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Friday, January 16, 2015

Photometry with AIP4WIN: a Tutorial
Part 5 – Setting up and performing differential photometry



To do the differential photometry, AIP4WIN’s Magnitude Measurement Tool was used.  If tracking error did not cause a star’s position in a frame to shift more than 40 pixels from what it was in the previous frame, this tool is intelligent enough to locate that particular star in the current frame that it is working on.  However, some of the star position shifts from image-to-image in the frames I had to use were in excess of 50 pixels!  This meant that I had to adjust images to where they aligned better with each other for the MMT to function properly.  I went ahead and did precise alignment of the stars (even for images where the shift was minor) because this preparation would make it easier to see what was going on at a later time when I was watching MMT display its automatic series of photometric observations.  If your drive tracks well enough so that no star’s position in any frame varies more than 40 pixels from the previous frame, there is no need to do this extra step.  Should that be the case, you can skip to the section of this article called Setting Up for Magnitude Measurements.

Alignment of each star image was done using one image as a base reference image and then making all stars in other images have the same relative positions in the frame as the stars in the base image.  Next will be illustrated the procedure for doing this alignment.


Figure 25: Getting the pixel coordinates of the reference star in the reference image.

As shown in the above illustration of the base image, the mouse pointer is placed over the centre of one reference star that will be used for this purpose in all images.  Its pixel coordinates (X=408, Y=284) can then be read in the Current Pixel frame of the ever-present Image Display Control that is located to the left and bottom of the screen.  From then on, one image after another is loaded and the same procedure is used to note the pixel location of the reference star in the frame.  After the pixel location of the reference star has been recorded for each image, the images are again loaded one right after the other and each image is adjusted so that the pixel coordinates of its stars match the pixel coordinates of the stars in the reference image.  In the case of the immediately following image, the coordinates of the reference star are X=352 and Y=248.  Considering the pixel coordinates from the base image, these results mean that in this frame the reference star is to be shifted to the left 408-352 = 56 pixels and downward 284-248 = 36 pixels.

Figure 26: Translating star positions to match positions in the reference image.

In Figure 26 the Translate-Rotate-Scale dialog box has been invoked from the Transform menu.  To move the star images in the X and Y directions, translation amounts are entered into the appropriate input boxes in the Translate frame and then the Apply button is clicked to yield results shown in the next illustration.

Figure 27: The image after it has been shifted.

Note that the actual exposed area is now shifted in the image frame.  The T-R-S:[5] image name shown in the title bar was automatically generated by the software.  The user should change the computer generated image name to something more meaningful when he/she saves it to disk.  As the reader will see in later depicted images, I added a ‘T’ to the original filename to indicate it had been translated.

I employed AIP4WIN’s blink comparator function to compare the reference image and the translation image to make sure that they were properly aligned.  This Blink function can be invoked from the Multi-Image menu.

Figure 28: The blink comparator dialog box.

As shown in Figure 28, a dialog box allows the selection of the two images used in the comparison, then, the Manual button is clicked to instantaneously switch between the two superimposed images.  If when switching between these two images the stars do not appear to move, then the image translation was properly done. 


Setting Up for Magnitude Measurements

When the user is sure all the stars in all of the frames are at good pixel coordinates from one image to the next, the user would apply the Magnitude Measurement Tool to automatically accumulate and process photometric data from all of the images.  However, before explaining the steps of how a successful photometric analysis of a series of images is done, it is important to mention a pitfall that could be hidden within the header of each of the images if your camera does not properly follow the FITS data standard.  It should be pointed out that this is a highly unlikely problem that most users will never encounter, but I am including a description of the problem and its solution for the sake of completeness. 

The first time I used the Magnitude Measurement Tool or MMT on the University of Tasmania images, data in the charts and graphs that were produced by the tool showed incorrect Julian dates for the observation times.  In fact the images were scrambled out of chronological order in the final data report.  This situation was most peculiar because I had earlier tried MMT with images from sources other than the University of Tasmania and the dates shown for them were correct.  What was different about the University of Tasmania data that caused MMT to malfunction in this manner?

After scrutinizing the headers of the FITS files, it became apparent that one difference between the University of Tasmania files and the files that gave no problems was the manner in which observation dates and times were specified.  The University of Tasmania files had the times and dates defined thusly within the header:
UT      = '18:37:36.0000'      / UT read from telescope                         
UTDATE  = '24-JUL-04'          / UT date read from telescope                   
DATE-OBS= '24/07/04'           / UT date read from telescope                   
Whilst the images from other sources had the time specified in the manner that is recommended in the official FITS file specifications.  With date, the capital letter ‘T’ and the time:
DATE-OBS= '2004-07-24T18:37:36'
Under said official FITS file specifications, time is assumed to be UT unless otherwise specified.  The specifications state that this time assignation method has been recommended since 1996. (FWG 96)  It is strange that the University of Tasmania exposures do not follow this convention given that they were shot in 2004.

When the time assignation in the header of the University of Tasmania image files was altered to follow the official data format of “DATE-OBS= 'YYYY-MM-DDTHH:MM:SS'”, then MMT produced correct Julian dates with points plotted in correct chronological order.  After such corrections were made to all frames, I reran the Magnitude Measurement Tool to do the final data processing.  It was during an earlier run of the Magnitude Measurement Tool (using steps described below) that I initially discovered the time stamp problem I discussed before -- as I mentioned, that is a problem that most users will never encounter.

In order to perform an analysis of a series of photometric images, one clicks the Magnitude Measurement Tool option under the Measure menu.  This action brings up the following dialog box.

Figure 29: The Observer tab of the Magnitude Measurement Tool.

Under the Observer tab, one can either type in the data relating to the observing location and save it for later retrieval with the Save File As button or recall the data for a previously defined location by clicking a location name in the list in the upper right corner and then clicking the Recall File button.  In the image above is displayed the relevant data for the University of Tasmania’s Mount Canopus observatory.  Once the observing location is defined, the Images tab is clicked to designate the images to be photometrically analysed.

Figure 30: Selecting the images to be analysed.

In Figure 30, the Select Disk Files button has been clicked causing the Select Files to Measure file navigation box to appear.  The user has navigated to the appropriate directory and has selected all of the files he/she wishes to analyse.  Clicking the Open button will then cause a return to the Images tab of the MMT.

Figure 31: An image to be used for star selection has been chosen.

In Figure 31 the Pick an Image for Star Selection button has been clicked after the clicking of a filename in the list box.  This action caused the image associated with the filename to appear.  The purpose of pulling up this image is to make optimal adjustments so that MMT can give photometric results with the highest possible accuracy.  Given this fact, it seems logical that one should choose one of the highest quality images available for this purpose, generally by finding an image with high contrast and more detail.

Moving on to the Settings tab will allow the user to set an appropriate size for the star aperture and annulus that is to be used in the photometric measurements.

Figure 32: The light profile of a chosen star.

When a user clicks a star in the image, a yellow aperture circle that is centred on the star engulfs the stellar image and an annulus is exhibited between two red concentric circles.  The yellow circle should be made just large enough such that it encompasses as much of the light emitted by the star as possible and still yield a signal-to-noise ratio that is near the maximum it can possibly be.  The area within the annulus should be placed far enough from the aperture such that it will essentially be illuminated only by normal ambient sky glow.  I entered values for the radius of the aperture circle and also values for the inner and outer radius of the annulus; they are 4.5 pixels, 9 pixels, and 15 pixels respectively.

A light profile plot showing the intensity of incident light is initially displayed by default.  It is easy to see that most of the intensely illuminated pixels reside within the aperture (whose radius is graphically represented between the left hand side of the plot and the leftmost green vertical line).  The middle green line represents the inner radius of the annulus, whilst the rightmost corresponds to the outer radius of the annulus.  Notice the plot is of very low intensity and flat between the rightmost two lines: a good indication that the annulus approximates the ambient illumination of the background sky.

The plotting software algorithms adjust their output data based on data that was input by the user into the CCD Properties frame.  All of this input data was supplied to me either within the project assignment documentation or was gotten from the header of one of the given image files from the University of Tasmania, with the exception of the datum labelled Zero Point.  The purpose of this quantity will be discussed shortly.  All that will be said at this juncture is that 25 is the program’s default value for the zero point and it will probably have to be changed to some other number by the user.  But before that happens, the user needs to determine whether or not the radius of the aperture should be further adjusted.  For this process the SNR radio button is clicked.

Figure 33: Checking the aperture radius with the SNR plot.

Judging by the above plot of signal-to-noise ratio versus radius, it appears that the aperture is already set very close to the size that yields the maximum signal-to-noise ratio.  Therefore, it is OK to leave the aperture set with a 4.5 pixel radius.

Clicking the Curve of Growth radio button causes a plot of stellar magnitude versus radius from the stellar centre to be plotted.


Figure 34: Curve of magnitude growth before the zero point is adjusted.

As can be seen from the curve, the default estimation for the instrumental magnitude of the comparison star is about 17.395.  Consulting the aforementioned USNO database, this star (designated 0757-0840610 in the NOMAD catalogue) has a listed V magnitude of 14.36. The zero point is used to bring the instrumental magnitude down to near its accepted true magnitude.  Figure 35 reveals what the curve looks like after the zero point is adjusted.

Figure 35: After the zero point has been adjusted.

Notice that the zero point has been changed to 21.96 and the instrumental magnitude of the comparison star now reads very close to the database value of 14.36.

The next step is to indicate to the software what stars in the frame are to be used for photometric purposes.  The Stars tab is clicked.

Figure 36: Star data and stars were chosen for photometry.

The data for the five prominent stars shown in the frame were gotten from the USNO database mentioned previously and entered by the user into the dialog box.  This block of data was then saved for later retrieval under the filename of GLIESE 876-V, with the -V in the filename indicating that the brightness data is in V filter magnitudes.  The magnitude of Gliese 876 was left blank because that is the datum to be photometrically determined.  However, remember the only two stars that have a high enough signal-to-noise ratio for accurate photometry are Gliese 876 and 0757-0840536; therefore, only these two will be employed for now.  Another use for the data belonging to the other stars will be discussed later.

First, the subject of the research, Gliese 876, is clicked to cause yellow aperture and annulus circles to surround it along with a “V” to indicate it is the potential variable star being studied.  A click of the star that sits to the right makes it become encircled in a green aperture ring and blue-green annulus rings along with the designation “C1 to indicate that it is the first star (in this case the only star to be used) from the comparison star list.

It is regrettable that the other stars are too noisy for use in the photometric run because they could have been used to check the accuracy of the photometric measurements.  It is important that C1 has no intrinsic short-term variability or the photometric measurements will be for naught.  No catalogue information could be found relating to whether or not it is a variable.  If the other stars had been usable, clicking each star in the image would have caused aperture and annulus circles to appear around the star and the computer program would have labelled them: C2, C3, and C4 in succession.  During the photometric run, the program would have figured one cumulative magnitude to represent all three of the other comparison stars and the difference between this cumulative magnitude and C1 would have been computed for each image. (Berry and Burnell 294) Large fluctuations in this magnitude difference from one frame to the next would reveal the degree to which the chosen C1 might have varied from a constant value.  As it is, the only way that the user will find out if the variation of C1 is too great is if there are large amounts of variation in the magnitude difference between the target star and C1 in the final data that cannot be accounted for otherwise.

There is the additional issue of Gliese 876’s known inherent variability, but this is only a very small long term variability with a maximum range of 0.05 magnitudes over approximately a 1000 day period. (Shankland et. al. 703)  The transition time between maximum and minimum or vice versa should be half the period or about 500 days.  This fact would imply a crude average estimate of 0.05 magnitude / 500 day ≈ 0.0001 magnitude/day rate of change. Given this fact, it seems reasonable me that there would probably be no detectable change in magnitude induced by Gliese 876’s inherent variability over the one night in which the University of Tasmania images were shot.

With the required stars chosen, the user moves to the Report tab.

Figure 37: Choosing the type of data report.

Under this tab, the user can choose the content and format of the final data produced from the photometric measurements.  A couple are special formats for organizations such as the AAVSO (American Association of Variable Star Observers) and the CBA (Centre for Backyard Astrophysics), while some others may have fairly uncommon or even esoteric uses.  The one that best fits the purpose of this project is invoked by clicking the Raw Instrumental Mags radio button.   With the report format chosen, it is time to move to the Execute tab.


Figure 38: Preparing to execute photometric measurements.

The default settings under the Guiding Properties frame will serve the user well in this case.  For each image that is brought up, the software will locate the chosen stars.  Because of the earlier implemented image translations intended to make the position of all stars coincide from image to image, the default 20-pixel search radius is irrelevant and can be left alone.  None of the guide star options need to be used at all.

At this point the only thing to do is to begin the series of photometric measurements by clicking the Run Photometry button.  The computer then displays each image in the image list and looks for the appropriate stars, does the photometric measurements, then repeats the process with the next image.  All of the data gets stored to a window called the Data Log.  The reader may notice from the following depicted data log that the images to be excluded were included in the photometric run.  The reason is that it is more convenient to delete their corresponding data lines after the run than to exclude them from the photometric series. 

It should be noted that from here on out, I will employ capital letters to denote actual standard magnitudes and lower case letters to stand for raw instrumental magnitudes.  Differences in magnitudes will be preceded by the Greek letter Δ.  In other words, Δv will represent the difference between two stars’ raw V filter magnitudes, whilst ΔV will represent the difference between the two stars’ standard V filter magnitudes.

Figure 39: The Data Log.

The Data Log for this photometric run may be saved to disk via that window’s Save to File menu.  AIP4WIN data logs are formatted to be loaded into Microsoft Excel.  Therefore, I turned the data log file into a spreadsheet under Excel, then added a column to figure the difference in measured magnitudes of the comparison star and Gliese 876 under the column heading Δv.

Since the comparison star is of a dimmer/higher magnitude and it is assumed to be of constant brightness, having the target star’s magnitude subtracted from the comparison star’s magnitude (rather than the other way around) causes Δv to increase when the target star brightens (magnitude decreases) and decrease when the target star dims (magnitude increases).  This fact makes unnecessary the extra step of inverting the plot under Excel in order to have increasing brightness of Gliese 876 in the upward direction of the Δv plot.  The following Table 1 reveals the results after the data for the inadequate images have been deleted.

Remember in differential photometry, the most significant thing is the magnitude difference between the comparison star and the target star (Berry and Burnell 279, 292) and thus any wide fluctuation of the actual raw magnitude of each of these stars from image to image is irrelevant.   One can see a demonstration of this fact in the grey area at the bottom of the table showing that the value of Δv ranges over a width of only 0.047 magnitudes, whilst variations in the target star and C1 instrumental magnitudes each span a range of one full magnitude!

Image #
Julian Day

Δv

Gliese 876

0757-0840536
(C1)
Noise Error ± Mag  Image Name
2
2453211.229
4.375
10.504
14.879
0.007
X2253V2470003CFT.fits
4
2453211.241
4.381
10.141
14.522
0.006
X2253V2470005CFT.fits
5
2453211.245
4.375
10.198
14.572
0.006
X2253V2470006CFT.fits
6
2453211.250
4.363
10.146
14.509
0.006
X2253V2470007CFT.fits
7
2453211.255
4.374
10.068
14.442
0.006
X2253V2470008CFT.fits
8
2453211.260
4.362
10.233
14.595
0.006
X2253V2470009CFT.fits
9
2453211.266
4.363
10.064
14.427
0.006
X2253V2470011CFT.fits
10
2453211.270
4.376
10.149
14.525
0.006
X2253V2470012CFT.fits
11
2453211.276
4.371
10.116
14.487
0.006
X2253V2470016CFT.fits
12
2453211.281
4.386
10.131
14.517
0.006
X2253V2470017CFT.fits
13
2453211.285
4.366
10.275
14.641
0.006
X2253V2470018CFT.fits
14
2453211.288
4.371
10.269
14.640
0.006
X2253V2470019CFT.fits
15
2453211.291
4.372
10.119
14.491
0.006
X2253V2470020CFT.fits
16
2453211.296
4.373
10.157
14.530
0.006
X2253V2470023CFT.fits
17
2453211.298
4.372
10.192
14.564
0.006
X2253V2470024CFT.fits
18
2453211.302
4.366
10.545
14.911
0.007
X2253V2470025CFT.fits
19
2453211.305
4.371
10.932
15.303
0.009
X2253V2470026CFT.fits
20
2453211.309
4.350
11.070
15.420
0.010
X2253V2470027CFT.fits
21
2453211.313
4.386
10.193
14.579
0.006
X2253V2470029CFT.fits
22
2453211.316
4.369
10.258
14.627
0.006
X2253V2470032CFT.fits
23
2453211.319
4.366
10.236
14.602
0.006
X2253V2470033CFT.fits
24
2453211.324
4.374
10.297
14.671
0.006
X2253V2470034CFT.fits
25
2453211.327
4.368
10.284
14.652
0.006
X2253V2470035CFT.fits
26
2453211.330
4.375
10.314
14.689
0.007
X2253V2470036CFT.fits
27
2453211.333
4.383
10.332
14.715
0.007
X2253V2470037CFT.fits
28
2453211.337
4.397
10.265
14.662
0.006
X2253V2470040CFT.fits
29
2453211.341
4.369
10.375
14.744
0.007
X2253V2470041CFT.fits
30
2453211.344
4.386
10.366
14.752
0.007
X2253V2470042CFT.fits
31
2453211.348
4.355
10.464
14.819
0.007
X2253V2470043CFT.fits
32
2453211.351
4.386
10.770
15.156
0.010
X2253V2470044CFT.fits

Avg
4.373
10.315
14.688



Max
4.397
11.070
15.420



Min
4.350
10.064
14.427



Range
0.047
1.006
0.993



Max - Avg
0.024
0.755
0.732



Avg - Min
0.023
0.251
0.261


Table 1: Photometric data from an Excel spreadsheet.

According to AIP4WIN’s built-in help documentation, the MMT automatically figures an error range for each star by converting the standard deviation of the noise part of the signal-to-noise ratio of each star into magnitudes.  Since the value of Δv is derived from a calculation using the magnitudes of the comparison star and the target star, it makes sense that the errors of magnitude for both of these quantities would apply to Δv.  Therefore, I computed error bar values for Δv by adding the noise errors of the subject stars and the results are shown in the Noise Error column of the above table.

Figure 40: Plot of photometric data.

In the above plot of Δv, there is some variation, but even the maximum variation from the green line denoting the mean value is no more than ±0.024 magnitudes (see Table 1)!    Shortly there will be a discussion of why and how the error bars in the plot should be extended.   Only the raw Δv magnitude differences shown in Table 1 and plotted in Figure 40 are needed for obtaining valid final research results in cases where all of the research data comes from one observatory using one telescope/CCD setup. (Warner 28)   However, colour transformation calculations to change the raw magnitude difference to a standard magnitude difference must be done if this data is to be mixed with data obtained from other observatories, and that issue is what will be covered next. (Warner 28; Benson 42)

The aforementioned USNO online database was used to obtain the colour characteristics of the stars that are most prominent within the images.  Here is a table with all prominent stars labelled in decreasing brightness (increasing magnitude) order C1, C2, C3 and C4.  I added columns for colour indices titled (B-V), (V-R) and (V-I) that may be used later.

Star
B
V
R
I
(B-V)
(V-R)
(V-I)
Gliese 876
11.77
10.27
9.39
8.57
1.50
2.38
1.70
C1
14.89
14.36
14.22
13.38
0.53
0.67
0.98
C2
17.58
16.32
15.56
15.10
1.26
2.02
1.22
C3
18.05
16.83
16.56
16.46
1.22
1.49
0.37
C4
17.25
16.97
16.65
16.61
0.28
0.60
0.36
Table 2: Standard magnitudes of stars in images and calculated differences.

It is not unusual for one observatory to want to blend its data with the data obtained from one or more other observatories to obtain greater confidence in a research outcome.  However each observatory will typically have a different filter/CCD combination.  Each filter will differ from a perfect band pass match for the ideal U, B, V, R, or I filters and no two CCD chips will have exactly the same response over their entire spectrum of sensitivity; therefore, certain procedures have been developed that are meant to compensate for this fact.  They involve calculations that will effectively convert raw magnitude differences to standard magnitude differences and allow direct comparison of data from different observatories. (Benson 42)  A discussion of these calculations will begin in the next paragraph.  If you are not going to be blending your observation data with observations done at another location, you can skip to the part of this documentation labelled Analysis of the Photometric Data.

Since there are only 5 of the near infrared I filter images of 2 seconds integration, it is obvious that they were not included to gather transit data on the target star.  Instead, they must be there solely to obtain a transformation coefficient to be used in the above-mentioned colour related calculations. (Benson 44)  The low band optical R filter images were also too few for transit analysis and therefore must have been supplied for the same reason as the I filter images.  However, the R filter images were rejected as unsuitable because their FITS headers revealed that they were of varying integration times and at least one of the images whose filename indicated it was an R filter image was actually taken with an I filter according to its header.

The idea behind the colour related calculations is to adjust for differences in a real world filter/CCD combination that stray away from the theoretical ideal filter/CCD combination and to compensate for the normal decrease in redness to which any star is subjected as it travels from the direction of the eastern horizon to the meridian and the reddening that occurs as the star transits from the meridian toward the western horizon. (Benson 42; Warner 29)   The same master bias frame and scaling master dark frame used with the V filter images were employed to calibrate the I filter images, but another flat-field image had to be made from raw I filter flat images of 3 second integration.  After calibrating the I filter images in the manner described earlier, several problematic issues arose.  Remember that it was mentioned earlier that comparison stars should ideally have an SNR of at least 100 which should give an error of about ±0.01 magnitudes.  The brightest star (other than Gliese 876 itself) in the I filter images is C1 and its best SNRs in that band were in the upper 50s, an SNR that offers at best an error of around ±0.02 magnitudes.  The best SNR for C2 was around 20 for an error of ±0.05 magnitudes and it was much worse for C2, C3, and C4.  Given this fact, the only stars that could be of real use in colour compensation would be Gliese 876 itself, C1 and possibly C2.  The above magnitude errors were obtained using the formula: σ = 1.0857 / SNR. (Berry and Burnell 278)

It appears that this is a situation where there is no choice but to do colour compensation by observing only a few stars through two filters (V and I) with one or two of those stars having low SNRs, when it is standard practice to perform this operation on a sizeable number of high SNR stars with several filters. (Benson 48)  If the R filter images had not been of such questionable quality, this may not have been a major issue, but as it is the I filter images are the only ones that can possibly be used for the transform calculations.  The situation is such that the colour transformation operation could yield less than optimal results.

Again, remember that in differential photometry, the difference in magnitudes is what is most important.  The following equation can be used for the colour transformation to allow V raw magnitude differences to be converted to true V standard magnitude differences:
ΔV = Δv + Tvi * Δ(V-I)  (Benson 45; Warner 49)
In the above formula, Tvi is the previously mentioned transformation coefficient.  Δ(V-I) is the difference between the V-I colour indices of the comparison and target stars.  Benson mathematically proves on the same page cited above that no terms involving airmass need to be considered in the above equation because they cancel out during the formula’s derivation.
 
However to use this formula requires that one first know Tvi.  If one plots the officially accepted values for V-I of each star on the X axis and the instrumental difference v-i on the Y axis, then Tvi will be the inverse of the slope of the best-fit regression line through the points. (Benson 46)  To derive the slope it was first necessary to find out which of the usable V and I images were contemporaneous.  Only three of the I filter images were used because the other two were of very poor quality as defined by stellar SNRs.  Table 3 reveals this information.

V Image
UT
I Image
UT
X2253V2470020
18:59:20
X2253I2470022
19:01:38
X2253V2470029
19:30:13
X2253I2470031
19:32:21
X2253V2470037
19:59:52
X2253I2470039
20:01:53
Table 3: Usable contemporaneous V and I images.

Using the methods outlined earlier, instrumental magnitudes were found for the I filter images (V filter image magnitudes were already available from the photometric run).  An optimum aperture radius of 3.8 was determined to give the best SNR for the I filter images.  Yet another zero point of 21.52 was obtained for the I filter images in the manner previously described for the V filter images.  Raw instrumental magnitudes were gotten for the three stars using MMT.  After that a plot of the necessary data was made in Excel.  In the table below, C2 is the third brightest star in the images with the third highest SNR that, because of SNR values in the 40s in the V band and 20s in the I band, was still a bit iffy to use.  However, I felt uneasy about using only two stars to derive the coefficient since a large number of stars are supposed to be used (Benson 48); therefore, C2 was included despite its less than optimal SNRs.  As was mentioned earlier, SNRs of other stars were far worse, in the 20s even for the V band and in the teens and single digits for the I band.

Star
V-I
v-i
v
i
G876
1.704
2.930
10.119
7.189

1.704
2.826
10.193
7.367

1.704
2.903
10.332
7.429
C1
0.980
0.855
14.491
13.636

0.980
0.783
14.579
13.796

0.980
0.849
14.715
13.866
C2
1.22
1.304
16.429
15.125

1.22
1.131
16.506
15.375

1.22
1.138
16.645
15.507
Table 4: V filter and I filter data for colour compensation.


Figure 41: Plot and equation for the regression line used to derive Tvi.

Therefore, the above Excel spreadsheet plot of the data from Table 4 means that Tvi is 1 / 2.9373 or approximately 0.340.  Now that Tvi is known, one can apply the equation for ΔV to each previously determined Δv within an Excel spreadsheet.

Table 5 shows each of the newly figured ΔV difference values along side of the old raw magnitude difference, Δv.  The source of the larger error bar values in that table will be explained shortly.

Image_#
Julian Day
ΔV
Δv
± Error
 (SNR+σ)
2
2453211.229
4.129
4.375
0.017
4
2453211.241
4.135
4.381
0.016
5
2453211.245
4.128
4.374
0.016
6
2453211.250
4.117
4.363
0.016
7
2453211.255
4.128
4.374
0.016
8
2453211.260
4.116
4.362
0.016
9
2453211.266
4.117
4.363
0.016
10
2453211.270
4.130
4.376
0.016
11
2453211.276
4.125
4.371
0.016
12
2453211.281
4.140
4.386
0.016
13
2453211.285
4.120
4.366
0.016
14
2453211.288
4.125
4.371
0.016
15
2453211.291
4.126
4.372
0.016
16
2453211.296
4.127
4.373
0.016
17
2453211.298
4.126
4.372
0.016
18
2453211.302
4.120
4.366
0.017
19
2453211.305
4.125
4.371
0.019
20
2453211.309
4.104
4.350
0.020
21
2453211.313
4.140
4.386
0.016
22
2453211.316
4.123
4.369
0.016
23
2453211.319
4.120
4.366
0.016
24
2453211.324
4.128
4.374
0.016
25
2453211.327
4.122
4.368
0.016
26
2453211.330
4.129
4.375
0.017
27
2453211.333
4.137
4.383
0.017
28
2453211.337
4.151
4.397
0.016
29
2453211.341
4.123
4.369
0.017
30
2453211.344
4.140
4.386
0.017
31
2453211.348
4.109
4.355
0.017
32
2453211.351
4.140
4.386
0.020
Avg

4.127
4.373

Max

4.151
4.397

Min

4.104
4.350

Range

0.047
0.047

Max-Avg

0.024
0.024

Avg-Min

0.023
0.023

Table 5: Δv magnitude differences adjusted to ΔV magnitude differences.

As can be plainly seen from the above table, though the ΔV values are somewhat lower than their Δv counterparts, the range between the minimum and maximum values stayed at 0.047 magnitudes, and the greatest distance above or below the average did not change, as one would expect since a constant value of Tvi * Δ(V-I) = 0.340 * (0.980 – 1.704) = 0.340*(-0.724) = -0.246 was applied to each raw Δv.  In the computing of Δ(V-I), the target star’s V-I colour index was subtracted from the comparison star’s colour index (rather than the other way around) because when the Δv values were figured the target star’s raw magnitude was subtracted from the comparison star’s raw magnitude. To emphasize the lack of change in variability, the new ΔV data is plotted in Figure 42 in order that it can be compared to the Δv plot in Figure 40.

Figure 42: Plot of photometric data converted to true V band magnitude differences.

So the variation in both charts is identical, as expected.  The only advantage this conversion has is that it supposedly yields true ΔV values (actual V standard magnitude differences) that can be directly compared with other observatories’ true ΔV values obtained using the same method. (Benson 42)  Again, in cases such as this project where all of the data is coming from a single observatory, such conversion to a standard magnitude difference is not necessary because the raw Δv plot reveals the same amount of information about how much the light curve varied. (Warner 28)  The ΔV plot was done mainly for the sake of completeness of coverage of useful photometry techniques.

The way things stood with the low quality of the data obtained from the images that were used for figuring the transformation coefficient; I would not trust the conversions to standard magnitudes for direct comparison with results from other observatories.  However, since such a comparison was not to happen in this project and since the conversion would not affect the all-important variation, the ΔV plot was still usable for the purposes of this paper with no sacrifice of accuracy in measured magnitude changes.

By the way, to get truly reliable colour transformations to standard magnitudes, I might have made a few special frames for both the V and I filters where the image of Gliese 876 was purposely saturated so that the exposure would be long enough for the remainder of the stars to have SNRs above 100 though still be unsaturated.  I then would have had four quality stars to use for finding a transformation coefficient.  I would not have used a magnitude reading from the pixels of Gliese 876 at all during calculation of the colour transformation coefficient because of its requisite saturation, but only used the remaining unsaturated stars.  Of course, none of the V filter images with a saturated target star would have been used in the actual photometric run to determine Δv values.

The above colour compensation calculations were for what is called first-order extinction.  There is also a second-order extinction colour compensation that may be done, but that is primarily used for making measurements from B filter and Clear filter images because of their higher sensitivity to airmass induced differential extinction across the transmitted wavelength band. (Warner 29-30)  Since there were no B filter or Clear filter images supplied with this project, second order colour correction was not carried out.

Analysis of the Photometric Data

Given that most of the variation revealed in either Figure 40 or Figure 42 has a random look to it, I am loath to say that there is any variation in the plot that is of other than systematic origin.  This idea is bolstered by the fact that, on much of the chart, the variation seems to go down a relatively large amount below the average, then usually goes up above the average to an amount of nearly equal distance, or vice versa.  In statistical analysis, such features are the type one normally sees associated with systemic errors.  Recall that it has already been proven that there were episodes of varying cloud haze during the observation period.  The relatively flat area could be a short episode of relatively constant haze thickness lessening the change in ΔV or Δv due to the large difference in colour between the comparison star and the target star.  To test the constant cloud thickness hypothesis for the relatively flat part of the plot, I checked the images represented in the flat section and found that they had a long continuing series of C1 SNRs hovering around 180 and they also consistently had C1 light curves of similar thickness.  To top it off, the most extreme low values in the plot typically had much lower C1 SNRs in the low 100’s and light curves that were somewhat thicker than their less extreme counterparts.

As indicated earlier, of all of the photometric methods, differential photometry is the least sensitive to airmass extinction related problems.  However, given that B-V colour indices of two stars are denoted as CI1 and CI2, then a colour difference of  (CI1 - CI2) which is ≥ 1.0 is “… a fairly large color difference that should be avoided even with differential photometry.” (Warner 78)  The bold italics are not from Warner but from me.  Berry and Burnell generally concur with this assessment on page 292.  However, it should be pointed out in this case that even if there is a significant difference between the colour indices of Gliese 876 and the chosen comparison star, the use of that particular comparison star could not be avoided because it is the only star other than the target star with the requisite SNR of greater than 100.

So is there a large difference in colour?  Using data from the USNO database, the B-V colour index (see Table 2) for Gliese 876 is 11.771-10.274 = 1.497, whilst for the comparison star it is 14.89-14.36 = 0.53.  The difference between the indices of 0.967 just barely qualifies as OK by Warner’s definition, but is probably less than optimal.  There are other stars in the field that were almost perfect colour index matches for the target star, but they were much fainter and their low SNRs indicated that they were unusable.   A large colour index difference between two stars coupled with rapid small changes in airmass should cause noticeable changes in shorter wavelength extinction.  That condition is the reason why, “Working with comparisons that have a large color difference from the target is pushing your luck too much. Sometimes you have no choice.” (Warner 69)  And this situation appears to be a case where there was no choice; hence, some accuracy will be lost due to a radical colour index difference.  I was concerned that the amount of induced error from this effect might possibly be enough to disallow reliable evidence of a planetary transit to be extracted from the photometric observations.  I was determined to find a way to compensate for this particular systematic error and any other systematic errors for which there was initially no compensation.

The error bar ranges that were seen in Figure 40, though good as far as measuring error induced by signal noise problems, do not address issues related to the large star colour difference.  It should be further pointed out that I was unable to find out whether or not the comparison star has any short-term variability; hence, yet another portion of the systematic error could come from that.  Therefore, it seems that the true error for each ΔV or Δv should be the error induced by signal noise plus the error induced by whatever the total remaining systematic problem is.  According to standard statistical procedure, a reasonable estimate of the unaccounted systematic error should be obtainable by calculating the standard deviation of the values.  Using Excel’s standard deviation function within the spreadsheet that produced the last plot gave a standard deviation for ΔV of approximately 0.00987 or 0.01 magnitudes for all intents and purposes.  Thus one should add 0.01 magnitudes to the already existing signal noise error on each side of each error bar and this change is reflected in Figure 42.  As the reader can see from that plot, the error bars came to jibe fairly well with the variations after this method was applied!

There is one datum above the mean line (remember, ideally, a rise in ΔV indicates an increase in brightness of the target star assuming a constant C1 magnitude) whose error bar does not touch the mean line. But the points immediately before and after it are much lower than it is, indicating that it is an unreliable point whose error bar is not quite long enough.  If it had signified the ending of a transit, any lower points that followed it on the right would be higher than most of the other low points to the left of it in the plot.  Instead, two of the four points to its right are well below the mean line and one of those two points is almost as low as the most extreme low point.  So the fact that the highest point’s error bar does not quite touch the mean probably indicates that the added compensation for systematic error is still off by a very small amount for this particular point.

Of the values under the mean (the ones that are possible dimming of the target star) there is only one value whose error bar doesn’t quite reach the green average line. That point’s error bar may be slightly shorter than it should be because the point plotted immediately after it is far above the average line.  Furthermore, this datum is associated with image number 20 in Table 1, whose Δv signal noise magnitude error (as shown in that table) is higher than in all other images save one and that fact only reinforces my impression that it’s error bar may be a hair-width too small.

Judging by the low end of the error bar belonging to the lowest plunging point, it appears that one can be fairly confident that no transit occurred that had a magnitude plunge of greater than about 0.042 magnitudes below the mean.  A transit with a lesser plunge might have occurred, but that is not possible to know with any certitude because it would be hidden within the error range of the plot.

For a typical extrasolar planetary transit investigation, the range of variation that is here would be too great to make any valid conclusions, since the variations typically produced by such a transit would be around three hundredths of a magnitude (Gary 3) and thus would be hidden within the error range of the plot.  However, it is about to be shown that Gliese 876 is a special case and some very definite conclusions are about to be drawn as regards to whether or not a planetary transit took place!

To see how this can be the case, one should look at the following plot of possible light curves for a planet c transit.  (Shankland et al. 702) The Julian dates are those for a possible transit that was predicted to occur on another day than the one considered here, but the curves should apply for any transit of planet c on any other date.


Figure 43: Possible dimming curves for a transit by planet c. (Shankland et al. 702)

In the above plot, solid curves represent periastron transits whilst dotted curves stand for apastron transits.  The deepest dip for each type of curve represents an orbital inclination of i=90º with each successive line upward being a tenth of a degree less.  I added the red horizontal line to mark the lower end of the uncertainty that was obtained from the end of the lowest error bar in the plot of Figure 42 to allow the determination of which curves may be definitely excluded. The very darkest peristron curve is really the two curves for i=90º and i=89.9º that are plotted almost on top of each other. The maximum possible dimming due to a transit of planet c is 0.13 magnitudes, well beyond the range of the values that were photometrically obtained in this project!

Why such a large possible dip?  As was mentioned before, Gliese 876 is a very small class M dwarf and if planet c were at a transit inclination it would be roughly of Jupiter’s size.  (Shankland et al. 701-702)  From this fact it is obvious that the relative size of the planet to the primary would be much greater than normal; therefore, the transiting planet would cut off a greater fraction of the primary’s light.

Recall that the maximum possible dip below average in the photometrically obtained plot of the University of Tasmania data is 0.042 magnitudes.  Going by Figure 43 one may definitely conclude that no periastron transits occurred for inclinations in the range of around 89.26º to 90º and no apastron transits happened with inclinations of about 89.52º to 90º.  By implication, transits occupying the gamut of inclination ranges in between these two orbital positions are also ruled out instantly.  Of course, logic dictates that the apastron exclusion range of 89.52º to 90º implies that indeed no transit occurred in that particular range of inclinations regardless of the planet’s relative orbital position at the time!   Again, the error range in the ΔVs preclude any certain conclusions concerning the smallest inclinations shown above the red line in Figure 43.

One can only imagine how much more information could have been gleaned from the data if the subject images had been more meticulously produced.  It is possible that all of the curves in the Shankland et al. plot may have been excluded to prove conclusively that no transit occurred under any conditions or it might even have been shown that one of the shallower transits occurred.  However both of these scenarios are unknowable conjecture.


Summary of conclusions drawn from the data
The FITS image headers indicate that the images were shot on July 24, 2004.  This date coincides with a possible transit of Gliese 876 by planet c. (Shankland et al. 702)  Hence, I assumed that the detection of a transit by planet c was the object of the observations.

Evidence of some degree of a lack of stringent attention to detail by the original producers of the images initially shook my confidence as to whether or not useful data could be gotten from the images.

For instance, certain comments in the header of the FITS files were for Eta Bootis rather than Gliese 876.  However, subsequent examination of actual data stored in the header along with a chart obtained from the USNO database proved that the images were indeed of the field surrounding Gliese 876.

Another disturbing thought is the possibility that there may have been less than careful handling of the CCD camera by an operator whose sneeze particle may exist on the camera window.  Whether from a sneeze or not, this blotch was caused by something from the external environment that came to be adhered to the window.  Before I attach my own CCD camera to my telescope, I always examine the condition of the window and clean it if necessary.  Furthermore, a less than optimal number of flat-field frames were shot and no dark flats were made, making signal-to-noise ratios in the final images lower than they had to be. Also, the lack of tightly controlled drive guiding necessitated that I translate the images into better alignment so that AIP4WIN’s Magnitude Measurement Tool could perform its automated functionality properly. Finally, the specified time in FITS files did not follow the officially recommended specification, necessitating that I make the necessary adjustment to the image files so that Julian dates would be displayed correctly and the images plotted in correct chronological order under AIP4WIN.

It was determined that the CCD camera used was of the temperature-controlled variety.  This meant that the production of a scaling dark frame was necessary.

On the plus side, the bias and dark frames supplied were of fine quality and allowed the creation of high quality master bias and scaling master dark frames.  Also, what flat-field frames that were shot were exposed to the requisite half maximum of saturation.  These facts kept the lack of a first rate master flat-field frame from being a worse problem than it was.

All image frames were checked for star saturation with the Single Star Photometry function, and any that contained a saturated image of Gliese 876 were excluded from the photometric study.  Conclusive evidence of a thin layer of haze in the sky that night was obtained when the Star Image Tools light profile function revealed that cloud particles were scattering light away from the central star disk.  A few images that revealed scattering at a detrimental level were rejected under the criterion that the comparison star should not have a signal-to-noise ratio less than 100. (Berry and Burnell 296; Warner 34)

Because there were no comparison images of parts of the sky other than the field containing the target, it is obvious that the University of Tasmania observers intended differential photometry to be used in their analysis. There are three reasons why the choice of this method was a good decision:
1)      The fact that accurate differential photometry may be done through thin clouds. (Berry and Burnell 293)
2)      It is simpler to implement than other methods. (Warner 32)
3)      Atmospheric extinction issues, such as cloud variation and airmass, affect it less than any of the other photometric methods.  However, a radical difference in the star colour of the target and comparison star can sometimes introduce a certain amount of a special type of airmass induced extinction for, not just differential photometry, but  other methods of photometry as well. (Berry and Burnell 279; Warner 32-33)  It is recommended that a comparison star of radically contrasting colour be avoided, since there is little that may be done to counter the detrimental effect because some extinction beyond normal proportions will occur with small airmass changes. (Warner 69)

Stating point 3 above in other words, because the comparison star is assumed to be of constant brightness, any variation in the difference in magnitudes of the target star and comparison star is presumed to be solely from changes in the true brightness of the target star.  The relative insensitivity of the difference between the magnitudes of the comparison star and the target star to atmospheric extinction means that variations in this difference are a better indicator of real changes in target star brightness than the instrumentally measured magnitude of the target star.  By contrast, the instrumentally measured magnitude of a star can change over a wide range of magnitudes due to variations in cloud thickness and airmass and thus would not be a reliable measurement of a real change in brightness of the star.

The Magnitude Measurement Tool was used to do the following:
1)      Enter and store relevant information about the observatory and instrumentation used to obtain the images.
2)      Choose images to be photometrically measured.
3)      Set optimum aperture and annulus radii.
4)      Adjust the magnitude zero point so that raw instrumental magnitudes would automatically come out close to their true standard magnitude counterparts.
5)      Enter relevant catalogue data on the target and comparison stars.
6)      Indicate to the software where the subject stars were to be found in an image.
7)      Choose the type of output report in which the final analysis data is to appear.
8)      Initiate execution of the automated photometric measurement of the entire series of images with the resulting data instantly placed into a log file.

Only one star other than Gliese 876 had a high enough signal-to-noise ratio for it to be used as a comparison star for differential photometry.  Thus, the extra accuracy check that additional comparison stars would normally provide was not available because of the poor SNRs of the remaining stars.

After converting the log file containing the photometric results into an Excel spreadsheet, a graph was made of the difference in raw magnitude between the comparison star and the target star- a quantity called Δv.  These values may have been somewhat adversely affected by the fact that the target and companion stars were of significantly different colour.  However, as mentioned before, the extremely low signal-to-noise ratios of the other stars in the image field makes it impractical to use a different comparison star.

Also, the situation was further complicated by the fact I could not find out whether or not the one comparison star that was used has a short-term variability.   This issue could have been directly addressed if more than one comparison star could have been chosen, because the difference between the cumulative magnitude of the other comparison stars and the main comparison star would have revealed such variability. (Berry and Burnell 294)  Nevertheless, I was able to address this issue and other possible systematic errors via statistical means in the final phases of data analysis.

Gliese 876’s meagre inherent maximum variability of about 0.05 magnitudes over a period of approximately 1000 days (Shankland 703) implies a rough change over the 500 day transition from maximum to minimum or vice versa of somewhere near 0.0001 magnitude/day.  Hence, this inherent variability should be of no concern because such a miniscule difference would be undetectable over the one night during which the research images were acquired.

Of the V, R, and I filter images, the R filter images had to be rejected out of hand because their FITS file headers revealed conflicting characteristics between the images.  These problems included inconsistent integration times and even one frame that was supposed to be R filter that was actually I filter.

Only raw Δv magnitude differences need be done to obtain accurate final results. (Warner 28)  However, if results are going to be compared to results obtained from other observatories, colour transformation calculations should be done. (Warner 28; Benson 42)  I applied the techniques that are supposed to compensate for filter/CCD induced colour differences and the reddening effects of the atmosphere due to increasing airmass by using the I filter images and V filter images that were included for that purpose.   This procedure was supposed to produce standard ΔV magnitude differences from the raw instrumental Δv magnitude differences via the production and application of a transformation coefficient.  Normally, the ΔV magnitude differences should be directly comparable to other observatories’ ΔV observations of the same target and comparison star during the same time period.  However, because of the lack of quality data for the transformation calculations, using them for comparing data with other observatories could prove problematic.  I suggested a method that could have prevented this problem that involves making a few V filter and I filter images solely for colour transformation purposes with intentionally saturated Gliese 876 pixels and longer exposed unsaturated other stars with resulting higher and more usable SNRs.  Gliese 876 would then not be used to get a colour transformation coefficient, just the other stars.  The few special V filter images obtained to figure the transformation coefficient would not be used with the other V filter images during the main photometric run because of the saturation of Gliese 876.

Since the colour transformation calculations only add a constant value to each Δv to obtain ΔV and the variations are the same for a plot of either ΔV or Δv (Warner 28), any inaccuracy in that constant value does not affect the accuracy of the observed variation.  The validity of this logic is bolstered by the ΔV results in Table 5 and Figure 42 which show the same variation as in the Δv results in Table 1 and Figure 40. Therefore, the accuracy of any conclusions about change in magnitude of the target star that would be made from the ΔV plot would be no different from what they would be for the Δv plot, even if the ΔV values were derived with a suboptimal transformation coefficient caused by the problems mentioned in the previous paragraph.

The relatively random appearance of the amplitude of the magnitude difference curve above and below the average value is indicative of a systematically induced error and statistical calculations were done to compensate for this error.  They revealed a maximum probable distance below the mean of the light curve of 0.42 magnitudes.

A plot of possible transit magnitude drops outlined a predicted maximum permissible dip of as much as 0.13 magnitudes for a periastron transit of planet c! (Shankland et al. 702)  After drawing the conclusion from statistical calculations that all of the variation in the plot is probably from nontransit sources, it can be soundly established that the following ranges of transits of planet c can be ruled out:  periastron transits with orbital inclinations of 89.26º to 90º, apastron transits between inclinations of 89.52º to 90º, and the series of ranges for relative orbital positions in between periastron and apastron.  By implication, it is established with certitude that no transit occurred with an inclination greater than the extreme end of the ascertainable apastron range; i.e., above 89.52º.  Of course, transits involving the inclination/orbital position values whose curves reach an extreme less than 0.042 magnitudes cannot be ruled out, but would be unobservable within the systematic error.

No one can ascertain how much more information could have been obtained had more stringent methodology been followed in the production of the subject images.  It might have been possible to prove conclusively that no transit happened at all under any circumstances or that one of the shallower transits occurred that is now hidden amongst the error in the data plot.  As things stand, both of these scenarios are unknowable conjecture.

I learned a lot about issues related to CCD photometry during the course of this research.  I am almost glad that many of the data sources were less than optimal because I feel I learned more from overcoming the associated problems than I would have otherwise.  It is my hope to try another transit photometry project at some time in the future with more meticulously produced data sources.

This marks the end of the final instalment of my stellar photometry tutorial.  It is my hope that the documented steps of my work on Gliese 876 will be of aid to both amateur and professional astronomers who are about to try stellar photometry for the first time.

References

Benson, Priscilla, “Transformation Coefficients for Differential Photometry”, I.A.P.P.P. Communications, 72 (1998) 42-52

Berry, Richard and James Burnell, Handbook of Astronomical Image Processing, (2006) Willmann-Bell, Inc., Richmond, Virginia, USA

Brown, Michael, Flat Fielding Dithered Data, http://www.ph.unimelb.edu.au/~mbrown/ccd/pfrancis/node14.html  (1996)

FWG (FITS Working Group), Definition of the Flexible Image Transport System (FITS), International Astronomical Union, (2008) 1-131

Gary, Bruce, Exoplanet Observing for Amateurs, (2007) Reductionist Publications, Hereford, Arizona, USA

Shankland, P.D., E.J. Rivera, G. Laughlin, D.L. Blank, A. Price, B. Gary, R. Bissinger, F. Ringwald, G. White, G.W. Henry, P. Mc Gee, A.S. Wolf, B. Carter, S. Lee, J. Biggs, B. Monard, and M.C.B. Ashely, “On the Search for Transits of the Planets Orbiting Gliese 876”, The Astronomical Journal, 653 (2006) 700-707

Walker, E. Norman, CCD Photometry, http://britastro.org/vss/ccd_photometry.htm (2007)

Warner, Brian D., A Practical Guide to Light Curve Photometry and Analysis, (2006) Springer Science + Business Media, Inc, New York, New York, USA


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