Towards precision astrometry of scattered images of compact radio sources: scintillometry theory and prospects

TL;DR

This work reframes interstellar scintillation around the instantaneous spatial wavefield $V(\nu, {\bm b})$ as the fundamental observable, unifying pulsar and FRB scintillometry under a common theoretical framework. By connecting $V(\nu, {\bm b})$ to its conjugate $\tilde{V}(\tau, {\bm k})$, the authors show how image positions ${\bm\theta}_j$ and effective distances $D$ are encoded in $(\tau, {\bm k})$-space, with dynamic spectra providing a lower-dimensional perspective. For FRBs, the paper outlines prospects for full lensing geometry reconstructions in repeating sources and notes that single bursts will primarily constrain the scattering disc with large baselines; it also highlights the novel possibility of measuring DM gradients across ~100 au scales via differential dispersive delays, offering a diagnostic of tiny-scale CGM turbulence. The analysis delineates the relative contributions of MW ISM, host ISM, CGM, and IGM to dispersive delays and emphasizes the practical limits and potential pathways (e.g., multi-baseline pulsar FRB studies, lower frequencies, and phase-retrieval techniques) toward a imaging-era of FRB scintillometry with implications for CGM physics and cosmology.

Abstract

Compact radio sources such as pulsars and FRBs undergo scintillation in the interstellar medium (ISM) when scattered images interfere at the observer. ``Scintillometry'' refers to the range of techniques to extract astrometric information -- such as the angular positions of the images and distances to the scattering screen and source -- from scintillation observations. Pulsar scintillometry has proven to be a powerful technique, revealing rich and unexpected scattering phenomenology in the ISM and also shedding light on the emission physics of pulsars. FRB scintillometry stands to be a similarly powerful probe of FRB emission, as well as structure on tiny scales in ionized media beyond our galaxy, such as the circumgalactic medium (CGM). However, nascent FRB scintillation studies are far from the sophisticated lensing geometry reconstructions that have been performed for scintillating pulsars. In this paper, we introduce a novel theoretical framework for scintillometry, demonstrating that the full astrometric content of scintillation observations is contained within a single underlying observable: the instantaneous spatial wavefield. We relate the instantaneous spatial wavefield to more familiar concepts from the pulsar scintillometry literature, such as the dynamic spectrum. Using this framework, we discuss prospects and limitations for FRB scintillometry, towards the goal of full astrometric reconstructions of FRB lensing geometries. We show how key degeneracies in two-screen scattering measurements can be ameliorated. In addition, we discuss the possibility of inferring dispersion measure gradients across scintillation screens, which may shed light on the highly unconstrained physics of the cool CGM phase on tiny ($\sim 100\,{\rm au}$) scales.

Figures (8)

• Figure 1: Diagram of the lensing geometry through a scattering screen. Fixed images (green circles) are formed on the scattering screen at angles $\theta_j$ defined relative to the optical axis. Eq. \ref{['eq:phase']} gives the phase of rays arriving at the observer passing through image $j$ relative to the ray arriving at the observer plane through the optical axis. The phase has three components: 1. the quadratic $\theta_j^2$ term due to the images' offset from the optical axis, 2. the ${\bm b} \cdot {\bm \theta_j}$ term due to the observer's offset, ${\bm b}$, from the optical axis, and 3. the accumulated phase delay $\zeta_j$ due to the different dispersion measure (${\rm DM} = \int n_e dl$) along the each ray. The distances $D_l$, $D_s$, and $D_{ls}$ refer to the distances from the observer plane to the scattering screen, the source, and the distance between screen and source, respectively. They appear in Eq. \ref{['eq:phase']} through the effective distance, $D = D_l D_{s}/D_{ls}$. For sources at infinity, $D \approx D_l$.
• Figure 2: The instantaneous spatial wave field and its conjugate for an isotropic scattering disc (top row) and an anisotropic scattering disc (bottom row). The left panel shows the angular position of the scattered images on the sky, with the colour indicating their magnification $\mu_j$. A radial Gaussian profile is chosen for the magnifications with a width of $\theta_{\rm sc} = 20\,{\rm mas}$, and $N_{\rm im} = 100$ images are uniformly distributed in a disc of radius $2 \theta_{\rm sc}$ (in the isotropic case) and a line of width $4 \theta_{\rm sc}$ (in the anisotropic case). The middle panel shows the instantaneous spatial wavefield that arises from these scattered images, computed via Eq. \ref{['eq:ISW_sum']}, where we choose $D = 1\,{\rm kpc}$, $\lambda = 75\,{\rm cm}$, and $\zeta_j = 0$. We compute the spatial wavefield over a grid of observer positions with dimension $10^4 \,{\rm km}\times 10^4\,{\rm km}$. We plot the spatial wavefield for different values of $\nu$ to show its evolution in frequency. The right panel shows the points of non-zero power (blue) in the resulting conjugate wavefield (Eq. \ref{['eq:CSW']}). The purple points show the blue points projected onto the ${\bm k}$-plane and the mesh paraboloid shows the quadratic relationship between the delay $\tau_j$ and ${\bm k}_j$ (Eq. \ref{['eq:tau_v_sigma']}).
• Figure 3: Diagram showing the relationship between the instantaneous spatial wavefield, the conjugate spatial wavefield, the dynamic wavefield, and the conjugate dynamic wavefield. (Top left) The 3D instantaneous spatial wavefield shown as a function of observer position ${\bm b}$ for a single frequency $\nu$ corresponding to $\lambda = 75\,{\rm cm}$. The scattering is due to an anisotropic line of $N_{\rm im} = 100$ images with a length $\theta_{\rm sc} = 20\,{\rm mas}$ and an effective distance $D = 100\,{\rm pc}$. (Top right) The non-zero points of the full conjugate spatial wavefield (blue). The purple points show the blue points projected onto the ${\bm k}$ plane, and the orange points show the blue points projected onto a vertical plane at an angle $\delta$ from the scattering direction. (Bottom left) The 2D dynamic wavefield that would be observed for an effective velocity of $V_{\rm eff} = 20\,{\rm km / s}$ at an angle $\delta$ from the scattering direction. That is, the dynamic wavefield is obtained by sampling the spatial wavefield along the white trajectory in the top left panel. (Bottom right) The conjugate dynamic wavefield obtained by taking the 2D Fourier transform of the bottom left panel. The conjugate dynamic wavefield is related to the conjugate spatial wavefield by taking the projected orange points in the top right panel and re-scaling the horizontal axis by $V_{\rm eff}$.
• Figure 4: (Top) The cross-secondary spectrum (Eq. \ref{['eq:Wftfv']}) for a one-dimensional scattering screen at $D = 100\,{\rm pc}$ with $N_{\rm im} = 100$ images randomly distributed over $40\,{\rm mas}$, with a Gaussian brightness distribution with width $\theta_{\rm sc} = 20\,{\rm mas}$ and $\lambda_0 = 75\,{\rm cm}$. (Bottom) The symmetric part of the cross-secondary spectrum phase, computed via Eq. \ref{['eq:symmetric-phase']}.
• Figure 5: (Top) The burst time series, $V(\tau, {\bm b})$, of a single FRB scattered into 100 images randomly placed on an isotropic scattering disc of angular size $0.015"$ at an effective distance $D=3\,{\rm kpc}$. The colour shows the phase of the electric field and the opacity represents the amplitude (see Appendix \ref{['app:gaussian']} for more details on how this is calculated). For visualization purposes, we only plot the burst time series for a single slice in baselines, ${\bm b} = (b_x,0)$. (Bottom) The conjugate spatial wavefield computed from the burst shown in the top panel. The instantaneous spatial wavefield, $V(\nu, {\bm b})$, is the spectrum obtained from the single burst time series for fixed baseline, ${\bm b}$. Then, the 3D Fourier transform is computed to obtain the conjugate spatial wavefield (shown in red). The opacity of the conjugated spatial wavefield in each voxel is set by the ratio $|V(\tau,\bm{k})|/{\sf max}(|V(\tau,\bm{k})|)$ and voxels with opacity $<0.05$ are not plotted. One can see that the power is centred at the predicted positions of the scattered images (black crosses) in $(\tau, \bm{k})$-space, which form a paraboloid specified by Eq. \ref{['eq:tau_v_sigma']}.