Differential Rotation Law Research Articles

Equilibriums of extremely magnetized compact stars with force-free magnetotunnels

We present numerical solutions for stationary and axisymmetric equilibriums of compact stars associated with extremely strong magnetic fields. The interior of the compact stars is assumed to satisfy ideal magnetohydrodynamic (MHD) conditions, while in the region of negligible mass density the force-free conditions or electromagnetic vacuum are assumed. Solving all components of Einstein's equations, Maxwell's equations, ideal MHD equations, and force-free conditions, equilibriums of rotating compact stars associated with mixed poloidal and toroidal magnetic fields are obtained. It is found that in the extreme cases the strong mixed magnetic fields concentrating in a toroidal region near the equatorial surface expel the matter and form a force-free toroidal magnetotunnel. We also introduce a new differential rotation law for computing solutions associated with force-free magnetosphere, and present other extreme models without the magnetotunnel.

Analytic model for photometric variation due to starspots on a differentially rotating star

Abstract We present an analytic model of the light-curve variation for stars with non-evolving starspots on a differentially rotating surface. The Fourier coefficients of the harmonics of the rotation period are expressed in terms of the latitude of the spot, ℓs, and the observer’s line-of-sight direction, ℓo, including the limb-darkening effect. We generate different realizations of multi-spots according to the model, and perform mock observations of the resulting light-curve modulations. We discuss to what extent one can recover the properties of the spots and the parameters for the differential rotation law from the periodogram analysis. Although our analytical model neglects the evolution of spots on the stellar surface (dynamical motion, creation, and annihilation), it provides a basic framework to interpret the photometric variation of stars, in particular from the existing Kepler data and the future space-born mission. It is also applicable to photometric modulations induced by rotation of various astronomical objects.

Models of binary neutron star remnants with tabulated equations of state

ABSTRACT The emergence of novel differential rotation laws that can reproduce the rotational profile of binary neutron star merger remnants has opened the way for the construction of equilibrium models with properties that resemble those of remnants in numerical simulations. We construct models of merger remnants, using a recently introduced 4-parameter differential rotation law and three tabulated, zero-temperature equations of state. The models have angular momenta that are determined by empirical relations, constructed through numerical simulations. After a systematic exploration of the parameter space of merger remnant equilibrium sequences, which includes the determination of turning points along constant angular momentum sequences, we find that a particular rotation law can reproduce the threshold mass to prompt collapse to a black hole with a relative difference of only $\sim 1{{\ \rm per\ cent}}$ with respect to numerical simulations, in all cases considered. Furthermore, our results indicate a possible correlation between the compactness of equilibrium models of remnants at the threshold mass and the compactness of maximum-mass non-rotating models. Another key prediction of binary neutron star merger simulations is a relatively slowly rotating inner region, where the angular velocity Ω (as measured by an observer at infinity) is mostly due to the frame dragging angular velocity ω. In our investigation of the parameter space of the adopted differential rotation law, we naturally find quasi-spherical (Type A) remnant models with this property. Our investigation clarifies the impact of the differential rotation law and of the equation of state on key properties of binary neutron star remnants and lays the groundwork for including thermal effects in future studies.

Evolution of brightness and magnetic features of young solar-type stars – I. The young G star HIP 89829

ABSTRACT The evolution in latitude of sunspots is a key feature of the cyclic solar dynamo. Here, we present the results of a spectroscopic and spectropolarimetric monitoring campaign on the young (∼20 Myr old) early G star HIP 89829, in order to investigate potential evolution in the distribution of the star’s spots and magnetic features. Our analysis of this G5V star spans eight epochs, from June 2010 to August 2015. The techniques of Doppler imaging and Zeeman–Doppler imaging were used to create brightness maps for each epoch and magnetic maps for two epochs. The brightness images show the star to have stable spot features with two main spot latitudes – a polar spot, often seen on young rapidly rotating stars such as this, and another highly unusual group of large spot features around the 20° and 30° latitudes. These lower spot latitudes appear to be rather stable over the 5 yr of observations. We included a solar-type differential rotation law into the imaging process and measured near-solid-body rotation for epochs where sufficient data exist for this analysis. The magnetic features show a dominant poloidal and a weaker toroidal magnetic field for both Stokes V epochs, which is unusual for a star with a rapid rotation period of 0.57 d. We conclude that HIP 89829 is an active young solar-type star with long-lived spots and near-solid-body rotation.

Long–Term Variations in Solar Differential Rotation and Sunspot Activity, II: Differential Rotation Around the Maxima and Minima of Solar Cycles 12 – 24

Studies of variations in the solar differential rotation are important for understanding the underlying mechanism of solar cycle and other variations of solar activity. We analyzed the sunspot-group daily data that were reported by Greenwich Photoheliographic Results (GPR) during the period 1874 – 1976 and Debrecen Photoheliographic Data (DPD) during the period 1977 – 2017. We determined the equatorial rotation rate [ $A$ ] and the latitude gradient [ $B$ ] components of the solar differential rotation by fitting the data in each of the 3-year moving time intervals (3-year MTIs) successively shifted by one year during the period 1874 – 2017 to the standard law of differential rotation. The values of $A$ and $B$ around the years of maxima and minima of Solar Cycles 12 – 24 are obtained from the 3-year MTIs series of $A$ and $B$ and studied the long-term cycle-to-cycle modulations in these coefficients. Here we have used the epochs of the maxima and minima of the Solar Cycles 12 – 24 that were recently determined from the revised Version-2 international sunspot-number series. We find that there exits a considerably significant secular decreasing-trend in $A$ around the maxima of solar cycles. There exist no secular trends in both $A$ and $B$ around the minima of solar cycles. The secular trend in $B$ around the maxima of solar cycles is also found to be statistically insignificant. We fitted a cosine function to the values of $A$ , and also to those of $B$ , after removing the corresponding linear trends. The cosine-fits suggest that there exist ≈54-year (≈94-year) and ≈82-year (≈79-year) periodicities in $A$ ( $B$ ) around the maxima and minima of solar cycles, respectively. The amplitude of the cosine-profile of $A$ ( $B$ ) around the minima of solar cycles is about 41% (65%) larger than that of $A$ ( $B$ ) around the maxima. In addition, the cosine profiles of $A$ and $B$ suggest a large (up to $180^{\circ }$ ) phase difference between the long-term variations of $A$ , and also between those of $B$ , around maxima and minima of solar cycles. Implications of all these results are discussed briefly.

A Differential Rotation Law for Stars and Fluid Planets

We derive, supported on a generalization of Bernoulli’s equation, a law of rotation for any axial-symmetric, self-gravitating fluid mass. For a homogeneous mass, the law depends solely on the derivative of the potential with respect to the distance to the rotation axis, implying generally differential rotation, the Maclaurin spheroids representing the only case of solid-body rotation. We turn then to a heterogeneous mass consisting of any number <i>l</i> of concentric layers, each of constant density, finding that the angular velocity profile of a given layer depends on that of the layer immediately above it. Finally, we let <i>l</i> tend to infinity to convert our model into continuous mass distribution, the result being a certain rotation profile for the surface, and law of differential rotation change at its interior. To support the fundamentals of our approach, we write the potential integrals for the three mass distributions. The aim of a continuous distribution is that it may facilitate a comparison---to be carried out in a future paper---between our results and those of other researchers who employ structure equations. We point out that the distribution of angular velocity is a consequence of the equilibrium, rather than being imposed <i>ad initio</i>. The law was used in a past paper to construct a Jupiter multi-layer model adopting the spheroidal (a distorted spheroid) shape for each of the layers, taking as reference the gravitational data surveyed by the Juno mission. The procedure used here is not restricted to axial-symmetric cases.

Differentially rotating strange star in general relativity

Rapidly and differentially rotating compact stars are believed to be formed in binary neutron star merger events, according to both numerical simulations and the multi-messenger observation of GW170817. The lifetime and evolution of such a differentially rotating star, is tightly related to the observations in the post-merger phase. Various studies on the maximum mass of differentially rotating neutron stars have been done in the past, most of which assume the so-called $j$-const law as the rotation profile inside the star and consider only neutron star equations of state. In this paper, we extend the studies to strange star models, as well as to a new rotation profile model. Significant differences are found between differentially rotating strange stars and neutron stars, with both differential rotation laws. A moderate differential rotation rate for neutron stars is found to be too large for strange stars, resulting in a rapid drop in the maximum mass as the differential rotation degree is increased further from $\hat{A}\sim2.0$, where $\hat{A}$ is a parameter characterizing the differential rotation rate for $j$-const law. As a result the maximum mass of a differentially rotating self-bound star drops below the uniformly rotating mass shedding limit for a reasonable degree of differential rotation. The continuous transition to the toroidal sequence is also found to happen at a much smaller differential rotation rate and angular momentum than for neutron stars. In spite of those differences, $\hat{A}$-insensitive relation between the maximum mass for a given angular momentum is still found to hold, even for the new differential rotation law. Astrophysical consequences of these differences and how to distinguish between strange star and neutron star models with future observations are also discussed.

Fractal Theory of Saturn’s Ring. II: Electromagnetic Phenomena

This article is a continuation of the author’s previous paper of 2015. It deals with electromagnetic phenomena in a model of a flat two-dimensional ring with a given law of differential rotation. The reasons of high electromagnetic activity in Saturn's ring are discussed. In parallel with the discussion, it is shown that similar reasons can clarify other processes such as the charging of thunderclouds. New aspects in the theory of magnetic fields of planets are also proposed.

Effects of mass variation on structures of differentially rotating polytropic stars

A method is proposed for determining equilibrium structures and various physical parameters of differentially rotating polytropic models of stars, taking into account the effect of mass variation inside the star and on its equipotential surfaces. The law of differential rotation has been assumed to be the form of ω2(s)=b1+b2s2+b3s4. The proposed method utilizes the averaging approach of Kippenhahn and Thomas and concepts of Roche-equipotential to incorporate the effects of differential rotation on the equilibrium structures of polytropic stellar models. Mathematical expressions of determining the equipotential surfaces, volume, surface area and other physical parameters are also obtained under the effects of mass variation inside the stars. Some significant conclusions are also drawn.

Gravity darkening in stars with surface differential rotation

Context. The interpretation of stellar apparent fundamental parameters (viewing-angle dependent) requires that they be treated consistently with the characteristics of their surface rotation law. Aims. We aim to develop a model to determine the distribution of the effective temperature and gravity, which explicitly depend on the surface differential rotation law and on the concomitant stellar external geometry. Methods. The basic assumptions in this model are: a) the external stellar layers are in radiative equilibrium; b) the emergent bolometric flux is anti-parallel with the effective gravity; c) the angular velocity in the surface obeys relations like Ω(θ) = Ωo [ 1 + αΥ(θ,k) ] where Υ(θ,k) = coskθ or sinkθ, and where (α,k) are free parameters. Results. The effective temperature varies with co-latitude θ, with amplitudes that depend on the differential-rotation law through the surface effective gravity and the gravity-darkening function (GDF). Although the derived expressions can be treated numerically, for some low integer values of k, analytical forms of the integral of characteristic curves, on which the determination of the GDF relies, are obtained. The effects of the quantities (η,α,k) (η = ratio between centrifugal and gravitational accelerations at the equator) on the determination of the Vsini parameter and on the gravity-darkening exponent are studied. Depending on the values of (η,α,k) the velocity V in the derived Vsini may strongly deviate from the equatorial rotational velocity. It is shown that the von Zeipel’s-like gravity-darkening exponent β1 depends on all parameters (η,α,k) and that its value also depends on the viewing-angle i. Hence, there no unique interpretation of this exponent determined empirically in terms of (i,α). Conclusions. We stress that the data on rotating stars should be analyzed by taking into account the rotational effects through the GDF, by assuming k = 2 as a first approximation. Instead of the classic pair (η,β1), it would be more useful to determine the quantities (η,α,i) to characterize stellar rotation.

Solar differential rotation in the period 1964–2016 determined by the Kanzelhöhe data set

The main aim of this work is to determine the solar differential rotation by tracing sunspot groups during the period 1964-2016, using the Kanzelh\"ohe Observatory for Solar and Environmental Research (KSO) sunspot drawings and white light images. Two procedures for the determination of the heliographic positions were applied: an interactive procedure on the KSO sunspot drawings (1964 - 2008, solar cycles nos. 20 - 23) and an automatic procedure on the KSO white light images (2009 - 2016, solar cycle no. 24). For the determination of the synodic angular rotation velocities two different methods have been used: a daily shift (DS) method and a robust linear least-squares fit (rLSQ) method. Afterwards, the rotation velocities had to be converted from synodic to sidereal, which were then used in the least-squares fitting for the solar differential rotation law. For the test data from 2014, we found the rLSQ method for calculating rotational velocities to be more reliable than the DS method. The best fit solar differential rotation profile for the whole time period is $\omega(b)$ = (14.47 $\pm$ 0.01) - (2.66 $\pm$ 0.10) $\sin^2b$ (deg/day) for the DS method and $\omega(b)$ = (14.50 $\pm$ 0.01) - (2.87 $\pm$ 0.12) $\sin^2b$ (deg/day) for the rLSQ method. A barely noticeable north - south asymmetry is observed for the whole time period 1964 - 2016 in the present paper. Rotation profiles, using different data sets (e.g. Debrecen Photoheliographic Data, Greenwich Photoheliographic Results), presented by other authors for the same time periods and the same tracer types, are in good agreement with our results. Therefore, the KSO data set is suitable for the investigation of the long-term variabilities in the solar rotation profile.

Rotational properties of hypermassive neutron stars from binary mergers

Determining the differential-rotation law of compact stellar objects produced in binary neutron stars mergers or core-collapse supernovae is an old problem in relativistic astrophysics. Addressing this problem is important because it impacts directly on the maximum mass these objects can attain and, hence, on the threshold to black-hole formation under realistic conditions. Using the results from a large number of numerical simulations in full general relativity of binary neutron star mergers described with various equations of state and masses, we study the rotational properties of the resulting hypermassive neutron stars. We find that the angular-velocity distribution shows only a modest dependence on the equation of state, thus exhibiting the traits of ``quasiuniversality'' found in other aspects of compact stars, both isolated and in binary systems. The distributions are characterized by an almost uniformly rotating core and a ``disk.'' Such a configuration is significantly different from the $j$-constant differential-rotation law that is commonly adopted in equilibrium models of differentially rotating stars. Furthermore, the rest-mass contained in such a disk can be quite large, ranging from $\ensuremath{\simeq}0.03\text{ }\text{ }{M}_{\ensuremath{\bigodot}}$ in the case of high-mass binaries with stiff equations of state, up to $\ensuremath{\simeq}0.2\text{ }\text{ }{M}_{\ensuremath{\bigodot}}$ for low-mass binaries with soft equations of state. We comment on the astrophysical implications of our findings and on the long-term evolutionary scenarios that can be conjectured on the basis of our simulations.

Equilibrium Configurations of Classical Polytropic Stars with a Multi-Parametric Differential Rotation Law: A Numerical Analysis

AbstractIn this paper we analyze in detail the equilibrium configurations of classical polytropic stars with a multi-parametric differential rotation law of the literature using the standard numerical method introduced by Eriguchi and Mueller. Specifically we numerically investigate the parameters’ space associated with the velocity field characterizing both equilibrium and non-equilibrium configurations for which the stability condition is violated or the mass-shedding criterion is verified.

Time-series Doppler images and surface differential rotation of the effectively single, rapidly rotating K-giant KU Pegasi

According to most stellar dynamo theories, differential rotation (DR) plays a crucial role for the generation of toroidal magnetic fields. Numerical models predict surface differential rotation to be anti-solar for rapidly-rotating giant stars, i.e., their surface angular velocity could increase with stellar latitude. However, surface differential rotation has been derived only for a handful of individual giant stars to date. The spotted surface of the K-giant KU Pegasi is investigated in order to detect its time evolution and quantify surface differential rotation. We present altogether 11 Doppler images from spectroscopic data collected with the robotic telescope STELLA between 2006--2011. All maps are obtained with the surface reconstruction code iMap. Differential rotation is extracted from these images by detecting systematic (latitude-dependent) spot displacements. We apply a cross-correlation technique to find the best differential rotation law. The surface of KU Peg shows cool spots at all latitudes and one persistent warm spot at high latitude. A small cool polar spot exists for most but not all of the epochs. Re-identification of spots in at least two consecutive maps is mostly possible only at mid and high latitudes and thus restricts the differential-rotation determination mainly to these latitudes. Our cross-correlation analysis reveals solar-like differential rotation with a surface shear of $\alpha=+0.040\pm0.006$, i.e., approximately five times weaker than on the Sun. We also derive a more accurate and consistent set of stellar parameters for KU Peg including a small Li abundance of ten times less than solar.

The Solar Rotation in the 1930s from the Sunspot and Flocculi Catalogs of the Ebro Observatory

The tables of sunspot and flocculi heliographic positions included in the catalogs published by the Ebro Observatory in the 1930s have recently been recovered and converted into digital format by using optical character recognition (OCR) technology. We here analyzed these data by computing the angular velocity of several sunspot and flocculi groups. A difference was found in the rotational velocity for sunspots and flocculi groups at high latitudes, and we also detected an asymmetry between the northern and southern hemispheres, which is especially marked for the flocculi groups. The results were then fitted with a differential-rotation law [ $\omega=a+b \sin^{2} B$ ] to compare the data obtained with the results published by other authors. A dependence on the latitude that is consistent with former studies was found. Finally, we studied the possible relationship between the sunspot/flocculi group areas and their corresponding angular velocity. There are strong indications that the rotational velocity of a sunspot/flocculi group is reduced (in relation to the differential rotation law) when its maximum area is larger.

Anti-solar differential rotation on the active sub-giant HU Virginis,

Measuring surface differential rotation (DR) on different types of stars is important when characterizing the underlying stellar dynamo. It has been suggested that anti-solar DR laws can occur when strong meridional flows exist. We aim to investigate the differential surface rotation on the primary star of the RS CVn binary HU Vir by tracking its starspot distribution as a function of time. We also aim to recompute and update the values for several system parameters of the triple system HU Vir (close and wide orbits). Time-series high-resolution spectroscopy for four continuous months was obtained with the 1.2-m robotic STELLA telescope. Nine consecutive Doppler images were reconstructed from these data, using our line-profile inversion code iMap. An image cross-correlation method was applied to derive the surface differential-rotation law for HU Vir. New orbital elements for the close and the wide orbits were computed using our new STELLA radial velocities (RVs) combined with the RV data available in the literature. Photometric observations were performed with the Amadeus Automatic Photoelectric Telescope (APT), providing contemporaneous Johnson-Cousins $V$ and $I$ data for approximately 20 years. This data was used to determine the stellar rotation period and the active longitudes. We confirm anti-solar DR with a surface shear parameter $\alpha$ of -0.029 $\pm$ 0.005 and -0.026 $\pm$ 0.009, using single-term and double-term differential rotation laws, respectively. The best fit is achieved assuming a solar-like double-term law with a lap time of $\approx$ 400 d. Our orbital solutions result in a period of 10.387678 $\pm$ 0.000003 days for the close orbit and 2726 $\pm$ 7 d ($\approx$ 7.5 yr) for the wide orbit. A Lomb-Scarge (L-S) periodogram of the pre-whitened $V$-band data reveals a strong single peak providing a rotation period of 10.391 $\pm$ 0.008 d.

New code for equilibriums and quasiequilibrium initial data of compact objects. III. Axisymmetric and triaxial rotating stars

We introduce new code for stationary and axisymmetric equilibriums, as well as for triaxial quasiequilibrium initial data, of single rotating relativistic stars. The new code is developed as a part of our versatile initial data code for compact objects, Compact Object CALculator (cocal). In computing strong gravitational fields, the waveless formulation is incorporated into the cocal code on top of the previously developed Isenberg-Wilson-Mathews formulation (conformally flat thin-sandwich formulation). Also introduced is a new differential rotation law that contains two parameters to control an angular velocity profile and a transition from uniform to differential rotation. We present convergence tests and solution sequences for both uniformly and differentially rotating equilibriums of stationary axisymmetric compact stars, as well as for quasiequilibrium initial data of uniformly rotating triaxial (nonaxisymmetric) compact stars. We also show comparisons of uniformly rotating axisymmetric solutions computed with three different codes: cocal, lorene, and the RNS code.

The Solar Rotation in the Period 1853 – 1870 from the Sunspot Catalogues of Carrington, Peters, and de la Rue

R.C. Carrington, C.H.F. Peters, and W. de la Rue observed the sunspots in the second half of the nineteenth century, determining their heliographic positions between 1853 and 1870, before the solar program of the Royal Greenwich Observatory was established. The large tables of sunspot positions included in the catalogues published by these observers have recently been converted into a machine-readable format. The present work analyses these data by calculating the sunspot group velocities for each observer. The results are then fitted with a differential-rotation law to compare the data of the three observers with each other and with the results published by others. Finally, we study the possible relationship between the sunspot-group areas as determined by de la Rue and the corresponding sunspot-group velocities.

Periodicity in some light curves of the solar analogue V352 Canis Majoris

Our aim was to study the light curve periodicity of the solar analogue V352 CMa and in particular show that the presence or absence of periodicity in low amplitude light curves can be modelled with the Continuous Period Search (CPS) method. We applied the CPS method to 14 years of V-band photometry of V352 CMa and obtained estimates for the mean, amplitude, period and minima of the light curves in the selected datasets. We also applied the Power Spectrum Method (PSM) to these datasets and compared the performance of this frequently applied method to that of CPS. We detected signs of a $11.7 \pm 0.5$ year cycle in in the mean brightness. The long-term average photometric rotation period was $7.24 \pm 0.22$ days. The lower limit for the differential rotation coefficient would be $|k| > 0.12$, if the law of solar surface differential rotation were valid for V352 CMa and the period changes traced this phenomenon. Signs of stable active longitudes rotating with a period of $7.157 \pm 0.002$ days were detected from the epochs of the light minima with the Kuiper method. CPS performed better than the traditional PSM, because the latter always used a sinusoidal model for the data even when this was clearly not the correct model.

Antisolar differential rotation of the K1-giantσ Geminorum revisited

Context. Surface differential rotation and other global surface flows on magnetically active stars are among the observable manifestations of the stellar dynamo working underneath. Therefore, such observations are important for stellar dynamo theory and useful constraints for solar dynamo studies as well. Aims. The active K1-giant component of the long-period RS CVn-type binary system sigma Gem and its global surface flow pattern is revisited. Methods. We refine the differential rotation law from recovering the spot migration pattern. We apply a detailed cross-correlation technique to a unique set of 34 time-series Doppler images recovered using data from 1996/97. By increasing the number of the available cross-correlation function maps from the formerly used 4 to 17 we expect a more robust determination of the differential surface rotation law. In addition, we present a new time-series Doppler imaging study of sigma Gem using our advanced surface reconstruction code iMap for a dataset collected in 2006/07. Results. Results from the reprocessed cross-correlation study confirm that the star performs antisolar-type differential rotation with a surface shear alpha of -0.04 +/- 0.01, i.e., almost a factor of two stronger compared to the previously claimed value. We also confirm the evidence of a global poleward spot migration with an average velocity of 0.21 +/- 0.03 km/s, in accordance with theoretical predictions. From the new observations we obtain three subsequent Doppler images. The time evolution of these images confirms the antisolar-type differential rotation of the same amount.