Average Light Curve Research Articles

Two Models for the Orbital Modulation of Gamma Rays in Cyg X-3

We model the currently available γ-ray data on Cyg X-3 from the Fermi Large Area Telescope. Thanks to Cyg X-3’s very strong γ-ray activity during 2018–2021, the data quality has significantly improved. We study the strong orbital modulation of the γ-rays observed at high γ-ray fluxes. The modulation, as found earlier, is well modeled by anisotropic Compton scattering of the donor blackbody emission by relativistic electrons in a jet strongly misaligned with respect to the orbital axis. We confirm that this model fits well both the average γ-ray modulation light curve and the spectrum. However, we find that if the jet were aligned with the spin axis of a rotating black hole, it would undergo geodetic precession with a period of ∼50 yr. However, its presence is ruled out by both the γ-ray and radio data. Therefore, we consider an alternative model in which the average jet direction is aligned, but it is bent outside the orbit owing to the thrust of the donor stellar wind, and thus precesses at the orbital period. The γ-ray modulation then appears as a result of the variable Doppler boosting of synchrotron self-Compton jet emission. This model also fits the data well. However, the fitted bending angle is much larger than the theoretical one based on the binary and wind parameters as currently known. Thus, both models disagree with important aspects of our current theoretical understanding of the system. We discuss possible ways to find the correct model.

Secondary eclipse and day-night modulation of exoplanets observed by transiting exoplanet survey satellite

Astronomers have identified over five thousand extrasolar planets since the discovery of the first solar-like star in 1995. These systems exhibit greater diversity than our solar system, providing insights into their formation and evolution. To understand these exoplanets and their habitability, it is crucial to examine their atmosphere composition. The Transiting Exoplanet Survey Satellite (TESS) can detect reflected light from exoplanets with high accuracy, providing valuable insights into their formation and potential for life in the Universe. Planetary reflection can be detected by monitoring the light curve, which is the sum of stellar flux and planetary reflection. Orbiting planets may also reveal their presence by modulating reflected starlight, showing day-night variations in observers sight lines. However, these flux variations are extremely small, at a level of 0.01%, making detection difficult. This study developed an efficient Python program to search for secondary eclipse and day-night modulation on the TESS light curves of over four hundred known short-period transiting exoplanets. The program automatically identifies and rejects low-quality photometric data points in light curves, folding high-quality light curves in the orbital phase determined by primary transits. The phase-folded average light curves have high photometric precision, suitable for detecting secondary eclipse and day-night modulation. The program successfully identified planetary reflection in seventeen transiting exoplanetary systems, larger than the current TESS sample by twice. The program plans to upgrade to search for orbital phase modulation in non-transit exoplanets, taking advantage of the high precision of TESS photometry.

TELAMON: Effelsberg monitoring of AGN jets with very-high-energy astroparticle emission

Aims. We introduce the TELAMON program which is using the Effelsberg 100-m telescope to monitor the radio spectra of active galactic nuclei (AGN) under scrutiny in astroparticle physics, specifically TeV blazars and candidate neutrino-associated AGN. Here, we present and characterize our main sample of TeV-detected blazars. Methods. We analyzed the data sample from the first ∼2.5 yr of observations between August 2020 and February 2023 in the range from 14 GHz to 45 GHz. During this pilot phase, we observed all 59 TeV-detected blazars in the Northern Hemisphere (i.e., Dec > 0°) known at the time of observation. We discuss the basic data reduction and calibration procedures used for all TELAMON data and introduce a sub-band averaging method used to calculate average light curves for the sources in our sample. Results. The TeV-selected sources in our sample exhibit a median flux density of 0.12 Jy at 20 mm, 0.20 Jy at 14 mm, and 0.60 Jy at 7 mm. The spectrum for most of the sources is consistent with a flat radio spectrum and we found a median spectral index (S(ν)∝να) of α = −0.11. Our results on flux density and spectral index are consistent with previous studies of TeV-selected blazars. Compared to the GeV-selected F-GAMMA sample, TELAMON sources are significantly fainter in the radio band. This is consistent with the double-humped spectrum of blazars being shifted towards higher frequencies for TeV-emitters (in particular for high-synchrotron peaked BL Lac type objects), which results in a lower radio flux density. The spectral index distribution of our TeV-selected blazar sample is not significantly different from the GeV-selected F-GAMMA sample. Moreover, we present a strategy to track the light curve evolution of sources in our sample for future variability and correlation analysis.

Understanding the nature of the optical emission in gamma-ray bursts: analysis from TAROT, COATLI, and RATIR observations

ABSTRACT We collected the optical light-curve data of 227 gamma-ray bursts (GRBs) observed with the TAROT, COATLI, and RATIR telescopes. These consist of 133 detections and 94 upper limits. We constructed average light curves in the observer and rest frames in both X-rays (from Swift/X-Ray Telescope) and the optical. Our analysis focused on investigating the observational and intrinsic properties of GRBs. Specifically, we examined observational properties, such as the optical brightness function of the GRBs at T = 1000 s after the trigger, as well as the temporal slope of the afterglow. We also estimated the redshift distribution for the GRBs within our sample. Of the 227 GRBs analysed, we found that 116 had a measured redshift. Based on these data, we calculated a local rate of ρ0 = 0.2 Gpc−3 yr−1 for these events with z < 1. To explore the intrinsic properties of GRBs, we examined the average X-ray and optical light curves in the rest frame. We use the afterglowpy library to generate synthetic curves to constrain the parameters typical of the bright GRB jet, such as energy (〈E0〉 ∼ 1053.6 erg), opening angle (〈θcore〉 ∼ 0.2 rad), and density (〈n0〉 ∼ 10−2.1 cm−3). Furthermore, we analyse microphysical parameters, including the fraction of thermal energy in accelerated electrons (〈ϵe〉 ∼ 10−1.37) and in the magnetic field (〈ϵB〉 ∼ 10−2.26), and the power-law index of the population of non-thermal electrons (〈p〉 ∼ 2.2).

ВПЛИВ ЕЛІПТИЧНОСТІ ДЖЕРЕЛА НА КРИВІ БЛИСКУ В ПОЗАГАЛАКТИЧНИХ ГРАВІТАЦІЙНО-ЛІНЗОВИХ СИСТЕМАХ

In extragalactic gravitational lens system (GLS) several macro-images of a remote source (quasar) are observed. Gravitational microlensing leads to uncorrelated brightness variations of the macro-images. These variations can be separated from proper brightness changes of the source, which are repeated in each image with some relative delay. In most papers the gravitational microlensing effects in extragalactic GLS typically deal with centrally symmetric sources, though the real form of the source can be more complicated. In this paper we consider statistical signals of the source ellipticity in light curves of microlensed objects, which arise as a result of observations of many high amplification events. As a first step, we consider a qualitative version of the problem, when an elliptic source with Gaussian brightness distribution is crossed by a number of simple (fold) caustics with different orientations. Also we assume that the caustic network is isotropic, that is all orientations of the caustics and directions of their velocities are equally possible. We use the linear fold caustic approximation for the amplification of a small point source, when two critical images appear/disappear after the caustic crossing. For a rough estimate, the strength and velocity of the caustic are fixed by certain characteristic values. We estimate the averaged light curve after many caustic crossing events and compare maxima of brightness on the light curves for different eccentricities of the source. The ellipticity signal in the light curves is noticeable for a considerable oblateness of the source. Though we expect that the effect will be smeared out after taking into account the distribution of the caustic velocities and caustic strengths. Possible complications of the problem are indicated due to interplay of effects due to velocity distribution of stars in the lensing galaxy and bulk motion of the lensing galaxy as a whole.

Discovery of deep eclipses in the cataclysmic variable IPHAS J013031.89+622132.3

I performed photometric observations of the poorly studied cataclysmic variable IPHAS J013031.89+622132.3 and discovered very deep eclipses. I obtained observations over 14 nights for a total time of 50 hours during a time span of 6 months. Thanks to the long observation interval, I determined the orbital period with high precision, Porb=0.14935014±0.00000020 d. I derived the eclipse ephemeris, which, thanks to the precision of the orbital period, has a formal validity of 300 years. The average eclipse depth was 1.88 ± 0.07 mag. The prominent parts of the eclipses were smooth and symmetrical. The average eclipse width, including extended asymmetric eclipse wings, was 0.18 ± 0.01 phases or 40 ± 2 min. The average orbital light curve did not show a prominent orbital hump. Because no dwarf nova outburst occurred during the 6 months of monitoring, this cataclysmic variable is likely to be a nova-like variable.

A cumulative search for hard X/γ-ray emission associated with fast radio bursts in Fermi/GBM data

Context. Fast radio bursts (FRBs) are millisecond-long bursts uniquely detected at radio frequencies. FRB 131104 is the only case for which a γ-ray transient positionally and temporally consistent was claimed. This high-energy transient had a duration of ∼400 s and a 15–150 keV fluence Sγ ∼ 4 × 10−6 erg cm−2. However, the association with the FRB is still debated. Aims. We aim at testing the systematic presence of an associated transient high-energy counterpart throughout a sample of the FRB population. Methods. We used an approach like that used in machine learning methodologies to accurately model the highly-variable Fermi/GBM instrumental background on a time interval comparable to the duration of the proposed γ-ray counterpart of FRB 131104. A possible γ-ray signal is then constrained considering sample average lightcurves. Results. We constrain the fluence of the possible γ-ray signal in the 8–1000 keV band down to 6.4 × 10−7 (7.1 × 10−8) erg cm−2 for a 200-s (1-s) integration time. Furthermore, we found the radio-to-gamma fluence ratio to be η > 108 Jy ms erg−1 cm2. Conclusions. Our fluence limits exclude ∼94% of Fermi/GBM detected long gamma-ray bursts and ∼96% of Fermi/GBM detected short gamma-ray bursts. In addition, our limits on the radio-to-gamma fluence ratio point to a different emission mechanism from that of magnetar giant flares. Finally, we exclude a γ-ray counterpart as fluent as the one possibly associated with FRB 131104 to be a common feature of FRBs.

Precise measures of orbital period, before and after nova eruption for QZ Aurigae

For the ordinary classical nova QZ Aurigae (which erupted in 1964), we report 1317 magnitudes from 1912--2016, including four eclipses detected on archival photographic plates from long before the eruption. We have accurate and robust measures of the orbital period both pre-eruption and post-eruption, and we find that the orbital period decreased, with a fractional change of -290.71+-0.28 parts-per-million across the eruption, with the orbit necessarily getting smaller. Further, we find that the light curve outside of eclipses and eruption is flat at near B=17.14 from 1912--1981, whereupon the average light curve starts fading down to B=17.49 with large variability. QZ Aur is a robust counter-example against the Hibernation model for the evolution of cataclysmic variables, where the model requires that all novae have their period increase across eruptions. Large period decreases across eruptions can easily arise from mass imbalances in the ejecta, as are commonly seen in asymmetric nova shells.

Relations between the Spectral Indices and Flux Densities of Eight Blazars

Relations between the flux densities (F) and spectral indices (α) can help us analyze the emission process. In this paper, we choose 8 blazars (0235+164, 0430+052, 1156+295, 3C345, 1308+326, 1413+135, 3C454.3, and 1749+096) from the University of Michigan Radio Observatory (UMRAO) database to study the relations between the spectral indices (α) and flux densities at 14.5GHz (F14.5). The main results are the following. (1) There are strong anticorrelations between α and F14.5, with the correlation coefficient r in the range from -0.33 to -0.87. (2) The α-F14.5 distributions show elliptic appearance, which have been fitted by elliptic curves. (3) For most of the sources, the time intervals of α-F14.5 elliptic circle (Tc) are consistent with the quasi-periodicities calculated by the averaged light curves and spectral variances (PαF¯).

Maser flare simulations from oblate and prolate clouds

We investigated, through numerical models, the flaring variability that may arise from the rotation of maser clouds of approximately spheroidal geometry, ranging from strongly oblate to strongly prolate examples. Inversion solutions were obtained for each of these examples over a range of saturation levels from unsaturated to highly saturated. Formal solutions were computed for rotating clouds with many randomly chosen rotation axes, and corresponding averaged maser light curves plotted with statistical information. The dependence of results on the level of saturation and on the degree of deformation from the spherical case were investigated in terms of a variability index and duty cycle. It may be possible to distinguish observationally between flares from oblate and prolate objects. Maser flares from rotation are limited to long timescales (at least a few years) and modest values of the variability index ($\lesssim 100$), and can be aperiodic or quasi-periodic. Rotation is therefore not a good model for H$_2$O variability on timescales of weeks to months, or of truly periodic flares.

Chromatic transit light curves of disintegrating rocky planets

Context. Kepler observations have revealed a class of short-period exoplanets, of which Kepler-1520 b is the prototype, which have comet-like dust tails thought to be the result of small, rocky planets losing mass. The shape and chromaticity of the transits constrain the properties of the dust particles originating from the planet’s surface, offering a unique opportunity to probe the composition and geophysics of rocky exoplanets. Aims. We aim to approximate the average Kepler long-cadence light curve of Kepler-1520 b and investigate how the optical thickness and transit cross section of a general dust tail can affect the observed wavelength dependence and depth of transit light curves. Methods. We developed a new 3D model that ejects sublimating particles from the planet surface to build up a dust tail, assuming it to be optically thin, and used 3D radiative transfer computations that fully treat scattering using the distribution of hollow spheres (DHS) method, to generate transit light curves between 0.45 and 2.5 μm. Results. We show that the transit depth is wavelength independent of optically thick tails, potentially explaining why only some observations indicate a wavelength dependence. From the 3D nature of our simulated tails, we show that their transit cross sections are related to the component of particle ejection velocity perpendicular to the planets orbital plane and use this to derive a minimum ejection velocity of 1.2 km s−1. To fit the average transit depth of Kepler-1520 b of 0.87%, we require a high dust mass-loss rate of 7−80 M⊕ Gyr−1 which implies planet lifetimes that may be inconsistent with the observed sample. Therefore, these mass loss rates should be considered to be upper limits.

Precursors of short gamma-ray bursts detected by the INTEGRAL observatory

We have analyzed the light curves of 527 short gamma-ray bursts (GRBs) registered by the SPI-ACS, SPI and IBIS/ISGRI experiments of INTEGRAL observatory totally to search for precursors. Both the light curves of each 527 individual burst and the averaged light curve of 372 brightest SPI-ACS bursts have been analyzed. In a few cases, we have found and investigated precursor candidates using SPI-ACS, SPI and IBIS/ISGRI of INTEGRAL, GBM and LAT of Fermi data. No convincing evidence for the existence of precursors of short GRBs has been found. A statistical analysis of the averaged light curve for the sample of brightest short bursts has revealed no regular precursor. Upper limits for the relative intensity of precursors have been estimated. We show that the fraction of short GRBs with precursors is less than 0.4% of all short bursts, detected by INTEGRAL.

Mapping the accretion disc of the short period eclipsing binary SDSS J0926+3624

We report the analysis of time-series of optical photometry of SDSS J0926+3624 collected with the Liverpool Robotic Telescope between 2012 February and March while the object was in quiescence. We combined our median eclipse timing with those in the literature to revise the ephemeris and confirm that the binary period is increasing at a rate $\dot{P}=(3.2 \pm 0.4)\times 10^{-13} \, s/s$. The light curves show no evidence of either the orbital hump produced by a bright spot at disc rim or of superhumps; the average out-of-eclipse brightness level is consistently lower than previously reported. The eclipse map from the average light curve shows a hot white dwarf surrounded by a faint, cool accretion disc plus enhanced emission along the gas stream trajectory beyond the impact point at the outer disc rim, suggesting the occurrence of gas stream overflow/penetration at that epoch. We estimate a disc mass input rate of $\dot{M}=(9 \pm 1)\times 10^{-12}\,M_\odot \,yr^{-1}$, more than an order of magnitude lower than that expected from binary evolution with conservative mass transfer.

Photometry of the long period dwarf nova GY Hya

Although comparatively bright, the cataclysmic variable GY Hya has not attracted much attention in the past. As part of a project to better characterize such systems photometrically, we observed light curves in white light, each spanning several hours, at Bronberg Observatory, South Africa, in 2004 and 2005, and at the Observatório do Pico dos Dias, Brazil, in 2014 and 2016. These data permit to study orbital modulations and their variations from season to season. The orbital period, already known from spectroscopic observations of Peters and Thorstensen (2005), is confirmed through strong ellipsoidal variations of the mass donor star in the system and the presence of eclipses of both components. A refined period of 0.34723972 (6) days and revised ephemeries are derived. Seasonal changes in the average orbital light curve can qualitatively be explained by variations of the contribution of a hot spot to the system light together with changes of the disk radius. The amplitude of the ellipsoidal variations and the eclipse contact phases permit to put some constraints on the mass ratio, orbital inclination and the relative brightness of the primary and secondary components. There are some indications that the disk radius during quiescence, expressed in units of the component separation, is smaller than in other dwarf novae.

SDSS J105754.25+275947.5: a period-bounce eclipsing cataclysmic variable with the lowest-mass donor yet measured

We present high-speed, multicolour photometry of the faint, eclipsing cataclysmic variable (CV) SDSS J105754.25+275947.5. The light from this system is dominated by the white dwarf. Nonetheless, averaging many eclipses reveals additional features from the eclipse of the bright spot. This enables the fitting of a parameterised eclipse model to these average light curves, allowing the precise measurement of system parameters. We find a mass ratio of q = 0.0546 $\pm$ 0.0020 and inclination i = 85.74 $\pm$ 0.21$^{\circ}$. The white dwarf and donor masses were found to be M$_{\mathrm{w}}$ = 0.800 $\pm$ 0.015 M$_{\odot}$ and M$_{\mathrm{d}}$ = 0.0436 $\pm$ 0.0020 M$_{\odot}$, respectively. A temperature T$_{\mathrm{w}}$ = 13300 $\pm$ 1100 K and distance d = 367 $\pm$ 26 pc of the white dwarf were estimated through fitting model atmosphere predictions to multicolour fluxes. The mass of the white dwarf in SDSS 105754.25+275947.5 is close to the average for CV white dwarfs, while the donor has the lowest mass yet measured in an eclipsing CV. A low-mass donor and an orbital period (90.44 min) significantly longer than the period minimum strongly suggest that this is a bona fide period-bounce system, although formation from a white dwarf/brown dwarf binary cannot be ruled out. Very few period-minimum/period-bounce systems with precise system parameters are currently known, and as a consequence the evolution of CVs in this regime is not yet fully understood.

Precursors of short gamma-ray bursts in the SPI-ACS/INTEGRAL experiment

We have analyzed the light curves of 519 short gamma-ray bursts (GRBs) detected in the SPI-ACS/INTEGRAL experiment from December 2002 to May 2014 to search for precursors. Both the light curves of 519 individual events and the averaged light curve of 372 brightest bursts have been analyzed. In a few cases, we have found and thoroughly studied precursor candidates based on SPI-ACS/INTEGRAL, GBM/Fermi, and LAT/Fermi data. A statistical analysis of the averaged light curve for the entire sample of short bursts has revealed no regular precursor. Upper limits for the relative intensity of precursors have been estimated. No convincing evidence for the existence of precursors of short GRBs has been found. We show that the fraction of short GRBs with precursors is less than 0.4% of all short bursts.

Eclipsing SU UMa-Type Dwarf Nova 1RXS J003828.7+250920 During thE “Period Gap”. I. Multiperiodicity and Color Features in 2011-2012

Results are reported from multicolor photometric observations of the SU UMa-type dwarf nova 1RXS J003828.7+250920 in 2011, roughly a year after the super-outburst of 2010, and in 2012, when the object was in a quiescent state and underwent an outburst. Partial eclipses were observed in the system with depths of about 0m.6 in all color bands, both in the quiescent state and during the outburst. The orbital period of the system, 0.09451001(4) days, was determined for the first time and identifies the object as a dwarf nova in the “gap” in the orbital-period distribution of the cataclysmic variables. The ephemerides for the middle of the eclipse are determined. The average light curve profile resembles the “classical” light curve of U Gem. The orbital periodicity in the brightness variations of 1RXS J003828.7+250920 coexists with another signal with a period of 0.092 days, which we interpret as a period of negative superhumps. The color temperatures derived from the color indices may indicate multicomponent emission from 1RXS J003828.7+250920 by sources which appear to be hot and cold parts of the accretion disk and a hot spot on the accretion disk.

DETECTING THE COMPANIONS AND ELLIPSOIDAL VARIATIONS OF RS CVN PRIMARIES. II.oDRACONIS, A CANDIDATE FOR RECENT LOW-MASS COMPANION INGESTION

To measure the stellar and orbital properties of the metal-poor RS CVn binary o Draconis (o Dra), we directly detect the companion using interferometric observations obtained with the Michigan InfraRed Combiner at Georgia State University's Center for High Angular Resolution Astronomy (CHARA) Array. The H-band flux ratio between the primary and secondary stars is the highest confirmed flux ratio (370 +/- 40) observed with long-baseline optical interferometry. These detections are combined with radial velocity data of both the primary and secondary stars, including new data obtained with the Tillinghast Reflector Echelle Spectrograph on the Tillinghast Reflector at the Fred Lawrence Whipple Observatory and the 2-m Tennessee State University Automated Spectroscopic Telescope at Fairborn Observatory. We determine an orbit from which we find model-independent masses and ages of the components (M_A = 1.35 +\- 0.05 M_Sun, M_B = 0.99 +\- 0.02 M_Sun, system age = 3.0 -\+ 0.5 Gyr). An average of a 23-year light curve of o Dra from the Tennessee State University Automated Photometric Telescope folded over the orbital period newly reveals eclipses and the quasi-sinusoidal signature of ellipsoidal variations. The modeled light curve for our system's stellar and orbital parameters confirm these ellipsoidal variations due to the primary star partially filling its Roche lobe potential, suggesting most of the photometric variations are not due to stellar activity (starspots). Measuring gravity darkening from the average light curve gives a best-fit of beta = 0.07 +\- 0.03, a value consistent with conventional theory for convective envelope stars. The primary star also exhibits an anomalously short rotation period, which, when taken with other system parameters, suggests the star likely engulfed a low-mass companion that had recently spun-up the star.

DETECTING THE COMPANIONS AND ELLIPSOIDAL VARIATIONS OF RS CVn PRIMARIES. I.σGEMINORUM

To measure the properties of both components of the RS Canum Venaticorum binary σ Geminorum (σ Gem), we directly detect the faint companion, measure the orbit, obtain model-independent masses and evolutionary histories, detect ellipsoidal variations of the primary caused by the gravity of the companion, and measure gravity darkening. We detect the companion with interferometric observations obtained with the Michigan InfraRed Combiner at Georgia State University's Center for High Angular Resolution Astronomy Array with a primary-to-secondary H-band flux ratio of 270 ± 70. A radial velocity curve of the companion was obtained with spectra from the Tillinghast Reflector Echelle Spectrograph on the 1.5 m Tillinghast Reflector at Fred Lawrence Whipple Observatory. We additionally use new observations from the Tennessee State University Automated Spectroscopic and Photometric Telescopes (AST and APT, respectively). From our orbit, we determine model-independent masses of the components ( , ), and estimate a system age of Gyr. An average of the 27 year APT light curve of σ Gem folded over the orbital period ( days) reveals a quasi-sinusoidal signature, which has previously been attributed to active longitudes 180° apart on the surface of σ Gem. With the component masses, diameters, and orbit, we find that the predicted light curve for ellipsoidal variations due to the primary star partially filling its Roche lobe potential matches well with the observed average light curve, offering a compelling alternative explanation to the active longitudes hypothesis. Measuring gravity darkening from the light curve gives , a value slightly lower than that expected from recent theory.

PHL 1445: an eclipsing cataclysmic variable with a substellar donor near the period minimum

We present high-speed, three-colour photometry of the eclipsing dwarf nova PHL 1445, which, with an orbital period of 76.3 min, lies just below the period minimum of ~82 min for cataclysmic variable stars. Averaging four eclipses reveals resolved eclipses of the white dwarf and bright spot. We determined the system parameters by fitting a parameterised eclipse model to the averaged lightcurve. We obtain a mass ratio of q = 0.087 +- 0.006 and inclination i = 85.2 +- 0.9 degrees. The primary and donor masses were found to be Mw = 0.73 +- 0.03 Msun and Md = 0.064 +- 0.005 Msun, respectively. Through multicolour photometry a temperature of the white dwarf of Tw = 13200 +- 700 K and a distance of 220 +- 50 pc were determined. The evolutionary state of PHL 1445 is uncertain. We are able to rule out a significantly evolved donor, but not one that is slightly evolved. Formation with a brown dwarf donor is plausible; though the brown dwarf would need to be no older than 600 Myrs at the start of mass transfer, requiring an extremely low mass ratio (q = 0.025) progenitor system. PHL 1445 joins SDSS 1433 as a sub-period minimum CV with a substellar donor. These existence of two such systems raises an alternative possibility; that current estimates for the intrinsic scatter and/or position of the period minimum may be in error.