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 Tool’s 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
AAVSO, CCD Camera Skills, http://www.aavso.org/observing/programs/ccd/manual/3.shtml
(2009)
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
