Testing the multi-epoch luminosity function of asymptotic giant branch stars in the Small
Magellanic Cloud with VISTA
R´ ois´ın O’Rourke Brogan June 2020
1 Abstract
The physics pertaining to the asymptotic giant branch (AGB) phase of stellar evolution has been studied for many years. However, the mechanics behind many characteristics displayed at this stage are still not fully understood. As a member of the Long Period Variable class of stars, AGB stars are invaluable in creating three-dimensional maps of the Milky Way, the Magellanic System and other galaxies with resolved stellar populations. Variable stars can be used to determine radial distances from Earth using their periodic luminosity variations.
As this type of star has unknown qualities, models of AGB populations need to be calibrated with observed data. Previous research has derived a best- fitting model using the TRILEGAL code (a TRIdimensional modeL of thE GALaxy). This model was calibrated against single-epoch luminosity functions (LFs) calculated from resolved stellar populations in the Small Magellanic Cloud (SMC). With multi-epoch data now available from the VISTA survey of the Magellanic Clouds (VMC), this best-fitting model can now be compared with the LFs as they vary with time. Firstly, statistical tests are completed to measure the extent of the LF variation between epochs and from the mean LF for both the full VMC AGB catalogue and for the oxygen-rich, carbon-rich and extreme AGB classes. Statistical tests are then performed to measure the similarity between the LFs from different epochs and the simulated LFs, again for the entire sample and the three classes. This investigation shows that, while the current best-fitting model is a good approximation of many individual epochs’
AGB LFs in the SMC to within 3σ, inclusion of multi-epoch data would make for a more robust analysis. In order to do this, it would be desirable to have more epochs with deeper and regular observations that could cover full lightcurves of some of the sources. There also seems to be a statistical difference between the inner and outer areas of the SMC, perhaps due to tidal disruptions. It would be interesting to see the results of a similar study using the LMC, which is less affected by the gravitational influence of its smaller companion.
2 Popul¨ arvetenskaplig sammanfattning
Mot slutet av sitt liv g˚ar stj¨arnor med en l˚ag till medelstor startmassa in i en fas som kallas den asymptotiska j¨attegrenen (kallad AGB-stj¨arnor fr˚an den engelska termen Asymptotic Giant Branch). Vid detta utvecklingsstadium har de stora radier och ljusstyrkor och ¨ar variabla i b˚ada dessa parametrar. Des- sa stj¨arnor ¨ar ocks˚a en b¨ordig milj¨o f¨or k¨arnkraftsyntes. Dessutom inneb¨ar den extrema massf¨orlusten som f¨orekommer i AGB-stj¨arnor och f¨orekomsten av termiska och radiella pulser att denna typ av stj¨arna ¨ar en av de mest dynamiska och komplexa att unders¨oka. F¨orutom att observera enskilda pro- ver kan mycket l¨aras fr˚an storskalig statistisk analys av tydligt observerade stj¨arnpopulationer. Ett anv¨andbart verktyg f¨or att studera stora grupper av ljusemitterande f¨orem˚al ¨ar en ljusfunktion som ger antalet ljusk¨allor per ljus- styrkaintervall. Detta kan till¨ampas p˚a hela gruppen av AGB-stj¨arnor f¨or att unders¨oka ljusutg˚angen av denna klass av stj¨arnor. Provet kan ocks˚a delas in i underklasser f¨or att separat unders¨oka AGB-stj¨arnor med olika ¨overfl¨od av kemikalier i sin atmosf¨ar. Provet som anv¨andes i denna studie ¨ar h¨amtat fr˚an en av tv˚a satellitdv¨arggalaxer som kallas de Stora och Lilla magellanska mol- net som kretsar kring Vintergatan, v˚ar galax. Dessa n¨ara galaxer utg¨or en unik m¨ojlighet att studera en tydligt observerade stj¨arnpopulation. I denna rapport, data h¨amtas fr˚an VISTA-teleskopunders¨okningen av Lilla magellanska molnet.
Tidigare studier har fokuserat p˚a detta omr˚ade och en simulering av AGB- befolkningen i Lilla magellanska molnet har skapats. Denna modell kalibrerades med hj¨alp av luminosty-funktionerna som skapats med hj¨alp av data fr˚an en annan unders¨okning - SAGE SMC. Dessa data togs emellertid endast under en observation och eftersom dessa ¨ar variabla stj¨arnor kanske ljusfunktioner- na inte ¨ar konsekventa ¨over tiden, vilket inneb¨ar att den kalibrerade modellen kanske inte alltid ¨ar den b¨ast passande. Denna rapport anv¨ander data som ta- gits ¨over flera observationer f¨or att j¨amf¨ora AGB-stj¨arnornas ljusfunktioner vid olika tidpunkter, i olika delar av galaxen och mot den nuvarande b¨ast passan- de modellen med hj¨alp av en m¨angd olika statistiska verktyg och funktioner.
P˚a detta s¨att kan det visas att modellen ¨ar ganska robust, men inf¨orlivandet av data som tagits vid olika tidsperioder ¨ar viktigt f¨or en fullst¨andig utred- ning. Det har ocks˚a blivit uppenbart att de inre och yttre AGB-populationerna i det Lilla magellanska molnet kan ha n˚agra grundl¨aggande skillnader mellan dem. M¨ojliga f¨orklaringar inkluderar en annan stj¨arnbildningshistoria eller tid- vatteneffekter som h¨arr¨or fr˚an gravitationseffekterna av det Stora magellanska molnet och Vintergatan.
3 Background
3.1 The Small Magellanic Cloud
The SMC is the smaller of two interacting satellite dwarf galaxies in the vicinity of the Milky Way, approximately 190 000 light years from Earth [1]. It is about half the angular diameter and one fifth of the mass of its companion; the Large Magellanic Cloud (LMC). Due to its smaller mass, it has been more obviously deformed than the LMC due to the gravitational interactions between the SMC, LMC and the Milky Way and as such is classified as an irregular dwarf galaxy.
However, bar structures can also be seen in both galaxies (more so in the LMC), which could indicate that the SMC was formally a barred spiral [10]. It is highly inclined and may be as much as 20 kpc along the line of sight [7]. It seems to be a more complex dwarf galaxy than its partner and, much like other dwarf galaxies, has a low metallicity - a quarter that of the Sun. The tidal interactions between the two galaxies and the Milky Way may also explain some of the other features in the Magellanic Cloud structure, such as the Magellanic Stream, a trail of neutral hydrogen connected to the galaxies [21]; and the Magellanic Bridge, another band of neutral hydrogen which links the SMC and LMC [23].
This structure may have arisen from the tidal interactions between the LMC and SMC [7].
Due to the proximity of these galaxies, a unique insight into galactic and stellar evolution is available as the features and stellar populations can be re- solved, unlike galaxies at higher redshifts. In [10] data from the VISTA survey of the Magellanic system (the VMC survey, used in this study) is used to look at the morphology of the SMC and LMC. By tracing differing populations of stars, various substructures can be seen in both. The SMC is highly elongated and has an eastern protrusion called the Wing. Young stars seem to congregate in the central regions and the Wing, whereas older stars are distributed more evenly. Variable stars such as Cepheid variables, RR Lyrae variables and AGB stars have been used to identify the three dimensional structure of the SMC.
Inspection of the Magellanic Cloud system can not only answer questions about the galaxies themselves but could enhance knowledge on dwarf galaxies, inter- acting galaxies, binary systems, satellite galaxies, tidal effects on star formation and bar formation in general [7].
3.2 Asymptotic giant branch stars
3.2.1 Stellar evolutionary phase
The lifetime of a star begins once a gas cloud that has condensed and reached hydrostatic equilibrium starts to convert hydrogen into helium [6]. The core hydrogen-burning phase is known as the main sequence and stars spend the largest amount of time in this phase. For stars of low to intermediate mass (about 0.6 to 10 M ) once the central hydrogen is exhausted, H-burning con- tinues in a shell around a helium core. The stellar envelope expands, the tem- perature decreases and the star moves off the main sequence and reaches the base of the red giant branch (RGB). During the RGB phase, the envelope expan- sion continues and the luminosity of the star increases. For intermediate-mass stars, the ignition and burning of helium in the core is stable, but for low-mass stars the helium is ignited in a degenerate core, resulting in a thermonuclear
Figure 1: Hertzsprung-Russell diagram of a complete 2M evolutionary track for solar metallicity from the main sequence to the white dwarf evolution phase.
In the cooler section of the post-AGB phase, wiggles in the track are caused by numerical convergence difficulties. The blue track shows a born-again evolution (triggered by a very late thermal pulse) of the same mass, however, shifted by approximately ∆ log Tef f = 0.2 and ∆ log L/L = −0.5 for clarity. The red and green stars mark the position of the central stars of planetary nebulae. The number labels for each evolutionary phase indicates the log of the approximate duration for a 2M case. Larger or smaller mass cases would have smaller or larger evolutionary timescales, respectively.
runaway known as the ”helium flash” at the tip of the RGB. Once the helium fuel is exhausted in the centre of the star, the core is now composed of carbon and oxygen and He-burning occurs in a shell around this C-O core. This marks the beginning of the AGB phase experience by both low- and intermediate-mass stars. One feature of this phase is strong mass-loss which progressively removes the stellar envelope. When the envelope is almost completely removed, the evolution continues through the post-AGB phase where rapid mass-loss occurs.
During this time part of the circumstellar envelope becomes ionised by the ultra- violet (UV) flux and the star appears as a planetary nebula. When the hydrogen burning shell is finally extinguished, the bare C-O core cools as a white dwarf.
Figure 1 show the typical evolution of a 2 M star in the Hertzsprung-Russel (HR) diagram [17].
If the AGB phase is looked at in more detail, it becomes clear that the struc- ture of the star at this point is quite complex. As shown in fig. 2, the internal structure consists of different layers, starting with the core, the smallest and
densest part of the star. Around this, there is a helium burning shell then a layer of helium fuel which is produced by the surrounding hydrogen burning shell. Further out there is a radiative layer and then convective envelope, which has the largest contribution to the radius of all the sections. In the outer layers where the star is cooler, dust can form which absorbs the internally produced ra- diation and re-emits it at infrared wavelengths. If the dust is especially opaque, the stellar output may be completely obscured at visible wavelengths [11]. Radi- ation pressure on these dust grains are also thought to be the driving mechanism behind most of the mass loss prevalent in the later stages of the AGB phase once dust is able to form. The mass-loss at the early AGB (E-AGB) stage is not fully understood but cannot be driven by radiation pressure as the luminosity is still low and the effective temperature is still high. Plausible explanations in- clude Alfv´en waves [9] (oscillation of ions due to tension in magnetic field lines) and turbulence in the cool extended chromosphere [26]. In the later, thermally pulsating stage (TP-AGB) the star will expand and contract radially, however the energy transferred is not thought to be enough to accelerate the stellar mat- ter to escape velocities [16]. Key characteristics of stars moving through this evolutionary phase are high luminosities, typically between 500 and 10000 L and low temperatures ranging from 2000 to 3500 Kelvin [16]. However, shock waves induced by the radial pulsations can bring gas out to larger radii, where dust particles can condense and can be accelerated by the radiation pressure resulting from the high stellar luminosity. AGB stars are extremely variable and the radii, temperatures and luminosities of these stars cycle throughout the phase. In the E-AGB phase, the bulk of the energy generated comes from the helium burning shell, however as the fuel is depleted, the hydrogen burning shell takes over. Whereas in the TP-AGB stage, the star is highly luminous and suffers rapid mass loss which eventually occur at a greater rate than the nuclear burning. Once this happens a ”superwind” develops and much of the stellar envelope is ejected.
3.2.2 Pulsations and mass loss
The AGB stage is when the third dredge-up (3DU) takes place, an important event in low to intermediate mass stellar evolution [16]. The term ”dredge-up”
refers to the appearance of nuclear burning ashes close to the outermost regions of the star after convections cells have been able to reach far down into the core [6]. The first such event takes place at the main sequence turn-off when transitioning to the RGB. The second is at end of helium core-burning for stars in the range of 4 to 8 M . Although labelled sequentially, all three dredge- ups are not necessarily experienced by every star. The energy generated in the core is at first absorbed by the outer layers as the radius expands and effective temperature decreases. The convective layer can extend to the core for stars with masses conducive to the second dredge-up and move some internal material to the surface. As the H-burning shell produces helium fuel, the helium burning shell turns on and off quasi-periodically. The bursts of activity are known as helium shell flashes and cause the hydrogen-burning shell to expand, cool and turn off temporarily. These are the cause of the thermal pulses, however the excitation mechanism of AGB radial pulses is as yet unknown.
The instability and sudden increase in internal energy production means that a convection zone can appear between the hydrogen- and helium-burning shells
Figure 2: Internal structure of a 1 M star at the beginning of the AGB phase.
The left part shows the different regions scaled according to mass-fraction, and to the right, scaled with radius. Image from [16].
whose reach increases with the strength of the flashes [6]. If the stellar mass is greater than approximately 2 M (the exact value depends on initial stellar metallicity), the convection zones will merge, extending further into the central regions and dredging up carbon to the surface. Multiple dredge-ups, depending on efficiency, can change the ratio of oxygen to carbon on the surface. If the carbon to oxygen ratio (C/O) exceeds unity, the star is classed as a carbon star.
The surface chemistry of AGB stars is important as this indicates the type of molecules and dust that will form in the cool (around 3000 K) outer regions [16]. The carbon and oxygen atoms tend to combine to form the extremely stable CO molecule, leaving the rest of the excess element to form dust grains.
In case of oxygen, silicates are formed and amorphous carbon is the type of dust found in carbon AGB stars. The latter is more opaque and this gives carbon stars a redder colour as the internal radiation is absorbed and re-emitted in the infrared. The mass loss is important as it enriches the interstellar medium with nuclei produced through AGB nucleosynthesis, with implications for galactic chemical evolution and the cosmic matter cycle as well. Elements that are created within AGB stars include helium, carbon and oxygen, as mentioned, and also slow neutron-capture process elements such as lead, xenon and barium [2].
3.2.3 Types of AGB stars
One method of classifying AGB stars is according to their observed pulsation properties, i.e. period and amplitude [16]. They are part of the broad group of Long-Period Variables (LPVs) which is subdivided into different variability types (Mira variables, semi-regular variables (SRVs) and irregular variables).
Mira variables have very regular periods and large amplitudes at visible wave- lengths and are thought to be more evolved than semi-regular variables. The semi-regular types can be split further into type a and type b; both have small
Figure 3: A Ks vs. J − Ks colour-magnitude diagram of the SAGE SMC (Surveying the Agents of Galaxy Evolution in the Tidally-Disrupted, Low- Metallicity Small Magellanic Cloud) sample of evolved stars. The grey lines are the photometric criteria adopted to separate red supergiants from AGB stars and O-rich from C-rich stars (K0, K1 and K2, defined in the section Com- parison with model data), the horizontal dashed line is the tip of the RGB in the Ksband. Taken from [22]
amplitudes but type a has a more regular periodicity than type b.
Another important distinction, and one which will be used in this study, is the atmospheric composition. In the previous section it was mentioned that the efficiency of the third dredge-up can alter the composition from oxygen-rich to carbon-rich. This is quantified through the C/O ratio; if this ratio is less than unity the star is oxygen-rich O-AGB and greater than unity indicates a carbon-rich star C-AGB. If the ratio is approximately unity this means the star is labelled S-type. The classification is based on specific features in the stellar spectra, associated either with oxygen- or carbon-based compounds. However, a spectral classification is not always available for every star in a given stellar system. In these cases, the classification can be performed using the observed photometry in specific combinations of filters. In particular, the populations of O-AGB and C-AGB can be distinguished in the Ks vs. J-Ks color-magnitude diagram (CMD). An example of this can be seen in fig. 3.
Further to carbon and oxygen AGB stars, this study also deals with extreme (X-AGB) and anomalous (a-AGB) AGB stars. The anomalous type is thought to be highly evolved and dusty, with low initial masses of less than 1.25 M in the SMC [3]. These include both oxygen-rich and carbon-rich atmospheric chemistries which infers that the third dredge-up can create carbon stars even at these low masses. The Boyer et al. (2015) paper [3] states that they are straddling the critical mass where the dredge-up is efficient enough to produce C-AGB stars. In the SMC, as it is a metal-poor environment, 50% of a-AGB stars are carbon-rich - higher than the less metal-poor LMC. The X-AGB stars are the most dusty and exhibit extreme mass-loss. Their colours tend to be
Figure 4: Filter transmission curves for the VMC survey (Y , J , Ks - black continuous lines) compared with the transmission of the 2MASS (J , H, Ks - red dashed lines) and DENIS (I, J , Ks - blue dotted lines) surveys. From [7].
redder than both the O- and C-AGB stars as most of this class have C-rich atmospheres [27].
3.3 The VMC survey
The data in this study is taken from VISTA (Visible and Infrared Survey Tele- scope for Astronomy), the current largest wide-field near-infrared imaging tele- scope in the world [7]. The VISTA Magellanic Clouds survey (VMC) takes data across three broadband filters: Y , J and Ks. As shown in figure 4, these are slightly different from the filters used for 2MASS (Two Micron All-Sky Survey) and DENIS (Deep Near Infrared Survey of the Southern Sky). The images are created with a series of six offsets so that the space between detectors is not ignored. The offset positions are named pawprints and cover 0.59 deg2 of the sky. These images are then combined to form one tile that covers approximately 1.5 deg2. During this process each area is observed at least twice.
This study specifically takes SMC Ks data in the VMC, which uses twenty seven tiles to cover the galaxy [7]. The tiles in the SMC have a position angle of 0 deg, whereas the LMC tiles have one of 90 deg, as can be seen in figure 5.
The tiles can also be divided into subregions which will become relevant in this study when comparing data to models. The tiles were positioned with reference to 2MASS, DENIS and SuperCOSMOS data to ensure that they covered a desirable section of the stellar population. The filters were chosen by bearing in mind the uses intended for the data, i.e. investigation of the star formation history (SFH), and filters less likely to be attenuated or have a large risk for error. For example, the J -band is less sensitive to atmospheric effects than the H-band and the Y -band comes out more favourably when comparing the
Figure 5: Magellanic system area tiled for VMC observations. From [7].
confusion limit to that of the Z-band (the confusion limit refers to the limiting flux density beyond which no new sources can be detected [31]). Ks is used in this case as at this wavelength variable stars have a well-defined period- magnitude relation. Moreover, there is a larger amount of epochs in this band - ideal for the study of variable stars like AGB stars. The photometric calibration for VMC uses the 2MASS catalogue and in the VMC data various aperture flux magnitudes are given. Ks”aper3” is used in this study which is quantified by a core radius of 1” or 3 pixels and observes 75% of the total stellar flux in a 0.8”
seeing observation [7].
Most previous Magellanic Clouds studies have used data from surveys with different original goals, such as the infrared sky survey 2MASS, used in [22].
The VMC survey is important as it offers extremely sensitive near-infrared data over multiple epochs [7]. In this way it can be used to investigate the SFH and shape of the system in three spatial dimensions. The wavebands chosen are suited for observation over a large range of stellar evolutionary phases, from the main sequence to supernovae remnants and are useful for study of those in the late stages of main sequence turn-off. When looking at these as part of the overall structure, conclusions can be drawn about how the system evolved to its current state and may continue to evolve in the future. The SFH in particular can be derived using CMD fitting techniques. Similar studies have already been completed for AGB stars (as will be mentioned later), however the increased sensitivity and availability of multi-epoch data can hopefully improve upon models already fit to less sensitive, single-epoch data.
Figure 6: SMC tiles and subregions included in this thesis and in [22]. Taken from [22].
4 Previous research
4.1 SAGE SMC data
The previously mentioned paper - Pastorelli et al. (2019) [22], provides a start- ing point for this study. It also focuses on thermally-pulsing AGB stars in the SMC. The paper uses survey data to find a best-fitting model that repro- duces the observed Ks-band luminosity functions for the AGB stars, and the subclasses of O-, C, and X-AGB. Scientists currently do not have a complete physical picture of the processes that occur during the AGB stage of evolution as the physical mechanisms driving the AGB evolution are complex and inter- connected. Previously, models were calibrated using large star clusters, which still may not contain enough AGB stars to lower the uncertainties in observed parameters to an acceptable level. A more significant problem with using large SMC clusters is the so-called ”AGB boosting effect” [12]. This occurs at in- termediate age, about 1.6 Gyrs, and causes the percentage of the population at the TP-AGB stage to be larger than expected. As the largest SMC clusters are found at around this age, this creates an issue with an over-estimation of the flux contribution from TP-AGB when using SMC cluster observations to calibrate stellar population models.
Models can also be calibrated using regions of nearby galaxies with reliable measurements of their SFH, which solves the problem of a small sample size for a large enough region. The drawback in using this method is that the AGB stars used have a large range of initial masses meaning that there is a larger degree of uncertainly for this parameter. However, the latter method seems to have better results statistically.
The catalogue used in [22] was created with SAGE SMC data (Surveying the Agents of Galaxy Evolution in the Tidally-Disrupted, Low-Metallicity Small
Figure 7: Number counts of RSG/AGB populations classified by Srinivasan et al. (2016) [29]. The star counts refer to the selected SMC area used in [22].
Magellanic Cloud) [14]. This is collated from several catalogues including DE- NIS, 2MASS and teh Spitzer Space Telescope. The AGB stars were first iden- tified and classified by Boyer et al. (2011) [4]. Srinivasan et al. (2016) [29]
then refined the AGB selection methods used in the catalogue, which is the ver- sion that was used in [22]. Figure 6 shows the VMC tiles overlaid on a density map of the AGB catalogue compiled in [29]. These are split into O-, C-, a- and X-AGB stars, as mentioned in the section on subclasses. As the a-AGB subgroup is thought to be made up of approximately 50% carbon-rich stars and 50% oxygen-rich stars in the SMC [3] and it is not possible to separate them by photometry, the counts in the luminosity function for this group was split evenly between the O- and C-AGB classes. Most of the AGB stars have been selected by photometry, however 81 have been spectroscopically confirmed by Ruffle et al. (2015) [25] and 273 by Boyer et al. (2015) [3]. As well as AGB stars, the catalogue includes red supergiants (RSGs), as shown in table 7, and far infrared objects.
This catalogue is used as a calibration for the model created in [22], com- paring star counts and luminosity functions in the Ks-band. The LF test is shown for each of the 2MASS and Spitzer filters in figure 8. The fit for the LFs was mainly concentrated in the Ks filter as AGB stars are generally less affected by circumstellar dust at this wavelengths, which reduces uncertainties in the circumstellar dust models [20]. These are single-epoch LFs and therefore it is uncertain whether the best-fit model is robust enough to be a good ap- proximation as the AGB stars pulsate with time. The two processes calibrated in [22] that are as yet not fully understood are the mass-loss processes and the third dredge-up. The C-AGB LF is particularly useful for calibration of the occurrence and efficiency of the 3DU.
4.2 The current best fit model
In order to create a model which can accurately replicate the stellar populations in the SMC, an accurate SFH should be used as an input. Pastorelli et al. (2019) [22] use the SFH derived in Rubele et al. (2018) [24]. This paper uses VMC data as it has low extinction in near-infrared wavebands and the photometry used includes the oldest main sequence turn-off points, important to estimate ages. The SFH from paper [24] uses the photometric data for 14 SMC tiles, however in [22] not all SMC tile subregions were included as some of them do not overlap with the SAGE SMC catalogue. Those included can be seen in
Figure 8: Comparison between the synthetic LFs obtained from the best-fitting model S 35 and the observed LFs in the 2MASS and Spitzer filters, going from shorter (top panels) to longer wavelengths (bottom panels). The y-axis indicates the number of sources in each bin. From [22].
Figure 9: Central coordinates of the VMC tiles used in [22] and number of AGB stars identified by [29] for each tile.
the table 9 under the column ”subregions”, which can be cross-referenced with figure 6. Another input for the stellar population simulation is the data from the stellar evolution code PARSEC [5], which calculates pre-TP-AGB stages and COLIBRI [19], which models the TP-AGB phase. The code used to create the models is the population synthesis code TRILEGAL (a TRIdimensional modeL of thE GALaxy) [13]. This is run several times for each set of input parameters and an average is taken of the runs for comparison.
The important aspect considered when selecting a best-fit model was that the luminosity functions (see figure 8) were closely approximated for each group of AGB stars: O-, C- and X-AGB. When comparing the LFs, a χ2 test was performed, as shown in figure 8, for the three classes and the whole catalogue.
Each value is an average of four χ2 values weighted by sample size. Firstly, the model sets which can replicate the star counts to within 3σ are chosen and from these the set with the lowest χ2 value is deemed best-fit. This yielded two best-fit model sets: S 07 and S 35. The model S 35 was then chosen as the best-fit overall as it better reproduced the initial-final mass relation, i.e. the masses of the white dwarf remnants.
5 The multi-epoch VMC AGB catalogue
The catalogue used in this investigation has approximately the same number of AGB sources as the catalogue used in [22]. Not every source is included as it was necessary to have the same amount of sources in each epoch for comparison and some stars may not have been observed in every epoch. The completeness is
Figure 10: Variation of completeness with magnitude for VMC tile SMC 4 3 in the Ksfilter.
unknown for brighter magnitudes, figure 10 shows the VMC completeness study for tile SMC 4 3 in the Ksband with a cut-off point at about 11.5 magnitudes.
However, there are significantly less sources at this brightness range than in [22]
and this is demonstrated in fig. 11 where the Ks versus J − Ks CMDs of the two catalogues are compared. Additionally, the VMC data LFs tend to agree with the current best-fit model better when this region is excluded.
Table 1 in [22] shows the number of stars found in the SAGE SMC catalogue.
The coordinates of each star were used in order to retrieve the most recently updated VMC counterparts from the online database. This includes the multi- epoch data needed in order to compare the LFs of this population at different time periods. In the VMC catalogue, the table vmcSource contains data for each source in the Y, J and Ksfilters and the vmcSynoptic tables contain information
Figure 11: Ks vs. J − Ks colour-magnitude diagrams for two different cata- logues. The diagram created from SAGE SMC data used in [22] is on the left and the VMC catalogue used in this study is on the right.
Tile No. of epochs No. of AGB stars
SMC 3 2 13 136
SMC 3 3 13 466
SMC 3 4 12 226
SMC 3 5 14 52
SMC 4 2 14 196
SMC 4 3 13 1644
SMC 4 4 13 941
SMC 4 5 15 78
SMC 5 3 14 111
SMC 5 4 29 179
SMC 5 5 14 64
SMC 6 3 13 52
SMC 6 4 13 64
SMC 6 5 13 28
Table 1: Number of epochs available for each tile used in this study and number of AGB stars identified by [29] which appear in every epoch.
on colour and multi-epoch observations [7]. The SAGE SMC catalogue contains a column denoting each subclass so, when downloading data for O-, C-, X- and a-AGB stars, the co-ordinates for each were found from the SAGE SMC catalogue and used to query VISTA Science Archive separately. This gave five separate data sets to use, including the full catalogue.
The full SAGE SMC catalogue contains not only AGB stars but also red supergiants and far infrared objects. These were removed in order to focus solely on the asymptotic giant branch. For each data set the sources were separated into their respective SMC tiles, which are shown in figure 6. This is important as the observation periods as well as amount of observations vary from tile to tile. Table 1 contains the number of epochs for each tile, varying from 12 to 29, as well as the number of sources, which have a wide range and can be compared with table 9. The data for each tile were then cleaned to remove any epochs with high seeing or a low number of stars and the LFs calculated for each tile and each epoch, readying the data for analysis.
6 Lightcurve analysis for variable stars
When observing a large population of AGB sources, not every star will be at the same phase in their light curve. When VISTA observes the SMC popula- tion at irregular time intervals there is no way of knowing without an in-depth light curve analysis whether the observing periods have managed to capture the maximum or minimum brightness. Perhaps the data only shows some of the stars at their brightest or each observation is taken when the star is below its true mean magnitude. In this way, it is unclear if the mean LFs calculated later on in this report, as well as those created with the maximum and minimum Ks magnitudes, are truly representative of the stellar outputs.
In order to gain an idea of how close the LFs are to reality, a very basic
Figure 12: Left : An example of results from the OGLE-III database of an AGB star from this study. I and V are the average magnitudes in these wavebands, P 1, P 2 and P 3 indicate the primary, secondary and tertiary periods in the I-band and A 1, A 2 and A 3 the amplitudes in the I-band. Right : A simple lightcurve plotted over multi-epoch VMC observations for this source, created using the primary OGLE period and 0.5 of the primary amplitude in the OGLE I-band. The VMC source ID for this star is 558371290749, as shown at the top of the plot. Also noted are the mean and standard deviation of the VMC data and the classification and tile of the star.
lightcurve analysis is performed as part of this study. In the parameters given for each source in the SAGE SMC catalogue, source identifiers for the OGLE-III catalogue [28] (Optical Gravitational Lensing Experiment) were included when available. This meant that the OGLE-III Variable Stars database could be queried in order to give the I-band light curve amplitude and primary period for the relevant source. An example of the result for one of the stars in the catalogue is shown in figure 12. Performing this procedure for every available source identifier is prohibitively time-consuming but investigating a few sources in this manner gave an indication of the study’s accuracy.
The first sources to be investigated were chosen from tile SMC 5 4 as this has the most epochs of any tile: twenty-nine. This meant that twenty-nine data points were available in the magnitude versus epoch plot with which to fit the basic lightcurve. The co-ordinates for this source were noted and matched with the counterpart in the SAGE SMC catalogue in order to find the OGLE ID. This was used to query the database which then shows the light curve data in the I-band: amplitude, period and, in the case of the checked sources, secondary and tertiary amplitudes and periods. As the OGLE survey uses optical data and the VMC uses infrared, the amplitude must be scaled down according to Groenewegen et al. (2020) [15]. In these simple plots the amplitude was scaled down by a half, however in reality the difference may be less than this as [15] deals with large amplitude Mira variables and these AGB stars have lower amplitudes.
With all the information needed for a simple light curve analysis, the VMC epoch data were plotted, along with the errors, the mean and standard devia- tion of these points. The primary amplitudes and periods were used to create sinusoidal functions which were fit as closely as possible to the data. It can be seen from figure 13 that, while the data occasionally follows the curve, the over-
Figure 13: Simple lightcurves plotted using the OGLE primary period and 0.5 of the primary I-band amplitude, fitted over multi-epoch VMC Ksmagnitudes.
The errors used are VMC errors but it is only for the source with VMC source ID 558371291376 that the errors are large enough to be visible. The mean and standard deviation for the points are shown in the plot and the classification and tile of the source is also shown.
whelming impression is that the functions created are not a great approximation of the observational data.
The conclusion that can be drawn from this investigation is that, at this time, it is not possible to say whether the means, minima and maxima in the VMC are representative of the stellar cycles. In order to confirm or deny this in-depth light curve analyses of the sources would be necessary. Unfortunately, this is outwith the remit of this particular study.
7 The luminosity functions
7.1 Variation between epochs
Once the data had been cleaned, the LFs at different epochs could be plotted for the fourteen tiles and for the three subclasses. As this created many plots, the most densely populated tile - SMC 4 3 - will be used for reference throughout this report and the rest will be included in the appendix. Figure 14 shows the multi-epoch LFs for this tile. The bins chosen are of 0.2 magnitudes in keeping with the LFs in [22].
These distributions were then compared to check how similar each epoch is
10.0 10.5 11.0 11.5 12.0 12.5 13.0
13.5 Ks Magnitude
0 50 100 150 200
No. of AGB sources out of 1644
Day, month, year:(18, 9, 2011)
10.0 10.5 11.0 11.5 12.0 12.5 13.0
13.5 Ks Magnitude
0 50 100 150 200
No. of AGB sources out of 1644
Day, month, year:(19, 9, 2011)
10.0 10.5 11.0 11.5 12.0 12.5 13.0
13.5 Ks Magnitude
0 50 100 150 200
No. of AGB sources out of 1644
Day, month, year:(19, 9, 2011)
10.0 10.5 11.0 11.5 12.0 12.5 13.0
13.5 Ks Magnitude
0 50 100 150 200
No. of AGB sources out of 1644
Day, month, year:(11, 10, 2011)
10.0 10.5 11.0 11.5 12.0 12.5 13.0
13.5 Ks Magnitude
0 50 100 150 200
No. of AGB sources out of 1644
Day, month, year:(16, 11, 2011)
10.0 10.5 11.0 11.5 12.0 12.5 13.0
13.5 Ks Magnitude
0 50 100 150 200
No. of AGB sources out of 1644
Day, month, year:(8, 12, 2011)
10.0 10.5 11.0 11.5 12.0 12.5 13.0
13.5 Ks Magnitude
0 50 100 150 200
No. of AGB sources out of 1644
Day, month, year:(10, 12, 2011)
10.0 10.5 11.0 11.5 12.0 12.5 13.0
13.5 Ks Magnitude
0 50 100 150 200
No. of AGB sources out of 1644
Day, month, year:(24, 8, 2012)
10.0 10.5 11.0 11.5 12.0 12.5 13.0
13.5 Ks Magnitude
0 50 100 150 200
No. of AGB sources out of 1644
Day, month, year:(24, 10, 2012)
10.0 10.5 11.0 11.5 12.0 12.5 13.0
13.5 Ks Magnitude
0 50 100 150 200
No. of AGB sources out of 1644
Day, month, year:(17, 11, 2012)
10.0 10.5 11.0 11.5 12.0 12.5 13.0
13.5 Ks Magnitude
0 50 100 150 200
No. of AGB sources out of 1644
Day, month, year:(4, 12, 2012)
10.0 10.5 11.0 11.5 12.0 12.5 13.0
13.5 Ks Magnitude
0 25 50 75 100 125 150 175 200
No. of AGB sources out of 1644
Day, month, year:(30, 6, 2013)
10.0 10.5 11.0 11.5 12.0 12.5 13.0
13.5 Ks Magnitude
0 25 50 75 100 125 150 175 200
No. of AGB sources out of 1644
Day, month, year:(18, 8, 2013)
¨
Figure 14: Multi-epoch VMC luminosity functions for tile SMC 4 3.
Tile χ2 p-value d.o.f.
SMC 3 2 44.2 1.00 96
SMC 3 3 83.2 1.00 120
SMC 3 4 34.7 1.00 99
SMC 4 2 28.8 1.00 91
SMC 4 3 294 1.39×10−16 120
SMC 4 4 96.9 0.77 108
SMC 4 5 24.0 1.00 84
SMC 5 3 25.7 1.00 91
SMC 5 4 104 1.00 224
SMC 5 5 19.2 1.00 78
Table 2: Results of a χ2 test on the contingency tables for all full catalogue epochs in each tile. Values are given to three significant figures, apart from the degrees of freedom - which are integers.
to the others. In [22] the χ2 test was used to compare data with the model, however for this test to be appropriate, at least five data points need to be included in each bin [30]. Due to the smaller sample size in this study and the variation between epochs of the bins at the outer edges of the luminosity functions, this test was not always suitable, i.e. for analysing the different AGB classes. For the full catalogue, the bin range needed to be altered in order that each epoch contained no bins with less than five sources. As a result, different tiles have LFs in different ranges. This does not affect the χ2 test as the epochs are only being compared with other epochs of the same tiles. As in this study many epochs were being compared, as opposed to a two sample test involving the observed and simulated data, a contingency table was created for the test. This meant that the χ2 test of independence of variables could be used for the contingency table. This code computes expected frequencies for the data under the assumption of independence as well as degrees of freedom for the calculation[18]. The distributions are then compared to the expected frequencies to test the null hypothesis that each LF comes from the same distribution. This returns both a χ2 test statistic, a p-value, as well as the degrees of freedom calculated and expected frequencies computed. The results (excluding expected frequencies) for the full catalogue are shown in table 2. The subclasses do not have a large enough sample size for each tile for this test to be appropriate.
The table shows a good agreement for most of the tiles; the p-values are above a significance level of 0.7 for all except SMC 4 3. This is the most central tile and has the largest sample size and the extremely low p-value suggests that the null hypothesis that the epochs come from the same distribution can be rejected.
7.2 Comparison with the mean
7.2.1 Comparisons for the full AGB catalogue
Further to the individual epoch data, the mean magnitudes over all epochs for each source was also found for each tile. If these values are then binned
and plotted, it gives the mean LF. Again, bins of 0.2 magnitudes were chosen and a range of 10 - 13.4 magnitudes to be sure to encompass the full range of values. As the χ2 test was not applicable to every LF in this study (i.e.
the subclasses), it was decided that a two sample Kolmogorov-Smirnov test would be a more relevant statistical test. However, a K-S test on a contingency table is not possible [18]. Therefore, when attempting to quantify the variation between epochs with a K-S test, it was thought that comparing each epoch with the mean was a more effective measure of variation. In this way the variation is quantified but does not take an abundance of time and yield an excessive amount of p-values.
Figure 15 shows each SMC 4 3 epoch’s LF, as seen in figure 14, against the SMC 4 3 mean LF. Figure 16 shows the varying p-value for each epoch when the K-S two sample test is applied to compare with the mean. The p-value plot for SMC 4 3 can be compared with figure 15 to check which epochs reject the null hypothesis. If the full set of p-values for each tile is considered and cross-reference with figure 6, it can be that the more densely populated tiles - SMC 3 3, SMC 3 4, SMC 4 3, SMC 4 4, and SMC 5 4 - have the least consistent p-values. Extreme variation is in particular in SMC 4 3 and SMC 4 4. However, it is important to note that the observations periods for each tile are at different times and irregularly spaced, which may cause the change in p-value to seem more drastic than it is.
10.0 10.5 11.0 11.5 12.0 12.5 13.0
13.5 Ks Magnitude
0 50 100 150 200 250
No. of AGB sources out of 1644
Day, month, year:(18, 9, 2011) epoch mean
10.0 10.5 11.0 11.5 12.0 12.5 13.0
13.5 Ks Magnitude
0 50 100 150 200 250
No. of AGB sources out of 1644
Day, month, year:(19, 9, 2011) epoch mean
10.0 10.5 11.0 11.5 12.0 12.5 13.0
13.5 Ks Magnitude
0 50 100 150 200 250
No. of AGB sources out of 1644
Day, month, year:(19, 9, 2011) epoch mean
10.0 10.5 11.0 11.5 12.0 12.5 13.0
13.5 Ks Magnitude
0 50 100 150 200 250
No. of AGB sources out of 1644
Day, month, year:(11, 10, 2011) epoch mean
10.0 10.5 11.0 11.5 12.0 12.5 13.0
13.5 Ks Magnitude
0 50 100 150 200 250
No. of AGB sources out of 1644
Day, month, year:(16, 11, 2011) epoch mean
10.0 10.5 11.0 11.5 12.0 12.5 13.0
13.5 Ks Magnitude
0 50 100 150 200 250
No. of AGB sources out of 1644
Day, month, year:(8, 12, 2011) epoch mean
10.0 10.5 11.0 11.5 12.0 12.5 13.0
13.5 Ks Magnitude
0 50 100 150 200 250
No. of AGB sources out of 1644
Day, month, year:(10, 12, 2011) epoch mean
10.0 10.5 11.0 11.5 12.0 12.5 13.0
13.5 Ks Magnitude
0 50 100 150 200 250
No. of AGB sources out of 1644
Day, month, year:(24, 8, 2012) epoch mean
10.0 10.5 11.0 11.5 12.0 12.5 13.0
13.5 Ks Magnitude
0 50 100 150 200 250
No. of AGB sources out of 1644
Day, month, year:(24, 10, 2012) epoch mean
10.0 10.5 11.0 11.5 12.0 12.5 13.0
13.5 Ks Magnitude
0 50 100 150 200 250
No. of AGB sources out of 1644
Day, month, year:(17, 11, 2012) epoch mean
10.0 10.5 11.0 11.5 12.0 12.5 13.0
13.5 Ks Magnitude
0 50 100 150 200 250
No. of AGB sources out of 1644
Day, month, year:(4, 12, 2012) epoch mean
10.0 10.5 11.0 11.5 12.0 12.5 13.0
13.5 Ks Magnitude
0 50 100 150 200 250
No. of AGB sources out of 1644
Day, month, year:(30, 6, 2013) epoch mean
10.0 10.5 11.0 11.5 12.0 12.5 13.0
13.5 Ks Magnitude
0 50 100 150 200 250
No. of AGB sources out of 1644
Day, month, year:(18, 8, 2013) epoch mean
Figure 15: Multi-epoch VMC luminosity functions for tile SMC 4 3. The mean luminosity function is shown alongside in orange.
56600 56800 57000 57200 57400 57600 Epoch, MJD
0.0 0.2 0.4 0.6 0.8 1.0
p-value
K-S test p-values for epochs/mean LFs in SMC 3_2
55800 55850 55900 55950 56000 56050 56100 Epoch, MJD
0.0 0.2 0.4 0.6 0.8 1.0
p-value
K-S test p-values for epochs/mean LFs in SMC 3_3
56600 56800 57000 57200 57400 57600
Epoch, MJD 0.0
0.2 0.4 0.6 0.8 1.0
p-value
K-S test p-values for epochs/mean LFs in SMC 3_4
55800 55900 56000 56100 56200
Epoch, MJD 0.0
0.2 0.4 0.6 0.8 1.0
p-value
K-S test p-values for epochs/mean LFs in SMC 3_5
56200 56300 56400 56500 56600 56700 56800 Epoch, MJD
0.0 0.2 0.4 0.6 0.8 1.0
p-value
K-S test p-values for epochs/mean LFs in SMC 4_2
55800 55900 56000 56100 56200 56300 56400 56500 Epoch, MJD
0.0 0.2 0.4 0.6 0.8 1.0
p-value
K-S test p-values for epochs/mean LFs in SMC 4_3
56200 56300 56400 56500 56600 56700 56800 Epoch, MJD
0.0 0.2 0.4 0.6 0.8 1.0
p-value
K-S test p-values for epochs/mean LFs in SMC 4_4
55800 55900 56000 56100 56200 56300 56400 56500 56600 Epoch, MJD
0.0 0.2 0.4 0.6 0.8 1.0
p-value
K-S test p-values for epochs/mean LFs in SMC 4_5
56200 56300 56400 56500 56600 56700 56800 56900 Epoch, MJD
0.0 0.2 0.4 0.6 0.8 1.0
p-value
K-S test p-values for epochs/mean LFs in SMC 5_3
55500 56000 56500 57000 57500 58000
Epoch, MJD 0.0
0.2 0.4 0.6 0.8 1.0
p-value
K-S test p-values for epochs/mean LFs in SMC 5_4
56600 56800 57000 57200 57400 57600
Epoch, MJD 0.0
0.2 0.4 0.6 0.8 1.0
p-value
K-S test p-values for epochs/mean LFs in SMC 5_5
55800 55900 56000 56100 56200 56300 56400 56500 56600 Epoch, MJD
0.0 0.2 0.4 0.6 0.8 1.0
p-value
K-S test p-values for epochs/mean LFs in SMC 6_3
56600 56800 57000 57200 57400 57600
Epoch, MJD 0.0
0.2 0.4 0.6 0.8 1.0
p-value
K-S test p-values for epochs/mean LFs in SMC 6_4
55800 55900 56000 56100 56200 56300 56400 56500 56600 Epoch, MJD
0.0 0.2 0.4 0.6 0.8 1.0
p-value
K-S test p-values for epochs/mean LFs in SMC 6_5
Figure 16: Plots of the K-S test p-value gained comparing each epoch with the mean for each SMC tile used in this paper.
7.2.2 Comparisons for the three AGB classes O-AGB stars
In addition to the full catalogue, the same tests were run for O-, C- and X-AGB stars. As mentioned before, it has been found that 50% of a-AGB stars in the SMC are carbon-rich [3], so LFs were plotted for the anomalous AGB samples and added to the oxygen-rich and carbon-rich samples’ LFs with a 0.5 weighting. Not all tiles were used in this study as each class had very few sources in some of the outer tiles. This is why fig. 17 shows plots for twelve tiles, rather than fourteen. The O-AGB stars are the most numerous, so a larger amount of tiles were used in comparing them, followed by the C-AGB stars and, least numerous, the X-AGB sample. The cut-off number of sources was taken to be 30 stars in total, through visual inspection of the LFs.
Fig. 17 shows the variation for SMC 4 3 and fig. 18 displays the plotted p-values. From fig. 18 it can be seen that the p-values are lower on average than for the entire sample. On the other hand, they do not drop to very low values quite as much and the null hypothesis cannot be rejected at a 0.1 significance level for most of the epochs, apart from some dips in SMC 3 2, SMC 4 2 and SMC 4 3.
10.0 10.5 11.0 11.5 12.0 12.5 13.0
13.5 Ks Magnitude
0 25 50 75 100 125 150 175
No. of AGB sources out of 933.0
Day, month, year:(18, 9, 2011)
10.0 10.5 11.0 11.5 12.0 12.5 13.0
13.5 Ks Magnitude
0 25 50 75 100 125 150 175
No. of AGB sources out of 933.0
Day, month, year:(19, 9, 2011)
10.0 10.5 11.0 11.5 12.0 12.5 13.0
13.5 Ks Magnitude
0 25 50 75 100 125 150 175
No. of AGB sources out of 933.0
Day, month, year:(19, 9, 2011)
10.0 10.5 11.0 11.5 12.0 12.5 13.0
13.5 Ks Magnitude
0 20 40 60 80 100 120 140 160
No. of AGB sources out of 933.0
Day, month, year:(11, 10, 2011)
10.0 10.5 11.0 11.5 12.0 12.5 13.0
13.5 Ks Magnitude
0 25 50 75 100 125 150 175
No. of AGB sources out of 933.0
Day, month, year:(16, 11, 2011)
10.0 10.5 11.0 11.5 12.0 12.5 13.0
13.5 Ks Magnitude
0 25 50 75 100 125 150 175
No. of AGB sources out of 933.0
Day, month, year:(8, 12, 2011)
10.0 10.5 11.0 11.5 12.0 12.5 13.0
13.5 Ks Magnitude
0 20 40 60 80 100 120 140 160
No. of AGB sources out of 933.0
Day, month, year:(10, 12, 2011)
10.0 10.5 11.0 11.5 12.0 12.5 13.0
13.5 Ks Magnitude
0 25 50 75 100 125 150 175
No. of AGB sources out of 933.0
Day, month, year:(24, 8, 2012)
10.0 10.5 11.0 11.5 12.0 12.5 13.0
13.5 Ks Magnitude
0 20 40 60 80 100 120 140 160
No. of AGB sources out of 933.0
Day, month, year:(24, 10, 2012)
10.0 10.5 11.0 11.5 12.0 12.5 13.0
13.5 Ks Magnitude
0 25 50 75 100 125 150 175
No. of AGB sources out of 933.0
Day, month, year:(17, 11, 2012)
10.0 10.5 11.0 11.5 12.0 12.5 13.0
13.5 Ks Magnitude
0 25 50 75 100 125 150 175
No. of AGB sources out of 933.0
Day, month, year:(4, 12, 2012)
10.0 10.5 11.0 11.5 12.0 12.5 13.0
13.5 Ks Magnitude
0 20 40 60 80 100 120 140 160
No. of AGB sources out of 933.0
Day, month, year:(30, 6, 2013)
10.0 10.5 11.0 11.5 12.0 12.5 13.0
13.5 Ks Magnitude
0 20 40 60 80 100 120 140 160
No. of AGB sources out of 933.0
Day, month, year:(18, 8, 2013)
Figure 17: Multi-epoch VMC luminosity functions for O-AGB stars in tile SMC 4 3. The mean luminosity function is shown alongside in orange.
56600 56800 57000 57200 57400 57600 Epoch, MJD
0.0 0.2 0.4 0.6 0.8 1.0
p-value
K-S test p-values for epochs/mean LFs in SMC 3_2 (O-AGB)
55800 55850 55900 55950 56000 56050 56100 Epoch, MJD
0.0 0.2 0.4 0.6 0.8 1.0
p-value
K-S test p-values for epochs/mean LFs in SMC 3_3 (O-AGB)
56600 56800 57000 57200 57400 57600
Epoch, MJD 0.0
0.2 0.4 0.6 0.8 1.0
p-value
K-S test p-values for epochs/mean LFs in SMC 3_4 (O-AGB)
56200 56300 56400 56500 56600 56700 56800 Epoch, MJD
0.0 0.2 0.4 0.6 0.8 1.0
p-value
K-S test p-values for epochs/mean LFs in SMC 4_2 (O-AGB)
55800 55900 56000 56100 56200 56300 56400 56500 Epoch, MJD
0.0 0.2 0.4 0.6 0.8 1.0
p-value
K-S test p-values for epochs/mean LFs in SMC 4_3 (O-AGB)
56200 56300 56400 56500 56600 56700 56800 Epoch, MJD
0.0 0.2 0.4 0.6 0.8 1.0
p-value
K-S test p-values for epochs/mean LFs in SMC 4_4 (O-AGB)
55800 55900 56000 56100 56200 56300 56400 56500 56600 Epoch, MJD
0.0 0.2 0.4 0.6 0.8 1.0
p-value
K-S test p-values for epochs/mean LFs in SMC 4_5 (O-AGB)
56200 56300 56400 56500 56600 56700 56800 56900 Epoch, MJD
0.0 0.2 0.4 0.6 0.8 1.0
p-value
K-S test p-values for epochs/mean LFs in SMC 5_3 (O-AGB)
55500 56000 56500 57000 57500 58000
Epoch, MJD 0.0
0.2 0.4 0.6 0.8 1.0
p-value
K-S test p-values for epochs/mean LFs in SMC 5_4 (O-AGB)
56600 56800 57000 57200 57400 57600
Epoch, MJD 0.0
0.2 0.4 0.6 0.8 1.0
p-value
K-S test p-values for epochs/mean LFs in SMC 5_5 (O-AGB)
56600 56800 57000 57200 57400 57600
Epoch, MJD 0.0
0.2 0.4 0.6 0.8 1.0
p-value
K-S test p-values for epochs/mean LFs in SMC 6_4 (O-AGB)
Figure 18: Plots of the K-S test p-value gained comparing each epoch with the mean for each SMC tile used to study O-AGB stars in this paper. Tiles with number of sources per epoch < 30 are excluded.
C-AGB stars
The procedure was then repeated and multi-epoch LFs plotted for the carbon- rich stars alongside the mean for this group. Half of the a-AGB star bin counts were also added and these plots are in figure 19. It is evident that C-AGB stars are on average brighter in the Ks filter, this is due to the opacity of the amor- phous carbon dust reprocessing the internal radiation to infrared wavelengths.
There are nine tiles which contain more than thirty sources in each epoch and nine plots in fig. 20. The limit on the y-axes was adjusted for a maximum of 0.45 as the p-values in this group are extremely low. If a significance level of 0.01 is assigned, the null hypothesis is not rejected for some of the outer tiles. However tiles SMC 3 3, SMC 4 2 and SMC 4 3 all definitively show a distribution dissimilar to their mean LFs. This follows the pattern appearing in this investigation whereby the more densely populated tiles seem to have more variable LFs.
