sine wave to this small data set).  The observing interval represented by each plotted point is 20 minutes, the time it takes to expose four 5-minute images.

The following graph is for another night, showing "blazar magnitude residual" versus time.

Figure 9Blazar brightness versus time for 2003.06.13 UT. The population S.E. is 5.6 mMag.

The unaccounted for blazar magnitudes exhibit approximately the same population S.E. as for the earlier night's observations, 5.6 versus 4.7 mMag.

Other dates have now been analyzed, and the following is a table of all "population S.E." for the night:

    June   8    8.1 mMag
    June 10    6.9    "
    June 11    4.7    "
    June 12  10.8    "
    June 13    5.6    "
    June 14    5.8    "
    June 15    6.3    "
    June 16   14.1   "
    June 18   15.0   "
    June 19    5.3   "
    June 20   13.3   "

Graphs for all of these W Com brightness versus time during a night's observing session can be found at PopSE.

The "median" population S.E. in the above list is 6.8 mMag.  Therefore...

I conclude that an amateur, using a small telescope with a good CCD, is able to monitor blazar brightness changes with a precision of ~7 mMag at intervals of ~30 minutes.
It remains to be seen if blazars typically vary at these low levels on such short time scales. (Maybe the professionals, using larger apertures, have established levels of variations on these time scales, but I'm not aware of it.)

Aperture Photometry Caution

This section illustrates the effect that an interfering star can have if it's located within the reference annulus when performing aperture photometry of an object.

Refer back to Fig. 2, the zoom image of the blazar with an interfering 19th magnitude star 10"arc to the south.  When doing aperture photometry a reference annulus is used to determine an average data number that is to be subtracted from the average of aperture circle.  This difference (aperture circle average data number minus reference annulus average data number) is the intensity for whatever is in the aperture annulus.  The 19th magnitude interfering star just happens to be in a typical reference annulus for my preferred choices of aperture/gap/annulus values, as the next figure shows.

Figure 10Aperture measurement with "aperture/gap/annulus" choice of 6,2,6 centered on the blazar, showing the interfering 19th magnitude star within the reference annulus (at the 5:00 PM location).

In performing an aperture photometry measurement the user can specify three parameters:  aperture radius, gap width and reference annulus thickness.  I'll refer to this set of three parameter values with the term "aperture/gap/annulus," or AGA.  The units for these parameters is pixels.  For this blazar the 19th magnitude star is within the reference annulus for the chosen AGA values.  The amount of apparent dimming of the blazar due to the interfering star will depend on the reference annulus thickness and its radial location - which depends on the other two parameter values.  In other words, we should expect to occasionally encounter an interfering star that causes slight changes in measured magnitude that depend on the values chosen for any of the 3 AGA parameters.  The W Com blazar is one such case, as shown in the next figure.

Figure 11Dependence of apparent blazar magnitude depends on aperture/gap/annulus values.

The range of blazar magnitudes for the AGA values used is 7 mMag.  Since the AGA choice of "727" will have the smallest effect, the corresponding blazar magnitude is presumably the better one.  Different observing sessions will produce different "point spread functions" because of "atmospheric seeing" and guiding differences.  When FWHM is good, small AGA values can be employed.  This will improve the signal-to-noise, which is good.  But the radii that define the reference annulus depend on all three AGA values, and a smaller aperture radius will shrink the reference annulus (for fixed gap and annulus thickness values).  Thus, if an interfering star is present in the vicinity of the reference annulus it is possible to encounter either brightenings or dimmings of the blazar due merely to the interfering star entering or leaving the reference annulus.  Generally, the wider the reference annulus, the better - both from a S/N standpoint and interfering star dilution standpoint.  It is therefore prudent to know if an interfering reference star is present, which requires a "deep" image taken on a night with good "seeing."  If an interfering star is present then this should be a guide in setting AGA values.

Precision Versus Object Brightness

The fainter the object, the poorer the precision, obviously.  Since stochastic uncertainties (also mistakenly called "errors") scale as the inverse of the object's intensity, we should expect a "1/itensity" relation of precision versus brightness.  However, this ignores systematic shortcomings of the analysis procedure.  Even for bright strs, where stochastic issues are less important than systematics, there should exist a non-zero precision.  Thus, a "precision floor" model should be fit to observed precision data.

Figure 12.  RMS scatter of 20-minute data averages after correction for extinction and image bias.  The fitted model is of the form "C1 + C2 / Intensity".

As expected, real data shows that the residual scatter (after allowing for extinction versus air mass and image biases) involves what I will refer to as a "procedure constant" plus the stochastic term that depends as 1/I, where I is star intensity.  Every observer will have a performance plot like the one shown, but with different constants.  When my 14-inch Celestron is returned from repair I expect to have the same 5 mMag "procedure constant" term (for bright stars) but a stochastic curve that's shifted to the right by ~0.8 magnitudes.  The stochastic curve is a fundamental limitation imposed by the number of photons available at the CCD, and the CCD's sensitivity (coldness, etc).  The "procedure constant" term, made apparent by a bright star, is a function of data analysis procedure cleverness, and should remain as a future goal for improvement.

5. Filtered Observations of Long-Term Stability

In what follows, each night's observations underwent the same processing in order to preserve systematic errors for all observing dates.  The same analysis procedure were also adhered to.  Detailed descriptions of these two procedures are found at Observing&Analysis.

So far the blazar "W Com" brightness changes can be categorized as belonging to a fading phase and a rising phase, with an abrupt transition between them.  The fading lasted for ~9 days, and the rise (still underway)  is only 2 days old.  The following graph shows nightly averages.

Figure 13Average brightness (negative of V-magnitude) versus UT date for air mass = 1.2.  The S.E. uncertainties for each night's average are indicated by the height of the plotted symbol. The residual RMS of pre-June 19 data with respect to the fitted polynomial is 17.2 mMag (allowing for 4 free parameters of the polynomial fit).  "Date bias" corrections, based on reference stars, has been applied to the blazar data plotted here.

A ~0.42 magnitude fading (32%) occured during the first phase, and part of this has already been recovered in just 2 days.  The rate of fading reached 59 mMag/day, whereas the brightening is occurring at the rate of 156 mMag/day, or ~3 times faster.  All check stars show no variation.  Here's an example of check star "135" (the "best behaved" check star).

Figure 14Average brightness (negative V-magnitude) versus UT date for air mass = 1.2 for one of the six reference stars. The residual RMS of data with respect to the average value is 7.2 mMag. This graph uses the same magnitude range and date range as in Fig. 13.

All six reference stars exhibit only small departures from their average value (these charts, and some others leading up to them can be found at RMS_residuals).  None show any hint of the trends exhibited by the blazar.  Comparing Fig.'s 13 and 14, it appears that the blazar has larger departures from the fitted trace than the scheck star.  Using only the blazar's fading phase data, the RMS departures from the 3rd-order polynomial fit is 17.2 mMag.  If the check star "135" is used to predict day-to-day departures from a fitted trace, then one would conclude that the blazar exhibited day-to-day departures of ~17.0 mMag.

For this calculation to be acceptable there wil have to be an explanation for the fact that the other check stars have RMS residuals from their "all dates average" that are systematically different from each other.  This is summarized in the following table:

    *131    14.5 mMag
    *148    13.8    "
    *135      7.2   "
    *101    14.9   "
    *147    12.2   "
    *168  156.0   "
    *177  148.0   "

The last two stars are faint, and they weren't used for establishing "date biases."  They were included in the analysis merely to illustrate that the analyses used here breaks down when stars become too faint.  They were used, for example, in constructing Fig. 12 (scatter of magnitudes within one night's observing versus magnitude).  They can also be used in constructing a plot similar to Fig. 12, but showing "night-to-night RMS scatter" versus star brightness.

Figure 15. Night-to-night RMS scatter of check stars (with respect to their all-date average), versus their brightenss.  The black dotted line is a suggested "best possible" performance (i.e., no systematic errors) that has the correct theoretical slope and is adjusted vertically to fit the best points.

Notice that there's a 7.8 magnitude range in star brightnesses in this graph; the brightest star is 1300 times brighter than the faintest.  The blazar brightness is ~14.7, and if it were plotted here it would be among the group of points in the middle.  A rule-of-thumb states that an object star's behavior can be best monitored using reference stars slightly brighter than the object star, but not too much brighter.  This is beacuse there are categoreis of systematic errors that are more likely to be present for bright stars (non-linearity, undetected cosmic ray defects within the bright star PSD, etc) and others that are more likely to be present for faint stars (described below).  The next paragraph argues that indeed the 4 middle stars should be used for estimating the expected "night-to-night RMS scatter" behavior for the blazar.

Notice that the brightest star, star "101" (having a V-magnitude of 10.1, but measured consistently to be 9.95 with my system), has an RMS scatter of 14.9 mMag (1.4%), night-to-night, which is 10 times the expected value.  Yet, the star next to it, star "147" (refer back to Fig. 1 for an image showing these star locations), has a slightly better RMS scatter of 12.2 mMag.  This difference can't be blamed on the notion that the "101" region of my CCD FOV is unreliable, since the two stars are close together. Rather, I suspect that the explanation has to do with the fact that star "101" is bright.  A bright star is subject to non-linearity problems, and it's also more likely to have cosmic ray blemishes that are overlooked (since visual inspection for blemishes is done quickly, at one brightness/contrast setting, usually).

The higher than expected scatter for the faintest stars has to have a different explanation.  The most likely one that I can think of is that the automatic photometry routine used by MaxIm DL positions the aperture pattern at a pixel location that correspons to a maximum intensity.  This method of positioning an aperture pattern for photometry will usually produce anomalously bright results, since random noise will always produce what appears to be a star where there is none if you adjust pixel location a few times.  When many such images are analyzed, some will suffer more from this effect than others, causing a greater than otherwise expected scatter in apparent magnitude.  I therefore discount the two faint star scatter values as being high due to the bias on photometry results due to randome noise.

These various speculations for systematic error sources are summarized in the following graph.  You may consider this graph and the paragraph following it to be an "aside," after which I'll resume discussion of blazar accuracy.

Figure 16.   "Stochastic uncertainty" (green and red) and the several components of "calibration error uncertainty" (dashed blue lines) are orthogonally add to produce "total uncertainty", also called "accuracy."  The 4 components of "calibration error uncertainty" are given letter symbols and are explained in the text.  The exact location of the various calibration plots are appoximately correct for this specific observing system, observing strategy and analysis procedure.

Note that in this figure the brightest stars would correspond to the right side, not the left as in Fig. 15.  It summarizes the inferences described for Fig. 15, and shows how a model could be constructed to account for the several sources of systematic errors.  Line "A" is caused by non-linearity.  Line "B" is caused by cosmic ray defects near the peak of bright stars that are not noticed and edited before combining images.  Line "C" is associated with flat field imperfections.  Line "D" is caused by the use of maximum intensity for locating the aperture pattern for faint stars.  A more complete description of this figure can be found at SE..

Returning to Fig. 15, we are left with 4 check stars for determining what level of night-to-night scatter to expect for the blazar.  Check star "131" is father from the image center than the other three check stars.  It is possible that it is therefore more likely to suffer from flat field imperfections.  This speculation is difficult to verify.  I will take the position that the 14.5 mMag RMS scatter value for "131" may be valid.  If it's accepted as valid, the fitted line would be shifted upward 26%.

We are now able to predict blazar night-to-night RMS scatter. If the "131" datum is omitted, the expected night-to-night RMS scatter for a 14.7 magnitude star, like the blazar, is 12.8 mMag; if the "131" datum is accepted, the expected night-to-night RMS scatter for a the blazar would be 16.1 mMag.  The measured blazar night-to-night RMS scatter (with respect to the fading model) is 17.2 mMag.  Orthogonal subtraction of the two expected values from the blazar measured value yields a pair of values for "unaccounted-for night-to-night RMS scatter of the blazar":  6.1 and 11.5 mMag.  All that can be said, based on this analysis, is that the blazar may exhibit a component of night-to-night variablity, in addition to the longer term smooth variation, that is approximately 10 mMag.

Link to my Sliding Roof Observatory web page
Return to Bruce's AstroPhotos web page


This site opened:  June 7, 2003 Last Update:  June 22, 2003