We present the results of TRIFFID simultaneous V- and B-band observations of the cores of the globular clusters M15, M92 and NGC 6712. A variability search of their dense centres was made feasible through performing post-exposure image sharpening on the images, increasing the image resolution by a factor of ∼2. The isis implementation of the image subtraction technique developed by Alard & Lupton was then used to detect flux variations in our image sets. We have obtained periods for all observable variables (in our field of view) in NGC 6712 and we have found two new RR Lyrae variables (an RRab and an RRc). We have confirmed three variables in our field of view of the M92. For M15, we detect 48 variables in our field of view, 23 of which are new discoveries. We obtain periods and amplitudes for all variables and classify new ones based on the light-curve shape, the most significant period and the mean magnitude in the V band. Among the detected RR Lyrae we find 19 RRc, 12 RRab and two RRd types. In the subsequent analysis we find a marked increase in RRc over RRab variables in the core. In a refined procedure to search for fainter objects we find no dwarf novae in our field of view of M15. Simulations performed on the data set to quantify our sensitivity to such objects indicate that an upper limit of 10 dwarf novae (at 92 per cent probability) exist in our field of view. The implications this result has on globular clusters are discussed.
Stellar variability studies of globular clusters (GCs) are of fundamental importance to our understanding of both stellar and cluster evolution. Both intrinsic pulsators, such as RR Lyrae and type II Cepheids, and binary systems (cataclysmic variables) can provide clues to varied themes such as the history of the cluster, the state of its stellar population and the distance scale, provided one can accurately assess the variable population of a cluster. Previous variability detection techniques in globular clusters have been based on rms scatter around a set of photometric measurements of individual stars. Although this method is adequate for the less-crowded outer regions of GCs, photometric errors increase in the denser cores. The recent development of image subtraction techniques (Alard & Lupton 1998; Alard 2000) has greatly assisted variability studies right into GC cores and it is now feasible to conduct such in-depth investigations using modest-sized telescopes and instruments (see, for example, Kopacki 2001; Strader, Everitt & Danford 2002). In this paper we describe a photometric study of three globular clusters using TRIFFID/MAMA1 images.
M5 and its variable population
At a distance of 11.5 kpc, M15 (mass ∼106 M⊙, Oosterhoof type II, metallicity of −2.15, Zinn 1985) is one of the most thoroughly researched globular clusters in recent years. This is mostly due to its status as the prototypical core-collapsed cluster — it has one of the highest central densities known (Webbink 1985; Djorgovski & King 1986). It contains a large population of observed variable stars, the majority of which are RR Lyrae types. Until the advent of the Hubble Space Telescope's (HST) superior resolving power over ground-based telescopes most previous variable searches have concentrated on the outer reaches of M15. The earliest discoveries are those listed in the Catalogue of Globular Cluster Variables (Sawyer Hogg 1973; Clement 1997; Clement et al. 2001, hereafter C01). Most of these variables (over 100) are outside the field of view of the M15 data set that we will present here. Accurate periods and amplitudes of several RR Lyrae stars were recorded by Silbermann & Smith (1995), although these are also outside our field of view. All of those catalogue variables seen in our data were originally noted by Rosino (1950) using the 82-in reflector at the McDonald Observatory, and are poorly described.
The Faint Object Camera (FOC) and Wide Field Planetary Camera 2 (WFPC2) on-board HST, for which the high resolution and excellent photometric properties make them perfect tools for detecting stellar flux variations, have been used extensively to probe the dense core of M15. However, the extended observations required to recover complete light curves can be difficult to obtain. Despite this, Ferraro & Paresce (1993, hereafter FP93) used the FOC in conjunction with F140W, F220W and F342W ultraviolet (UV) filters to image, for the first time with high resolution, a 22 × 22 arcsec2 section of the core of M15. They used the rms frame-to-frame scatter to identify 19 variables, 15 of which they noted were probable RR Lyrae stars.
A high-resolution search of the core of M15 using a similar TRIFFID/MAMA arrangement to that described here was instigated by Butler et al. (1998, hereafter B98). They imaged a section of the core similar to that seen by FP93 above, and managed to obtain periods and light curves for all but four of the known variables in that region. 16 suspected new variables were also identified and periods were estimated where possible. Finally, in the course of studying the low-mass X-ray binary (LMXB) AC 211 (discussed below) using archival WFPC2 images in the F336W filter (WFPC2 U-band equivalent), Charles, Clarkson & van Zyl (2002) noted three new variables. Two of these appear to be positionally close to known variables, but the third (their V1) was a transient. The authors suggested it to be a dwarf nova, based on the magnitude swing (>5 mag) and chance of occurrence.
Fig. 1 is a finding chart for all previously known or suspected variables in the core of M15, which should be visible in our images of the cluster. The image was created using archival HST/WFPC2 exposures. Although this field is slightly smaller than our field of view, those areas outside this region and inside our own contain no other known variables. The variables beginning with ‘F’ are those listed in Ferraro & Paresce's (1993) table 2; those beginning with ‘B’ are those listed in Butler et al.'s (1998) table 3; those beginning with ‘V’ are Catalogue (Clement et al. 2001) variables.
M92 is one of the most metal-poor clusters with [Fe/H]=−2.24 (Zinn 1985) and, as such, it is an important empirical constraint in developing stellar evolutionary models for low-metallicity stars. Photographic plate-to-plate scatter enabled the first identification of M92 cluster variables (see, for example, Arp, Baum & Sandage (1953) and references therein). Walker (1955) found three of the brightest red giants in the cluster to be variable, although Welty (1985) saw no evidence of variability in a search for RGB and AGB variables in six globular clusters, including M92. Later, Kheylo (1965) obtained periods and amplitudes for several of the brightest cluster variables. Carney et al. (1992) presented BV photometry and light curves of seven RR Lyrae stars in the hope of using two of them (catalogue variables V1 and V3) for distance determination using Baade–Wesselink (BW) analysis. Cohen & Matthews (1992) followed this up with photometry in V and i for five RR Lyrae stars, updating periods and deriving mean magnitudes again for BW analysis, as did Storm et al. (1992) using K-band light curves. A search for erupting dwarf novae (DNe) in M92 was initiated by Shara, Bergeron & Moffat (1994). They used a series of CCD images with a limiting magnitude of B∼ 21.5. Although they found none, they obtained, for the first time, a quantitative upper limit in a globular cluster of less than seven DNe. Later, Fox et al. (1996) detected an X-ray source in the cluster using the ROSAT High Resolution Imager (HRI), although this source exhibited no sign of variability. Using HST-WFPC2 F336W and F225W images Ferraro et al. (2000) identified three UV-dominant objects in M92 that appeared to be associated with X-ray sources. They argued, based on their X-ray flux, UV colour and absolute visual magnitude that these are a form of cataclysmic variable, possibly of a different type from those normally found in the field.
Kopacki (2001) (hereafter K01), using CCD data and performing difference imaging on the images, completed the most comprehensive survey to date of variability in M92. All but two of the 28 objects listed in C01 were observed. Six new variables were discovered. Four of these are RR Lyrae stars, the other two are SX Phoenicis types. Variable V7 turned out to be of BL Herculis type. One of the more interesting results of this paper was the lack of variability seen in previously suspected candidates (half of them exhibited no variability). Several of these latter stars were observed in photographic plates by Kadla et al. (1983). They had only a small number of exposures and their derived magnitudes were only slightly larger than the sensitivity of the photometry (0.1–0.2 mag). This suggests that Catalogue ‘variables’ in other clusters might be equally suspect.
As one of the galactic globular clusters with the lowest inclination (at , Cudworth 1988), photometric studies of NGC 6712 have always been hampered by contaminating field stars. A comparatively open, metal-rich cluster, with [Fe/H]=−1.01 (Zinn 1985), it is also unusual in that of the clusters with an X-ray source, it is the only one that is not centrally condensed (Peterson & King 1975) give it a value of log(rt/rc) = 1.0). The cluster has also presented a difficult target in detecting variable stars. Although it is one of the loosest of all globular clusters it is also relatively faint compared with some of the larger, better-studied clusters. Early variable discoveries included Smith et al. (1963) (13 variables including eight RR Lyrae and one long-period red giant) and Sandage, Smith & Norton (1966) (16 variables, 10 of which were of RR Lyrae type). C01 attributes a total of 22 known or suspected variables to NGC 6712 including an RV Tauri (with a period of 104.6 d).
OBSERVATIONS AND DATA REDUCTION
TRIFFID Images and PEIS
The observations were taken using the TRIFFID/MAMA system on the 1-m Jacob Kapteyn Telescope (JKT) part of the Isaac Newton Group (ING) of telescopes, at La Palma between 1997 July 1 and 14 (see Table 1). TRIFFID, which was mounted on the Cassegrain focus of the JKT, is an optical system including dichroic beamsplitters which simultaneously focuse B- and V-band images on to separate halves of the MAMA detector. The instrument was developed to allow post-exposure image sharpening (PEIS, involving shift-and-add software to compensate for the tip-tilt effect imposed by the turbulent atmosphere on incoming wavefronts), to be performed on the data. These have been discussed at length in B98. Briefly, a pupil mask was used to reduce the effective telescope aperture to match the prevailing seeing conditions at (3.5–4)R0, where R0 is Fried's seeing parameter (Fried 1966). The M15 exposures were taken between 1:30 and 4:30 ut each night, when the cluster was at its highest altitude and thus optimizing the seeing conditions (the average seeing was 0.8 arcsec). The adverse consequences of such a rigid observing schedule are discussed in Section 3.
PEIS resulted in an increase of a factor of ∼2 in the seeing. Exposures of over 2400 s were split into two (at the expense of photon counts per exposure) to ensure the RR Lyrae variable light curves would be sufficiently sampled. Flat-fields were taken to correct for pixel-to-pixel sensitivity variations across the detector. Deep dome flat-fields were convolved with sky flat-fields to model the shape of the night sky flat-fielding response. After rejecting substandard images (due to focusing, tracking problems, etc.) we were left with 75, 33 and 23 images of M15, M92 and NGC 6712, respectively, although not all images covered the same region of the core. Table 2 (see Section 4) shows the number of exposures taken during each night of the run. The long baseline of 14 d enabled us to obtain a sufficient number of points to accurately describe the horizontal branch variable population, despite the relatively poor throughput of the TRIFFID/MAMA system (in this MAMA throughput is particularly low in most of the half occupied by the B-band image). In this variant of TRIFFID, ∼60 per cent of both the B and V images covers a common field of view. This explains why some stars were observed in B or V only, while the majority of stars were observed in both bands.
Image subtraction and variable detection
The isis code (Alard & Lupton 1998; Alard 2000) was used to perform image subtraction. After image registration, a selection of the best images were combined to produce a high signal-to-noise reference image. The convolution kernels needed to transform the reference frame to each science frame were obtained. The convolved reference frame was then subtracted from each science frame, giving a series of difference images. Any residual in a difference image is indicative of a flux variation between the two images. The flux deviations in such an image can be due either to authentic flux variations (due to variable stars) or to spurious fluctuations due to some systematic source [e.g. flat-field or point spread function- (PSF-) related problems]. To allow a greater chance of detecting variable stars, a median difference image, in which each pixel is the median of the set of corresponding pixels in the difference image set, was created. All objects 6σ above the local background were selected as candidate variables. The reference and corresponding median difference images for all three clusters are shown in Fig. 2. The annular (or square) patterns seen in the median difference images at the positions of certain bright, isolated stars are the artefacts of the use of these stars as reference stars during the PEIS step. No candidate variables were selected at these positions.
Photometry on the candidate variables was performed once accurate positions were obtained. Although simple centring algorithms are usually sufficient to do this, an even better solution, used here, is to use HST-WFPC2 positions, providing the stars are visible on these frames. The reference image PSF was created by fitting a profile to several bright stars. A PSF model for each particular image was generated by convolving the reference PSF with the kernel solution. This PSF model was then fitted to the position of a candidate variable on the difference image, giving a flux measurement out to a specific radius (which is a measurement of the flux difference of the variable relative to the flux of the variable in the reference image). To find the absolute flux of the variable (out to the same radius) profile-fitting photometry was performed on the reference image, using a deep star list obtained from archival WFPC2/HST images. Photometry was then performed on archival WFPC2/HST images of the core of the three clusters and the offsets required to transform the instrumental magnitudes of the candidate variables to the WFPC2 Standard Flight Magnitude system (Holtzman et al. 1995) were obtained. Although some clusters have been extensively observed in almost all filter bands of the WFPC2, this has not been case for M92, which although heavily researched throughout the years, has not been observed using F439W (the Johnson B-band equivalent) filter. Thus it has not been possible to convert the M92 B-band data to the WFPC2 system. It was decided instead to leave the light curves for all B-band variables seen in M92 in their original (isis) flux formats.
Period determination and aliasing
The phase dispersion minimization (PDM) technique (Stellingwerf 1978), included in the standard iraf (Tody 1986) distribution as the task pdm, was used to calculate variable periods. A period aliasing problem arises because the variable light curves were sampled at a specific and systematic sampling rate (all images were observed during a narrow temporal window between 1:30 and 4:30 ut each night).
To illustrate the period aliasing effect, Fig. 3 shows the theta plots from the pdm task for the M15 variable V135 (see Table 4 in Section 6). This has a roughly sinusoidal light curve of unknown period. The plots are split into three sections for easier visualization ([0, 2], [0, 1] and [0, 0.5] d). There exists a pattern in the forest of possible periods, represented by theta minima. If one ignores the harmonics, a series of possible periods P1, P2, P3, …, Pn present themselves. The longest period of interest here is at ∼1. 29 d, the next is at ∼0.56 d and so on. The series can be represented by
Which period is the true one?Fig. 4 shows light curves for V135 folded on six different periods, starting at ∼0.17 d and increasing. From the scatter in the light curve we can probably rule out the shortest period, and possibly the next period at ∼0.21 d using the same argument. It is more difficult to eliminate all but one of the remaining four periods. In a situation like this we have made an informed decision of the appropriate choice of period based upon the scatter about the light curve, the morphology of the curve, the position of the variable on the colour–magnitude diagram (CMD) and its mean magnitude.
There is another discriminatory factor at work in the TRIFFID data. In the case described above the true amplitude of the variable was accurately determined using pdm. However, problems arise when dealing with variables with periods (in days) that are divisors of one. The sampling of our data is such that the curve of the variable is nearly always sampled at or near the same magnitude and the derived light curve gives little or no information on the true shape of the light curve and, by extension, the variable type. Even with knowledge of the true period, the resulting light curve yields little information. Any variable with a period (in days) close to a divisor of 1 will appear only as a faint signal (or not at all) in the median difference image because the observed amplitude will be small, regardless of the real amplitude. A longer temporal observing window would provide a sampling that is staggered enough relative to the period of the variable to allow a passable reconstruction of the true light curve.
Results for NGC 6712
Fig. 5 shows the total coverage afforded by the B and V bands of the TRIFFID/MAMA images of NGC 6712 with orientation axes superimposed. The origin is the (0, 0) point (in units of arcsec) defined by the X−Y positions of known variables in NGC 6712 given in the Catalogue (C01). Three previously known variables appear in our field of view. V6 has coordinates (18, −41)2 according to the Catalogue. V20 is at (1, 9). Also shown is the location of the optical counterpart of the X-ray source (too faint to be seen in our images). According to the Catalogue V6 is probably an RR Lyrae with a period of 0.510 84 d. V20 is of the same type with a period of 0.330 87 d. Four known variables (V12, V17, V18 and V19) lie just off our field of view. The latest addition to the Catalogue (V22, or KC460 from Cudworth 1988) is also nearby but just off the field of view. Finally, star V21 in the Catalogue has no associated position relative to the cluster centre.
It is clear from simple visual inspection of Fig. 2 that NGC 6712 is not awash with detectable variable stars, although a striking feature is the strong variable signature near the centre of the median difference image. This is surprising as it appears not to have been seen before. The PEIS reference star signatures stand out in this image because the open nature of the cluster, coupled with a relatively large contamination of field stars, meant bright, isolated stars were easier to come by than in the more crowded cores of the other clusters.
Table 2 gives details on all of those variables seen towards NGC 6712 in the TRIFFID/MAMA images. We only expected to see two (possibly three, depending on the position of Catalogue variable V21) previously known variables in NGC 6712. The optical counterpart to the X-ray source is too faint to appear on these images. V6 data is in the V band only. V20 data is in the B band only. Fig. 6 shows the light curves of catalogue variables V6 and V20. The light curve of the former was transformed to F555W magnitudes using the procedure discussed in Section 2. Because of the poor B-band flat-field, the latter was kept in the original difference flux units. The most significant period of 0.521 40 d for V6 differs somewhat from that in the Catalogue (P= 0.51085 d). There is also a slightly less significant period at ∼0.9 d. Not identified in the literature as any particular type, it is possibly an RR Lyrae of type Rab. V20, visible in the B band, and is considerably noisier than V6. Its most significant period, according to pdm, is ∼0.51 d. The period from the Catalogue is 0.330 87 d. We find a period of 0.338 83 d to be only slightly less significant than the value in Table 2 and it is no coincidence that these numbers are consistent with equation (1) above.
Two other candidates for variability were noted in NGC 6712. The strongest of these, provisionally called P2, appears clearer than any other in the median difference image. The light curves of this and the other variable (P1) are shown in Fig. 7. The y-axis has been kept in flux units to avoid misrepresenting the shape of the light curve. P1 and P2 show less noisy curves than V6 and V20. P1 has the characteristic shape and period of an RRc light curve. P2 is probably an RRab variable. For both variables, the B- and V-band periods agree well with each other. P1 and P2 have coordinates (J2000) and , respectively.
Finally, it should be noted that the Catalogue variable V21 (Lloyd Evans & Menzies 1973), the position of which in the cluster is not listed by the Catalogue, could be either of these two newly discovered variables. Those authors attributed a value of V−IK= 2.40 to this variable. A colour for P2 of F439W − F555W = 0.895 was obtained from archival HST/WFPC2 images (P1 was not observed in both wavelength bands).
Variables in M92
We had expected to see three of the previously detected variables (V29, V30 and V31) in the field of view afforded by our TRIFFID/MAMA images of M92 (see Table 3). Two of them are common to both B- and V-band images, whereas the third (V31) is only visible in the B band. Fig. 8 displays the light curves of V29 and V30 in both B and V bands. They are folded on the most significant period found by the pdm task in iraf. The B-band light curves are in flux units. Fig. 9 shows the light curve of V31 (in B only).
V31 has a very poor light curve, which is slightly surprising considering the quality of the other two. This can probably be explained by the fact that the variable lies close to the edge of the B-band images. Stars on the edge of images suffer the inevitable consequences of scattered light and incomplete PSFs. Poor throughput (requiring larger flat-fielding corrections) in much of the B-band half of the MAMA is also no doubt a contributing factor. K01 attributes an amplitude in V of 0.52 mag to the variable, so there is no question that it would reveal a better light curve were it more favourably placed on the detector. The data is folded on a period of 0.398 000 d, similar to K01's value of 0.398 1648 d. In contrast, V29 and V30 have superior light curves in both the B and V bands, though V29's curve suffers slightly from incompleteness. All curves are folded on the most significant periods which agree well with K01's estimates.
K01 attributes magnitude swings (ΔV) of 0.95 and 1.59 mag to variables V29 and V30, respectively. They did not attempt to measure the magnitude of the appropriate variable in the reference image. Rather, they used the fact that all RR Lyrae stars in a given cluster have similar mean brightness, which they estimated from photometry on RR Lyrae variables far from the core of M92. They then obtained the linear transformation necessary to convert these daophot fluxes to the differential flux of isis. The small field of view in our images ruled out this technique. Our value of ΔV is 1.00 mag for V29, which is in reasonable agreement with K01.
On the other hand, our value of 0.60 mag for V30 is considerably different from K01's estimate. This is easily explained. The mean magnitude of V29 is 15.16 mag, which is in excellent agreement with the measurement of RR Lyrae variables in the outer regions of the cluster by K01. However, the mean magnitude for V30 is 13.73 mag (!). This is obviously not the HB magnitude. The reason for this discrepancy can be seen in Fig. 10. This shows a close-up image of the two variables (V29 and V30) in the isis reference image, median difference image and an archival WFPC2 image of the same field. The reference image should reflect the mean magnitude of a particular variable star because it is basically an average of all the different epochs of the cycle of the variable, provided the light curve is reasonably sampled. However, the isis reference image clearly shows V30 to be brighter than V29. This is also the case in the WFPC2 image. The WFPC2 image actually seems to suggest that V30 is a pair (or more) of unresolved stars. Thus, the reason for the bright mean magnitude and erroneous amplitude (when compared with K01) is an incorrect zero-point flux estimate when converting the difference image fluxes to magnitudes. The calculated zero-point flux is a measurement of the combined flux of at least two, if not more, unresolved stars. One could say that this is a drawback of the difference imaging technique. However, without an accurate (high-resolution) star list, this problem could not have been avoided using traditional profile-fitting techniques either.
Results for M15
All objects rising 6σ above the local background on the V-band median difference image were noted as possible variable candidates, with the exception of glaring and obvious artefacts. For example, the reference stars used in the PEIS procedure are oversharpened and so have a slightly different shape of PSF to the other stars. Reference stars can be identified by the particular signature they leave on the summed residual image. In most cases these do not cause problems as they do not rise above the background to the extent that bona fide variables do.
In total 53 variable candidates were detected in the observed region of M15. Five of these were found to be false detections due to the nearby presence of a sharpening reference star. Light curves for the remaining 48 variables were created in the manner described above. Careful examination of their light-curve morphology and most significant period allowed us to ascribe a known variable type to all but 11 variables. These include two type II Cepheids/AHB1 stars (V142 and V161), one long-period W Vir type Cepheid (V86), one LMXB optical counterpart (AC 211) and 33 RR Lyrae variables. Table 4 lists these variables.
The first and second columns give the variable identification in accordance with the Catalogue of Variable Stars in Globular Clusters (C01) and previous variable identifiers from the discovery papers, respectively. Currently, the Catalogue attributes 158 variables to M15. The new discoveries described in this paper start at V160. Variable V159 is FP93's V17, V18 and V19. We see these three stars to be one variable and although those authors saw variability in the three stars which make up this object in our images, we cannot confirm this due to our poorer resolution. Until future observations can resolve this variable triumvirate, we shall deal with it as a single star.
The F555W mean magnitude, as measured on the TRIFFID/MAMA frames, is given in the third column of Table 4. A colour measurement (in F439W − F555W, the HST/WFPC2 equivalent of Johnson B−V), obtained from WFPC2 archival images is listed in the fourth column. The V-band period and amplitude are in the fifth and sixth columns, respectively. The following two columns give the right ascension and declination of variables (2000), obtained using the positional information contained in the WFPC2 FITS3 file headers. When possible, a suggestion of variable type is listed in the final column. Fig. 11 shows the location of all 48 variables superimposed on an image of the cluster created using a number of HST/WFPC2 exposures taken at several different epochs.
The characteristic mean magnitude and period of RR Lyrae stars made it relatively easy to identify them from other types of variable. However, it was more difficult to separate the RR Lyrae stars into the different subtypes. RRab and RRc variables differ in light-curve shape, amplitude and period, although the Blazhko effect can sometimes result in similarities between RRc and RRab types in noisy data. It is also difficult to distinguish RRc from RRd variables, which tend to have light curves containing traits of both RRc and RRab types. Although the data for the suspected RRd stars are quite noisy, we do see a hint of a double structure in the PDM theta plots (not shown) at where one might expect the fundamental frequency to be, based on the characteristics of the rest of the M15 RRab variables (the second structure accounts for the aliasing already discussed). Taking these considerations into account, 12 stars are identified as RRab (RR0) variables based on their light-curve shape and most significant periods. 19 stars are classed as RRc (RR1) variables and two are classed as RRd (RR01) variables. The RRd variable light curves (shown in Fig. 12) show more scatter than that allowed for by the error bars. Note that the light curve of V167 is similar in shape to the curve of an RRab star, and could conceivably be a Blazhko variable. Figs A1 and A2 in the Appendix show the light curves of the RRab and RRc variables, respectively. Fig. A3 in the Appendix shows the light curves of those detected variables for which identification must await further observations.
The W Vir-type Cepheid, V86 (see Fusi Pecci, Voli & Rosino 1980 for more details), is the brightest variable in this region of the cluster. The recorded period for this variable (C01) is 16.83 d. The most significant period obtained here is ∼14 d. The discrepancy is due to the fact that the light curve was not fully sampled during our 14 nights of observations. The light curve shown in Fig. 13 bears little resemblance to that of Fusi Pecci et al. Their wide maxima and sharp minima are difficult to discern. The incompleteness of the curve probably has some bearing on this, although some saturation effects due to the brightness of the star cannot be ruled out. The aforementioned authors suggested the star to be related to the LMXB AC 211 on the basis of rapid outbursts near the maximum (which have since been noted in other W Vir-type Cepheids, Alcock et al. 1998). It is difficult to attribute such fluctuations to this variable on the basis of the current data.
V142 was noted by B98 to be a short-period type II Cepheid. Their derived period (1.28 d) is consistent with our period of 1.24 d. The amplitude of 1.51 mag derived here differs from their value of 1.38 mag. Small bandpass differences may be responsible for this, but a more likely explanation is the conversion of isis differential fluxes to apparent magnitudes. V161, a newly detected variable, is identified as a short-period Cepheid based on similarities with V142 in period (1.27 d), amplitude and light-curve morphology. Fig. 14 shows the light curves of these two variables.
The LMXB optical counterpart, AC 211
The low-mass X-ray binary AC 211 lies very close (∼2 arcsec) to the core of the cluster, and attempts to investigate it have been hampered by the close crowding conditions therein. The star was found by Auriere, Le Fevre & Terzan (1984) to be within the Einstein HRI4 error circle of the X-ray source X2127+119 (Hertz & Grindlay 1983) and due to its variability it was identified as its optical counterpart. An understanding of AC 211, which has the highest optical to X-ray luminosity ratio of all LMXBs, is considered to be extremely important in furthering our knowledge of the dynamical evolution of M15 in particular and globular clusters as a whole. Its period in the optical range has been the subject of much debate. Original period estimates using X-ray (Hertz 1987), optical (Ilovaisky et al. 1987) and spectroscopic data (Naylor et al. 1988) hovered around 8.5 h, indicating that the secondary was a ∼0.9-M⊙ star. However, a revision of archival data using different period estimate techniques as well as new data (Ilovaisky et al. 1993) have favoured a period of 17.11 h and to date, this is the generally accepted value.
Our light curve of AC 211 is among the poorest of all the variables seen in M15. It is a relatively faint object in V and lies only ∼2 arcsec away from the dense centre of the cluster. An accurate magnitude estimate is difficult to obtain for stars this close to the core and an error in a magnitude estimate of the star would manifest itself in the resulting period estimate. For this reason the relative differential flux values were used to obtain the most likely period and construct the resulting light curve. Fig. 15 displays the theta-period plot produced by pdm. The period search window is between 0 and 1 d. From the diagram the most significant period lies at around 17.1 h, consistent with the more recent literature discussed above. The 8.5-h period is locally significant, but this can also be said for a few other test periods. Fig. 15 shows the light curves obtained when the AC 211 data is folded on the above two periods. They suggest that the 17.1 h period is more significant than the 8.5-h period for AC 211. Even so, the 17.1-h light curve shown in Fig. 15 is far from smooth. In this case, the light curve of AC 211 is probably more affected by the crowding of variables in this part of the cluster than by the crowding of non-variable stars. Star-subtraction photometry on the difference images would probably provide a smoother light curve. Nevertheless, there is clear evidence of the double-dip shape reported by Ilovaisky et al.
Undetected variables, period variations and false detections
Although this comprehensive survey of the central region of M15 has revealed a wealth of data on its variable star population, there were nevertheless several previously suspected variables that went undetected. Table 5 lists those variables previously mentioned in the literature and which were not seen in the 1997 JKT TRIFFID/MAMA M15 data. These are either real variables which on this occasion went undetected due to observational circumstances or they were originally false detections.
Several of those listed are variables suspected by B98. The detection method they used, identifying scatter in an the set of PSF-fitting magnitudes of an individual star, is a viable one. However, the extreme crowding of the core, coupled with the low throughput of the TRIFFID/MAMA detector, makes this choice of technique inferior to the image subtraction method, for which case crowding of non-variable stars is generally not a problem in obtaining confirmation of variability. The smaller field of view of the earlier version of the TRIFFID camera, and the constant repositioning of that field of view around the cluster core, had three further adverse affects on the B98 study; namely stronger edge effects, fewer observations per star (on average) and a necessary reliance on accurate geometric mappings to model the locally changing field distortion in each image and to apply it to the master HST star list prior to each PSF-fitting run. Any centring errors or omissions in the master HST star list would also have induced photometric noise in the B98 TRIFFID photometry. Here again we see the advantage of image subtraction over conventional photometry for variability detection. We conclude that the non-detection of some B98 variable star candidates implies that most or all of these cases were false detection. For some of these same reasons, some of our periods differ from B98's estimates. Where this happened, we applied the PDM technique to their data and, in many cases, found significant periods and light curves (although less significant than those listed in B98) similar to our values. The exceptions, where we find no evidence of a significant period near ours, in their data, are V141, V130 and V132.
Most of the variables of FP93 were seen in this survey. Exceptions are their V17, V18 and V19, which are located right in the core of M15 (one of the ‘triangle’ stars), and are barely resolved even in their FOC/HST images. Thus there is no hope of resolving them in our ground-based data. It was difficult to convert the relative flux difference to an accurate magnitude estimate for this variable because of its multiple-star nature. The light curve for this variable is thus shown in Fig. 16 with the flux difference on the y-axis, rather than a WFPC2 filter magnitude. Very little can be said concerning the nature of this triumvirate other than that the dominant period on which the curve is folded here (0.6385 d) is typical of the M15 RRab stars and the overall morphology of the noisy light curve is also similar to that of an RRab star. They are included in Table 4 as V159.
Finally, the possible dwarf nova seen by Charles et al. (2002) cannot be confirmed with this data. If it is a DN the chances of seeing it in outburst again are quite small. We cannot hope to see it if it is in a quiescent state. However, the region in which it lies is generally well studied (it is very close to AC 211 and the core of the cluster) and if it is a DN it will probably make another appearance in the literature in the future.
RR Lyrae Stars in M15
Amplitude detection limits
The sensitivity and completeness of the survey of variability in M15 was assessed by adding artificial stars to the images and examining the results of the image matching and subsequent photometry. To ensure identical image PSFs and added-star PSFs the M15 image set was recreated using a star and a magnitude list and the extracted PSFs of the images. All stars (detectable in the TRIFFID/MAMA images) in M15 were added to blank template images, according to the stellar fluxes on the real images of M15. An appropriate (and different) PSF was used for each image. A sky background, also representative of the 1997 observational data, was added to each of the 50 images. The core of M15 has a varying (differential) background across the field of view as a result of the sea of unresolved stars present on the images. The shape of this background was estimated by subtracting all the detectable stars from the TRIFFID/MAMA images. The resultant star-subtracted images thus only contained the sky background, including the unresolved stars. These images were co-added and a polynomial surface fitted to the result. This surface was scaled according to flux levels on the real images and added to the artificial images. Poisson noise was added to all the artificial images.
RR Lyrae variable light curves were simulated using selected Fourier coefficients from Petersen (1994) and scaled to the mean F555W magnitude of the horizontal branch of the M15 colour–magnitude diagram. Each variable was assigned an amplitude of x magnitudes, where 0.05 is a divisor of x and x is in the range [0.05, 0.50] (for RRc stars) and [0.1, 0.5] (for RRab stars), and was added at a random position (on the condition that the variables would not interfere with each other) to the simulated set of images. The appropriate variable magnitude for addition to each image was obtained by sampling the theoretical light curves of these variables at the Julian date of each original image. After image matching was performed on this set of images, the median difference image was examined and the significance of the variable signature was noted. This procedure was performed 100 times for every value of x for both RRab and RRc variable types.
Fig. 17 shows the sensitivity to which the image matching approach applied to our simulated images is to the RRab and RRc amplitude swings. The chances of detecting an RR Lyrae variable in M15 of amplitude swing 0.15 mag is ∼80 per cent. This figure drops rapidly to only ∼20 per cent at 0.10 mag. The reason for the increased sensitivity for RRc over RRab for a given amplitude limit is because an RRc variable spends more time away from the mean magnitude. The fast, sharp peak of an RRab contrasts with the slower, smoother RRc curve.
There are some points to note concerning these plots. They could benefit from a higher resolution (in amplitude on the x-axis). Note, however, that a single point consists of statistics performed on 800 measurements of a single variable star (eight stars of a given amplitude on 100 image sets, each containing 50 images). Thus obtaining even a single point was a computationally expensive procedure, even for the 40-processor SGI-Origin 3800 used for the calculations. Positional dependence of the variables on the detector (i.e. flat-fielding considerations) have not been explored. If included these would add some scatter to the simulated light curves, and probably affect the drop-off. Sharpening reference stars have not been taken into account. These would have been difficult to model in our simulated images. They affect a variability search by adding spurious signatures, as opposed to suppressing the signature of a true variable as flat-fielding errors do. However, these are localized and easily identified because the positions of the reference stars are known. Finally, it should be noted that none of the added variables interfere with each other. Should they lie within each other's flux-summing radius, the photometry would have additional errors.
The colour–magnitude diagram of M15
Fig. 18 (upper panel) shows the horizontal branch (HB) of a colour–magnitude diagram constructed using archival HST/WFPC2 images taken through F555W and F439W filters. The mean times of exposure were 00:57:17 and 00:48:17 (1994 April 7) for F439W and F555W images, respectively. The slight (and unavoidable) time delay between the two images results in a colour lag in the B−V colours. For instance, a variable captured at a rising light epoch in its cycle through the F555W filter will have brightened slightly by the time it is observed through the F439W filter (in this case 9 min later). Thus the calculated B−V colour will be slightly lower than the true value. Given that we are dealing with periods of between 7 and 18 h, this 9 min represents between 2 and 0.8 per cent of the period and a colour correction would therefore have a negligible impact on the CMD.
The CMD was populated with only 41 of the 48 detected variables because of insufficient WFPC2 coverage of the cluster. The F439W − F555W colour has been used. However, instead of the F555W magnitude, the mean magnitude of the variable, as measured by performing photometry on the reference M15 images and transforming to the F555W filter of the WFPC2 Standard Flight Magnitude system, has been used. The scatter in mean magnitudes, which was already apparent from looking at the light curves, can be seen. Although the average of the variable mean magnitudes is ∼16 mag, variables with a mean magnitude as bright as 15.5 mag can be seen. The faintest of these variables seems to be around 16.2 mag. There is no doubt that some of this scatter is due to the quality of the data and of the photometry, even though every attempt has been made to reduce all sources of error. The difficulty in obtaining accurate magnitudes in the cluster core, where crowding is a severe problem even for profile-fitting techniques, has already been discussed. The problematic variable(s) V159 (previously discussed) can be seen to be much brighter than the HB. This is because of the unresolved nature of the (at least) three stars that comprise V159.
The lower panel of Fig. 18 shows the location of the different RR Lyrae variable types on the same portion of the CMD. RR Lyrae variables on the CMD are expected to arrange themselves in colour according to type (Sandage 1990). RRab stars are redder than RRc variables. This observational fact provides us with a further constraint on the variable type. The triangles indicate those variables for which the period and light-curve shape unambiguously identify them as RRab-type stars. The boxes similarly indicate those variables that are very likely to be RRc types. Diamonds indicate RRd-type variables. The crosses indicate those variables for which the identification is uncertain.
Fig. 18 shows that although most of the RRc and RRab variables keep to their respective sides of the HB, there is some overlap. The two RRd variables lie near the division. The variable V174, identified as an RRc is unusual. It is redder than most of the RRc or RRab variables. It does not, however, exhibit an RRab-type curve. Its mean magnitude is ∼15.2, which is very bright for an RR Lyrae. This could reflect an incorrect flux estimate for the star in the reference image, although this would not affect its colour here (obtained from a different image set). It is close in colour and brightness to V142, which is a short-period Cepheid. However, there is no evidence of a period (which is not an alias) longer than that attributed to it. On the other side of the instability strip, V162 is classed as an RRab type but is snug amongst the RRc types. Its light-curve shape is definitely RRab-like and nothing in the WFPC2 photometry suggests any large errors.
Coverage and completeness
Fig. 19 shows the inner core of M15. Approximately 14 000 stars were identified from the HST/WFPC2 images and are plotted as filled circles, the sizes of which reflect their magnitudes. The rectangular outline is that region of M15 which was searched for variability. RR Lyrae variables are indicated according to known or suspected type. RRab stars are enclosed with triangles, RRc with boxes and RRd with diamonds. The r∼ 10, 20 and 30 arcsec limits are shown. The studied region in this work is strongly skewed eastward of the centre (0, 0). There is no reason to believe more variables exist in this region than westward of the centre. Indeed, simple radial symmetry in the distribution of RR Lyrae stars indicates that up to 10 RR Lyrae stars could exist in this unstudied region. Future observations will no doubt confirm this.
Is the observed population of RR Lyrae stars within the region studied the complete sample of RR Lyrae stars residing there? There may exist, because of an aliasing effect, certain variables for which the particular periods work against our detecting them. Light curves of variables close to these periods also reflect erroneous (lower) amplitudes.5 Other than these limitations, the simulations in Fig. 17 show that we are complete for variables with an amplitude limit down to ∼0.15 mag for RRc and RRab variables (at the 80 per cent confidence limit). Since RRab variables are not expected to have such low-amplitude limits, we can say with reasonable confidence that the recovered population of RRab variables represent the total population of RRab variables in the region of the core of M15 which we have studied. That we have easily recovered all known variables in the field and obtained accurate periods for them is testament to this. However, we see a few RRc variables that do exhibit small amplitude swings (∼0.2 mag). Therefore, although it is unlikely that RRc variables exist in the region studied which have gone undetected, the possibility remains.
We have not conducted sensitivity simulations for RRd variables (because of the difficulty in modelling the light curves) and this population is very likely to be underestimated in this search. Poor observations of an RRd variable, when folded on a significant period, can mimic a noisy RRc light curve. In our case most of the variables classified as RRc types do show relatively smooth curves. Some, however (for example, V137 or V139), show some scatter around the curve which may or may not be the signature of an RRd. As we discuss below, our periods for RRc stars included several periods that are longer than most previously recorded RRc variables in M15, suggesting that they might be double-mode pulsators.
Analysis of RR Lyrae types
Table 6 details the breakdown in type and mean period of the RR Lyrae population of M15. Data from three separate sources is included. The first, taken from C01, includes those variables that lie outside our coverage. The average periods were obtained after excluding those values of B98 (including FP93) that were included by C01. The second source is that of Silbermann & Smith (1995), who observed an arbitrary sample of RR Lyrae variables in the outer reaches of M15. Our averaged periods for RRab, RRc and RRd constitute the third source. In this table, one should probably assume that Silbermann & Smith, whose data were not hampered by severe crowding conditions, obtained the most accurate values.
Considering that we are dealing with two distinct and separate populations of stars (ours being more centrally concentrated), there is a noted similarity in average periods of the different types of RR Lyrae stars. Smith (1995) attributes values of and , where the latter includes both RRc and RRd variables. These values are in good agreement with ours (Table 6). Silbermann & Smith (1995) record an average period of 0.359 d for RRc variables and 0.641 d for RRab stars. Interestingly, no RRc variable recorded by Silbermann & Smith has a period of over 0.406 d (which, incidentally, is their average period of RRd variables). We, however, have several, indicating that we may have misclassified some RRd stars as RRc variables.
The fact that so many variables have been found in the central core relative to the outer parts is not so surprising. A large proportion of the stellar (luminous) mass of M15 resides within 30 arcsec of the core, and the large RR Lyrae population within this radius reflects this.
What is surprising is the large number of RRc variables relative to RRab inside the r∼ 30 arcsec boundary. For the outer parts of the cluster (which until now applied to the whole cluster) Nc/Nab+c≈ 0.41.6 Inside the r∼ 30 arcsec radius this value is 0.61. This rises to 0.64 if one includes the RRd types with the RRc types. Since we believe that the RRc variables are more likely to be underrepresented in the region of our search (see below), this figure is a lower estimate of what it could be. The number of RRab variables must be increased by ∼16 if we are to recover the value of Nc/Nab+c≈ 0.41 that is representative of the RR Lyrae population situated further from the core.
One explanation is that 16 RRab variables have been falsely identified as RRc. This is obviously not the case from looking at our variable light curves. A more realistic reason could be due to the natural detection bias against low-amplitude RRc variables in the outer parts of the cluster (especially given that the most recent variable discoveries in M15 took place c. 1995 and that large regions of the cluster have only been observed on photographic plates), which would cause them to be underrepresented in the Catalogue. It could also be that C01 contains variables that have been misclassified. We have already seen a few examples of this in our data.
We have also seen that the studied region is skewed slightly eastwards of the core. Thus a potentially large section of the variable population within 30 arcsec is undiscovered. However, there is no reason to believe that this region should have enough RRab types to compensate for the lack of them in our studied region. Alternatively, we might have missed a significant population of RRab variables in our search, although we have shown our survey to be more complete in RRab than RRc variables. However, this fact does not rectify the discrepancy between these populations inside and outside r∼ 30 arcsec.
Indeed, it seems that Nc/Nab+c could more likely lean towards even higher values rather than smaller ones. We have identified two variables as RRd stars based on the scatter around the light curve. The relatively large number of RRd variables in the outer parts of M15 (17 out of 85) is not reflected in our inner-cluster data (two out of 33). We have already mentioned above that although our average periods for RRc and RRab are in relatively good agreement with Silbermann & Smith, their RRc variables all have periods of below 0.406 d. We see several RRc variables with longer periods than this, suggesting that perhaps our RRd population should be larger. If four RRd stars were misclassified as RRc variables, the ratio would be similar to that for the outer parts of the cluster. However, this still would not change the predominance of RRc stars over RRab stars. Admittedly, we are dealing with small numbers here and a slight change can significantly alter the statistics. For instance, taking these statistics into account, the value of 19 ± 4.3/31 ± 5.5 has a considerable range. The high end is ∼0.90 and the low end is ∼0.41 (which is the value obtained by C01). A more complete description of the variable population of the inner region would help to narrow this range.
The question must also be asked, how did previous observers distinguish RRd from RRc types and did they do so in a reliable and consistent fashion? In other words, how sure are we that the total of 17 RRd stars given by C01 is correct? Assuming it is, then mass segregation could be a reason for the lack of RRd variables in the core of M15. The mean mass of RRc variables in M15 is 0.80 M⊙ (Clement, Bezaire & Giguere 1995) or 0.73 M⊙ (Clement & Shelton 1996). The mean mass of RRd variables in Oo II clusters is 0.65 M⊙ (Simon & Cox 1991). However, we would require a better description of the total RRd population from deeper observations of the centre, to be more certain of the role of mass segregation in altering these population ratios.
Dwarf Novae in Our Data Set
Cataclysmic variables in globular clusters
The dense nature of globular clusters coupled with the large population of LMXBs compared with the field led Hertz & Grindlay (1983) to suggest that cataclysmic variables (CVs) should be relatively common in GCs. This is also to be expected on the simple basis that white dwarfs are more common than neutron stars. However, the only concrete evidence of CVs in a GC until recently were two classical novae (Sawyer 1938; Hogg & Wehlau 1964) and a dwarf nova, V101 in M5 (Margon, Downes & Gunn 1981; Shara, Potter & Moffat 1987). Narrow-band Hα imaging of GCs (CVs are expected to emit in this region) by Shara, Moffat & Hanes (1985) and Grindlay, Cool & Bailyn (1991) revealed no evidence of CVs.
di Stefano & Rappaport (1994) predicted, through simulations of two-body tidal capture coupled with detailed consideration of binary evolution, that there should be ∼100 CVs in both 47 Tuc and ω Cen, and several thousand in the galactic globular cluster system as a whole. Their calculations predicted a large number with accretion luminosities under 1032 erg s−1, and forecast that X-ray and optical studies (especially using the HST) should begin to reveal more. Though their results were in contrast with Verbunt & Meylan (1988), who predicted that ∼36 CVs should have formed via tidal capture in 47 Tuc, Di Stefano & Rappaport had nevertheless provided a prediction which could be observationally tested using the HST. Paresce & de Marchi (1994) used the FOC on the HST to identify a DN with an amplitude of 4.3 mag, in the core of 47 Tuc. This was, according to them, the first positively identified variable of this type in a globular cluster. Meanwhile Shara et al. (1994) observed M92 for 10 consecutive nights without detecting any DNe. Through simulations they found that they were 90 per cent complete down to 21.5 mag in B. A DN at its maximum in M92 would hit B= 19 mag, leading them to conclude that there are fewer than seven DNe outside 6rc in M92. This was the first observational quantitative limit on the CV population of a globular cluster.
Further evidence of a CV population in NGC 6397 was found by Cool et al. (1995) (see also Grindlay et al. 1995) in the form of three Hα bright stars, for which the absolute magnitudes (MR= 5.6–7.3) and inferred emission-line strengths made them candidate CVs. Meanwhile Shara et al. (1995) constrained the number of CVs and contact binaries in NGC 6752 to less than or equal to 10 in total (with 95 per cent probability). This led them to suggest that the cluster had fewer tidal capture binaries than the simple tidal capture theory predicts. This was confirmed by Shara & Drissen (1995) when HST (WFPC2) images of the home of the first classical nova in a globular cluster, M80, revealed only one other CV candidate. The authors expected to see 75 CVs with MB < 5 and 200 with MB < 7 on the basis of the calculations by di Stefano & Rappaport (1994). These results suggested that either most GC binaries could be quickly ejected or destroyed or that they merge or maintain a large enough separation to avoid mass transfer.
A thorough search of 47 Tuc using archival HST images covering 12 separate epochs found one possible variable candidate, as well as seeing the CV reported by Paresce & De Marchi in eruption (Shara, Zurek & Rich 1996b). Detailed simulations led them to believe that they should have detected one-third of all DNe in the cluster (as predicted above). An upper limit of DNe resulted.
In the years that followed, Bailyn et al. (1996), Shara et al. (1996a) and Edmonds et al. (1999) confirmed the detection of two, one and one CVs in NGC 6752, 6624 and 6397 (bringing its total to four), respectively. Neither M15, M92 nor NGC 6712 have yet revealed the presence of a CV. However, Pooley et al. (2002) have recently reported the discovery of 10 likely CVs in NGC 6752 using Chandra X-ray observations coupled with HST archival images, indicating that perhaps these elusive objects are finally beginning to be found.
A search for DNe in M15
The method of searching for variables described above has revealed no sources of variability other than pulsating variables (except for AC 211). In outburst, which can last from a few to tens of days, DNe are known to jump several magnitudes in brightness and have been seen to appear as bright as the main-sequence turn-off. This spread in periods/cycle durations can manifest itself in our data in several ways. A DN in outburst for over 14 d (the duration of our observations) might not exhibit variability despite being seen in our images. This could only be interpreted as such by comparing each star on our images with images taken during some earlier epoch when the DN was in quiescence. All bona fide variable detections in M15 have been found to have both an optical counterpart (of magnitudes around the HB magnitude) in WFPC2 images of the cluster and a regular period (of less than 2 d), ruling out the chance that any of these variables could be DNe. On the other hand, a DN in quiescence in our images might never be seen to flare, and would thus remain undiscovered. The most interesting case, from our point of view, is when a DN outbursts during the observing run. To find how sensitive we are to such objects, a DN has been simulated to appear in only some of our simulated images, representing an outburst that lasts a number of days only. The typical light curve of a DN has not been modelled. Instead, a simple step function has been used — from zero flux (quiescence) to a particular magnitude. Under this assumption, any fall-off happens outside the observational epochs of our individual images.
Eight DNe of a particular magnitude were simulated and placed at different and random locations in 10 sequential artificial images, representing a duration of outburst of 2–3 d. In this instance, because the added stars are not smoothly variable we are not interested in obtaining light curves. Rather, we are only interested in whether a significant detection signature can be obtained in the median difference image. In addition, we have fine-tuned our detection technique to search lower down the CMD. Since a DN in outburst is likely to feature in all images on a given night (because of its long ‘period’ relative to, say, an RR Lyrae star) combining all images on a given night and processing them should increase the chances of detection. The set of artificial images were processed in the search for variability and the significance of their signature in the median difference image was recorded. This was repeated 100 times for DNe of maximum V magnitudes of 18, 18.5, 19, 19.5, 20, 20.5, 21 and 21.5 mag
Fig. 20 shows the results of these simulations. There is a 60 per cent chance of detecting a DN of magnitude 19.5 (or three magnitudes below the horizontal branch) in outburst. At this level it would be very difficult to obtain a light curve, as the photometric errors would be large. Nevertheless, even a single detection of a DN would be a very significant result in M15. The probability drops to ∼20 per cent at magnitude 20 (V). It should also be noted that several DNe might last for longer than our test DN and so could appear in more than 10 of our images, increasing the chance of detection. The limit here would be 25 images. If a DN appears in over 25 images it would begin to lose its significance in the residual image because of its greater significance in the reference image.
The M15 images were searched for DNe as described above. Because the 14-image (night-by-night) median difference image is more likely to reveal DNe than the 75-image (frame-by-frame) median difference image, a subtraction of the two images provides a robust means of distinguishing DNe from the other variables. Fig. 21 shows the M15 cluster (top) and the resultant image when the median difference image created using individual images is subtracted from the median difference image created by combining all images in a given night (bottom). This contains some variable signatures. However, closer inspection and comparison with the positions of known variables already detected reveal these signatures to be of the very same variables. Although combining images taken during the same night is expected to reduce the significance of shorter-period variables (by averaging out the light curve), some variables (depending on their period and the epoch at which the images were taken) will not be as strongly affected. What is important in this case is that there are no significant variable signatures which do not correspond to previously seen variables. Hence, we see no signature corresponding to a DN in outburst in the cluster.
The case of M92
Fig. 22 shows the results of a similar search instigated in M92. The signatures of the two prominent central variables can be seen in the M92 median difference image. There is also an object above these stars that is close to a bright star. Fig. 23 shows three images. The first (left-hand) image is a portion of the reference image of M92 centred on the object. The bright star is clearly dominating the image. The central image shows the corresponding region in the median difference image. The object can be seen. The image to the right is the same region of the cluster but this time imaged using the WFPC2 (PC1, through the F555W filter). Note that this is near the edge of the PC1 detector. This image shows that the location of the residual seen in the central image is not centred on the bright star, but rather above and slightly to the left. Thus the object might have one of the faint stars to the left and slightly above the bright star on the PC1 image as a possible optical counterpart.
WFPC2 photometry performed on F555W and F336W (V- and U-band equivalent) images give the two stars to the left of the bright star a magnitude of 18.106 (top) and 18.155 (bottom) in V and U−V colours of 0.126 and 0.175, respectively. This would place those two stars on or slightly below the TO. These stars are the nearest optical counterparts to the object. However, this photometry has uncertainties, mostly due to the nearby bright star and the high proportion of scattered light. Profile-fitting photometry cannot be used to subtract this bright star because it is saturated in the CCD images. Fig. 24 shows the results of our difference photometry on the object in question. Examination of the constituent images of the first point of the plot (corresponding to JD 245 0632) did not indicate a large flux difference between them at the location of the star, eliminating the possibility of a cosmic ray or other artefact contributing to the signature.
We have several reasons for thinking that this object seen in M92 could be a possible DN. First, the object only appeared when the detection method was fine-tuned to optimize a DNe search. The photometry of the difference images seem to suggest that the object was initially observed in a higher state of flux, which lessened over 2–3 d. The object does lie very close to a bright star, suggesting that possible saturation effects resulted in a poor match to the PSF and hence artefact residuals in the difference images. However, this bright star is not the brightest in the field.7 Furthermore, this star was not used as a reference star during the sharpening process. Thus it should not have a different PSF from its neighbours.
However, Fig. 24 could benefit from a data point at JD = 245 0633. The point at JD = 245 0634 has a large error bar (because the images taken during this night had a smaller exposure time). At the low end of this error bar, the only significantly bright data point is the first. The rest could be said to represent a flat distribution (no flux change). On this basis, we will wait for further observations of this region of M92 before concluding that this object is real.
We also searched for DNe in NGC 6712 but found none. We do not go into the details of this as the small number of images we had, coupled with the narrow field of view of the cluster afforded to us by TRIFFID, would not enable us to obtain a reasonable upper limit value. In any case, NGC 6712 is not core-collapsed, and so would be less likely to harbour DNe.
Discussion of results
Dwarf novae are expected to have a maximum magnitude of V=+4.3 in outburst, corresponding to a magnitude of 19.67 (given the distance modulus of M15 is (m−M)V= 15.37). At this magnitude we have a 62 per cent chance of detection, based on our simulations. In reality, this is probably slightly higher, because 75 images were used to search for DNe in the cluster as opposed to the 50 images used in the simulations. An upper limit on the number of DNe in the cluster can be found by the arguments of Shara et al. (1994). According to Szkody & Mattei (1984), the average length of time between successive outbursts is 39 d. Our images of M15 cover 14 nights, so the chances of observing a DN while in outburst is 14/39 × 0.62 = 22 per cent. The chances of missing a given DN is then 0.78. The probability of missing N DNe is then (0.78)N. We can therefore say that there is a 92 per cent probability of there being less than 10 DNe in the portion of the cluster of M15 covered by our DNe search.
These upper limits of DNe apply only to those that:
erupt every 39 d;
have an absolute magnitude of +4.3 (in the V band);
appear in 10 of our images, representing a duration of 2–3 d.
DNe exhibit a wide range of physical and observational parameters. For example, the upper limit of 10 of a particular type of DN in M15 would necessarily have to be increased for fainter maximum magnitudes and longer periods of eruption. Our present detection limits do not cover DNe with these traits. It would be computationally intensive to produce detection limits for several DN populations exhibiting a range of observational traits. Therefore, we cannot say that these objects are underrepresented in globular clusters relative to the field.
The upper limit of 10 DNe with MV < 4.5 in the centre of M15 is the first such observational quantitative limit for the cluster. What significance does this result have? Production of these relatively short-period DNe could be suppressed in globular clusters. The white dwarf and secondary star in short-period DNe are expected to have small separations, allowing mass transfer to occur at regular, short intervals. Our lack of detection of these variables could therefore mean that DNe in GCs have wider separations. However, binaries in the dense cores of GCs are subject to two-body encounters which will either harden them (leading to tighter orbits) or disrupt them. It is therefore difficult to reconcile these facts in light of our results.
Alternatively, very tight binaries might not need to wait the ∼40 d to transfer enough matter from the secondary on to the primary dwarf. It could be that mass transfer is occurring with more frequent regularity in GC DNe compared with the field. If this were the case one would not expect such large magnitude swings because there would not have been enough time for the necessary mass to ‘build up’. This scenario might instead exhibit a flickering effect on much shorter time-scales. Assuming this is the case in M15, one would not expect this flickering to be visible in our data. This observational signature might be visible in data from superior instruments.
On the other hand, wider separations (which were not broken dynamically) between DNe components would mean longer time-scales between outbursts. The possibility of white dwarf–main-sequence binary systems existing in CVs but rarely, if ever, outbursting was discounted by Shara et al. (1996) among others. This scenario is not predicted by the disc instability models used to model DNe and there is no reason why GC DNe should be suppressed in this way.
We should also note one final result by Davies (1997), who considered the production of CVs in globular clusters and investigated whether the binaries which produce CVs in the field (called potential CVs, PCVs) would affect a GC population of CVs to the same extent. These PCVs would find a GC a hostile place. A soft system is likely to be disrupted. A hard system could be hardened to such an extent as to inhibit CV production. Alternatively, an exchange encounter could remove one of the PCV members, replacing it with a star not conducive to CV formation. The author found that PCVs are likely to be broken up (through single and hard binary collisions) in high-density regions (stellar number density, N, less than 6 × 103 pc−3, i.e. in the cores of globular clusters). PCVs should, however, be more prevalent in the outer halo of the cluster. He concludes: ‘in clusters with a high central density, where PCVs are destroyed in the cores, a relative paucity of CVs in their centres will indicate that tidal capture and encounters between binaries and single stars are relatively inefficient pathways to produce CVs’. Although we have no evidence to confirm or deny the existence of CVs in the outer reaches of M15, we have firm evidence of a paucity of at least one particular type of CV in the inner regions of the cluster.
We have presented a thorough variable search of the cores of the globular clusters NGC 6712, M92 and M15 using the difference imaging technique applied to post-exposure image sharpened data obtained using the TRIFFID/MAMA optical system. We have obtained periods and amplitudes for variables in M92 and NGC 6712, although our research has concentrated on M15, in which we have recorded the presence of 48 variables in the core, including a total of 23 new discoveries. The vast majority of the detected variables are of RR Lyrae type. The exceptions are a W Vir type II Cepheid and two short-period type II Cepheids. The most significant period of the LMXB AC 211 (derived using the phase dispersion minimization technique) is at ∼17.1 h. Some variables previously detected in M15 have not been detected in this search. Most of these candidates had been tentatively identified as variables, and as such a non-detection in this data probably discounts their existence. The subsequent analysis of the variable distribution in the cluster indicates a dearth of RRab stars in comparison to RRc variables. Future observations of the core region will help to more accurately quantify this discrepancy. Furthermore, based on coverage considerations of our data we anticipate the detection of several more variables in the core region. Although our search for DNe in M15 proved fruitless, our upper limit for M15 is the first such result for the cluster. Deeper observations will allow for searches of different types of DNe, including those with fainter outbursts and DNe which erupt less frequently. Finally, in a search for DNe in M92, we have noted a possible candidate DN variable.
This work was completed with the assistance of funding from Enterprise Ireland. We are grateful to A. Golden (NUI, Galway) for his critical reading of the text. We also wish to thank C. Alard (Institut d'Astrophysique de Paris) for assistance regarding the image subtraction code, C. Clement for useful comments in relation to the Catalogue and G. Kopacki regarding M92. We would like to thank the anonymous referee for constructive criticism which improved the quality of this work.
Figures A1, A2 and A3 show the light curves of the RRab, RRc and unidentified variables, respectively, seen toward M15. The curves have been folded on the most significant period obtained using the PDM technique. Note that each panel does not necessarily cover the same magnitude range.