Milliarcsecond X-ray Astrometry

• Milliarcsecond X-ray astrometry is a technique that achieves milli-arcsecond precision by combining high-resolution imaging with gravitational lensing and external Gaia references.
• It employs parametric lens modeling alongside Bayesian and likelihood-based inference to constrain source positions far below Chandra’s native resolution.
• Reported applications have localized lensed AGN within 9–11 mas and achieved sub-mas uncertainties, advancing investigations of co-spatiality and dual/offset AGN.

Searching arXiv for papers on milliarcsecond X-ray astrometry, including lensed AGN and Chandra astrometric methods. Milliarcsecond X-ray astrometry denotes the localization of X-ray emission with positional precision at the milli-arcsecond scale, typically by combining high-resolution X-ray imaging with external astrometric references or with the geometric amplification supplied by strong gravitational lensing. In the contemporary literature, the method has been developed most explicitly for strongly lensed AGN and quasars, where parametric lens modelling and Bayesian or maximum-likelihood inference are used to infer source-plane X-ray positions far below the native angular resolution of Chandra. Reported applications include the localization of X-ray emission in CLASS B0712+472 and CLASS B1608+656 within 11 mas and 9 mas from the radio source, respectively (Spingola et al., 2022), the measurement of a 1$\sigma$ quasi-elliptical uncertainty region of $0.5 \times 1.3$ milli-arcsecond for HE 0435-1223 (Rogers et al., 12 May 2025), and a 99.7% confidence constraint of $0.\!\!''030 \times 0.\!\!''020$ for the radio-quiet quasar GraL J065904.1+162909 (Sisk-Reynes et al., 26 Sep 2025). Related Chandra work on pulsars shows that Gaia-referenced frame alignment can also deliver absolute positions with typical astrometric precision $\sim 10$ mas and proper-motion statistical uncertainties down to $1.3$ mas yr$^{-1}$ (Dinsmore et al., 20 Feb 2026).

1. Strong lensing as an X-ray astrometric amplifier

The gravitational-lensing formulation used in this area is the standard lens equation

$\beta = \theta - \nabla \psi(\theta),$

or equivalently, in one HE 0435-1223 analysis,

$\theta = \beta + \nabla \psi(\theta),$

with $\theta$ the image-plane coordinate, $\beta$ the source-plane coordinate, and $0.5 \times 1.3$0 the scaled two-dimensional lens potential (Rogers et al., 12 May 2025, Sisk-Reynes et al., 26 Sep 2025). In the J0659 analysis, the deflector is modeled with two mass components, including a Singular Isothermal Ellipsoid plus external shear for “deflector 1” and a singular isothermal sphere for “deflector 2” (Sisk-Reynes et al., 26 Sep 2025).

The operative principle is that strong lensing is achromatic but highly astigmatic: near the caustics, small separations in the source plane are stretched by factors of order $0.5 \times 1.3$1 along one direction and compressed along the orthogonal direction (Rogers et al., 12 May 2025). In mathematical terms, the local source-to-image mapping is described by the magnification matrix

$0.5 \times 1.3$2

whose eigenvalues are $0.5 \times 1.3$3 (Rogers et al., 12 May 2025). This lens-mediated anisotropic amplification is the basis for obtaining source-plane constraints that are much smaller than the $0.5 \times 1.3$4 Chandra PSF.

In the lensed-AGN literature, this framework is used to address a specific observational limitation: high-$0.5 \times 1.3$5 AGN are too distant to be spatially resolved with current or upcoming X-ray facilities (Rogers et al., 12 May 2025). A plausible implication is that lensing functions not merely as a flux amplifier but as a geometry-dependent astrometric transform, enabling relative localization of emission components that would otherwise remain unresolved.

2. Parametric lens modelling and external astrometric anchors

Milliarcsecond X-ray astrometry in lensed AGN depends on a lens model that reproduces the observed image configuration to very high precision. For GraL J065904.1+162909, Gaia DR3 and HST observations are used to infer a mass model for the deflector that reproduces the positions of the quasar lensed images to milliarcsecond precision (Sisk-Reynes et al., 26 Sep 2025). The analysis employs LENSMODEL under a flat $0.5 \times 1.3$6CDM cosmology with $0.5 \times 1.3$7 and $0.5 \times 1.3$8, minimizes

$0.5 \times 1.3$9

and reports a best-fit yielding $0.\!\!''030 \times 0.\!\!''020$0, reproducing each lensed image to $0.\!\!''030 \times 0.\!\!''020$1 mas (Sisk-Reynes et al., 26 Sep 2025).

For HE 0435-1223, optical astrometry comes from Gaia DR3, which detects all four lensed images with sub-mas precision, $0.\!\!''030 \times 0.\!\!''020$2–$0.\!\!''030 \times 0.\!\!''020$3 mas (Rogers et al., 12 May 2025). The best Gaia-constrained lens model is an SIE+external shear, and the “zero-point” of co-registration is set by the four Gaia image positions mapped back to a single source position in the source plane (Rogers et al., 12 May 2025). In this construction, Gaia supplies the absolute relative geometry against which X-ray image centroids can be registered.

This reliance on external astrometric anchors also appears outside the lensing context. In the Chandra pulsar proper-motion study, unresolved X-ray flux from stars in the Gaia catalog is used in addition to X-ray bright point sources for alignment, improving uncertainties (Dinsmore et al., 20 Feb 2026). That work selects “classical” X-ray point sources and Gaia DR3 stars, assigns each source a PSF model using a precomputed MARX-based library, and fits simultaneously for frame translations, source positions, pulsar position, and proper motion (Dinsmore et al., 20 Feb 2026). Although its primary target class is different, it establishes the general importance of Gaia as an absolute astrometric reference frame for milliarcsecond-regime X-ray analyses.

3. Statistical inference, PSF forward modelling, and confidence regions

A central methodological feature of the lensed-quasar studies is pixel-level forward modelling of Chandra data. For HE 0435-1223, each ObsID is modeled by simulating high-fidelity Chandra PSFs of four point sources at the lens-predicted image positions using SAOTrace + MARX with the actual aspect solution and energy spectrum of each image (Rogers et al., 12 May 2025). The data are cut to a $0.\!\!''030 \times 0.\!\!''020$4 region around the target and binned to $0.\!\!''030 \times 0.\!\!''020$5 pixels. If $0.\!\!''030 \times 0.\!\!''020$6 denotes the observed counts in pixel $0.\!\!''030 \times 0.\!\!''020$7 and $0.\!\!''030 \times 0.\!\!''020$8 the model prediction, the fit maximizes the Poisson likelihood or equivalently minimizes the Cash statistic

$0.\!\!''030 \times 0.\!\!''020$9

At each trial $\sim 10$0, the image flux normalizations are adjusted to minimize $\sim 10$1 (Rogers et al., 12 May 2025).

For J0659, the Bayesian formulation is explicit. A triangular grid of 200 trial source positions is selected near the inner caustic. For each source position, 1,000 MARX ray-traces are run, yielding predicted counts $\sim 10$2 in pixel $\sim 10$3 of image $\sim 10$4. The likelihood is

$\sim 10$5

with per-image normalizations $\sim 10$6, and with a flat prior the posterior satisfies

$\sim 10$7

Defining $\sim 10$8, the 2D contours $\sim 10$9 correspond to 68.3%, 95.5%, and 99.7% confidence levels under 2 d.o.f. (Sisk-Reynes et al., 26 Sep 2025).

Confidence regions are then extracted either from Wilks’s theorem or from the Hessian of the objective function. In HE 0435-1223, the residual uncertainty in $1.3$0 for each ObsID is derived from Wilks’s theorem, so that $1.3$1 is distributed as $1.3$2 with 2 degrees of freedom (Rogers et al., 12 May 2025). In J0659, around the posterior peak the $1.3$3 covariance matrix $1.3$4 is computed from the Hessian of $1.3$5, and in the rotated principal frame the 99.7% contour satisfies

$1.3$6

Its semi-axes are

$1.3$7

with orientation

$1.3$8

(Sisk-Reynes et al., 26 Sep 2025).

The same likelihood-centric design appears in Gaia-referenced Chandra astrometry for pulsars. There, events are modeled by

$1.3$9

with

$^{-1}$0

and the total likelihood $^{-1}$1 is maximized, after which the Hessian of $^{-1}$2 is inverted to yield a Gaussian covariance matrix $^{-1}$3 for all fit parameters (Dinsmore et al., 20 Feb 2026). This suggests that milliarcsecond X-ray astrometry is now defined less by a single instrument capability than by a coupled framework of forward PSF modelling, external registration, and global likelihood inference.

4. Observational implementations and reported localization performance

The best-documented lensed-AGN results span several source classes and redshifts.

System	Reported astrometric result	Principal implication
CLASS B0712+472	X-ray source spatially located within 11 mas from the radio source	X-ray emission co-spatial with radio and optical (Spingola et al., 2022)
CLASS B1608+656	X-ray source spatially located within 9 mas from the radio source	X-ray co-spatial with radio, but displaced with respect to optical emission at $^{-1}$4 level (Spingola et al., 2022)
HE 0435-1223	$^{-1}$5 quasi-elliptical region of $^{-1}$6 mas	Evidence for a projected 3 mas optical–X-ray offset (Rogers et al., 12 May 2025)
GraL J065904.1+162909	99.7% ellipse of $^{-1}$7	Maximum possible X-ray–optical distance 44.7 mas (Sisk-Reynes et al., 26 Sep 2025)

For CLASS B0712+472 and CLASS B1608+656, the 2022 study reports the localization of the X-ray emission from two strongly lensed AGN at $^{-1}$8 and $^{-1}$9, respectively, using a novel method that combines parametric lens modelling with a Bayesian analysis (Spingola et al., 2022). The paper states that this high astrometric precision improves on the limitations of existing X-ray instruments by two orders of magnitude and opens a path to search for offset and binary AGN at $\beta = \theta - \nabla \psi(\theta),$0 (Spingola et al., 2022).

For HE 0435-1223, the X-ray imaging consists of 11 ACIS-S observations on Chandra, totaling 285 ks and $\beta = \theta - \nabla \psi(\theta),$1 counts (Rogers et al., 12 May 2025). The X-ray source position is measured within a $\beta = \theta - \nabla \psi(\theta),$2 quasi-elliptical region of $\beta = \theta - \nabla \psi(\theta),$3 milli-arcsecond, corresponding to about $\beta = \theta - \nabla \psi(\theta),$4 pc$\beta = \theta - \nabla \psi(\theta),$5 at $\beta = \theta - \nabla \psi(\theta),$6 (Rogers et al., 12 May 2025). The authors state that, by referencing all X-ray images to the Gaia positions under the same lens model, they effectively achieve tens-of-$\beta = \theta - \nabla \psi(\theta),$7as relative astrometry in the source plane even though the raw Chandra PSF is $\beta = \theta - \nabla \psi(\theta),$8 (Rogers et al., 12 May 2025).

For J0659 at $\beta = \theta - \nabla \psi(\theta),$9, the inferred mass model reproduces the image positions to milliarcsecond precision, and archival Chandra observations are used to constrain the X-ray origin to be within a $\theta = \beta + \nabla \psi(\theta),$0 ellipse at the 99.7% level (Sisk-Reynes et al., 26 Sep 2025). The methodology is described as enhancing Chandra’s spatial resolution at high-$\theta = \beta + \nabla \psi(\theta),$1 by a factor of 6 (Sisk-Reynes et al., 26 Sep 2025).

Outside the lensing regime, the pulsar study reports absolute positions referenced to Gaia with typical astrometric precision $\theta = \beta + \nabla \psi(\theta),$2 mas and proper-motion statistical uncertainties down to $\theta = \beta + \nabla \psi(\theta),$3 mas yr$\theta = \beta + \nabla \psi(\theta),$4, which the abstract describes as the most precise X-ray PM achieved to date (Dinsmore et al., 20 Feb 2026). The paper’s scope is proper motion rather than source-plane deprojection, but its inclusion is relevant because it demonstrates that Gaia-tied Chandra astrometry can operate in the same milli-arcsecond scale, albeit with different observables and different error budgets.

5. Optical–X-ray offsets, co-spatiality, and astrophysical interpretation

The empirical motivation for milliarcsecond X-ray astrometry is the possibility that the X-ray and optical centroids of a distant AGN need not coincide. In CLASS B0712+472, the X-ray emission is reported to be co-spatial with the radio and optical emission (Spingola et al., 2022). In CLASS B1608+656, the X-ray emission is co-spatial with radio but displaced with respect to the optical emission at $\theta = \beta + \nabla \psi(\theta),$5 level, and the source is therefore positioned as an offset AGN candidate (Spingola et al., 2022).

HE 0435-1223 provides the clearest quantified optical–X-ray displacement. The best X-ray source position is reported as $\theta = \beta + \nabla \psi(\theta),$6 mas relative to the Gaia optical centroid at $\theta = \beta + \nabla \psi(\theta),$7, yielding

$\theta = \beta + \nabla \psi(\theta),$8

or $\theta = \beta + \nabla \psi(\theta),$9 pc (Rogers et al., 12 May 2025). To test coincidence, the difference in minimum Cash statistic between the global best fit and the best solution constrained to lie at the Gaia position is $\theta$0; since $\theta$1, the corresponding $\theta$2-value is $\theta$3, i.e. a $\theta$4 rejection of coincidence (Rogers et al., 12 May 2025). The offset is interpreted as most likely associated to a portion of the X-ray emission arising from an X-ray jet or outflow (Rogers et al., 12 May 2025).

For J0659, the most probable X-ray source coincides with the optical source predicted by the lens model (Sisk-Reynes et al., 26 Sep 2025). At 99.7% confidence, however, the maximum possible separation on the contour is

$\theta$5

which corresponds to $\theta$6 pc at $\theta$7 for $\theta$8 kpc (Sisk-Reynes et al., 26 Sep 2025). The same pipeline can be applied to energy-filtered events, giving soft $\theta$9–$\beta$0 keV$\beta$1 emission localized to $\beta$2 and hard $\beta$3–$\beta$4 keV$\beta$5 emission localized to $\beta$6, both consistent with the optical position though with peak positions differing by $\beta$7 mas (Sisk-Reynes et al., 26 Sep 2025).

These results bear directly on the interpretation of quasar inner structure. The authors of the lensed-quasar papers connect such offsets to X-ray jets, outflows, offset AGN candidates, and the possible presence of binary/offset AGN systems (Spingola et al., 2022, Rogers et al., 12 May 2025). A plausible implication is that positional non-coincidence can function as a morphological diagnostic when direct imaging of the underlying structure is unavailable.

6. X-ray varstrometry, dual-source hypotheses, and broader extensions

In HE 0435-1223, the astrometric framework is extended to “X-ray varstrometry,” a centroid-jitter test for unresolved multiple emitters. If two X-ray-emitting components are separated by $\beta$8 and at least one varies in flux, the centroid jitter has rms amplitude

$\beta$9

where $0.5 \times 1.3$00 is the mean flux ratio, $0.5 \times 1.3$01 the mean total flux, and $0.5 \times 1.3$02 its rms variability (Rogers et al., 12 May 2025). For HE 0435-1223, $0.5 \times 1.3$03 is measured after excluding two epochs strongly affected by microlensing, and the measured perpendicular jitter is $0.5 \times 1.3$04 mas (Rogers et al., 12 May 2025). Imposing $0.5 \times 1.3$05 mas and $0.5 \times 1.3$06 gives $0.5 \times 1.3$07 mas $0.5 \times 1.3$08, or $0.5 \times 1.3$09 pc at $0.5 \times 1.3$10 (Rogers et al., 12 May 2025). Thus X-ray varstrometry places upper limits on sub-kpc dual/offset AGN.

J0659 presents a complementary model-selection perspective. The paper states that, in future applications, single-source and two-source hypotheses can be compared through a Bayes factor

$0.5 \times 1.3$11

with the aim of detecting dual or binary AGN at projected separations $0.5 \times 1.3$12–$0.5 \times 1.3$13 pc at $0.5 \times 1.3$14 (Sisk-Reynes et al., 26 Sep 2025). This is presented as an extension rather than a demonstrated detection in that source.

The broader methodological extension is toward spectrally resolved astrometry. The J0659 work explicitly describes a novel approach that extends the methodology toward investigating the origin of the soft and hard X-ray emitting regions in lensed quasars (Sisk-Reynes et al., 26 Sep 2025). It further elaborates on the potential of upcoming broadband and spectrally resolved X-ray astrometric studies to probe complex quasar morphology at high-$0.5 \times 1.3$15 and to identify dual and binary AGN candidates (Sisk-Reynes et al., 26 Sep 2025).

7. Limitations, precision floors, and research prospects

The current literature shows that milli-arcsecond X-ray astrometry is not a property of the detector alone but of the entire inference chain: external astrometry, lens or PSF model fidelity, count statistics, and registration strategy. In the pulsar study, a point source with $0.5 \times 1.3$16 counts and PSF rms width $0.5 \times 1.3$17 has positional uncertainty

$0.5 \times 1.3$18

and the frame-registration precision in the background-free approximation satisfies

$0.5 \times 1.3$19

where the sum is over field point sources including Gaia stars (Dinsmore et al., 20 Feb 2026). This makes explicit that dense reference grids of faint sources can materially tighten the astrometric solution.

Systematics remain important. By comparison to the VLBI proper motion of PSR B2224+65, the pulsar analysis estimates a systematic floor $0.5 \times 1.3$20 mas yr$0.5 \times 1.3$21, while other tests indicate systematic errors $0.5 \times 1.3$22–$0.5 \times 1.3$23 mas yr$0.5 \times 1.3$24 (Dinsmore et al., 20 Feb 2026). For lensed quasars, the reported precisions and contours depend on the adopted lens model, the fidelity of MARX or SAOTrace-based PSF simulations, and the validity of the co-registration assumptions across epochs (Rogers et al., 12 May 2025, Sisk-Reynes et al., 26 Sep 2025). This suggests that the achieved milliarcsecond performance is an inferred source-plane or Gaia-referenced quantity, not a direct imaging resolution in the conventional sense.

The prospects articulated in the literature are correspondingly specific. The 2022 lensed-AGN study states that the demonstrated method opens a path to search for offset and binary AGN at $0.5 \times 1.3$25, and to directly test supermassive black hole formation models in a redshift range that has been mostly underconstrained to date (Spingola et al., 2022). The HE 0435-1223 paper states that the technique opens the door to sub-kiloparsec X-ray imaging at high redshift for hundreds to thousands of strongly lensed quasars (Rogers et al., 12 May 2025). The J0659 study points to upcoming broadband and spectrally resolved applications and notes that the methodology has been applied, so far, to five lensed quasars, including J0659 (Sisk-Reynes et al., 26 Sep 2025).

Taken together, these studies define milliarcsecond X-ray astrometry as an emerging high-precision regime in which Chandra data, Gaia astrometry, parametric lens models, and likelihood-based forward modelling are combined to test co-spatiality, measure offsets, and constrain unresolved multi-component X-ray structure in distant systems (Spingola et al., 2022, Rogers et al., 12 May 2025, Sisk-Reynes et al., 26 Sep 2025, Dinsmore et al., 20 Feb 2026).