The Rossiter-McLaughlin effect reloaded: Probing the 3D spin-orbit geometry, differential stellar rotation, and the spatially-resolved stellar spectrum of star-planet systems

When a planet transits its host star, it blocks regions of the stellar surface from view; this causes a distortion of the spectral lines and a change in the line-of-sight (LOS) velocities, known as the Rossiter-McLaughlin (RM) effect. Since the LOS velocities depend, in part, on the stellar rotation, the RM waveform is sensitive to the star-planet alignment (which provides information on the system’s dynamical history). We present a new RM modelling technique that directly measures the spatially-resolved stellar spectrum behind the planet. This is done by scaling the continuum flux of the (HARPS) spectra by the transit light curve, and then subtracting the infrom the out-of-transit spectra to isolate the starlight behind the planet. This technique does not assume any shape for the intrinsic local profiles. In it, we also allow for differential stellar rotation and centre-to-limb variations in the convective blueshift. We apply this technique to HD 189733 and compare to 3D magnetohydrodynamic (MHD) simulations. We reject rigid body rotation with high confidence (>99% probability), which allows us to determine the occulted stellar latitudes and measure the stellar inclination. In turn, we determine both the sky-projected (λ ≈ −0.4 ± 0.2◦) and true 3D obliquity (ψ ≈ 7+12 −4 ◦ ). We also find good agreement with the MHD simulations, with no significant centre-to-limb variations detectable in the local profiles. Hence, this technique provides a new powerful tool that can probe stellar photospheres, differential rotation, determine 3D obliquities, and remove sky-projection biases in planet migration theories. This technique can be implemented with existing instrumentation, but will become even more powerful with the next generation of high-precision radial velocity spectrographs.

Topics in differential geometry.

Based on lectures given in the spring of 1949; a few of the latest results of work done since that time have been included.

Lectures on classical differential geometry.

Bibliography: p. vii-viii.

Projective differential geometry of curves and ruled surfaces,

Available on demand as hard copy or computer file from Cornell University Library.

Concepts from tensor analysis and differential geometry.

Mode of access: Internet.

Introduction to differential geometry : [notes for Math. 140, Dept. of Mathematics, University of Chicago /

Caption title.

The elementary differential geometry of plane curves,

Mode of access: Internet.

Anwendung der Differential- und Integralrechnung auf die allgemeine Theorie der Flächen und der linien doppelter Krümmung.

Available on demand as hard copy or computer file from Cornell University Library.

Riemannian geometry.

Bibliography: p. [289]-300.

Axial symmetry and transverse trace-free tensors in numerical relativity

Transverse trace-free (TT) tensors play an important role in the initial conditions of numerical relativity, containing two of the component freedoms. Expressing a TT tensor entirely, by the choice of two scalar potentials, is not a trivial task however. Assuming the added condition of axial symmetry, expressions are given in both spherical and cylindrical coordinates, for TT tensors in flat space. A coordinate relation is then calculated between the scalar potentials of each coordinate system. This is extended to a non-flat space, though only one potential is found. The remaining equations are reduced to form a second order partial differential equation in two of the tensor components. With the axially symmetric flat space tensors, the choice of potentials giving Bowen-York conformal curvatures, are derived. A restriction is found for the potentials which ensure an axially symmetric TT tensor, which is regular at the origin, and conditions on the potentials, which give an axially symmetric TT tensor with a spherically symmetric scalar product, are also derived. A comparison is made of the extrinsic curvatures of the exact Kerr solution and numerical Bowen-York solution for axially symmetric black hole space-times. The Brill wave, believed to act as the difference between the Kerr and Bowen-York space-times, is also studied, with an approximate numerical solution found for a mass-factor, under different amplitudes of the metric.

MA158exam01

Exam questions and solutions in LaTex

Geometria diferencial das curvas planas

Pós-graduação em Matemática Universitária - IGCE

[Lecture notes]

Notes from lectures at the New York Institute for mathematics and mechanics?

[Lecture notes on pure mathematics. Convex sets]

Includes bibliographies.

Lezioni sulla teoria delle superficie.

Available on demand as hard copy or computer file from Cornell University Library.