Von Zeipel theorem

Abstract: KELT-9 b is an ultra hot Jupiter transiting a rapidly rotating, oblate early-A-type star in a polar orbit. We model the effect of rapid stellar rotation on KELT-9 b's transit light curve using photometry from the Transiting Exoplanet Survey Satellite (\tess) to constrain the planet's true spin-orbit angle and to explore how KELT-9 b may be influenced by stellar gravity darkening. We constrain the host star's equatorial radius to be $1.089\pm0.017$ times as large as its polar radius and its local surface brightness to vary by $\sim38$\% between its hot poles and cooler equator. We model the stellar oblateness and surface brightness gradient and find that it causes the transit light curve to lack the usual symmetry around the time of minimum light. We take advantage of the light curve asymmetry to constrain KELT-9 b's true spin orbit angle (${87^\circ}^{+10^\circ}_{-11^\circ}$), agreeing with \citet{gaudi2017giant} that KELT-9 b is in a nearly polar orbit. We also apply a gravity darkening correction to the spectral energy distribution model from \citet{gaudi2017giant} and find that accounting for rapid rotation gives a better fit to available spectroscopy and yields a more reliable estimate for the star's polar effective temperature.

Abstract: Context: Some secondary effects are known to introduce variations in spectra of massive binaries. These phenomena (such as the Struve--Sahade effect, difficulties to determine properly the spectral type,...) have been reported and documented in the literature. Aims: We simulate the spectra of circular massive binaries at different phases of the orbital cycle and accounting for the gravitational influence of the companion star on the shape and physical properties of the stellar surface. Methods: We use the Roche potential to compute the stellar surface, von Zeipel theorem and reflection effects to compute the surface temperature. We then interpolate in a grid of NLTE plan-parallel atmosphere model spectra to obtain the local spectrum at each surface point. We finally sum all the contributions (accounting for the Doppler shift, limb-darkening, ...) to obtain the total spectrum. The computation is done for different orbital phases and for different sets of physical and orbital parameters. Results: Our first models reproduce the Struve--Sahade effect for several lines. Another effect, surface temperature distribution is visible but the distribution predicted by our current model is not yet consistent with observations. Conclusions: In some cases, the Struve--Sahade effect as well as more complex line intensity variations could be linked to blends of intrinsically asymmetric line profiles that are not appropriatly treated by the deblending routine. Systematic variations of lines for (nearly) contact systems are also predicted by the model.

Abstract: We analyze a class of physical properties, forming the content of the so-called von Zeipel theorem, which characterizes stationary, axisymmetric, non-selfgravitating perfect fluids in circular motion in the gravitational field of a compact object. We consider the extension of the theorem to the magnetohydrodynamic regime, under the assumption of an infinitely conductive fluid, both in the Newtonian and in the relativistic framework. When the magnetic field is toroidal, the conditions required by the theorem are equivalent to integrability conditions, as it is the case for purely hydrodynamic flows. When the magnetic field is poloidal, the analysis for the relativistic regime is substantially different with respect to the Newtonian case and additional constraints, in the form of PDEs, must be imposed on the magnetic field in order to guarantee that the angular velocity $\Omega$ depends only on the specific angular momentum $\ell$. In order to deduce such physical constraints, it is crucial to adopt special coordinates, which are adapted to the $\Omega={\rm const}$ surfaces. The physical significance of these results is briefly discussed.

Abstract: The von Zeipel theorem is generalised to account for differential rotation in the case of a "shellular" rotation law (cf. Zahn 1992). We write this law in the form =(r) ,a simplification which does not apply to fast rotation. We find that von Zeipel's relation contains a small additional term, generally further increasing the radiative flux at the pole and decreasing it at the equator. We also examine the local Eddington factor in rotating stars and notice some significant differences with respect to current expressions. We examine the latitudinal dependence of the mass loss rates _ M(#) in rotating stars and find two main source of wind anisotropies: 1) the " ge " effect which enhances the polar ejec- tion; 2) the "opacity effect" (or " -effect"), which favours equa- torial ejection. In O-stars the ge effect is expected to largely dominate. In B- and later type stars the opacity effect should favour equatorial ejection and the formation of equatorial rings. We also examine the behaviour of the wind density and notice a strong enhancement at the equator of B- and later type stars. Possible relations with the polar ejections and the skirt of Carinae and with the inner and outer rings of SN 1987 A are mentioned. If _ M(#) has sharp extrema due to some peaks in the opacity law, non equatorial and symmetrical rings may be produced. We also show that the global mass loss rate of a star at a given location in the HR diagram is rapidly increasing with rotation, which is in good agreement with the numerical models by Friend & Abbott (1986). Anisotropic stellar winds remove selectively the angular momentum. For example, winds passing through polar caps in O-stars remove very little angular momentum, an excess of an- gular momentum is thus retained and rapidly redistributed by horizontal turbulence. These excesses may lead some Wolf- Rayet stars, those resulting directly from O-stars, to be fast spinning objects, while we predict that the WR-stars which have passed through the red supergiant phase will have lower rotation velocities on the average. We also show how anisotropic ejection can be treated in numerical models by properly modi- fying the outer boundary conditions for the transport of angular momentum. Finally, in an Appendix the equation of the surface for stars with shellular rotation is discussed.