Period–colour and amplitude–colour relations in classical Cepheid variables – V. The Small Magellanic Cloud Cepheid models

Period–colour (PC) and amplitude–colour (AC) relations at maximum, mean and minimum light are constructed from a large grid of full amplitude hydrodynamic models of Cepheids with a composition appropriate for the Small Magellanic Cloud (SMC). We compare these theoretical relations with those from observations. The theoretical relations are, in general, in good agreement with their observational counterparts, though there exist some discrepancy for short period  (log [ P ] < 1)  Cepheids. We outline a physical mechanism which can, in principle, be one factor to explain the observed PC/AC relations for the long and short period Cepheids in the Galaxy, Large Magellanic Cloud (LMC) and SMC. Our explanation relies on the hydrogen ionization front (HIF)–photosphere interaction and the way this interaction changes with pulsation period, pulsation phase and metallicity. Since the PC relation is connected with the period–luminosity (PL) relation, it is postulated that such a mechanism can also explain the observed properties of the PL relation in these three galaxies.     

Period–colour and amplitude–colour relations in classical Cepheid variables – IV. The multiphase relations

The superb phase resolution and quality of the Optical Gravitational Lensing Experiment (OGLE) data on the Large Magellanic Cloud (LMC) and Small Magellanic Cloud (SMC) Cepheids, together with existing data on Galactic Cepheids, are combined to study the period–colour (PC) and amplitude–colour (AC) relations as a function of pulsation phase. Our results confirm earlier work that the LMC PC relation (at mean light) is more consistent with two lines of differing slopes, separated at a period of 10 d. However, our multiphase PC relations reveal much new structure which can potentially increase our understanding of Cepheid variables. These multiphase PC relations provide insight into why the Galactic PC relation is linear but the LMC PC relation is non-linear. This is because the LMC PC relation is shallower for short  (log  P < 1)  and steeper for long  (log  P > 1)  period Cepheids than the corresponding Galactic PC relation. Both of the short- and long-period Cepheids in all three galaxies exhibit the steepest and shallowest slopes at phases around 0.75–0.85, respectively. A consequence is that the PC relation at phase ∼ 0.8 is highly non-linear. Further, the Galactic and LMC Cepheids with  log  P > 1  display a flat slope in the PC plane at phases close to the maximum light. When the LMC period–luminosity (PL) relation is studied as a function of phase, we confirm that it changes with the PC relation. The LMC PL relation in V and I band near the phase of 0.8 provides compelling evidence that this relation is also consistent with two lines of differing slopes joined at a period close to 10 d.     

Period–luminosity relations for type II Cepheids and their application

JHK _s magnitudes corrected to mean intensity are estimated for Large Magellanic Cloud (LMC) type II Cepheids in the OGLE-III survey the third phase of the Optical Gravitational Lensing Experiment (OGLE). Period–luminosity (PL) relations are derived in JHK _s as well as in a reddening-free VI parameter. Within the uncertainties, the BL Her stars  ( P < 4 d)  and the W Vir stars (   P = 4  to 20 d) are colinear in these PL relations. The slopes of the infrared relations agree with those found previously for type II Cepheids in globular clusters within the uncertainties. Using the pulsation parallaxes of V553 Cen and SW Tau, the data lead to an LMC modulus uncorrected for any metallicity effects of  18.46 ± 0.10  mag. The type II Cepheids in the second-parameter globular cluster, NGC 6441, show a PL( VI ) relation of the same slope as that in the LMC, and this leads to a cluster distance modulus of  15.46 ± 0.11  mag, confirming the hypothesis that the RR Lyrae variables in this cluster are overluminous for their metallicity. It is suggested that the Galactic variable κ Pavonis is a member of the peculiar W Vir class found by the OGLE-III group in the LMC. Low-resolution spectra of OGLE-III type II Cepheids with   P > 20  d (RV Tau stars) show that a high proportion have TiO bands; only one has been found showing C₂. The LMC RV Tau stars, as a group, are not colinear with the shorter period type II Cepheids in the infrared PL relations in marked contrast to such stars in globular clusters. Other differences between LMC, globular cluster and Galactic field type II Cepheids are noted in period distribution and infrared colours.     

The linearity of the Wesenheit function for the Large Magellanic Cloud Cepheids

There is strong evidence that the period–luminosity (PL) relation for the Large Magellanic Cloud (LMC) Cepheids shows a break at a period around 10 d. Because the LMC PL relation is extensively used in distance scale studies, the non-linearity of the LMC PL relation may affect the results based on this LMC calibrated relation. In this paper we show that this problem can be remedied by using the Wesenheit function in obtaining Cepheid distances. This is because the Wesenheit function is linear, although recent data suggest that the PL and the period–colour (PC) relations that make up the Wesenheit function are not. We test the linearity of the Wesenheit function and find strong evidence that the LMC Wesenheit function is indeed linear. This is because the non-linearity of the PL and PC relations cancel out when the Wesenheit function is constructed. We discuss this result in the context of distance scale applications. We also compare the distance moduli obtained from  μ₀=μ _V − R (μ _V −μ _I )  (equivalent to Wesenheit functions) constructed with the linear and the broken LMC PL relations, and we find that the typical difference in distance moduli is  ∼ ±0.03 mag  . Hence, the broken LMC PL relation does not seriously affect current distance scale applications. We also discuss the random error calculated with equation  μ₀=μ _V − R (μ _V −μ _I )  , and show that there is a correlation term that exists from the calculation of the random error. The calculated random error will be larger if this correlation term is ignored.     

Cyclic variations in the angular diameter of χ Cygni

We report the direct detection of cyclic diameter variations in the Mira variable χ Cygni. Interferometric observations made between 1997 July and 1999 September, using the Cambridge Optical Aperture Synthesis Telescope (COAST) and the William Herschel Telescope (WHT), indicate periodic changes in the apparent angular diameter at a wavelength of 905 nm, with amplitude 45 per cent of the smallest value. The star appears largest at minimum light. Measurements made at a wavelength of 1.3 μm over the same period suggest much smaller size changes. This behaviour is consistent with a model in which most of the apparent diameter variation at 905 nm is caused by a large increase in the opacity of the outer atmospheric layers (which is mostly owing to titanium oxide) near minimum light, rather than by physical motions of the photosphere. The 1.3-μm waveband is relatively uncontaminated by TiO, and so much smaller size changes would be expected in this band. The latest non-linear pulsational models predict maximum physical size close to maximum light, and increases in opacity near minimum light that are too small to reproduce the diameter variation seen at 905 nm. This suggests either that the phase-dependence of the model pulsation is incorrect, or that the opacities in the models are underestimated. Future interferometric monitoring in uncontaminated near-infrared wavebands should resolve this question.     

Confirmation of previous ground-based Cepheid P–L zero-points using Hipparcos trigonometric parallaxes

Comparisons show agreement at the 0.1-mag level between the calibration of the Cepheid period–luminosity (P–L) relation by Feast & Catchpole (FC) using the early release of Hipparcos data and four previous ground-based calibrations, three of which are either largely or totally independent of the distance to the Large Magellanic Cloud (LMC). Each of the comparisons has the sense that the FC calibration is brighter, but only at the level of ≲0.1 mag. In contrast, FC argue that their Hipparcos recalibration leads to a 0.2-mag revision in the distance to the LMC, and thereby to a 10 per cent decrease in the Hubble constant. We argue differently. The comparison of the Hipparcos recalibration with others should be made using only local Galactic Cepheids, not based on Cepheids in the LMC that require a set of precepts that are not germane to the direct Hipparcos recalibration. The comparison made here, using only Galactic Cepheids, gives a correction of ∼4 per cent or less to our value of H ₀ based on Type Ia supernovae, keeping all other factors and precepts the same.
  A second success of the Hipparcos mission is the calibration of the position of the main sequence in the Hertzsprung–Russell diagram as a function of metallicity using local subdwarfs. These data have been used by Reid and by Gratton et al. to obtain, similarly to FC, a brighter absolute magnitude of RR Lyrae stars by ∼0.3 mag from that often currently adopted. These new calibrations confirm the earlier brighter calibrations by Walker, by Sandage, and by Mazzitelli, D'Antona & Caloi, thereby reducing the ages of globular clusters by ∼30 per cent. This removes most of the cosmological time-scale problem if H ₀∼55 km s⁻¹ Mpc⁻¹. A similar conclusion, based on pulsation theory and MACHO data, has been reached by Alcock et al.     

Period–luminosity relations for red supergiant variables – I.The calibration

We present CCD photometry of red supergiant long-period variables (LPVs) in the Per OB1 association, the Large Magellanic Cloud (LMC) and M33. The photometry was obtained in the Kron–Cousins R and I bandpasses and in a narrow bandpass ( λ ₀=8250 Å, FWHM=300 Å) chosen to avoid TiO bands in the spectral energy distribution of the LPVs. Because the strength of the TiO bands varies greatly with temperature, which varies with the phase of an LPV, avoiding TiO reduces the amplitude of the photometric variations seen in LPVs. The result is a lower dispersion and a well defined period–luminosity (PL) relation.
For the LMC sample we find an rms dispersion of 0.27 mag in the narrow-band PL relation and slightly larger dispersions for the LPVs in Per OB1 and M33. This dispersion is comparable to that of the Cepheid PL relation at similar wavelengths. Adopting a distance modulus of 18.5±0.1 mag for the LMC, we obtain distance moduli of 11.68±0.15 mag for Per OB1 and 24.85±0.13 mag for M33. These distances agree well with those based on main sequence fitting for Per OB1 and the Cepheid distance for M33. Since LPVs are ∼ 5 times more common than Cepheids and have a well defined PL relation, LPVs provide a promising method for estimating Galactic and extra galactic distances.     

V440 Per: the longest-period overtone Cepheid

V440 Per is a Population I Cepheid with a period of 7.57 d and low-amplitude, almost sinusoidal light and radial velocity curves. With no reliable data on the first harmonic, its pulsation mode identification remained controversial. We obtained a radial velocity curve of V440 Per with our new high-precision and high-throughput Poznań Spectroscopic Telescope. Our data reach an accuracy of 130 m s⁻¹ per individual measurement and yield a secure detection of the first harmonic with an amplitude of   A ₂= 140 ± 15 m s⁻¹  . The velocity Fourier phase φ₂₁ of V440 Per is inconsistent at the 7.25σ level with those of fundamental-mode Cepheids, implying that the star must be an overtone Cepheid, as originally proposed by Kienzle et al. Thus, V440 Per becomes the longest-period Cepheid with securely established overtone pulsations. We show that a convective non-linear pulsation hydrocode can reproduce the Fourier parameters of V440 Per very well. The requirement to match the observed properties of V440 Per constrains the free parameters of the dynamical convection model used in the pulsation calculations, in particular the radiative loss parameter.     

Further empirical evidence for the non-linearity of the period–luminosity relations as seen in the Large Magellanic Cloud Cepheids

Recent studies, using OGLE data for LMC Cepheids in the optical, strongly suggest that the period–luminosity (PL) relation for the Large Magellanic Cloud (LMC) Cepheids shows a break or non-linearity at a period of 10 d. In this paper we apply statistical tests, the chi-squared test and the F -test, to the Cepheid data from the MACHO project to test for a non-linearity of the V - and R -band PL relations at 10 d, and extend these tests to the near-infrared ( JHK -band) PL relations with 2MASS data. We correct the extinction for these data by applying an extinction map towards the LMC. The statistical test we use, the F -test, is able to take account of small numbers of data points and the nature of that data on either side of the period cut at 10 d. With our data, the results we obtained imply that the VRJH -band PL relations are non-linear around a period of 10 d, while the K -band PL relation is (marginally) consistent with a single-line regression. The choice of a period of 10 d, around which this non-linearity occurs, is consistent with the results obtained when this 'break' period is estimated from the data. We show that robust parametric (including least-squares, least absolute deviation, robust regression) and non-parametric regression methods, which restrict the influence of outliers, produce similar results. Long-period Cepheids are supplemented from the literature to increase our sample size. The photometry of these long-period Cepheids is compared with our data and no trend with period is found. Our main results remain unchanged when we supplement our data set with these long-period Cepheids. By examining our data at maximum light, we also suggest arguments as to why errors in reddening are unlikely to be responsible for our results. The non-linearity of the mean V -band PL relation as seen in both of the OGLE and the MACHO data, using different extinction maps, suggests that this non-linearity is real.     

Cepheid parallaxes and the Hubble constant

Revised Hipparcos parallaxes for classical Cepheids are analysed together with 10 Hubble Space Telescope ( HST )-based parallaxes. In a reddening-free V , I relation we find that the coefficient of log  P is the same within the uncertainties in our Galaxy as in the Large Magellanic Cloud (LMC), contrary to some previous suggestions. Cepheids in the inner region of NGC 4258 with near solar metallicities confirm this result. We obtain a zero-point for the reddening-free relation and apply it to the Cepheids in galaxies used by Sandage et al. to calibrate the absolute magnitudes of Type Ia supernova (SNIa) and to derive the Hubble constant. We revise their result for H ₀ from 62 to 70 ± 5 km s⁻¹ Mpc⁻¹. The Freedman et al. value is revised from 72 to 76 ± 8 km s⁻¹ Mpc⁻¹. These results are insensitive to Cepheid metallicity corrections. The Cepheids in the inner region of NGC 4258 yield a modulus of 29.22 ± 0.03 (int.) compared with a maser-based modulus of 29.29 ± 0.15. Distance moduli for the LMC, uncorrected for any metallicity effects, are 18.52 ± 0.03 from a reddening-free relation in V , I ; 18.47 ± 0.03 from a period–luminosity relation at K ; 18.45 ± 0.04 from a period–luminosity–colour relation in J , K . Adopting a metallicity correction in V , I from Macri et al. leads to a true LMC modulus of 18.39 ± 0.05.     

The period–luminosity relation for type II Cepheids in globular clusters

We report the result of our near-infrared observations ( JHK _s) for type II Cepheids (including possible RV Tau stars) in galactic globular clusters. We detected variations of 46 variables in 26 clusters (10 new discoveries in seven clusters) and present their light curves. Their periods range from 1.2 d to over 80 d. They show a well-defined period–luminosity relation at each wavelength. Two type II Cepheids in NGC 6441 also obey the relation if we assume the horizontal branch stars in NGC 6441 are as bright as those in metal-poor globular clusters in spite of the high metallicity of the cluster. This result supports the high luminosity which has been suggested for the RR Lyr variables in this cluster. The period–luminosity relation can be reproduced using the pulsation equation     assuming that all the stars have the same mass. Cluster RR Lyr variables were found to lie on an extrapolation of the period–luminosity relation. These results provide important constraints on the parameters of the variable stars.
Using Two Micron All-Sky Survey (2MASS) data, we show that the type II Cepheids in the Large Magellanic Cloud (LMC) fit our period–luminosity relation within the expected scatter at the shorter periods. However, at long periods (   P > 40  d, i.e. in the RV Tau star range) the LMC field variables are brighter by about one magnitude than those of similar periods in galactic globular clusters. The long-period cluster stars also differ from both these LMC stars and galactic field RV Tau stars in a colour–colour diagram. The reasons for these differences are discussed.     

Pulsational MV versus [Fe/H] relation(s) for globular cluster RR Lyrae variables

We use the results from recent computations of updated non-linear convective pulsating models to constrain the distance modulus of Galactic globular clusters through the observed periods of first-overtone (RR _c ) pulsators. The resulting relation between the mean absolute magnitude of RR Lyrae stars 〈 M _V (RR)〉 and the heavy element content [Fe/H] appears well in the range of several previous empirical calibrations, but with a non-linear dependence on [Fe/H] so that the slope of the relation increases when moving towards larger metallicities. On this ground, our results suggest that metal-poor ([Fe/H]<−1.5) and metal-rich ([Fe/H]>−1.5) variables follow two different linear 〈 M _V (RR)〉−[Fe/H] relations. Application to RR Lyrae stars in the metal-poor globular clusters of the Large Magellanic Cloud (LMC) provides an LMC distance modulus of the order of 18.6 mag, thus supporting the 'long' distance scale. The comparison with recent predictions based on updated stellar evolution theory is briefly presented and discussed.     

AGB variables and the Mira period–luminosity relation

Published data for large-amplitude asymptotic giant branch variables in the Large Magellanic Cloud (LMC) are re-analysed to establish the constants for an infrared ( K ) period–luminosity relation of the form   M _K =ρ[log  P − 2.38]+δ  . A slope of  ρ=−3.51 ± 0.20  and a zero-point of  δ=−7.15 ± 0.06  are found for oxygen-rich Miras (if a distance modulus of 18.39 ± 0.05 is used for the LMC). Assuming this slope is applicable to Galactic Miras we discuss the zero-point for these stars using the revised Hipparcos parallaxes together with published very long baseline interferometry (VLBI) parallaxes for OH masers and Miras in globular clusters. These result in a mean zero-point of  δ=−7.25 ± 0.07  for O-rich Galactic Miras. The zero-point for Miras in the Galactic bulge is not significantly different from this value.
Carbon-rich stars are also discussed and provide results that are consistent with the above numbers, but with higher uncertainties. Within the uncertainties there is no evidence for a significant difference between the period–luminosity relation zero-points for systems with different metallicity.     

Carbon-rich Mira variables: kinematics and absolute magnitudes

The kinematics of Galactic C-Miras are discussed on the basis of the bolometric magnitudes and radial velocities of Papers I and II of this series. Differential Galactic rotation is used to derive a zero-point for the bolometric period–luminosity relation which is in satisfactory agreement with that inferred from the Large Magellanic Cloud (LMC) C-Miras. We find for the Galactic Miras,   M _bol=−2.54 log  P + 2.06(±0.24)  , where the slope is taken from the LMC. The mean velocity dispersion, together with the data of Nordström et al. and the Padova models, leads to a mean age for our sample of C-Miras of  1.8 ± 0.4 Gyr  and a mean initial mass of  1.8 ± 0.2 M_⊙  . Evidence for a variation of velocity dispersion with period is found, indicating a dependence of period on age and initial mass, the longer period stars being younger. We discuss the relation between the O- and C-Miras and also their relative numbers in different systems.     

Globular cluster systems in nearby dwarf galaxies – I. HST/ACS observations and dynamical properties of globular clusters at low environmental density

We investigate the old globular cluster (GC) population of 68 faint  ( M _V > −16 mag)  dwarf galaxies located in the halo regions of nearby (≲12 Mpc) loose galaxy groups and in the field environment based on archival Hubble Space Telescope ( HST )/Advanced Camera for Surveys (ACS) images in F606W and F814W filters. The combined colour distribution of 175 GC candidates peaks at  ( V − I ) = 0.96 ± 0.07 mag  and the GC luminosity function turnover for the entire sample is found at   M _{V ,TO}=−7.6 ± 0.11 mag  , similar to the old metal-poor Large Magellanic Cloud (LMC) GC population. Our data reveal a tentative trend of   M _{V ,TO}  becoming fainter from late- to early-type galaxies. The luminosity and colour distributions of GCs in dIrrs show a lack of faint blue GCs (bGCs). Our analysis reveals that this might reflect a relatively younger GC system than typically found in luminous early-type galaxies. If verified by spectroscopy, this would suggest a later formation epoch of the first metal-poor star clusters in dwarf galaxies. We find several bright (massive) GCs which reside in the nuclear regions of their host galaxies. These nuclear clusters have similar luminosities and structural parameters as the peculiar Galactic clusters suspected of being the remnant nuclei of accreted dwarf galaxies, such as M54 and ωCen. Except for these nuclear clusters, the distribution of GCs in dIrrs in the half-light radius versus cluster mass plane is very similar to that of Galactic young halo clusters, which suggests comparable formation and dynamical evolution histories. A comparison with theoretical models of cluster disruption indicates that GCs in low-mass galaxies evolve dynamically as self-gravitating systems in a benign tidal environment.     

Contribution of the Large Magellanic Cloud to the Galactic warp

Multi-scale interaction between the LMC, the Galactic halo, and the disk is examined with N-body simulations, and precise amplitudes of the Galactic warp excitation are obtained. The Galactic models are constructed most realistically to satisfy available observational constraints on the local circular velocity, the mass, surface density and thickness of the disk, the mass and size of the bulge, the local density of the halo matter at the solar radius, and the mass and orbit of the LMC. The mass of the halo within R=50 kpc is set to about 5×10¹¹ M_⊙. Since the observational estimate of the mass distributed in outer region has large ambiguity, two extreme cases are examined; M(<170 kpc)=2.1 and 0.9×10¹² M_⊙. LMC is orbiting in a ellipse with apocentric radii of 100 kpc, thus the main difference between our two models is the mass density in the satellite orbiting region, so that our study can clarify the role of the halo on excitation of the warp.By using hybrid algorithm (SCF–TREE) I have succeeded to follow the evolution with millions of particles. The orbiting satellite excites density enhancement as a wake, and the wake exerts a tidal force on the disk. Because of the additional torque from the wakes in the halo, the amplitudes of the induced warps are much larger than the classical estimate by Hunter and Toomre [ApJ 155 (1969) 747], who considered only the direct torque from the LMC. The obtained amplitudes of m=0, 1, 2 warps in the larger halo model show very good agreement with the observed amplitude in the Milky Way. This result revives the LMC as a possible candidate of the origin of the Galactic warp. Our smaller halo model, however, yield only weak warps in all the harmonic modes. Therefore, the halo still has significant influence on excitation of warp even in the interaction scenario for excitation of warps.     

On the use of Principal Component Analysis in analysing Cepheid light curves

We show how Principal Component Analysis can be used to analyse the structure of Cepheid light curves. This method is more efficient than Fourier analysis at bringing out changes in light curve shape as a function of period. Using this technique, we study the shape of fundamental and first-overtone mode Cepheid light curves in the Galaxy, LMC and SMC over a wide period range. For fundamentals, we find evidence for structural changes at     It is suggested that the feature at     is associated with a resonance in the Cepheid normal-mode spectrum. For overtones, we recover the Z shape in the R ₂₁ period plane and reproduce the metallicity dependence of this Z shape.     

The mid-infrared colours of Galactic bulge, disc and Magellanic planetary nebulae

We present mid-infrared (MIR) photometry for 367 Galactic disc, bulge and Large Magellanic Cloud (LMC) planetary nebulae (PNe), determined using data acquired with the Spitzer Space Telescope , and through the Legacy Programs GLIMPSE II (Galactic Legacy Infrared Mid-plane Survey Extraordinaire II) and SAGE (Surveying the Agents of the Galaxy's Evolution). This has permitted us to make a comparison between the luminosity functions of bulge and LMC PNe, and between the MIR colours of all three categories of source. It is determined that whilst the  3.6 μm  luminosity functions of the LMC and bulge sources are likely to be closely similar, the [3.6]–[5.8] and [5.8]–[8-0] indices of LMC nebulae are different from those of their disc and bulge counterparts. This may arise because of enhanced  6.2 μm  polycyclic aromatic hydrocarbon emission within the LMC sources, and/or as a result of further, and more radical differences between the spectra of LMC and Galactic PNe. We also determine that the more evolved disc sources listed in the Macquarie/AAO/Strasbourg (MASH) catalogues of Parker et al. and Miszalski et al. have similar colours to those of the less evolved (and higher surface brightness) sources in the catalogue of Acker et al., a result which appears at variance with previous studies of these sources.     

Globular clusters and the Mira period–luminosity relation

A globular cluster distance scale based on Hipparcos parallaxes of subdwarfs has been used to derive estimates of M _K for cluster Miras, including one in the Small Magellanic Cloud (SMC) globular cluster NGC 121. These lead to a zero-point of the Mira infrared period–luminosity (PL) relation, PL( K ), in good agreement with that derived from Hipparcos parallaxes of nearby field Miras. The mean of these two estimates together with data on LMC Miras yields a Large Magellanic Cloud (LMC) distance modulus of     in evident agreement with a metallicity-corrected Cepheid modulus     .
The use of luminous asymptotic giant branch (AGB) stars as extragalactic population indicators is also discussed.     

Dynamics of an interacting luminous disc, dark halo and satellite companion

This paper describes a method, based on linear perturbation theory, to determine the dynamical interaction between extended halo and spheroid components and an environmental disturbance. One finds that resonant interaction between a galaxy and passing interlopers or satellite companions can carry the disturbance inward, deep inside the halo, where it can perturb the disc.    Applied to the Milky Way for example, the LMC and SMC appear to be sufficient to cause the observed Galactic warp and possibly seed other asymmetries. This is a multi-scale interaction in which the halo wake has a feature at roughly half the satellite orbital radius owing to a 2:1 orbital resonance. The rotating disturbance then excites an m  = 1 vertical disc mode which has the classic integral-sign morphology. A polar satellite orbit produces the largest warp and therefore the inferred LMC orbit is nearly optimal for maximum warp production.   Both the magnitude and morphology of the response depend on the details of the disc and halo models. Most critically, a change in the halo profile will shift the resonant frequencies and response location and consequently alter the coupling to the bending disc. Increasing the halo support relative to the disc, a sub-maximal disc model, decreases the warp amplitude.   Finally, the results and prognosis for N -body simulations are discussed. Discreteness noise in the halo, similar to that arising from a population of 10⁶-M⊙ black holes, can produce observable warping.