Detecting and quantifying stellar magnetic fields -- Sparse Stokes profile approximation using orthogonal matching pursuit

In the recent years, we have seen a rapidly growing number of stellar magnetic field detections for various types of stars. Many of these magnetic fields are estimated from spectropolarimetric observations (Stokes V) by using the so-called center-of-gravity (COG) method. Unfortunately, the accuracy of this method rapidly deteriorates with increasing noise and thus calls for a more robust procedure that combines signal detection and field estimation. We introduce an estimation method that provides not only the effective or mean longitudinal magnetic field from an observed Stokes V profile but also uses the net absolute polarization of the profile to obtain an estimate of the apparent (i.e., velocity resolved) absolute longitudinal magnetic field. By combining the COG method with an orthogonal-matching-pursuit (OMP) approach, we were able to decompose observed Stokes profiles with an overcomplete dictionary of wavelet-basis functions to reliably reconstruct the observed Stokes profiles in the presence of noise. The elementary wave functions of the sparse reconstruction process were utilized to estimate the effective longitudinal magnetic field and the apparent absolute longitudinal magnetic field. A multiresolution analysis complements the OMP algorithm to provide a robust detection and estimation method. An extensive Monte-Carlo simulation confirms the reliability and accuracy of the magnetic OMP approach.Comment: A&A, in press, 15 pages, 14 figure

Spot evolution on the red giant star XX Triangulum. A starspot-decay analysis based on time-series Doppler imaging

Solar spots appear to decay linearly proportional to their size. The decay rate of solar spots is directly related to magnetic diffusivity, which itself is a key quantity for the length of a magnetic-activity cycle. Is a linear spot decay also seen on other stars, and is this in agreement with the large range of solar and stellar activity cycle lengths? We investigate the evolution of starspots on the rapidly-rotating ($P_{\rm rot}$ $\approx$ 24 d) K0 giant XX Tri, using consecutive time-series Doppler images. Our aim is to obtain a well-sampled movie of the stellar surface over many years, and thereby detect and quantify a starspot decay law for further comparison with the Sun. We obtained continuous high-resolution and phase-resolved spectroscopy with the 1.2-m robotic STELLA telescope on Tenerife over six years. For each observing season, we obtained between 5 to 7 independent Doppler images, one per stellar rotation, making up a total of 36 maps. To quantify starspot area decay and growth, we match the observed images with simplified spot models based on a Monte Carlo approach. It is shown that the surface of XX Tri is covered with large high-latitude and even polar spots and with occasional small equatorial spots. Just over the course of six years, we see a systematically changing spot distribution with various timescales and morphology, such as spot fragmentation and spot merging as well as spot decay and formation. An average linear decay of $D$ = $-$0.022 $\pm$ 0.002 SH/day is inferred. We found evidence of an active longitude in phase toward the (unseen) companion star. Furthermore, we detect a weak solar-like differential rotation with a surface shear of $\alpha$ = 0.016 $\pm$ 0.003. From the decay rate, we determine a turbulent diffusivity of $\eta_T$ = (6.3 $\pm$ 0.5) $\times$ 10$^{14}$ cm$^2$/s and predict a magnetic activity cycle of $\approx$ 26 $\pm$ 6 years

A fast method for Stokes profile synthesis -- Radiative transfer modeling for ZDI and Stokes profile inversion

The major challenges for a fully polarized radiative transfer driven approach to Zeeman-Doppler imaging are still the enormous computational requirements. In every cycle of the iterative interplay between the forward process (spectral synthesis) and the inverse process (derivative based optimization) the Stokes profile synthesis requires several thousand evaluations of the polarized radiative transfer equation for a given stellar surface model. To cope with these computational demands and to allow for the incorporation of a full Stokes profile synthesis into Doppler- and Zeeman-Doppler imaging applications as well as into large scale solar Stokes profile inversions, we present a novel fast and accurate synthesis method for calculating local Stokes profiles. Our approach is based on artificial neural network models, which we use to approximate the complex non-linear mapping between the most important atmospheric parameters and the corresponding Stokes profiles. A number of specialized artificial neural networks, are used to model the functional relation between the model atmosphere, magnetic field strength, field inclination, and field azimuth, on one hand and the individual components (I,Q,U,V) of the Stokes profiles, on the other hand. We performed an extensive statistical evaluation and show that our new approach yields accurate local as well as disk-integrated Stokes profiles over a wide range of atmospheric conditions. The mean rms errors for the Stokes I and V profiles are well below 0.2% compared to the exact numerical solution. Errors for Stokes Q and U are in the range of 1%. Our approach does not only offer an accurate approximation to the LTE polarized radiative transfer it, moreover, accelerates the synthesis by a factor of more than 1000.Comment: A&A, in pres

The Magnetic Topology of the Weak-Lined T Tauri Star V410 - A Simultaneous Temperature and Magnetic Field Inversion

We present a detailed temperature and magnetic investigation of the T Tauri star V410 Tau by means of a simultaneous Doppler- and Zeeman-Doppler Imaging. Moreover we introduce a new line profile reconstruction method based on a singular value decomposition (SVD) to extract the weak polarized line profiles. One of the key features of the line profile reconstruction is that the SVD line profiles are amenable to radiative transfer modeling within our Zeeman-Doppler Imaging code iMap. The code also utilizes a new iterative regularization scheme which is independent of any additional surface constraints. To provide more stability a vital part of our inversion strategy is the inversion of both Stokes I and Stokes V profiles to simultaneously reconstruct the temperature and magnetic field surface distribution of V410 Tau. A new image-shear analysis is also implemented to allow the search for image and line profile distortions induced by a differential rotation of the star. The magnetic field structure we obtain for V410 Tau shows a good spatial correlation with the surface temperature and is dominated by a strong field within the cool polar spot. The Zeeman-Doppler maps exhibit a large-scale organization of both polarities around the polar cap in the form of a twisted bipolar structure. The magnetic field reaches a value of almost 2 kG within the polar region but smaller fields are also present down to lower latitudes. The pronounced non-axisymmetric field structure and the non-detection of a differential rotation for V410 Tau supports the idea of an underlying $\alpha^2$-type dynamo, which is predicted for weak-lined T Tauri stars.Comment: Accepted for A&A, 18 pages, 10 figure

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.Comment: Accepted for publication in A&

Extremal black holes, gravitational entropy and nonstationary metric fields

We show that extremal black holes have zero entropy by pointing out a simple fact: they are time-independent throughout the spacetime and correspond to a single classical microstate. We show that non-extremal black holes, including the Schwarzschild black hole, contain a region hidden behind the event horizon where all their Killing vectors are spacelike. This region is nonstationary and the time $t$ labels a continuous set of classical microstates, the phase space $[\,h_{ab}(t), P^{ab}(t)\,]$, where $h_{ab}$ is a three-metric induced on a spacelike hypersurface $\Sigma_t$ and $P^{ab}$ is its momentum conjugate. We determine explicitly the phase space in the interior region of the Schwarzschild black hole. We identify its entropy as a measure of an outside observer's ignorance of the classical microstates in the interior since the parameter $t$ which labels the states lies anywhere between 0 and 2M. We provide numerical evidence from recent simulations of gravitational collapse in isotropic coordinates that the entropy of the Schwarzschild black hole stems from the region inside and near the event horizon where the metric fields are nonstationary; the rest of the spacetime, which is static, makes no contribution. Extremal black holes have an event horizon but in contrast to non-extremal black holes, their extended spacetimes do not possess a bifurcate Killing horizon. This is consistent with the fact that extremal black holes are time-independent and therefore have no distinct time-reverse.Comment: 12 pages, 2 figures. To appear in Class. and Quant. Gravity. Based on an essay selected for honorable mention in the 2010 gravity research foundation essay competitio

A Naturally Minute Quantum Correction to the Cosmological Constant Descended from the Hierarchy

We demonstrate that an extremely small but positive quantum correction, or the Casimir energy, to the cosmological constant can arise from a massive bulk fermion field in the Randall-Sundrum model. Specifically, a cosmological constant doubly descended from the Planck-electroweak hierarchy and as minute as the observed dark energy scale can be naturally achieved without fine-tuning of the bulk fermion mass. To ensure the stabilization of the system, we discuss two stabilization mechanisms under this setup. It is found that the Goldberger-Wise mechanism can be successfully introduced in the presence of a massive bulk fermion, without spoiling the smallness of the quantum correction.Comment: 5 page

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.Comment: 13 pages, 12 figures, accepted for publication in A&