Rotation of real spherical harmonics

Cóllado, J. R. Alvarez; Rico, J. Fernández; López, Rafael; Paniagua, Miguel; Ramı́rez, G.

doi:10.1016/0010-4655(89)90107-0

Cited by 28 publications

(10 citation statements)

References 5 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…ROTATION OF SPHERICAL HARMONICS C.1. Euler angles Collado et al (1989) derived expressions for the rotation matrices for the real spherical harmonics of a given degree l from the corresponding complex rotation matrices (Steinborn & Ruedenberg 1973):…”

Section: B Change Of Basismentioning

confidence: 99%

starry: Analytic Occultation Light Curves

et al. 2019

View full text Add to dashboard Cite

We derive analytic, closed form, numerically stable solutions for the total flux received from a spherical planet, moon or star during an occultation if the specific intensity map of the body is expressed as a sum of spherical harmonics. Our expressions are valid to arbitrary degree and may be computed recursively for speed. The formalism we develop here applies to the computation of stellar transit light curves, planetary secondary eclipse light curves, and planet-planet/planet-moon occultation light curves, as well as thermal (rotational) phase curves. In this paper we also introduce starry, an open-source package written in C++ and wrapped in Python that computes these light curves. The algorithm in starry is six orders of magnitude faster than direct numerical integration and several orders of magnitude more precise. starry also computes analytic derivatives of the light curves with respect to all input parameters for use in gradientbased optimization and inference, such as Hamiltonian Monte Carlo (HMC), allowing users to quickly and efficiently fit observed light curves to infer properties of a celestial body's surface map.

show abstract

Section: B Change Of Basismentioning

confidence: 99%

starry: Analytic Occultation Light Curves

et al. 2019

View full text Add to dashboard Cite

show abstract

“…The fact that the orientation of axes in Figure usually does not coincide with the molecular frame is not a real problem, since the rotation of regular real spherical harmonics can be performed also very efficiently . This rotation transforms each

r^{l} z_{l}^{m} (θ, φ)

into a linear combination of regular harmonics with the same l and several (usually all) compatible m values, what implies that radial factors in the rotated frame are just linear combinations of the corresponding unrotated factors.…”

Section: Methodsmentioning

confidence: 99%

Efficient algorithm for expanding theoretical electron densities in canterakis–zernike functions

2018

Self Cite

View full text Add to dashboard Cite

An algorithm for the efficient computation of Canterakis–Zernike moments of theoretically computed molecular electron densities and rotationally invariant Fingerprint indices derived from them is reported. The algorithm is suitable for any density expressed in terms of Gaussian‐ or Slater‐type functions within the Linear Combination of Atomic Orbitals framework at any level of computation. Electron density is expressed as a one‐center expansion of real regular spherical harmonics times radial factors by means of translation techniques, which facilitates the efficient computation of the moments in terms of a single one‐dimension numerical integration. The performance of the algorithm is analyzed showing that the computation of radial factors in the quadrature points is responsible for almost all computational time. The procedure is applicable to any density obtained with standard packages for molecular structure calculations. © 2018 Wiley Periodicals, Inc.

show abstract

“…where α, β, and γ are the Euler angles describing the rotation in the ẑ−ŷ−ẑ convention and i is the imaginary unit (Collado et al 1989). The terms of the d-matrix depend on powers of sin ( β /2) and cos ( β /2) (see Equation C15 in Luger et al 2019), but it is convenient to use the half-angle formula to express these terms instead as…”

Section: C21 Rotation Matricesmentioning

confidence: 99%

Mapping stellar surfaces II: An interpretable Gaussian process model for light curves

Luger,

Foreman-Mackey,

Hedges

2021

Preprint

View full text Add to dashboard Cite

The use of Gaussian processes (GPs) as models for astronomical time series datasets has recently become almost ubiquitous, given their ease of use and flexibility. GPs excel in particular at marginalization over the stellar signal in cases where the variability due to starspots rotating in and out of view is treated as a nuisance, such as in exoplanet transit modeling. However, these effective models are less useful in cases where the starspot signal is of primary interest since it is not obvious how the parameters of the GP model are related to the physical properties of interest, such as the size, contrast, and latitudinal distribution of the spots. Instead, it is common practice to explicitly model the effect of individual starspots on the light curve and attempt to infer their properties via optimization or posterior inference. Unfortunately, this process is degenerate, ill-posed, and often computationally intractable when applied to stars with more than a few spots and/or to ensembles of many light curves. In this paper, we derive a closed-form expression for the mean and covariance of a Gaussian process model that describes the light curve of a rotating, evolving stellar surface conditioned on a given distribution of starspot sizes, contrasts, and latitudes. We demonstrate that this model is correctly calibrated, allowing one to robustly infer physical parameters of interest from one or more stellar light curves, including the typical radii and the mean and variance of the latitude distribution of starspots. Our GP has far-ranging implications for understanding the variability and magnetic activity of stars from both light curves and radial velocity (RV) measurements, as well as for robustly modeling correlated noise in both transiting and RV exoplanet searches. Our implementation is efficient, user-friendly, and open source, available as the Python package starry_process.

show abstract

Rotation of real spherical harmonics

Cited by 28 publications

References 5 publications

starry: Analytic Occultation Light Curves

starry: Analytic Occultation Light Curves

Efficient algorithm for expanding theoretical electron densities in canterakis–zernike functions

Mapping stellar surfaces II: An interpretable Gaussian process model for light curves

Contact Info

Product

Resources

About