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5.2 Source B We created a light curve of source B in the 0.5–2.0 keV band with Suzaku, which is shown in Fig. The time bin size is 512 s. Although we detected no drastic flare event, the light curve seems to show flickering. In fact, Kolmogorov–Smirnov test indicates the probability of no variability in the light curve... |
Discussion 6.1 Thermal Component We have calculated the electron number density and the age of the diffuse thermal plasma of region 1 and 2 on the basis of the best-fit parameters summarized in table and |
. We assume that the plasma in region 1 distributes uniformly within a sphere with a radius of [MATH] , which is the Half-Width-of-Half-Maximum (HWHM) radius obtained from the Chandra image (Aharonian et al., 2008 . Assuming the distance to CTB37B of 10.2 kpc (Caswell et al., 1975 we obtain the real radius of region 1 ... |
[MATH] , the volume of region 2 is calculated to be [MATH] cm (the resultant electron density becomes smaller by a factor of [MATH] if we substitute [MATH] for |
[MATH] ). With the aid of the emission measures obtained from the spectral fitting, [MATH] [MATH] (region 1) and [MATH] (region 2 upper limit, see § 4.2), the electron number density of region 1 and 2 are |
[EQUATION] and [EQUATION] where we adopt the relation [MATH] for fully ionized plasma. The parameter region in parenthesis is those allowed at the 90% confidence level. Assuming the strong shock, we obtain the pre-shock densities to be [MATH] 0.43 cm and |
[MATH] 0.21 cm , respectively, which are significantly lower than the average density of the interstellar matter in the galactic plane. That means CTB37B exploded in a low density space. The density of region 1 together with the ionization parameter obtained from the fit of the region 1 spectra [MATH] [cm -3 s], enable... |
[EQUATION] CTB37B is one of the best candidates of SN393 in Chinese historical record (Stephenson & Green, 2002 . The plasma age calculated from the observed ionization parameter and emission measure supports this identification. |
The number of electrons ( [MATH] ) in region 1 and 2 are [MATH] and [MATH] . As a result, the total mass included in the two regions are 15 (11–20) [MATH] and [MATH] 3.7 [MATH] , respectively, and the thermal energy ( [MATH] ) are |
[EQUATION] under the assumption of energy equipartition between electrons and ions. The total thermal energy could be larger if other portions of the remnant are included and if the proton temperature is significantly larger than the electron temperature as is expected for supernova remnants with large shock velocities... |
6.2 The Nature of the Point Sources 6.2.1 Source A The best-fit spectral parameters of Chandra source A and those of Suzaku region 1 (source A and the diffuse thermal emission) are summarized in table . The measured hydrogen column densities [MATH] cm -2 ) are the same between Chandra and Suzaku. However, since the Suz... |
Fahlman & Gregory ( 1981 ); Kuiper et al. ( 2006 ). Young age of [MATH] 700 yr (§ 6.1) also supports this interpretation e.g. the AXP 1E1841-045 is associated with Kes 73 whose age is estimated to be 500-1000 yr; |
Tian & Leahy ( 2008 ). Since the spin period of AXP is in the range 6–12 s, it is natural that we have found no evidence of pulsation from the Chandra data, because the frame time of the ACIS-I is 2.3 s. In addition, the flux of the power law measured by Suzaku is apparently greater that that with Chandra by a factor o... |
(Gavriil & Kaspi, 2002 ; Kaspi et al., 2003 . We therefore conclude based on the Chandra and Suzaku observations that source A is probably a new AXP. We need a fast timing observation with an imaging detector with a time resolution of less than 1 s to confirm our conclusion. |
6.2.2 Source B From the spectrum fitting, The hydrogen column density obtained from the fit to the Suzaku region 3 spectra [MATH] (table ) is much smaller than that obtained from region 1 and 2. This result indicates that source B is a foreground source. The best-fit plasma temperature of [MATH] keV is reminiscent of a... |
6.3 Non-Thermal Component X-ray emission from CTB37B is composed of the diffuse thermal component (region 1) and the non-thermal component (region 2) as well as a point source (source A), as demonstrated in §3 and §4. Hence, CTB37B now is the third SNR after RCW86 and Cas A that possesses the thermal and non-thermal X-... |
[MATH] -ray bands. The fluxes of the non-thermal emission of these five non-thermal SNRs are compared in table The non-thermal diffuse component detected from region 2 has a remarkably flat X-ray spectrum with a photon index of 1.5. Since this photon index is equal to the typical radio photon index (energy index [MATH]... |
[MATH] 15 keV (table ). This roll-off energy is higher than any other SNR that is accompanied by the non-thermal X-ray and TeV [MATH] -ray emission, such as [MATH] 9 keV for RX J1713.7–3946 |
(Takahashi et al., 2008 [MATH] 0.23 keV for SN1006 (Bamba et al., 2008 [MATH] 0.87 keV for RCW 86 (Bamba et al., 2005 This indicates high electron acceleration efficiency in region 2. In addition, the density around region 2 is considered as being lower than in region 1, given that only the upper limit of the thermal e... |
It should be a matter of debate whether the TeV [MATH] -ray emission and the non-thermal X-ray emission from region 2 are produced by the same population of electrons, since the images of these two bands shown in Fig. are far from similar at first sight. The TeV [MATH] -ray image is compatible with a shell with a radiu... |
The maximum electron energy ( [MATH] can be evaluated by the shape of TeV [MATH] -ray spectrum. Using the H.E.S.S. spectrum whose photon index is 2.3 |
(Aharonian et al., 2006 shown in Fig. in blue, we determined [MATH] of 10 TeV. The red line in Fig. , on the other hand, is the X-ray power-law spectrum of region 2 with a photon index of 1.5. A series of the dashed plots are the model spectra calculated under the assumptions of [MATH] of 10 TeV, an index of the electr... |
Conclusion We have obtained with Suzaku the images and the high quality spectra of the supernova remnant CTB37B. The X-ray diffuse emission region coincides with that of radio and TeV [MATH] -ray. The X-ray emission consists of thermal and non-thermal diffuse components as well as a point source resolved by Chandra. CT... |
[MATH] -ray emission are detected all together, and the fifth SNR that is accompanied by non-thermal emission both in X-ray and TeV |
[MATH] -ray bands. The diffuse thermal emission can be best described by a non-equilibrium collisional ionization plasma model (NEI model) with a temperature, an ionization parameter ( [MATH] [cm -3 s]), and the abundances of 0.9 [MATH] keV, 3.5 [MATH] , and |
[MATH] 0.5 [MATH] (Mg, Si), respectively. The image size and the observed emission measure provides the number density of the thermal electrons before the shock to be 0.2–0.4 cm -3 which is significantly lower than that of the Galactic plane. This suggests that the supernova explosion associated with CTB37B took place ... |
[MATH] 650 [MATH] yr. This is consistent with the tentative identification of CTB37B with SN393 within the error. In contrast, the diffuse component occupying southern part of CTB37B (region 2) is non-thermal and represented by a power-law model or a srcut model. The photon index of 1.5 is significantly smaller than an... |
[MATH] -ray was emitted by 1-zone IC scattering, there are no solution for magnetic field strength that can reproduce the observed synchrotron spectrum in X-ray. This suggests that TeV [MATH] -ray is produced by multi-zone IC scattering, or by the decay of neutral pions proton impacts. |
Owing to the high spatial resolution of Chandra, a point source is resolved from the brightest part of the Suzaku image of CTB37B (region 1). Its association to the diffuse thermal emission indicated by |
[MATH] , the photon index of [MATH] 3, the X-ray luminosity of order [MATH] erg s -1 , and the long term flux variation evident from the Chandra and Suzaku observations all indicate that the point source is a new anomalous X-ray pulsar. A high speed photometric observation is encouraged. |
The authors are greatful to all of the Suzaku and H.E.S.S. team members. We also thank H. Yamaguchi, T. Tanaka and J. Vink for useful comments. This work was supported in part by Grant-in-Aid for Scientific Research of the Japanese Ministry of Education, Culture, Sports, Science and Technology, No. 19 [MATH] 4014 (A. B... |
# Source: arxiv 0808.2536 # Title: Lighthouses with two lights: burst oscillations from the accretion-powered millisecond pulsars # Sections: all # Downloaded: 2026-03-02T07:58:10.948288+00:00 |
Lighthouses with two lights: burst oscillations from the accretion-powered millisecond pulsars Abstract The key contribution of the discovery of nuclear-powered pulsations from the accretion-powered millisecond pulsars (AMPs) has been the establishment of burst oscillation frequency as a reliable proxy for stellar spin... |
Keywords: binaries: general, stars: neutron, stars: rotation, X-rays:bursts 95.85.Nv, 97.10.Kc, 97.60.Gb, 97.60.Jd, 97.80.Jp Introduction |
‘History’, as George Orwell once noted, ‘is written by the winners’ (Orwell, 1944 - or, in this case, by the workshop hosts. When we gathered in Amsterdam in April 2008, it was ostensibly to celebrate ten years since the discovery of the first Accreting Millisecond X-ray Pulsar (AMXP). We were, however, a full two year... |
Most of this volume focuses on the AMPs, where persistent pulsations are generated as accreting material is channeled by the magnetic field onto magnetic polar caps that are offset from the rotational poles. The NMPs, by contrast, show pulsations during Type I X-ray bursts (thermonuclear explosions on the stellar surfa... |
Section provides a brief overview of the relevant observational results. The bulk of the review focuses, however, on the astrophysical questions where these sources have made or are making a major contribution to our understanding. These include the spin distribution of AMXPs, torque modeling, and the burst oscillation... |
Observational summary Type I X-ray bursts have been observed from two of the seven persistent , and all three of the intermittent, AMPs. The AMPs for which no bursts have been detected are the four ultra-compact systems (XTE J1751-305, XTE J1807-294, XTE J0929-314 and SWIFT J1756.9-2508) and IGR J00291+5934. The latter... |
The first AMP to be detected as an NMP was SAX J1808.4-3658 (Chakrabarty et al., 2003 . The burst oscillations exhibit frequency drifts of a few Hz in the rising phase of the brightest bursts, settling down to a frequency that is within 0.1 Hz of the spin frequency in the burst tail. This result was followed by the dis... |
(Watts et al., 2005 Of the three intermittent AMPs only Aql X-1 has burst oscillations. Indeed this source was discovered to be an NMP |
(Zhang et al., 1998 long before its detection as an intermittent AMP (Casella et al., 2008 . The frequency inferred from the accretion-powered pulsations is offset by a small amount ( [MATH] 1 Hz) from the asymptotic frequency of the burst oscillations (as in many other sources, the burst oscillations from Aql X-1 drif... |
Neutron star spin 3.1 Spin distribution and the paucity of rapid rotators Identifying the spin distribution of the various classes of neutron star has been a major research goal for many years. Spin rates are important to our understanding of stellar and binary evolution: finding rapidly-rotating accreting neutron star... |
Identifying the most rapidly-rotating X-ray pulsars, however, has proved to be far more difficult than for the radio pulsars. The sources are far less numerous, and the vast majority of neutron stars in Low Mass X-ray Binaries (LMXBs) do not have accretion-powered pulsations. The discovery of the NMPs promised a substa... |
(Muno et al., 2002a , and the failure to identify the mechanism responsible for the nuclear-powered pulsations, had led to some caution - although repeated measurements of the same burst oscillation frequency for a given source (Muno et al., 2002a , and the presence of oscillations at the same frequencies in both Type ... |
The main consequence has been a doubling of the number of rapidly-rotating accreting neutron stars with a measured spin rate (Figure ). This has brought to light a new and interesting problem. Simple estimates of accretion-induced spin-up over the lifetime of an LMXB suggest that there should be a population of neutron... |
(Hartman et al., 2003 , while the fastest radio pulsar spins at 716 Hz (Hessels et al., 2006 . If the evolutionary estimates are sound, there is a requirement for a braking mechanism to halt the spin-up. Magnetic braking (due to interaction between the stellar field and the accretion disk) is one possibility (Ghosh and... |
It is of course worth remembering that we still do not understand the mechanism responsible for burst oscillations. The burst oscillation properties of SAX J1808.4-3658 and XTE J1814-338 are rather unusual compared to the rest of the NMPs (Section ), and it is still possible that the cause of the nuclear-powered pulsat... |
3.2 Spin variation and torques in neutron stars The spin histories of the AMXPs are valuable because of what they reveal about the properties of the star and the different torque mechanisms that may operate. In terms of stellar properties we are interested in the moments of inertia of, and degree of coupling between, t... |
Spin histories for the AMXPs are constructed using standard radio pulsar timing techniques that involve measuring phase shifts between folded pulse profiles. Pulse profiles in the AMXPs are however notoriously variable, since they are affected by fluctuations in the accretion flow. This additional noise leads to pulse ... |
Timing using burst oscillations could in principle provide an independent test of spin variation, since the emission mechanism for the nuclear-powered pulsations is thought to be quite different to the accretion-powered pulsations. Drifts in burst oscillation frequency, of course, complicate this task (making analysis ... |
and the discussion in Watts et al. ( 2008b ), but these need to be tested further. If they do not prove viable, this would support there being at least some genuine spin variation. |
Burst oscillation mechanisms The nature of the brightness asymmetry that causes nuclear-powered pulsations is still not understood for any source, AMP or otherwise. The status of the various different models may be summarized as follows: |
Hotspot spread : A localized growing hotspot is expected to exist in the burst rise, since ignition should not start simultaneously across the stellar surface (Shara, 1982 There is evidence that expanding hot spots are related to the presence of burst oscillations in the rising phases of some bursts (see for example St... |
Thermonuclear hurricanes : The Coriolis force can act to confine the burning area during the burst rise (Spitkovsky et al., 2002 (making this model to some degree a variant of the spreading hotspot). This may explain the presence of oscillations in the burst rise, although like the previous model it has yet to be put t... |
Surface modes : Global oscillations may be excited by flame spread, generating a brightness asymmetry that could persist throughout the burst tail. Attention has focused on buoyant r-modes in the neutron star ocean, since these have frequencies close to the stellar spin rate (Heyl, 2004 The mode model also provides a n... |
(Piro and Bildsten, 2005 , has now been shown to be inefficient (Berkhout and Levin, 2008 . Alternative mode types including photospheric |
(Heyl, 2004 and shearing oscillations (Cumming, 2005 also have shortcomings (Berkhout and Levin, 2008 , but magnetic effects may play a significant role in determining mode behaviour (contribution by Cumming, this workshop). |
The most significant contribution that the AMPs have made to our understanding of the burst oscillation mechanism is of course the requirement that the frequency should lie within a few Hz of the spin rate: all of the models listed above take this as a basic premise. However these sources are also valuable in other way... |
(Brown and Bildsten, 1998 but there are other local effects at the deposition point, such as higher temperature, that may be important. |
These factors have motivated efforts to measure the properties of the burst oscillations of the AMPs and compare them to both the accretion-powered pulsations and the other burst oscillation sources. Key results, drawn from many different references (non-AMPs Muno et al. ( 2002a 2003 2004 ); Galloway et al. ( 2008 ; in... |
1. For most NMPs, including Aql X-1, burst oscillations are only detectable in some bursts. They are more prevalent at high accretion rates and in short, He-rich bursts. SAX J1808.4-3658 and XTE J1814-338 show them in every burst despite low accretion rates and, for XTE J1814-338, long mixed H/He bursts. |
2. The amplitudes of burst oscillations in the non-AMPs lie in the range 2-20% RMS. The absolute amplitudes of the burst oscillations from the AMPs are similar, but never exceed the accretion-powered pulsation amplitudes. |
3. SAX J1808.4-3658 and XTE J1814-338 are the only sources where the burst oscillations have detectable harmonic content, albeit at a lower amplitude than in the accretion-powered pulsations. |
4. The burst oscillations for most NMPs, and Aql X-1, have an amplitude that rises with energy. For XTE J1814-338, and most of the bursts from SAX J1808.4-3658, amplitude falls with energy. This behaviour is inconsistent with both simple hotspot and surface mode models (Piro and Bildsten, 2006 . A fall in amplitude wit... |
5. Burst oscillations from the AMPs show no detectable phase lags. This behaviour is similar to that of the other NMPs, which show at most marginal hard lags. |
6. The frequency drifts seen in Aql X-1 are typical of the other NMPs: a slow rise during the burst to a saturation frequency. The two persistent AMPs with burst oscillations are very different. For XTE J1814-338 there are no detectable drifts except in the final brightest burst, which has a small drop in frequency in ... |
7. The nuclear-powered pulsations in XTE J1814-338 are completely phase-locked to the accretion-powered pulsations even though the latter show substantial phase wander over the course of the outburst. |
So what do these results tell us about the burst oscillation mechanism? Firstly, that existing models are no better at explaining AMP burst oscillation properties than they are at explaining the rest of the NMP population. The fall of amplitude with energy, and frequency overshoot, for example, are not predicted by any... |
The second important question is whether we are looking at a continuum of behaviour that could be attributed to one mechanism (with differences being set by varying magnetic field strength, for example). The burst oscillation properties of Aql X-1 sit comfortably within the general population, perhaps not surprisingly ... |
Chakrabarty et al. ( 2003 , the rapidity and magnitude of the frequency shift could be shown to depend strongly on magnetic field strength (or the degree of misalignment between the magnetic field and rotational pole). |
The remaining source, XTE J1814-338, is an oddball. The properties of its nuclear-powered pulsations differ in almost every way from the rest of the sample. The phase-locking of the two types of pulsation, however, suggests that the presence of nuclear-powered pulsations in this source may be related to premature ignit... |
Conclusions The discovery of nuclear-powered pulsations from the AMPs has cemented the link between burst oscillation frequency and spin frequency. In addition to confirming the absence of rapid rotators (now a major problem for evolutionary models), this has imposed the strongest single constraint on candidate burst o... |
# Source: arxiv 0808.2542 # Title: The Variable Stars of the Draco Dwarf Spheroidal Galaxy - Revisited # Sections: all # Downloaded: 2026-03-02T07:58:12.082814+00:00 |
The Variable Stars of the Draco Dwarf Spheroidal Galaxy - Revisited Abstract We present a CCD survey of variable stars in the Draco dwarf spheroidal galaxy. This survey, which has the largest areal coverage since the original variable star survey by Baade & Swope, includes photometry for 270 RR Lyrae stars, 9 anomalous... |
magnitudes, [MATH] amplitudes, and periods, have been derived. Photometric metallicities of the ab-type RR Lyrae stars were calculated according to the method of Jurcsik & Kovacs, yielding a mean metallicity of [MATH] . The well known Oosterhoff intermediate nature of the RR Lyrae stars in Draco is reconfirmed, althoug... |
Variable stars: RR Lyrae, anomalous Cepheids, long period variables — dwarf spheroidal galaxy: Draco (catalog ) Introduction The Draco dwarf spheroidal (dSph) galaxy ( [MATH] |
[MATH] ), a satellite of the Milky Way Galaxy, was first extensively studied by Baade & Swope ( 1961 (hereafter known as B&S). They reported discovering over 260 variable stars and obtained photometry for 138 variables in the central region of Draco, 133 of which were of RR Lyrae (RRL) type. Several subsequent studies ... |
Zinn & Searle ( 1976 reported new observations of the anomalous Cepheids in Draco. Nemec ( 1985a reanalyzed the B&S photometry and produced updated periods for the B&S variables. Both Nemec ( 1985a |
and Goranskij ( 1982 reported new double-mode RRL in Draco. Recently Bonanos et al. ( 2004 provided a photometric study of Draco which produced light curves for 146 RRL stars, four anomalous Cepheids, an SX Phe star, and a field eclipsing binary. In this work, we use CCD observations to update the census of variable st... |
This paper is organized in the following manner: Section 2 describes our data acquisition and data reduction processes. Section 3 covers our analysis techniques. Periods, light curves, and classifications of the variable stars are presented in Section 4. A re-discussion of the Oosterhoff classification of the Draco dwa... |
Data Acquisition and Reduction Our survey of the Draco dSph galaxy was conducted at two telescopes: the 1.0m at the US Naval Observatory in Flagstaff, AZ., and the 2.3m telescope at the Wyoming Infrared Observatory (WIRO), at Mt. Jelm, Wyoming. Combined, the two datasets cover a time interval of four years (1993-1996).... |
2.1 USNO observations Images of Draco were taken with the 1.0 m telescope of the U.S. Naval Observatory in Flagstaff, AZ, during the 1995 and 1996 observing seasons. A Tektronix 2048 [MATH] 2048 CCD was used with a pixel size of 0.68 arcsec, giving a field size of 23.2 arcmin. Four fields were observed, each covering o... |
The images were taken with a Johnson [MATH] filter throughout the 1995 and 1996 observing seasons, and with a Cousins [MATH] filter mostly during the 1996 season. The seeing was typically 2 [MATH] and the exposure times were 15-30 min depending on the seeing. Exposures were taken switching between quadrants, and altern... |
was used to measure all images. A small radial correction for image distortion in the corners of each image was applied for the data taken at USNO. |
For the goals of identifying variable stars and measuring accurate magnitudes and light curves, five sources contribute errors to these data. Some stars are crowded or near brighter stars and have erroneous measurements. The CCD has a few defects that produce spurious magnitudes for some stars that occasionally fall on... |
also lists those stars that B&S originally marked as variable candidates but which were found not to be variable in our survey. The instrumental magnitudes were shifted onto a common system, iteratively rejecting variable stars, using a method similar to that described by |
Honeycutt ( 1992 Finally the USNO instrumental magnitudes were transformed to standard Johnson [MATH] and Cousins [MATH] magnitudes as follows. On three photometric nights when Draco images were taken in all quadrants, Landolt ( 1992 standards were also observed and used to determine transformation coefficients of the ... |
[EQUATION] and similarly for [MATH] . On one additional photometric night, using a different Tektronix 1024x1024 CCD, images were taken centered on Draco, together with Landolt standards. Color coefficients were small, typically 0.01 and 0.03 in [MATH] and [MATH] respectively. These coefficients are presented in Table |
for both [MATH] and [MATH] bands and per photometric night. Three nights were used to determine mean [MATH] and [MATH] standard magnitudes for a subset of bright (16-18 mag) nonvariable stars in the Draco images. The transformation of instrumental magnitudes (after shifting onto the common system) to standard magnitude... |
shows good agreement. The resulting errors in photometry for a single observation are estimated to be 0.01 from calibration uncertainties, 0.02 from image distortion in the CCD corners, and photon noise that increases from 0.01 at [MATH] to 0.03 at [MATH] to 0.05 at [MATH] . After combining frames, the errors in the me... |
2.2 WIRO Observations The USNO dataset was combined with data obtained at WIRO during the summer quarter observing season of 1993 and 1994. An RCA [MATH] pix CCD camera was used, which had a 1.2 “/pix plate scale. The field of view was much smaller compared to the USNO dataset. The WIRO fields were [MATH] and overlappe... |
shows where the WIRO fields are in relation to the USNO fields. The data were obtained with Johnson [MATH] and Cousins [MATH] filters. From WIRO, a maximum of 28 [MATH] and 18 [MATH] -band images supplemented the USNO data. All available data from WIRO was used for light curve and period analysis of the variable stars.... |
images for stars found in both USNO and WIRO fields. The WIRO observations were placed on a standard system by using secondary standards from the USNO analysis. A total of 45 stars were used for the calibration, and the dominant source of uncertainty are from the original calibration done with the USNO dataset for each... |
and are the transformation equations for the WIRO dataset. The coefficients [MATH] and [MATH] were field dependent and were determined from a weighted mean of differences. The coefficients [MATH] and [MATH] |
were obtained from a linear least squares fit between [MATH] and [MATH] . The standard [MATH] magnitude was found through an iterative process, incorporating the standard [MATH] magnitude of that star. The values of the transformation coefficients for the WIRO dataset are given in Table |
[EQUATION] [EQUATION] Photometry was performed on the WIRO data using Stetson’s DAOPHOT II and ALLFRAME stand alone package (Stetson 1987 1994 |
2.3 Variable Star Identifications We have kept the original numbering system of B&S for the first 203 variable stars plus number 204 assigned by Zinn & Searle ( 1976 . All new variable star identifications, as well as the new long period variable stars and the QSOs, are an extension of B&S’s system, but organized by ri... |
Stars with high dispersion or high chi-square for their magnitude were considered to be potential variables and inspected further. For the USNO dataset, we used a plot of chi-squared vs. magnitude to identify potential variables. We did not use the Welch & Stetson variability index (Welch & Stetson 1993 because it is d... |
For the WIRO dataset, the variable stars were selected by using a simple variability index which compared the external to internal uncertainties of the observations. Our results were then compared to the B&S catalog and we identified the known variable stars. New variables were found and classified by their color, peri... |
A total of 270 RR Lyrae stars, 9 anomalous Cepheids, 12 semi-irregular or carbon stars, and 2 eclipsing binaries were discovered in this survey. We were able to recover all of the original B&S variable stars, and reclassified 7 stars. We discuss the variable stars of Draco in more detail in section 4. |
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