Analytic Interferometry of Rotating Stellar Surfaces

TL;DR

This work addresses the ill-posed problem of imaging rotating stellar surfaces with optical interferometry by deriving closed-form visibilities for arbitrary surface maps expressed in the real spherical-harmonic basis and introducing stellar rotation synthesis to add information through rotation. It provides two analytic solution families for the Fourier integral via hemispheric harmonics and complementary hemispheric harmonics, includes limb-darkening via a radial polynomial, and implements the model in the open-source package harmonix built on $JAX$. It then analyzes the information content with Fisher information and the crlb, showing that simultaneous space-based photometry and, in principle, dense intensity-interferometer arrays can dramatically improve surface-map precision beyond what $V(u,v)$ and closure phases alone offer. Together these developments enable sub-milliarcsecond mapping of nearby main-sequence stars with current facilities and guide the design of next-generation interferometric instruments.

Abstract

The surfaces of rotating stars serve as a window into their interiors, magnetic dynamos, and are important in other areas including exoplanet discovery and atmospheric characterization. While indirect techniques such as photometry and Doppler imaging have been studied for their ability to map stellar surfaces, the gold standard remains optical long-baseline interferometry. In this paper, we develop new closed-form solutions for the interferometric visibility of a rotating star with an arbitrary inhomogeneous surface. We introduce the concept of 'stellar rotation synthesis' in interferometry--an analog of Earth rotation synthesis--where stellar rotation adds information to the spherical harmonic modes representing the star's surface intensity. We implement these solutions in the open-source package harmonix, written in JAX with automatic differentiation, providing a rich ecosystem for fitting and inference. Inspired by similar studies for photometry and Doppler imaging, we use simulations of a fiducial star as observed by the CHARA Array and intensity interferometers to perform a comprehensive theoretical study of the information theory of the starspot mapping problem in interferometry. We show that adding simultaneous photometry from a space-based instrument such as TESS adds complementary spatial information to interferometry and can improve the precision on the map coefficients by over an order of magnitude, enabling the detailed mapping of nearby main-sequence stars with current facilities. Finally, we evaluate the performance of existing and proposed intensity interferometers for stellar surface mapping.

Figures (9)

• Figure 1: Two-dimensional projections of the set of spherical harmonics up to degree $l=5$. The hemispheric harmonics (which are linear combinations of Zernike polynomials) are plotted in red, and the complementary hemispheric harmonics in blue. https://github.com/shashankdholakia/analytic-interferometry-paper/blob/main/src/scripts/spherical_harmonics.py
• Figure 2: Fractional error on stellar angular diameter for simulated stars ranging from 0.1 to 1.0 milliarcseconds. The black line indicates the visibility curve for a uniform disk star at the baseline of the instrument as a reference; a uniform disk star of a given angular diameter would truncate at that point on the curve. The blue curve shows the fractional error (standard deviation over the angular diameter) with only the visibilities, and the red curve shows the same with visibilities and closure phases. As expected, closure phases provide no additional information in this radially symmetric example. https://github.com/shashankdholakia/analytic-interferometry-paper/blob/main/src/scripts/chara_sim_ld_radius.py
• Figure 3: The stellar rotation synthesis SPOT problem for a hypothetical observation with the chara Array. We paint the word SPOT on the northern hemisphere of the star using an $l_{\mathrm{max}}=15$ in spherical harmonics, following luger2021b (shown in a Mollweide projection at top left). The star is given a $60^{\circ}$ inclination, and 8 equally-spaced rotational phases are 'observed' (2nd panel from top). Each observation is assumed to have the same $uv$ coverage, using all 6 telescopes to produce 15 independent baselines (top right, each baseline is assigned a color). The interferometric visibility amplitudes are colored corresponding to the baseline and each rotational phase is overplotted. The 10 independent closure phases are shown (bottom panel) and are colored corresponding to the maximum baseline in each triangle. https://github.com/shashankdholakia/analytic-interferometry-paper/blob/main/src/scripts/spot_map_plot_chara.py
• Figure 4: Recovered surface maps at six different values of snr for the SPOT problem with the CHARA Array's visibility amplitudes and closure phases without regularization. The star is assumed to have a known inclination of $60^{\circ}$, and as a result, latitudes below $-60^{\circ}$ are never visible. https://github.com/shashankdholakia/analytic-interferometry-paper/blob/main/src/scripts/spot_map_plot_chara_optimization.py
• Figure 5: Information content for each spherical harmonic degree $l$ as a function of stellar angular diameter for the CHARA SPOT problem. The black line indicates the visibility curve for a uniform disk star with a given angular diameter with uniform $u, v$ coverage out to the maximum baseline, and is meant as a reference for which null is reached for a given angular diameter. The information content is plotted as the natural log of the determinant of the block matrix containing only terms of a given degree $l$, color coded in ascending order. For each $l$, the information saturates at some maximum baseline, providing an effective bandlimit for a given angular diameter on an interferometric array. https://github.com/shashankdholakia/analytic-interferometry-paper/blob/main/src/scripts/chara_sim_radius.py