IDENTIFICATION AND CHARACTERIZATION OF LONG PERIOD VARIABLE STARS IN THE GLOBULAR CLUSTER M69
Paul W. Husband Jr.
A Thesis
Submitted to the Graduate College of Bowling Green State University in partial fulfillment of the requirements for the degree of
MASTER OF SCIENCE
August 2017
Committee:
Andrew C. Layden, Advisor
John B. Laird
Dale W. Smith ii ABSTRACT
Andrew Layden, Advisor
Observations of the globular cluster M69 were taken from August 2009 to September
2014 using the 0.4-m PROMPT #4 telescope in Chile. This telescope took observations in V and
I bandpass filters, approximately once per week, for ten months of each year. Using the image
subtraction software suite ISIS, the cluster was examined for variable stars with an emphasis on
long period variable stars.
As a part of the overall project the long period variable stars were examined for multiple
period relationships, including long secondary periods. The long period variables stars were also
plotted in a color magnitude diagram, along with the rest of the stars in the images, to help
evaluate the location of the long period variable stars on the diagram. Plots of period vs. I
magnitude, period vs. I magnitude range, and I magnitude range vs I magnitude were also plotted
to assess whether measurements of one or two characteristics would provide insight into other
characteristics of the long period variable stars. The light curves, period examination, and tools
to conduct them, are presented in the body of this paper. Having 400+ images in V and I filters
over a 5-year time frame is an improvement over the previous work on M69, which was either
photographic or photoelectric. Based on these observations magnitudes, periods, and
classification of seven previously known long period variable stars were reconfirmed or
improved, eight new long period variable stars were classified as semi-regular, and three stars
were marked for further investigation. iii
To my wife, Tamara
Without whom, this project would still be
“in progress”
Love you iv ACKNOWLEDGMENTS
First, I have to acknowledge the guidance and support provided by Dr. Andrew C.
Layden. His availability, knowledge, and patience were limitless and I cannot adequately express
my appreciation for his help during this project. Dr. John B. Laird and Dr. Dale W. Smith, the
remaining members of the supervisory committee, were also very helpful through this process in ways too numerous to list. I also have to mention Dr. Farida Selim. Her tireless efforts
supporting my research as an undergraduate student took me places, and in directions, I doubt I
would have found on my own.
My parents, Paul and Dawnnene, made this possible. Without their love and support,
even when I squandered opportunities as a child, never wavered. The child they raised to be able
to achieve what I have so far, with so much left before me, is a testament to their resilience and
patience. My brother, Eric, helped prepared me for many adversities and I look forward to seeing
what he achieves.
Then there is Tamara, who made this happen. Without her, I would not have travelled
across the country to pursue a chance at happiness. Without her, I would not have enrolled in
school full time. Without her, I may not have made it to this far, and definitely not as quickly as
I have. “Thank you” seems too small a response, but it is a good down payment before the next
adventure begins. v
TABLE OF CONTENTS
Page
CHAPTER 1 INTRODUCTION ...... 1
CHAPTER 2 OBSERVATIONS ...... 6
2.1 Telescope ...... 6
2.2 CCD ...... 6
2.3 Filters ...... 7
2.4 Observing Seasons ...... 8
2.5 Observation Statistics ...... 9
CHAPTER 3 IMAGE ANALYSIS ...... 13
3.1 Image Preparation ...... 13
3.2 Interpolation...... 14
3.3 Subtraction ...... 14
3.3 Detection ...... 16
3.4 Light Curves, Flux Correction, and Apparent Magnitude ...... 18
CHAPTER 4 RESULTS ...... 20
4.1 Variable Star V5 ...... 22
4.2 Variable Star V4 ...... 29
4.3 Variable Star V1 ...... 32
4.4 Variable Star V3 ...... 34
4.5 Variable Star V6 ...... 36
4.6 Variable Star V7 ...... 38
4.7 Variable Star V8 ...... 40 vi
4.8 Variable Star NV12 ...... 42
4.9 Variable Star NV19 ...... 44
4.10 Variable Star NV101 ...... 45
4.11 Variable Star NV103 ...... 47
4.12 Variable Star NV104 ...... 48
4.13 Variable Star NV105 ...... 50
4.14 Variable Star NVa ...... 51
4.15 Variable Star NVb ...... 53
4.16 Variable Star NVc ...... 54
4.17 Variable Star NVd ...... 56
4.18 Variable Star NVe ...... 57
CHAPTER 5 CONCLUSION ...... 59
REFERENCES ...... 67 vii
LIST OF FIGURES
Figure Page
1 Air Mass vs. Heliocentric Julian Date...... 9
2 V Filter FWHM Values...... 11
3 I_long filter FWHM Values...... 12
4 I_short filter FWHM values...... 12
5 Reference images (“ref.fits”) from ISIS...... 15
6 Subtracted V filter image from 2010 Sep 18 ...... 16
7 Variable star locations...... 17
8 V filter light curve with single period of 196 days...... 25
9 V filter light curve with the 196 & 98 day model...... 26
10 V filter light curve with 196, 98, and 423 day periods...... 27
11 I_long light curve with 196-day model...... 28
12 I_long light curve with model created with 196, 96, and 454-day periods...... 28
13 Graph demonstrating the similarity between I_long and I_short...... 29
14 I_long light curve for V5 with model of 196.5, 97.1, and 440.4 days...... 29
15 V Filter Light Curve for V4...... 31
16 I filter light curve for V4...... 32
17 V Filter light curve for V1...... 34
18 I filter light curve for V1...... 34
19 V light curve for V3...... 36
20 V filter light curve for V6...... 38
21 V light curve for V7...... 40 viii
22 V light curve for V8...... 42
23 I light curve for V8...... 42
24 V filter light curve for NV12...... 44
25 V filter light curve for NV19...... 45
26 V filter light curve for NV101...... 47
27 V filter light curve for NV103...... 48
28 V filter light curve for NV104...... 50
29 V filter light curve for NV105 ...... 51
30 V filter light curve for NVa...... 53
31 V filter light curve for NVb...... 54
32 V filter light curve for NVc...... 56
33 V filter light curve for NVd...... 57
34 V filter light curve for NVe ...... 58
35 Color-Magnitude Diagram of M69...... 62
36 Log(P) vs. I Luminosity plot...... 63
37 I magnitude range vs. I magnitude plot...... 64
38 Plot of Range in I magnitude vs. log(P)...... 65 ix
LIST OF TABLES
Table Page
1 Table of minimum and maximum FWHM values ...... 11
2 Six highest power periods for V5...... 25
3 Periods for V4...... 31
5 Periods for V1...... 33
6 Periods for V3...... 36
7 Periods for V3...... 38
8 Periods for V7...... 39
9 Periods for V8...... 41
10 Periods for NV12...... 43
11 Periods for NV19...... 45
12 Periods for NV101...... 46
13 Periods for NV103...... 48
14 Periods for NV104...... 49
15 Periods for NV105...... 51
16 Periods for NVa...... 52
17 Periods for NVb...... 54
18 Periods for NVc...... 55
19 Periods for NVd...... 57
20 Periods for NVe...... 58
21 Table of basic data for each variable star ...... 61 1
CHAPTER 1
INTRODUCTION
Variable stars come in many varieties. The International Variable Star Index (VSX) [17] breaks them into two distinct classes, extrinsic and intrinsic. Extrinsic variables are broken into three categories: eclipsing, where the orbital plane is along the line of sight of the observer; rotating, where the variability is caused by spots, reflection, or ellipsoidal shape of the star; and microlensing events. Intrinsic variables are broken into four categories: pulsating, which is the star expanding and contracting; eruptive, which is chromospheric activity or mass ejection; cataclysmic, interacting binary systems with white dwarfs or stars showing large amplitude outbursts; x-ray, binary systems with neutron stars or black holes.
Common types of pulsating variables include Cepheids, RR Lyrae, Semi-Regular, and
Mira. RR Lyrae variable stars have periods of less than 2 days and a magnitude range of 0.2 to
2.0 in the V filter. Cepheid variable stars have periods between 1-135 days and a magnitude range of several hundredths to 2 mag in the V filter.
Semi-Regular (SR) and Mira (M) variables are the primary targets for this project and may be referred to as Long Period Variable (LPV) in general in this paper. Mira variable stars have a pronounced periodicity in the range of 80 to more than 1000 days. Mira have a magnitude range of between 2.5 to 11 mag in the V filter, and is greater than the magnitude range in the I filter, which can be less than 2.5 mag. Semi-Regular variables have periods of 20 to more than 2000 days, with a magnitude range of several hundredths up to 2 mag in the V filter. There are two sub-classes of SR variable that will be discussed, SRb variables have been observed to have two or more simultaneous periods, and Slow irregular variables (L) LPV 2 demonstrate variability with no recognizable periodicity, or the observation interval is insufficient to find the period(s) that may be present. Due to the need for accurate measurements at multiple points over an extended time frame, identifying and studying LPV has been difficult.
In 2013 Soszy´nski, Wood, and Udalski [22] published a paper that examined LPV in the
Large Magellanic Cloud (LMC). They compiled data from multiple projects and “observations in some stars exceeded 16 yr and typically it was 12 yr, with over 1000 observing points” [22].
They identified 91,995 LPV in the LMC. In their analysis of the LMC LPV they found and plotted five periods for SR variable stars and demonstrated that at least some of the SR variables pulsate in fundamental mode, first overtone, and second overtone.
After that work by Soszy´nski et al., Osborn et al. [21] used archival data and new observations to examine the bright red variables of globular cluster M13 in 2016. Part of that work included an investigation into the periods and modes of those red variable stars and concluded that SR LPV in M13 also exhibit multiple periods. Combining the results of
Soszy´nski et. al and Osborn et al. demonstrates that when studied in volume, LPV have multiple periods and that LPV in globular clusters have this same behavior.
In addition to exhibiting multiple periods some SR LPV appear to have a long secondary period (LSP). In a 2004 paper, Wood, Olivier, and Kawaler [23] describe the LSP as being approximately 400-1500 days and roughly ten times the length of the primary pulsation period.
Contrasting that is Osborn et al. [21] in their 2013 paper, on Table 8, list variable stars with suspected LSP of less than 300 days, and not approaching the multiple of the primary pulsation period that Wood et al. [23] describe. Perhaps our results will help to further refine the general definition of LSP. For an example of phase plots demonstrating the behavior of LSP we suggest
Soszy´nski and Udalski’s 2014 paper [24] which presents four LPV with LSP in Figure 2. 3
Another example of LSP can be found in the Abbas et al. [20] paper in Figure 5 in their light curve for V3 in GC NGC 6496. Wood et al. [23] investigated the possible causes of the LSP phenomenon, specifically testing Radial Pulsation, Nonradial Pulsation, Orbiting Companions,
Rotating Stars, and Chromospheric Activity. They conclude that there is no clear explanation for the LSP. However, they “… speculate that AGB stars with LSP are the precursors of the asymmetric planetary nebula.” [23] “… through a combination of low-order nonradial modes and star spot activity.” as described by Abbas et al. [20]. This project will not be exploring the causes of LSP, but will examine the periods of candidate LPV for periods that meet the proposed criterion for LSP.
One of the ways to make studying LPV more efficient is to look for regions where many
LPV can be observed simultaneously. Globular Clusters (GC) are a way to accomplish this goal.
In addition to potentially finding multiple LPV in a small area and at a single distance (where instrumental magnitude can be used as a proxy for luminosity), member stars in a specific GC were created from the same medium at approximately the same time. This means the LPV studied in a GC have the same age and metallicity, which leaves mass as a variable to be studied to determine its effect on the evolution of stars.
Metallicity is a measurement of the elemental content of a star. The equation for metallicity is
Fe N N = log − log H N N where N is the number of atoms of the subscripted element (Fe is iron, H is hydrogen), star refers to the observed star, and sun refers to Sol. The globular cluster M69 was initially determined to 4 have an [Fe/H] of -0.8 in 1995 by Minniti [2], but a more recent survey by Sloan et al. refined the [Fe/H] of the cluster to be -0.66 [3].
The globular cluster M69 (M69) is a cluster that has been observed on a limited basis. In her catalog [5], Clement includes a brief history of the observations of M69. Rosino discovered a number of variable stars in 1962 [1]. In the course of Rosino’s observations ten variable stars were initially identified, but that number has been revised since the initial work. Rosino’s V1,
V3, V4, and V10 are included in this project. The variable stars identified by Rosino as V2, V5,
V6, V7, V8, and V9 have since been reclassified as non-member stars by Hartwick and Sandage in 1968 [16] due to their distance from the center of the cluster and the magnitude dispersion of the variable stars from the expected location on the color magnitude diagram (CMD). In that same paper Hartwick and Sandage identified three new candidate variable stars which have since been labeled as V6, V7, and V8. These three stars are included in the results of this paper.
Rosino’s V10 was renamed to V5 by Sawyer Hogg in her 3rd catalogue [5] and is also included in this paper.
Of the LPV that have been studied in M69 only two have been classified: V5 is a Mira variable with a period of 198.0 days and mean apparent magnitude of 7.82 in the K filter, and V4 is also a Mira variable with a period of 200 days with mean apparent magnitude of 7.86 in the K filter. The most current data, in the catalog, for the remaining LPV is either photographic
(Rosino) or photoelectric (Hartwick & Sandage), and all of these LPV are still classified as L variable stars.
One recent observation of M69 by Escobar et al. [15] discovered many new potential
LPV while searching for short period variable stars. Based on the number of potential LPV in the cluster, a member of the project, Dr. Layden, decided to take a number of observations of 5
M69 over five-years with the goal of identifying and characterizing the LPV in the cluster. As part of this project the LPV will be examined for potential multiple periods, long secondary periods, mean magnitudes, range of variability, color, and some of the relationships between these characteristics. 6
CHAPTER 2
OBSERVATIONS
2.1 Telescope
Observations were taken at the Cerro Tololo Inter-American Observatory (CTIO) in
Chile. Images of the globular cluster M69 were taken by Panchromatic Robotic Optical
Monitoring and Polarimetry Telescope 4 (PROMPT4), a 16-inch (0.41 meter) telescope with an
F-Ratio of 11.2, providing a good plate scale and CCD resolution matched to the local seeing.
Observations on June 27, 2014 were taken with PROMPT5, an identical telescope, due to
PROMPT4 being unavailable.
The PROMPT telescopes are controlled by Skynet Robotic Telescope Network [7].
Observations are requested through a web portal where exposure times and filters are selected for the specific telescope. As part of the automated process each image is automatically bias, dark, and flat-field corrected and the corrected image is then stored by Skynet and is available for download from Skynet to a local storage device at BGSU.
2.2 CCD
The charge-coupled device (CCD) camera used to record the observations is an Apogee
Alta F47. This CCD is a 1024x1024 pixel (1 megapixel) chip with each pixel being 13 microns in width [8]. The field of view (FoV) of the images is 10x10 arcminutes. Taking the FoV and breaking it up into even parts that will be recorded by a corresponding pixel on the CCD results in a resolution of 0.59 arcseconds/pixel. An identical CCD is mounted on PROMPT5, which was used on June 27, 2014, as mentioned previously. 7
2.3 Filters
Images were taken in V and I filters. The V filter has a central wavelength of 551nm with a FWHM of 88nm giving it a range of 463nm to 639nm. The I filter has a central wavelength of 806nm with a FWHM of 149nm giving it a range of 657nm to 955nm [9]. Having images in different filters for the same night allows the color (temperature) of the star to be determined. Additionally, observations in two filters gives an independent internal evaluation for the accuracy of the time-series of each variable star. So long as the peak and trough phases occur at the same points in the plot for both filters, confidence can be high that the time-series is accurate.
With GC M69 being a fainter cluster with a low magnitude in the V filter it was decided to use 60 second exposures for the V filter. Red stars are fainter in the V filter so the CCD wouldn’t be saturated by the stars in the cluster. For the I filter the stars are brighter so it was possible the CCD would be saturated with too long of an exposure time. On the other hand, too short of an exposure would increase the likelihood of missing fainter stars in the cluster. Rather than trying to find a single “perfect” exposure time the images in the I filter were taken with two different exposure lengths: 8 seconds (I_short) and 30 seconds (I_long), determined by taking test exposures of this cluster. The V, I_short, and I_long images were processed as separate sets through each step to keep similar data apart from dissimilar data.
In 2013, it was determined that none of the stars in the I filter were saturating the CCD in the I_long images, so the I_short exposures were discontinued. At the same time, the I_long exposure time was decreased to 25 seconds, to minimize the chance of digital saturation during times of exceptional seeing. 8
2.4 Observing Seasons
Observations were taken approximately one night per week from Aug 2009 to Aug 2014.
From first observation to last observation a total of 1911 days elapsed. During the 2009 to 2012 seasons each night included four exposures in each set for twelve total images. For 2013 and
2014 each night had two exposures each in the V and I_long sets for four total images. By the end of the observation interval images were taken on 189 individual nights with a total of 1660 images.
Figure 1 below shows the air mass for each observation. An air mass of 1 indicates that
M69 was at zenith (90 degrees) in relation to PROMPT4 and the optical path through the atmosphere was shortest. At higher air mass the optical path is longer, which causes more absorption and scattering of the light before it reaches the telescope. This effect is noticeable when examining the full width at half max (FWHM) of the stars on the image. A low FWHM means that the majority of the light from the star is concentrated into a smaller area on the image.
While a low FWHM is both optimal and desirable, for this project observations were restricted to air mass of 3 because at higher air mass the telescope does not track as well and the amount of blurring of the image is increased. From the observing location at CTIO, M69 is either low in the sky or actually below the horizon during December and January so observations for these two months were not taken. These time frames are marked on Figure 1 with heavy black lines. 9
3 2.8 2.6 2.4 2.2 2
Air mass 1.8 1.6 1.4 1.2 1 5000 5200 5400 5600 5800 6000 6200 6400 6600 6800 7000 HJD - 2,450,000
Figure 1 - Air Mass vs. Heliocentric Julian Date. Black vertical lines mark breaks in observing schedule when M69 was too low in the sky for reliable results.
2.5 Observation Statistics
A non-crowded, non-variable star near the cluster center was selected to perform an initial evaluation on each image. Basic astrometric data was taken which included the FWHM of the star. A low FWHM means that the light of the star is contained in fewer pixels which pushes the value of those pixels higher above the background. In order to evaluate the data, the FWHM of each image was sorted into a list and divided between the three data sets. Table 1 shows the results of that list, where the minimum and maximum FWHM values of each data set was noted, then the list was divided into quarters. For example, in the V filter, the minimum observed
FWHM was 1.32 arcseconds and the maximum was 5.87 arcseconds. The median value, found at the half way point of the list, was 2.46 arcseconds. This demonstrates that 50% of the FWHM values were between 1.32 and 2.46 arcseconds. Table 1 shows the corresponding statistics for the three data sets. Three histograms were created by breaking the lists into twenty equally 10 spaced bins and each image was counted in the bin that matched the FWHM of that image. The counts were then plotted on Figure 2 (V filter), Figure 3 (I_long filter), and Figure 4 (I_short filter).
It is noteworthy that the V filter FWHM quartile values are larger than the equivalent values in the I filter. This is due to the V filter being shorter in wavelength than the I filter, and shorter wavelengths light is affected more by scattering in the atmosphere than longer wavelengths. Putting that together with the longer exposure times of the V filter means more light from the star has scattered and created a larger FWHM. An unfortunate effect of a larger
FWHM, in any filter, is that the light from stars that are even moderately crowded may interfere with the accuracy of flux counts for a single star. This will also effect the calculated instrumental magnitudes and any calculations based on those magnitudes.
The median FWHM values for our data are noticeably greater than desired. Preferably, images with FWHM < 1 arcsecond would be available and used, without the less well refined images. This would help to reduce the possibility of unresolved stars influencing results for a potential target. While using the PROMPT telescope there were nights with resolution approaching 1 arcsecond, but they were not often enough to provide sufficient data points to look for multiple periods in the LPV. The availability and accessibility of the PROMPT telescope makes it possible to take observations for the majority of the year, making it possible to get a much better idea of the behavior of the variable star, but at the cost of a median FWHM that is over 2 arcseconds. 11
V Filter I_long Filter I_short Filter
(arcseconds) (arcseconds) (arcseconds)
Min 1.32 1.18 1.07
1st Quarter 2.11 1.81 1.68
Median 2.46 2.09 2.06
3rd Quarter 2.89 2.59 2.58
Max 5.87 6.57 7.19
Table 1 – Table of minimum and maximum FWHM values for each data set, broken down further into the first 25%, the value at the mid-point of the data set, and the value at 75%. This table demonstrates that 75% of the V filter FWHM were less than 2.9 arcseconds and that 75% of the I_long and I_short filter were better than 2.6 arcseconds.
Figure 2 - V Filter FWHM Values. The greatest occurrence of values is between 2.23 and 2.68 arcseconds. The majority of images have a FWHM between 1.77 and 2.91 arcseconds. 12
Figure 3 - I_long filter FWHM Values. Most of the I_long images have a FWHM between 1.45 and 2.26 arcseconds.
Figure 4 - I_short filter FWHM values. The majority of this set have FWHM values between
1.38 and 2.30 arcseconds. 13
CHAPTER 3
IMAGE ANALYSIS
Since we are looking for stars that change in brightness over time, and the majority of stars in the cluster are essentially constant in brightness, it is possible to take images from separate observations and subtract them from a reference image to remove those constant stars and leave only the stars that have changed in brightness. These changes in brightness will show up as either “hills” (fainter than the reference) or “valleys” (brighter than the reference) on the subtracted image. This process was accomplished using ISIS, an image subtraction software suite created by Alard and Lupton [10]. There are a number of steps to ensure that results are accurate, which are described in the following sections. Each of these steps were applied to the
I_long, I_short, and V filter image sets separately.
3.1 Image Preparation
Images downloaded from Skynet may not all be aligned in the same direction and orientation, so the first step is to examine each image and to flip it in the x, y, or both axis to bring all images into agreement. Before analyzing the images with ISIS the full size images were trimmed from the original 1024x1024 pixel size to 620x620 pixels, with the center of the cluster offset. This offset allows ISIS to examine the, now, less crowded center of the trimmed image to find matching stars from one image to the next. Attempts without the offset were not successful during the interp step (see Sec. 3.2). An effect of this trimming is that potential variable stars outside of this trimmed area will not be included in the ISIS analysis, which includes the star Rosino identified as V2 in his 1962 paper [4]. Another effect of this trimming is that the programs in ISIS will have less “twisting” that happens around the edges of a 14 telescope image to deal with when matching one image to the next. This has an added benefit of reducing the processing time required to match the images.
3.2 Interpolation
A master image needs to be selected for each set to define the (x,y) coordinate system for that set. The master image should have good seeing, represented by a low FWHM, with low background sky values and low ellipticity, when a star is stretched by poor telescope tracking.
For this project, the night of 23 August 2009 was selected for the master images because it has images in each filter that fit the necessary criteria. Each image was then aligned with the master image. ISIS identifies a number of stars on the master image and then matches those to each image with three degrees of freedom. This allows ISIS to correct for direct shifts in the x and y axis (first degree), changes in scale and rotation (second degree), and non-linear terms (third degree).
3.3 Subtraction
A reference image was built from the four images taken on 23 Aug, 2009 in each set by taking the average count of each pixel from the four aligned images and compiling the results into a single image. Any pixel with a value outside of 3 sigma of the median was rejected in order to remove any cosmic rays or pixels that were not collecting light correctly. Reference images, named “ref.fits”, for each set are displayed below in Figure 5. 15
Figure 5 - Reference images (“ref.fits”) from ISIS in V filter (left), I_long (middle), and I_short
(right).
After the reference image was assembled, ISIS compared each image in the set against the reference to find the change in flux from the reference image to the current image. Due to differences in seeing, background, sky transparency, and exposure times between images it is necessary to modify the reference image to match the image before subtraction can take place.
This is accomplished by convolving the reference image with a transformation kernel, a mathematical function consisting of a sum of Gaussian curves with values of sigma that are found by least-squares minimization between the reference and current image [18]. This kernel will match the FWHM between the reference and current image. At the same time as the
FWHM is brought into agreement the kernel will also account for the variation in background levels between the images. The final step is ensuring that the total flux after convolving is equal to the total flux before.
A subtracted image was then created by ISIS for each image by taking the difference in flux between the convolved reference image and the current image. An example of a subtracted image is Figure 6 below. Most of the image has a uniform background of about zero counts. The dark and bright spots mark potential variable star locations on this particular image where the 16 flux of the star on the image deviated from the flux of the star on the reference image. For each image set (V, I_long, and I_short) there is now a subtracted image for each original image.
Figure 6 - Subtracted V filter image from 2010 Sep 18
3.3 Detection
The subtracted images are then combined to create a single image named “var.fits” for each set (V, I_long, and I_short) that shows where potential variable stars are located. The value of a particular pixel on each image is squared so that all values are positive, and the values of potential variable star locations are increased relative to the local background. Then the values of that pixel are combined, and this process is repeated for each pixel to create a single “var.fits” image. If pixel values were to be combined without squaring then variable stars would be harder 17 to detect because the total value would likely be closer to the background noise level of the image due to positive and negative values cancelling each other out. Since the majority of each image is either faint sky or non-variable stars the resulting image will show the location of potential variable stars with a higher signal value than any single subtracted image. The resulting files showing candidate variable star locations are shown in Figure 7.
Figure 7 - Variable star locations for V (left), I_long (middle), and I_short (right)
From the “var.fits” image ISIS will create a list of potential variable stars. Potential variables near the edge (~10 pixels) of the image can be rejected. These are most likely due to the shifting and trimming of the images throughout this process and not actual variable stars. A list of potential variable stars is created with the x,y position and a temporary identifier. Initial candidates for the three sets numbered 495 in the V set, 218 in the I_long set, and 1506 in the
I_short set. Candidates near the edge of the images were rejected, making the effective region of our variable star search 2.7 x 3 arcminutes roughly centered on the cluster. These three lists were then compared to find potential variables that only showed up in one list and those candidates were rejected; the handful of stars rejected by this criterion were all near the
(arbitrarily selected) variability threshold, and were unlikely to produce high quality light curves 18 from our data. Future studies may test these fainter or low amplitude variable candidates more effectively (also see the discussions of candidates NVa and NVe in Chapter 4).
The final number of potential variable stars for each set is 18 in the V and I_long, and 11 in the I_short. These reduced lists were run through one final step in ISIS to create initial light curve data for each potential variable in each filter. This list will contain the Heliocentric Julian
Date (HJD) of the image and the flux difference for the star on that image and the reported uncertainty from ISIS.
3.4 Light Curves, Flux Correction, and Apparent Magnitude
In order to find the actual flux, and then the instrumental magnitude, of the star on each image the flux difference from that image needs to be returned to the actual flux of that image.
This was accomplished by subtracting the non-variable stars from the “ref.fits” image for each set, leaving the variable stars with their original flux value. Dr. Layden performed this step to create images for the I_long and V filter sets in DAOPHOT. He then put these images through an aperture photometry process in the IRAF task DIGIPHOT to find the flux values and instrumental magnitudes of the variable stars on the reference images.
The instrumental magnitude of the star on each image is found by using:
m = −2.5 ∗ log f − Δf + 25
where reference flux is fr and flux difference is Δf. The value in the argument for the log is the corrected flux. As a check to make sure that the I_long and I_short values are comparable the ratio in exposure time also needs to be taken into account. This was accomplished by multiplying the corrected I_short flux by the ratio of exposure time between I_long and I_short of 3.75. After the correction, the magnitude-based light curves were good matches to each other. 19
To take the instrumental magnitude to standard magnitude a filter specific correction was added to the equation based on 16 non-variable stars whose standard V and I magnitudes are known from the Escobar et al. paper [1]. In the I_long and I_short filters the correction is -0.967 mag, and in the V filter the correction is 0.0434 mag. At this point we have calibrated light curves in V and I filters for the 18 candidate variable stars. The calibrated light curves consist of standard V and I magnitudes as a function of time and will be examined in the next chapter. 20
CHAPTER 4
RESULTS
To find the periods of the candidate variable stars the light curves for each star were analyzed using VStar. VStar is a program available from the American Association of Variable
Star Observers (AAVSO) website [19]. Using Date Compensated Discrete Fourier Transform
(DC DFT) [13] VStar is able to analyze the light curve for potential periods and evaluate their power and amplitude. As a part of this process VStar allows the user to designate a range of trial periods and a resolution level for the evaluation. Then the program does a Fourier analysis of the light curve to find a combination of sine waves that, when combined, will approximate the observed light curve. Each sine wave, with its period, amplitude, and phase shift, is then matched against the observed light curve and assigned a “power” value that is a measure of the goodness of fit. The results of this analysis are then presented in a table, with each potential period listed in order by “power”, or how well it matches the observed light curve. A Fourier transform normally assumes equally spaced observations/data points in the calculation. In astronomical observations, this is very unlikely to occur due to observational limitations from weather, location of the object in the sky, and other events. This has an impact on the accuracy of the Fourier transform and VStar utilizes the Date-Compensated form of the Discrete Fourier
Transform developed by Ferraz-Mello in 1981 [13], which is better able to account for the unequal observations inherent in astronomy.
Due to the observing schedule, and the resulting gaps, there is a lower limit on the periods that can be reliably determined, which is referred to as the Nyquist frequency. The
Nyquist frequency corresponds to twice the average gap between observations [14] and for this 21 project that limit is 22 days. While there may be periods of less than 22 days for any of these variable stars, using DC DFT means they are not able to be determined reliably.
There are many potential periods presented by VStar for each candidate LPV, but that is a mathematical construct, which may not be a result of the physical characteristics of the star.
Given the statistical nature of Fourier transforms it is difficult to determine how many periods should be included. In an effort to be consistent each star was only evaluated with the six highest-power periods, and only periods that matched between the V and I filter were used for the model. This is unlikely to create an exact match of the observed data, but still allows for a sense of the behavior of the star. Since the light curves for each candidate variable star were available in two separate filters a comparison of periods seen in both filters for each candidate
LPV is the first criterion. If a period is only seen in one filter and not in both it is less likely to be real.
Each of the 18 candidate variable stars is described below, starting with the seven previously known cluster member variable stars V1, V3-V5 discovered by Rosino in 1962 [4] and V6-V8 discovered by Hartwick and Sandage [16] in 1968. After the previously identified variable star descriptions there are six identified candidate variable stars from Escobar et al. [1] designated with NV and a number, and then five candidate LPV discovered in the current ISIS analysis, designated NVa through NVe. As mentioned in the introduction, multiple periods were examined by Soszy´nski [22] in the LMC, and used by Osborn et al. [21] to describe the light curves of LPV in globular cluster M13. With the intention of relating the LPV of M69 with LPV from other sources in the future, the following examinations will find the periods that best approximate the observed light curve. 22
4.1 Variable Star V5
With a large magnitude range of 2.7 magnitudes in the V filter and 3.2 magnitudes in the
I filter, V5 is easy to locate despite being near the center of the cluster. Based on the light curve,
V5 reaches maxima with a regular pattern. Using VStar we find the period with the largest power is 196.3 days in the V filter, 196.5 days in the I_long and 196.2 in the I_short (see Table
2). These values are consistent with Sloan’s [3] result of 198 days in the K filter. Given the regular observing schedule and longer time frame of observations this paper has a more complete light curve than Sloan’s that includes minimum brightness measurements not apparent in his reported phase plots. The data in this project re-confirms the Mira classification for V5.
The single highest power period results in a model that matches the timing of the light curve very well, but does not match the variability of the amplitude of the light curve, as shown in Figures 8 and 11 for the V and I_long filters, respectively. When the 98-day period is included into the V curve the troughs of the model light curve become flatter and match better with the observed light curve, but the variability of the peaks is still not well represented by the model. This is displayed in Figure 9 below. Introducing the 426-day period and making a combined model from all three periods, displayed in Figure 10, results in a good match for the trough and starts to show the pattern of changes in the peak values. Vstar has found more potential periods, but they are all of lower power, so have less likelihood of being real, and have lower amplitude so adding any one or two more periods still did not completely match the light curve, and including too many periods runs the risk of over correcting. Figure 10 shows the observed V light curve with the combined model of 196.4, 98.0 and 426.4 days.
Period analysis of the I_long data using VStar produces a very similar set of periods with
the relative importance (power) of the 96-day and 454-day periods reversed from their positions 23 in the V filter (see Table 1). Figure 12 shows the I_long light curve with the final model created from the 196.5, 454.4, and 96.1 day periods.
Figure 13 shows the similarity between I_short and I_long light curves where they overlap. However, due to the I_short time-series being approximately 65% of the I_long, it will have fewer cycles of data and carry less information about the period of the star. I_short has the same primary period of 196 days but only one of the two secondary periods that both the V and
I_long agreed upon. Given the similarity in the overlap and the close agreement of the periods, it will not be necessary to analyze the I_short data for the remaining candidate variable stars.
To find the period of the star itself, and not the separate filters, the average of each comparable model period was taken, along with the appropriate standard deviation, and a final model was created from those and is shown in Figure 14 with the I_long light curve to demonstrate how well it matches the data. The final periods used to create this model were
196.5 ± 0.09 days, 97.1 ± 1.4 days and 440.4 ± 19.8 days. This process is similar to the one used by Osborn et al. [21] in their examination of the LPV in M13. Future work in this area will include a more statistically rigorous examination utilizing RMS or chi-squared analysis to determine which periods are significant.
Examination of Osborn et al. [21] table 7 illustrates that LPV are observed to have a larger range in V than in I. Accordingly, the VSX definition [17] of Mira variables states that they usually have a larger range in V than in I, which is not what is seen for V5 with its range of
2.7 mag in V and 3.2 mag in I. This is likely caused by unresolved stars in the dense region of the cluster where V5 resides, influencing the measurement of the observed flux and propagates through into the instrumental magnitude through the equation in section 3.4. This influence doesn’t alter the Δf measurements by ISIS. Examining the light curves from V and I, the V light 24 curve demonstrates a flattening of the curve during minima but the I light curve more closely resembles the expected light curve of Mira variables. If the reference flux is measured to be larger than its actual value due to unresolved stars the median magnitude of V5 will be brighter, reducing the observed magnitude range and cause the flattening of the minima. Reducing the reference flux, to bring it closer to its probable value, would increase the magnitude range in V.
Based on the VSX definition for Mira LPV, where the magnitude range in V should be greater than I, and the measured I magnitude range is 3.2, the V magnitude range is probably greater than 4 mag. Using the median observed magnitude for each filter results in a (V-I) color of 2.83 mag. This (V-I) color is probably smaller than it should be, due to the brighter than expected V magnitude. Assuming the median I magnitude is accurate, this means the (V-I) color of V5 should be greater than 3.5 mag
Having five-years of data that match in both V and I, and with a long and regular period, and a high-amplitude, makes V5 a confirmed Mira variable pulsating in fundamental mode.
V Set I_long Set I_short Set
Period Amp Period Amp Period Amp
(Days) Power (mag) (Days) Power (mag) (Days) Power (mag)
196.39 222.81 0.90 196.52 203.66 0.91 196.18 195.06 1.00
98.04 22.14 0.29 454.43 24.53 0.31 436.32 27.20 0.38
426.37 16.51 0.25 96.11 14.71 0.24 258.27 14.22 0.27
131.69 9.40 0.18 171.21 8.06 0.18 160.31 10.88 0.24
158.70 8.24 0.17 262.61 7.86 0.18 94.61 6.95 0.19
61.84 6.45 0.15 61.77 6.86 0.16 132.15 5.98 0.18 25
Table 2 – Six highest power periods for V5, for each data set found by VStar.
11.5
12
12.5
13
Magnitude 13.5
14
14.5 5000 5200 5400 5600 5800 6000 6200 6400 6600 6800 7000
HJD Light Curve Model
Figure 8 - V filter light curve with single period of 196 days. This single-period model matches the sinusoidal timing of the observed light curve. The single-period model is a sinusoidal model with a single, fixed amplitude, which does not match the observed variability of the peaks. 26
11.5
12
12.5
13 V Magnitude 13.5
14
14.5 5000 5200 5400 5600 5800 6000 6200 6400 6600 6800 7000
Observed Light Curve HJD 196 & 98 day model
Figure 9 - V filter light curve with the 196 & 98 day model. This brings the troughs into better agreement with the observed light curve, but the maxima of the model are not showing the observed variability of the peaks. 27
11.5
12
12.5
13
Magnitude 13.5
14
14.5 5000 5200 5400 5600 5800 6000 6200 6400 6600 6800 7000
HJD Light Curve Model
Figure 10 - V filter light curve with 196, 98, and 423 day periods. This combination of sinusoidal waves still matches the timing of the cycles and the flatness of the minima, and also starts to describe how the maxima of each cycle is not constant, but changes from peak to peak. 28
9.5 10 10.5 11 11.5 12 12.5 Magnitude 13 13.5 14 14.5 5000 5200 5400 5600 5800 6000 6200 6400 6600 6800 7000
HJD Light Curve Model
Figure 11 - I_long light curve with 196-day model. Similar to the V light curve, the single wave model matches the timing of the peaks, but does not describe the change in maximum or minimum amplitudes from peak to peak.
9.5 10 10.5 11 11.5 12 12.5 Magnitude 13 13.5 14 14.5 5000 5200 5400 5600 5800 6000 6200 6400 6600 6800 7000
HJD Light Curve Model
Figure 12 - I_long light curve with model created with 196, 96, and 454-day periods. 29
9.5 10 10.5 11 11.5 12 12.5 Magnitude 13 13.5 14 14.5 5000 5200 5400 5600 5800 6000 6200 6400 6600 6800 7000
HJD I_long I_short
Figure 13 - Graph demonstrating the similarity between I_long and I_short for the interval where they overlap.
9.5 10 10.5 11 11.5 12 12.5 Magnitude 13 13.5 14 14.5 5000 5200 5400 5600 5800 6000 6200 6400 6600 6800 7000
HJD I_long Averaged Model
Figure 14 - I_long light curve for V5 with the final model of 196.5, 97.1, and 440.4 days.
4.2 Variable Star V4 30
Sitting outside of the cluster center V4 is easily identifiable as a variable star, especially with its magnitude range of 5.2 in the V filter and 2.4 in the I filter. The relative sizes of the V and I ranges are as expected for this star, indicating that our data for V4 are not affected by crowding as they were for V5. This is also the second star that Sloan [3] examined and for V4 found a period of 200 days in the K filter. Like V5, the phase plot in Sloan’s paper shows few data points during the minimum points of V4’s variability.
In the VStar analysis of our data, there is a high-power period of 203.2 days in the V filter and 203.3 days in the I filter that models the peak timing of the observed light curve very well. These values are also comparable to Sloan’s result of 200 days. Just like with V5 this single model does not do a good job of describing the variability of the peak amplitudes.
Similar to V5, there are secondary periods that begin to bring the model into better alignment with the observed light curve. The first of these is 161.2 days in the V filter and 160.9 days in the I filter. There are also matching periods that give the averages described in the next paragraph. Adding these periods to the model does not exactly match the observed data, but does start to bring the amplitudes into alignment as shown on Figure 15 (V filter) and Figure 16
(I filter).
The final periods used to build the model are 203.2 ± 0.08 days, 161.1 ± 0.2 days, 133 ±
2.2 days, and 95.4 ± 0.2 days. V4 has a range of 5.23 mag in the V filter and 2.4 mag in the I filter. The observed median (V-I) color is 3.56 mag. With the wide range in variability and consistent period, V4 is also a Mira variable star in fundamental mode. The magnitude values for V4 are more reliable than those for V5, given that V4 is in a much less crowded region of the cluster. Between the number of data points in the light curve, and its location being uncrowded, 31 these observations of V4 are more promising for inclusion into future work comparing the characteristic relationships of LPV in different GC based on metallicity.
V Set I_long Set
Period (days) Power Amp (mag) Period (days) Power Amp (mag)
203.15 208.93 1.65 203.30 198.39 0.69
161.22 31.78 0.64 480.91 37.60 0.30
130.78 17.80 0.48 160.91 25.58 0.25
433.38 15.72 0.46 135.21 12.78 0.18
662.45 15.22 0.43 95.59 9.75 0.15
95.25 12.66 0.40 274.97 9.50 0.16
Table 3 - Periods for V4 from VStar.
12
13
14
15
Magnitude 16
17
18 5000 5200 5400 5600 5800 6000 6200 6400 6600 6800 7000
HJD Observed Model
Figure 15 - V Filter Light Curve for V4 with final model. 32
9.5
10
10.5
11
Magnitude 11.5
12
12.5 5000 5200 5400 5600 5800 6000 6200 6400 6600 6800 7000
HJD Observed LC Model
Figure 16 - I filter light curve for V4 with final model.
4.3 Variable Star V1
The first non-Mira variable identified by Rosino (1968), V1 has a less well defined period and smaller range in brightness, which makes it a different kind of long period variable than V4 and V5 when examining the light curves in the V and I filters. Clement’s catalog has
V1 listed as an L variable based on the data from Rosino in 1968, which was on photographic plates. With our more current data it should be possible to either confirm the L classification or possibly reclassify V1 as an SR variable with multiple periods.
It has strong period matches between the V and I light curves (Table 5). The first five periods are very comparable, but the 70-day period is a likely alias1 of the 115-day period resulting from the observing schedule. The sixth period in both filters are unlike each other or any other period. This implies that, except for the 70-day period and the 103-day period which
1 Alias is a period artifact that occurs due to the observing schedule, not a real period that describes the behavior of the variable star. 33 doesn’t pair up with a potential period in the I filter, these are all real periods. The V light curve is displayed with the final model as Figure 17. The I light curve is presented with the same model in Figure 18.
The periods for V1 are 115.0 ± 0.4 days, 61.9 ± 0.02 days, 609.1 ± 0.5 days, and 56.5 ±
0.01 days. There are likely additional periods, but they have less significance and a smaller effect on the model. It has a magnitude range of 1.16 mag in the V filter and 0.62 mag in the I filter. Based on the median observed V and I magnitudes the (V-I) color is 2.50 mag. While the model doesn’t match the observed light curve exactly, the timing and slope is well approximated.
With the smaller range in magnitudes, and having multiple periods, V1 is an SRb LPV pulsating in at least the fundamental and first overtone.
V Set I_long Set
Period (days) Power Amp (mag) Period (days) Power Amp (mag)
114.65 60.97 0.14 115.39 73.30 0.08
61.90 44.73 0.13 61.93 33.08 0.05
609.66 20.82 0.08 70.70 27.69 0.05
70.13 19.51 0.08 608.59 26.66 0.05
56.48 14.34 0.07 56.49 15.57 0.04
103.58 11.25 0.06 404.31 12.24 0.03
Table 4 - Periods for V1 found by VStar 34
13
13.2
13.4
13.6
13.8 Magnitude 14
14.2
14.4 5000 5200 5400 5600 5800 6000 6200 6400 6600 6800 7000
HJD Observed Model
Figure 17 - V Filter light curve for V1 with final model.
10.8 10.9 11 11.1 11.2
Magnitude 11.3 11.4 11.5 11.6 5000 5200 5400 5600 5800 6000 6200 6400 6600 6800 7000 HJD Observed Model
Figure 18 - I filter light curve for V1 with final model.
4.4 Variable Star V3
The remaining variable star originally identified by Rosino (1968) is V3, and it is currently listed as L variable in Clement et al. where the most current data is photographic. This 35 paper will either confirm this classification or, preferably, be able to reclassify V3 as SR or SRb based on the new observations.
V3 has a number of matching periods between the V and I filters (see Table 6). The primary periods of 85.6 days in the V filter and 88.6 days in the I filter are very close. The next three periods in each list are reordered slightly, but they all appear in both filters and the fifth period in the V filter matches up nicely with the sixth period in the I filter. The V filter light curve with the final model is Figure 19 below.
Given that Soszy’nski plotted five periods for each of the LPV in the LMC, finding five periods that match between filters for V3 is promising. However, this will need to be further explored and investigated statistically through more rigorous methods, which will be done with the same method referenced on page 22 for variable star V5, that are unfortunately outside the scope of this current thesis. These results will remain here with the caveat that they may be modified after such an investigation occurs. This applies to all of the candidate LPV.
The final periods used for the model for V3 were 87.1 ± 1.5 days, 79.2 ± 0.2 days, 60 ±
0.1 days, 54.7 ± 0 days, and 106.6 ± 0.1 days. The observed (V-I) color for V3 is 2.18 mag. The range of magnitudes in V is 0.82 magnitudes and in I it is 0.49 magnitudes. As with V1 the strong power of the primary period, lower but still noticeable magnitude variance, and multi- periodicity identifies V3 as an SRb LPV. V3 is also likely pulsating in fundamental and first overtone modes.
V Set I_long Set
Period (days) Power Amp (mag) Period (days) Power Amp (mag)
85.63 92.29 0.14 88.57 73.17 0.06 36
79.43 26.51 0.07 60.03 24.49 0.03
59.93 19.36 0.06 54.69 20.86 0.03
54.63 18.80 0.06 78.94 20.14 0.03
106.42 14.90 0.06 96.20 18.02 0.03
277.53 14.29 0.05 106.70 13.57 0.02
Table 5 - Periods for V3 found by VStar.
12.7 12.8 12.9 13 13.1 13.2 13.3 13.4 13.5 13.6 13.7 5000 5200 5400 5600 5800 6000 6200 6400 6600 6800 7000
V Light Curve Model
Figure 19 - V light curve for V3 with final model.
4.5 Variable Star V6
The first of the LPV identified by Hartwick and Sandage [16], this star is also listed in
Clement’s catalog as an L LPV from photoelectric and photographic plate data.
There is not a single dominant period for V6 (see Table 7). While the highest power period from the table, 1067.28-day, in the V filter may be relevant, there is not a comparably long period in the I filter, so we exclude it from the model. For any LPV, as the range of 37 variation decreases with a fairly constant photometric error, VStar is increasingly likely to pick up spurious periodicity in the LPV. Essentially, as the magnitude range of an LPV decreases,
VStar is more likely to find periods that are less likely to be real. With that in mind there are still a number of periods shared between the V and I filters for V6. The 662.1-day (second highest power in V filter) and 713.06 day (most powerful in I filter) periods seem comparable. Then they have strong matches at 70.28 and 70.26 (third in V and second in I respectively), 73.69 and
73.53 (fifth most power in V and third in I respectively), and at 135.31 and 138.03 days (sixth in power for both filters). Due to the similarity between the light curves in the V and I filters in the previous stars, from this point forward only the V filter light curve will be displayed for the rest of the variable stars. The V filter light curve with the final model for V6 is displayed in Figure
20.
The final model of the period was built with a primary period of 70.3 ± 0.01 days, and secondary periods of 73.6 ± 0.08 days, 136.7 ± 1.4 days, and 687.6 ± 25.5 days. The V filter has a range of 0.88 magnitudes and the I filter has a range of 0.39 magnitudes. The median observed
(V-I) color is 3.04 mag. With a range of magnitudes comparable to V3 and multiple periods, V6 is an SR LPV.
V Set I_long Set
Period (days) Power Amp (mag) Period (days) Power Amp (mag)
1067.28 40.30 0.09 713.06 54.77 0.04
662.11 37.32 0.08 70.26 31.79 0.03
70.28 31.33 0.07 224.76 25.48 0.03
88.24 27.45 0.07 73.53 22.67 0.03 38
73.69 25.07 0.07 123.35 17.18 0.02
135.31 23.59 0.06 138.03 15.69 0.02
Table 6 - Periods for V3 found by VStar.
13.7 13.8 13.9 14 14.1 14.2 14.3
Magnitude 14.4 14.5 14.6 14.7 14.8 5000 5200 5400 5600 5800 6000 6200 6400 6600 6800 7000
HJD Observed Model
Figure 20 - V filter light curve for V6 with final model.
4.6 Variable Star V7
The second of the three LPV identified by Hartwick and Sandage, V7 is also classified as an L LPV and the most current data listed on Clement’s catalog is from photoelectric plates.
There are three strong periods that appear in both V and I filters (see Table 8). The first period of 72.4 in the V filter and 72.9 in the second filter match up very well. After that the appropriate periods show up in slightly different orders between the filters but the 325.1 in the I filter is tied to the 331.9 in the V filter, and the 229.3 in the I filter matches up with the 236.6 in the V filter. The near match of 212.9 in the V is a bit weaker and has less amplitude so it was not used. While the 90.5 in V and 90.8 in the I are a good match, they are also alias periods of 39
72 days, so may be created by the observing schedule and not a physical characteristic of the variable star. The observed light curve and the final model is displayed in Figure 21 below.
The periods for the final model are 72.6 ± 0.2 days, 328.5 ± 3.4 days, and 233.0 ± 3.6 days. V7 has a magnitude range of 0.6 mag in the V filter and 0.21 mag in the I filter. The observed median (V-I) color is 2.48 mag. The smaller, but still obvious, range in variability and multiple periods identified make V7 a good candidate for SRb status. It is different from the previously re-classified SRb in that it has one shorter period and two much longer ones.
V Set I_long Set
Period (days) Power Amp (mag) Period (days) Power Amp (mag)
72.41 47.67 0.07 72.87 46.09 0.02
90.45 23.04 0.05 325.12 40.11 0.02
331.87 22.18 0.05 229.32 21.20 0.02
43.94 19.27 0.05 90.83 17.61 0.01
236.60 18.22 0.05 52.40 14.39 0.01
212.92 17.13 0.04 68.52 14.33 0.01
Table 7 - Periods for V7 found by VStar. 40
13.4
13.5
13.6
13.7
13.8 Magnitude 13.9
14
14.1 5000 5200 5400 5600 5800 6000 6200 6400 6600 6800 7000
HJD Observed Model
Figure 21 - V light curve for V7 with final model.
4.7 Variable Star V8
The last of the Hartwick and Sandage LPV, V8 is the last of the stars in this paper identified on Clement’s catalog, and like the others it is an L LPV at this time.
Looking at the top six periods found by VStar for V8, all of them have a match between the V and I filters, and none of them appear to match an expected alias of the primary period of
134.1 days. Building the model using all six presents a model that does a good job describing the observed light curve. The periods are listed in Table 9. The light curves with the final model are displayed in Figure 22 (V filter) and Figure 23 (I filter). For this project, V8 is the strongest indication that multiple periodicity is occurring, given the model built from six matching periods matches so well with the observed data.
The periods for V8 are 134.7 ± 0.6 days, 121.4 ± 0.3 days, 66.9 ± 0.1 days, 71.3 ± 0.4 days, 625.4 ± 1.6 days, and 74.4 ± 0.01 days. The range of magnitudes for the V filter is 0.8 mag 41 and in the I filter it is 0.5 mag. The median observed (V-I) color is 2.94 mag. This is an SRb variable star. The magnitude variability, multi-periodicity, and agreement between the model and the observed data make for a compelling case. Given the number of agreeing matches in period this is definitely pulsating in fundamental and first overtone modes, and may also be active in the second overtone.
V Set I_long Set
Period (days) Power Amp (mag) Period (days) Power Amp (mag)
134.08 59.30 0.10 135.36 59.02 0.06
121.10 41.20 0.09 121.66 46.23 0.05
66.97 34.57 0.08 623.72 40.22 0.05
70.96 34.48 0.08 71.66 33.09 0.04
627.00 31.84 0.07 74.35 26.17 0.04
74.37 26.69 0.07 66.73 19.11 0.03
Table 8 - Periods for V8 found by VStar. 42
13.5 13.6 13.7 13.8 13.9 14 14.1 Magnitude 14.2 14.3 14.4 14.5 5000 5200 5400 5600 5800 6000 6200 6400 6600 6800 7000
HJD Observed Model
Figure 22 - V light curve for V8 with final model.
10.8 10.85 10.9 10.95 11 11.05 11.1
Magnitude 11.15 11.2 11.25 11.3 11.35 5000 5200 5400 5600 5800 6000 6200 6400 6600 6800 7000
HJD Observed Model
Figure 23 - I light curve for V8 with final model.
4.8 Variable Star NV12
The first of the candidate variable stars identified by Escobar et al. [1], NV12 has a primary period of 39.23 days in V and 39.20 days in I and those periods have the highest 43 amplitude so that will be designated as the primary period. Following that are three periods with intervals fairly similar to the primary period. This may help to explain the portion of the observed light curve around day 6400 where the amplitude goes nearly flat because the multiple periods may be experiencing destructive interference. There is another period that matches between the V and I filter at 35.3 days, but it is a likely alias of the 39.23-day primary period, so is not likely to be real and was not included in the final model. The V filter light curve with the final model is displayed in Figure 24 below.
The final model was created using 39.22 ± 0.02 days, 56.0 ± 0.05 days, 52.76 ± 0 days, and 62.43 ± 0.03 days. NV12 has a magnitude range of 0.36 mag in V filter and 0.19 mag in I filter. The observed median (V-I) color is 2.1 mag. With observable, but smaller, magnitude variability, NV12 is an SR LPV. Given the multiple periods it is likely an SRb.
V Set I_long Set
Period (days) Power Amp (mag) Period (days) Power Amp (days)
39.23 61.53 0.05 39.20 52.45 0.02
55.95 31.91 0.03 56.04 40.98 0.02
52.76 30.40 0.03 52.76 32.39 0.02
35.37 25.30 0.03 35.32 29.23 0.02
62.40 24.82 0.03 62.46 21.47 0.01
67.42 14.54 0.02 33.46 14.62 0.01
Table 9 - Periods for NV12 found by VStar. 44
13.5 13.55 13.6 13.65 13.7 13.75
Magnitude 13.8 13.85 13.9 13.95 5000 5200 5400 5600 5800 6000 6200 6400 6600 6800 7000
HJD Observed Model
Figure 24 - V filter light curve for NV12 with final model.
4.9 Variable Star NV19
With the 42.78-day period matching between both filters and it having the highest power in both it is a good candidate for the primary period for NV19. The periods with the next highest power in both filters match in V and I and are likely to be real, so will be the periods for the final model, while the periods with lower power in the table are not matched between V and I so were not included in the final model.
The final model was created with periods of 42.78 ± 0 days, 58.65 ± 0.1 days, 67.91 ±
0.06 days, and 381.34 ± 0.4 days. The V filter light curve with final model is displayed in Figure
25 below. The magnitude range for NV19 is 0.4 mag in V filter and 0.19 mag in I filter. The median (V-I) color is 2.18 mag. NV19 is another SR LPV with multiple periods.
V Set I_long Set
Period (days) Power Amp (mag) Period (days) Power Amp (mag) 45
42.78 47.46 0.04 42.78 33.06 0.02
58.55 26.71 0.03 58.74 24.18 0.01
67.96 21.93 0.03 67.85 19.53 0.01
381.73 21.07 0.03 380.95 19.25 0.01
36.55 18.54 0.03 200.25 18.66 0.01
119.44 14.65 0.02 40.30 17.12 0.01
Table 10 - Periods for NV19 found by VStar.
13.4 13.45 13.5 13.55 13.6 13.65 13.7 Magnitude 13.75 13.8 13.85 13.9 5000 5200 5400 5600 5800 6000 6200 6400 6600 6800 7000
HJD Observed Model
Figure 25 - V filter light curve for NV19 with final model.
4.10 Variable Star NV101
Of the variable stars in M69, NV101 may have the most interesting set of periods, shown in Table 12. For the primary period, the model will use the 65.17-day period in V, third in power overall, but with comparable power to the first and second periods, and the 65.40-day period in I.
A visual inspection of the light curve also shows a period of about 65 days. The other noticeable 46 part of the period analysis results from VStar is the number of periods greater than 100 days. For the final model the 65.17-day period in V and 65.40-day in I make up the primary period, supplemented by 172.78 days in V with 174.63 days in I, and 216.77 days in V with the 226.11 day period in I rounds out the final model. This model is displayed with the V filter light curve in Figure 26 below.
The actual values for the final model are 65.29 ± 0.12 days, 221.44 ± 4.67 days, and
173.71 ± 0.93 days. The magnitude range in V is 0.39 mag and in I it is 0.25 mag. The observed median (V-I) color is 2.34 mag. NV101 is an SR LPV with multiple periods and small amplitude in variability.
V Set I_long Set
Period (days) Power Amp (mag) Period (days) Power Amp (mag)
216.77 32.70 0.04 65.40 42.68 0.02
172.78 30.97 0.04 174.64 32.56 0.02
65.17 30.14 0.04 226.11 19.08 0.01
427.83 25.46 0.04 435.02 18.19 0.01
45.92 21.36 0.03 123.71 16.36 0.01
123.00 18.15 0.03 312.54 13.43 0.01
Table 11 - Periods for NV101 found by VStar. 47
13.6 13.65 13.7 13.75 13.8 13.85 13.9 Magnitude 13.95 14 14.05 14.1 5000 5200 5400 5600 5800 6000 6200 6400 6600 6800 7000
HJD Observed Model
Figure 26 - V filter light curve for NV101 with final model.
4. 11 Variable Star NV103
When examining the period analysis results from VStar, all six of the top periods have matches between V and I. Due to the exact period match of 38.60 in both filters this has been used as the primary period for the model. While it is possible, or even probable, that not all of these periods are real, deciding which ones to keep is beyond the tools used so far. With that in mind, all of the periods were used to build the final model for NV103 and is displayed in Figure
27.
The final model was built with 38.60 ± 0 days, 35.11 ± 0.02 days, 61.66 ± 0.03 days,
33.66 ± 0.03 days, 36.01 ± 0.04 days, and 68.22 ± 0.06 days. In V the magnitude range is 0.45 mag and in I it is 0.30 mag. The median observed (V-I) color is 1.81 mag. Like V8, NV103 actually matches in the 6 listed periods and the model does a good approximation of the observed light curve. With all of those things being true, NV103 is a likely SRb LPV. 48
V Set I_long Set
Period (days) Power Amp (mag) Period (days) Power Amp (mag)
35.09 35.17 0.04 38.60 32.90 0.02
38.60 32.40 0.04 35.12 27.99 0.02
61.69 30.42 0.04 61.63 26.50 0.02
33.63 28.11 0.04 68.28 25.26 0.02
68.16 24.40 0.03 33.68 22.14 0.02
35.97 23.15 0.03 36.05 12.24 0.01
Table 12 - Periods for NV103 found by VStar.
13.45 13.5 13.55 13.6 13.65 13.7 13.75
Magnitude 13.8 13.85 13.9 13.95 14 5000 5200 5400 5600 5800 6000 6200 6400 6600 6800 7000
HJD Observed Model
Figure 27 - V filter light curve for NV103 with final model.
4.12 Variable Star NV104
For NV104 there is a strong primary period of 36.61 days in V and 36.67 days in I.
There is also a strong short secondary period of 25.56 days in V and 25.61 days in I, and a long 49 secondary period of 452.04 days in V and 459.22 days in I. The 33.51 is an alias of 36.61 and the other periods have a small enough power compared to the primary that they don’t contribute much to the final model. The fit is displayed in Figure 28 below.
The final model for NV104 was built with 36.64 ± 0.03 days, 25.59 ± 0.03 days, and
455.63 ± 3.59 days. The magnitude range in V is 0.33 mag and in I it is 0.12 mag. The median observed (V-I) color is 2.18 mag. The magnitude range and multiple periods for NV104 make it an SR LPV.
V Set I_long Set
Period (days) Power Amp (mag) Period (days) Power Amp (mag)
36.61 73.40 0.04 36.67 53.33 0.01
25.56 34.96 0.03 459.22 31.20 0.01
33.51 33.48 0.03 25.61 25.55 0.01
452.04 29.61 0.02 33.50 20.59 0.01
35.31 24.10 0.02 66.97 14.38 0.01
67.04 11.10 0.01 30.59 11.23 0.01
Table 13 - Periods for NV104 by VStar. 50
13.45 13.5 13.55 13.6 13.65
Magnitude 13.7 13.75 13.8 13.85 5000 5200 5400 5600 5800 6000 6200 6400 6600 6800 7000
HJD Obse…
Figure 28 - V filter light curve for NV104 with final model.
4.13 Variable Star NV105
There are a number of periods that may be relevant, but with the low power and amplitude found for each filter it is difficult to assign anything definitively. The light curve is displayed in Figure 29, but no model was built.
Even without a model to describe the cycle of variability for NV105, it is still possible to put together basic photometric data for the star. The range in magnitude in V is 0.16 mag and
0.10 mag in I. The median observed (V-I) color is 1.98 mag. There is some variability in the brightness of NV105 but without a defined period, or periods, this is an irregular, or L class,
LPV. Additional observations will need to be taken to further refine, and classify, this star.
V Set I_long Set
Period (days) Power Amp (mag) Period (days) Power Amp (mag)
223.89 14.16 0.01 226.48 16.77 0.01 51
211.13 13.81 0.01 428.03 14.52 0.00
434.55 13.73 0.01 128.56 14.34 0.00
27.13 11.60 0.01 25.56 12.57 0.00
33.37 10.05 0.01 115.74 11.12 0.00
128.17 9.78 0.01 22.08 10.54 0.00
Table 14 - Periods for NV105 found by VStar.
13.7
13.72
13.74
13.76
13.78
13.8
Magnitude 13.82
13.84
13.86
13.88
13.9 5000 5200 5400 5600 5800 6000HJD 6200 6400 6600 6800 7000
Figure 29 - V filter light curve for NV105
4.14 Variable Star NVa
This is the first of the new potential variable stars identified in this project. Looking at the light curve in V (Figure 30) there appears to be some sort of variability in the magnitude, but it is less than 0.05 mag over the entire observation time frame. Additionally, the periods found in VStar only come up with two matches between the filters at approximately 174 days and 75 52 days. Combining the very minor range of magnitudes with the low power and non-existent amplitude from the VStar analysis, there is not a high confidence in this being a true variable star. Visual inspection of the “ref.fits” image shows that NVa may be a faint star located very close to a brighter star, so the observed variability may be a construct of the interp step during image processing. It has been identified for future projects to follow up on.
V Set I_long Set
Period (days) Power Amp (mag) Period (days) Power Amp (mag)
173.54 18.71 0.00 1001.70 21.56 0.00
326.81 18.33 0.00 174.43 15.33 0.00
944.35 9.97 0.00 317.86 8.34 0.00
944.37 9.97 0.00 479.70 6.94 0.00
944.39 9.97 0.00 74.61 5.46 0.00
75.19 7.76 0.00 60.06 5.32 0.00
Table 15 - Periods for NVa found by VStar. 53
12.72
12.73
12.74
12.75 Magnitude 12.76
12.77
12.78 5000 5200 5400 5600 5800 6000 6200 6400 6600 6800 7000 Heliocentric Julian Date
Figure 30 - V filter light curve for NVa.
4.15 Variable Star NVb
This variable star, NVb, is another candidate LPV discovered with image subtraction and a much better result than NVa. While the 303.89-day period in V is matched by the 302.31-day period in I, this is likely to be a long secondary period. Instead the 41.55 days in V and 41.56 days in I will be used as the primary, supplemented by the 39.66 days in V and 39.02 in I. The
303.89 and 445.29 in V and 302.31 and 447.14 in I are long secondary periods. The V filter light curve with the final model is displayed in Figure 31.
The values used to create the final model were 41.56 ± 0.01 days, 39.82 ± 0.16 days,
303.10 ± 0.79 days, and 446.22 ± 0.93 days. The magnitude range in V is 0.41 mag, and in I it is
0.19 mag. The median observed (V-I) color is 2.33 mag. With magnitude range and multiple periods, NVb is also an SR LPV. 54
V Set I_long Set
Period Power Amp Period Power Amp
303.89 49.76 0.04 302.31 52.10 0.02
41.55 30.11 0.03 447.14 31.42 0.01
37.29 24.40 0.03 39.97 21.94 0.01
39.66 23.49 0.03 41.56 17.16 0.01
164.94 22.54 0.03 39.02 16.35 0.01
445.29 19.39 0.03 63.42 16.23 0.01
Table 16 - Periods for NVb found by VStar.
13.55 13.6 13.65 13.7 13.75 13.8 13.85 Magnitude 13.9 13.95 14 14.05 5000 5200 5400 5600 5800 6000 6200 6400 6600 6800 7000
HJD Observed Model
Figure 31 - V filter light curve for NVb with final model.
4.16 Variable Star NVc
This variable star has a lower amplitude and the analysis in VStar resulted in table 18.
Like NVb the longer period is showing a higher power, but there is a shorter period acting on 55 this light curve. With that in mind the primary period is 34.58 days in V, and 35.34 days in I.
This is supplemented by the 24.07-day period in V and 22.46-day period in I, with the long secondary period of 393.14 in V and 418.52 in I. Despite the Nyquist limit an attempt to find shorter potential period was made, but the results between the V and I filter did not match so it was not used. The light curve with the final model is displayed in Figure 32.
The final values for the model were 34.96 ± 0.38 days, 23.27 ± 0.81 days, and 405.83 ±
12.69 days. The range in magnitudes is 0.15 mag in V and 0.12 mag in I. The median observed
(V-I) color is 2.08 mag. Despite the relatively low range in magnitudes, the multiple periods of
NVc are a strong implication of membership in the SR class of LPV.
V Set I_long Set
Period (days) Power Amp (mag) Period (days) Power Amp (mag)
393.14 26.35 0.01 418.52 26.83 0.01
34.58 12.21 0.01 35.34 14.93 0.01
24.91 11.19 0.01 92.24 12.44 0.01
24.07 10.05 0.01 197.84 10.77 0.00
31.40 9.46 0.01 121.39 10.55 0.00
1377.18 9.22 0.01 22.46 9.84 0.00
Table 17 – Periods for NVc from VStar 56
13.76 13.78 13.8 13.82 13.84 13.86
Magnitude 13.88 13.9 13.92 13.94 5000 5200 5400 5600 5800 6000 6200 6400 6600 6800 7000
HJD Obser…
Figure 32 - V filter light curve for NVc with final model.
4.17 Variable Star NVd
NVd is another low amplitude variable star found by image subtraction. Unfortunately, using the limitation on potential periods due to the Nyquist limit results in nothing definitive.
Acknowledging that the result may be a non-physical result a secondary analysis was done in
VStar with a lower limit of 9 days. In this secondary analysis a primary period of 14.44 days in
V matches well with 14.45 days in I. There is also a good long secondary period of 446.03 days in V and 431.21 days in I. The V filter light curve with final model is displayed in Figure 33.
The values used to build the final model are 14.45 ± 0.01 days, and 438.62 ± 7.41 days.
The range in magnitudes in the V filter is 0.15 mag and in the I filter it is 0.14 mag. The observed median (V-I) color is 1.87 mag. Assuming the period found is accurate, NVd is an SR
LPV.
V Set I_long Set 57
Period (days) Power Amp (mag) Period (days) Power Amp (mag)
15.18 41.25 0.01 431.21 26.99 0.01
14.44 40.47 0.01 14.45 19.07 0.01
446.03 28.47 0.01 149.88 14.88 0.01
11.81 21.57 0.01 15.19 14.58 0.01
13.74 15.55 0.01 15.02 12.70 0.00
11.52 14.63 0.01 11.75 11.20 0.00
Table 18 - Periods for NVd found by VStar.
13.8 13.82 13.84 13.86 13.88 13.9
Magnitude 13.92 13.94 13.96 13.98 5000 5200 5400 5600 5800 6000 6200 6400 6600 6800 7000
Observed Model HJD
Figure 33 - V filter light curve for NVd with final model.
4.18 Variable Star NVe
Similar to NVa this candidate LPV doesn’t have any good potential periods and is very low amplitude. The results of the VStar analysis are displayed in table 20, and the V filter light 58 curve is display in Figure 34. At this time NVe is going to be tagged as a suspected variable for future investigation.
V Set I_long Set
Period (days) Power Amp (mag) Period (days) Power Amp (mag)
341.46 16.74 0.00 28.97 11.63 0.00
32.21 12.41 0.00 32.00 10.44 0.00
393.39 12.09 0.00 27.73 8.71 0.00
30.75 11.98 0.00 119.72 7.99 0.00
585.65 10.37 0.00 102.36 7.10 0.00
121.54 9.31 0.00 303.02 6.52 0.00
Table 19 - Periods for NVe found by VStar.
13.32
13.34
13.36
13.38
13.4
Magnitude 13.42
13.44
13.46
13.48 5000 5200 5400 5600 5800 6000 6200 6400 6600 6800 7000 HJD
Figure 34 - V filter light curve for NVe 59
CHAPTER 5
CONCLUSION
This project to identify and characterize long-period variable stars in the globular cluster
M69 was conducted using images collected in the V and I filters over a five-year period. These images were taken approximately once a week for ten months of each year, with a two-month gap occurring when M69 was either too low or actually below the horizon during each January and December. The images were corrected automatically by Skynet with bias frames, dark frames, and flat fields to remove instrumental effects from the raw images.
The images were then processed through ISIS to find candidate variable stars. From this process 18 candidate variable stars were identified, and light curves were created for each.
These light curves were then analyzed in VStar to determine potential periods for each candidate variable star. During this process, the flux difference photometry was converted to instrumental magnitudes and the mean magnitude and magnitude range in the V and I filters were determined, and the (V-I) color was calculated. These data are in Table 21. Also, a color-magnitude diagram
(CMD) of V magnitude vs. (V-I) color for the cluster based on the variable star data and data received from DAOPHOT for the non-variable stars is displayed in Figure 35
Examining the CMD it becomes possible to make some determinations about the candidate variable stars in this project. The first region of note is the horizontal branch red clump, a dense clump of stars located at (V-I) just past 1 mag and V magnitude of 16 mag. From the clump, moving up and to the right is the red giant branch. The SR variables create a trail that moves from along the asymptotic giant branch (AGB) toward the Mira variables at the end.
Considering that V4 and V5 are Mira variables, and that a correction to the probable error in the 60
V magnitude of V5 would move it further to the right and lower on the plot, Mira variables appear to be a more evolved LPV than the SR LPV. This implies that as an SR LPV evolves it gets more luminous and cooler, and as a consequence, its pulsation becomes more regular.
Realizing NVa is too far to the left, and significantly brighter, than the other candidate LPV enhances the argument that NVa is not actually variable. By contrast NVe is also not listed as a confirmed SR LPV in this project, but given its position near the beginning of the AGB, NVe may be in the first phases of transition to becoming an SR variable. Another notable candidate, based on its position of the CMD, is V3. With a mean magnitude in V at least half a magnitude brighter than the other SR LPV, then it is likely that V3 is actually located between M69 and the observer and not actually a member of the cluster. Given that V3 is located in a less dense region of the cluster the increased brightness is unlikely to be the result of unresolved stars, as is suspected in the case of V5.
Of the 18 candidate variable stars identified as members of the globular cluster M69 in this project, four were previously discovered by Rosino [4] and are identified as V1, V3-V5; three were from Catchpole et al. and identified as V6-V8; six were previously identified by
Escobar et.al (2010) [1] and are NV12, NV19, NV101, NV103, NV104 and NV105; and five are newly identified candidate variable stars with the designations NVa, NVb, NVc, NVd, and NVe.
After evaluating the light curve and periods of each star V4 and V5 are confirmed as Mira LPV,
V1, V3 and V6-V8 are reclassed as SRb, and with the exceptions of NVa, NVe, and NV105, the
NV stars are all classified as SR variables. The observed variability in candidate NVa is likely an artifact of the image processing resulting from the proximity of a brighter star. Candidates
NVe and NV105 are potentially variable but require additional observation to determine their classification. 61
V Mean V I Mean I (V-I) Primary Period
Star ID Mag V err Range Mag I err Range color Period error
V8 13.99 0.0071 0.80 11.05 0.0029 0.45 2.94 134.72 0.64
V7 13.70 0.0062 0.61 11.22 0.0025 0.21 2.48 72.64 0.23
V6 14.17 0.0077 0.88 11.12 0.0027 0.39 3.04 70.03 0.01
V5 13.59 0.0055 2.68 10.76 0.0023 4.45 2.83 196.46 0.09
V4 14.17 0.0076 5.23 10.61 0.0023 2.41 3.56 203.23 0.08
V3 13.11 0.0045 0.82 10.93 0.0026 0.49 2.18 87.1 1.5
V1 13.65 0.0064 1.16 11.15 0.0030 0.62 2.50 115.02 0.37
NVE 13.41 0.0026 0.11 11.76 0.0026 0.14 1.66 322.24 19.22
NVD 13.87 0.0067 0.15 12.00 0.0042 0.14 1.87 14.45 0.01
NVC 13.86 0.0068 0.15 11.78 0.0041 0.12 2.08 34.96 0.38
NVB 13.78 0.0067 0.41 11.45 0.0035 0.19 2.33 41.56 0.01
NVA 12.73 0.0017 0.04 11.89 0.0017 0.00 0.85 173.99 0.45
NV19 13.60 0.0057 0.40 11.42 0.0032 0.19 2.18 42.78 0.00
NV12 13.66 0.0061 0.36 11.56 0.0037 0.19 2.10 39.22 0.01
NV105 13.79 0.0065 0.16 11.81 0.0041 0.10 1.98 225.19 1.30
NV104 13.70 0.0062 0.33 11.52 0.0035 0.12 2.18 36.64 0.03
NV103 13.64 0.0066 0.45 11.83 0.0046 0.01 1.81 38.60 0.00
NV101 13.78 0.0063 0.39 11.44 0.0035 0.25 2.34 65.29 0.12
Table 20 - Table of basic data for each variable star 62
12 Nva NVe NV103 V3 NVb
NV12 13 NV19 V1 V5 NVd V7 14 V8 V6 V4 NV105 NV101 NV104 15 NVc
16
17 V Magnitude
18
19
20
21
22 -0.5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5
V-I magnitude V mag V Mag Var
Figure 35 – Color-Magnitude Diagram of M69. Blue dots represent stars not identified as potential variable stars. Orange circles note the position of the variable stars based on the mean
(V-I) color and mean V magnitudes of the observed light curves. 63
Plotting the primary period of each star with the I magnitude in Figure 36 it appears that as the mean I luminosity becomes brighter the primary period of the star becomes longer. Due to the Mira stars V4 and V5 being at one end of this track, it would appear that as a LPV star evolves it will get brighter in the I filter, and its period will increase by a related amount. A trendline of Log(p) vs I magnitude was plotted on the Figure and has R2=0.81. This is a good indication that the period of the SR LPV is related to the magnitude in the I filter. As a test for future LPV the equation of this trendline is I mag =-1.13*log(p)+13.4. There does not appear to be a corresponding strong correlation between the long secondary periods and the I magnitude, other than to say that none of the LPV with an LSP have a luminosity greater than 11 mag.
Log(LSP) Log(P) Linear (Log(P)) R² = 0.80904 10.5 V4 10.7 V5
10.9 V3 V8 V8 11.1 V6 V1 V1 V6 V7 V7 11.3 NVBNV19 NV101 NVB NV19 I Magnitude 11.5 NV104 NV104 NV12 11.7 NVC NV105 NV103 NV105 11.9 NVD NVD 12.1 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3
Log(p)
Figure 36 - Log(P) vs. I Luminosity plot of the candidate long period variable stars in M69. A number of the LPV have a (LSP) which was also plotted against the mean I luminosity. 64
The I magnitude vs. I magnitude range was also plotted (Figure 37). It appears that as the
Semi-Regular LPV increase in brightness as they evolve up the giant branch the range of V magnitudes increases. A logarithmic trend line has been plotted in order to evaluate the relationship between I magnitude and I magnitude range of the LPV with an R2 = 0.70. This value indicates a weak relationship. Perhaps improved data or additional LPV may improve the fit of the trendline. More work needs to be done to further test this possible relationship. It also appears to be a fairly linear relationship of the evolution of the LPV until it reaches Mira stage.
This jump in magnitude range has been interpreted to be a “mode switch” from first overtone to fundamental by Soszy´nski et al., Abbas et al., and Osborn et al. in their respective papers.
While the I range increases noticeably, the Mira LPV do not become much more bright in the I filter when compared to the Semi-Regular LVP.
10.4 R² = 0.69807 10.6 V4
10.8 V5 V3 11 V8 V6 V1 11.2 V7 NV19 11.4 NV101 NVB NV104NV12
I Magnitude 11.6 NVC 11.8 NV103NV105 12 NVD 12.2
12.4 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 I Magnitude Range
Figure 37 - I magnitude range vs. I magnitude plot. 65
A final plot (Figure 38) of the I magnitude range vs. log(P) was plotted for the Semi-
Regular LPV. There is another visible trend of the evolution of the LPV where the range of I magnitude increases with the log of the period. This is another indication of a relationship between the evolution of the LPV and its variability. An exponential trend line has been plotted to demonstrate the relationship, and has an R2 = 0.52. This value includes the Mira LPV, V4 and
V5, which have much larger range in the I magnitude than the SR LPV and are not included on
Figure 38. A visual inspection of the trend line shows a good relationship between the I magnitude and log of the period. For reference by future researchers to test the I magnitude of
LPV by the log of the primary period, this equation of the trendline is I mag=0.01e3.50*log(p).
0.7 y = 0.0111x5.6147 V1 0.6 R² = 0.51861
0.5 V3 V8 0.4 V6
0.3 NV12 NV101 NV19 I Magnitude Range 0.2 NVD V7 NV105 0.1 NVB NV104 NV103 0 1.4 1.5 1.6 1.7 1.8 1.9 2 2.1 2.2 2.3 2.4 Log(P)
Figure 38 - Plot of Range in I magnitude vs. log(P) for the primary, pulsation periods. The vertical axis has been restricted to 0.7 mag range in order to demonstrate the trend of the Semi-
Regular LPV magnitude range and period.
Overall the results of this project are an improvement on the data previously available for
M69. Since no one has revisited the individual stars, with the exception of the known Miras V4 66 and V5 (Sloan et al. 2010), in the cluster for more than four decades. The Rosino 1962 paper [4] variables V1 and V3 had only photographic plates and insufficient time series and quality to recognize the multiple interacting periods in these stars. Hartwick & Sandage [16] identified the variables V6-V8 with photoelectric results in 1968 but also had limited quality and time series to recognize the multiple interacting periods in the variable stars. Using CCD images and a longer time series we have been able to not only confirm V4 and V5 as Mira variables, we have demonstrated that V4 and V5 also exhibit multiple interacting periods. Additionally, we have improved upon the classification of the other previously identified LPV in M69. With new data processing tools we have also identified and confirmed eight new SR LPV: NV12, NV19,
NV101, NV103, NV104, NVb, NVc, and NVd. This brings the number of LPV in M69 from the previously identified seven stars to fifteen confirmed LPV. We have also shown that LPV in GC exhibit multiple periods in their behavior. Our data provide almost 2000 days of observed light curves. Given the range of FWHM in our data the magnitudes of crowded stars, specifically V5,
NV105, and NVe, can probably be refined and made more accurate with a higher resolution telescope. We also provide some additional data to further explore the role metallicity plays in the evolution and characteristics of LPV. 67
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