Proper Motion Anomaly in Astrometry

• Proper motion anomaly is the quantified mismatch between observed angular motion and the expected motion derived from astrometric models.
• Techniques compare measurements from Gaia, Hipparcos, and URAT1 to detect deviations, revealing binary companions and field-sampling biases.
• The concept underpins corrections for systemic accelerations and variability-induced shifts, playing a key role in refining galactic kinematics.

Proper motion anomaly denotes a discrepancy between an observed proper motion and a reference proper motion defined by a kinematic model, a long-term astrometric baseline, or an assumed inertial frame. In the cited literature, the term is used in several distinct but related ways: as the residual between instantaneous and long-term stellar proper motions in binary detection, as a bias in a field-averaged center-of-mass proper motion, as a systematic aberrational term induced by Solar System or stellar acceleration, and as an apparent motion produced by photocenter variability rather than bulk translation (Kervella et al., 2018, Bekki, 2011, Liu et al., 2024). The common structure is a comparison between measured angular motion and an expected linear or bulk motion; the physical interpretation depends on which reference motion is adopted and on which processes are omitted from the model.

1. Operational definitions across astrometric subfields

In nearby-star multiplicity work, the proper-motion anomaly is usually defined as the difference between an instantaneous proper motion and a long-term center-of-mass estimate. Kervella and collaborators write

$$\Delta\boldsymbol\mu=\boldsymbol\mu_{\rm inst}-\boldsymbol\mu_{\rm long\text{-}term},$$

with the long-term motion derived from Hipparcos and Gaia positions and the instantaneous term taken from Hipparcos or Gaia catalog proper motions (Kervella et al., 2018). In the Gaia DR2 Cepheid and RR Lyrae analysis, the long-term motion is

$$\mu_{\rm HG}=(\vartheta_{\rm G2}-\vartheta_{\rm Hip})/(t_{\rm G2}-t_{\rm Hip}),$$

and the anomaly vector is

$\Delta\mu=\mu_{\rm G2}-\mu_{\rm HG},$

with signal-to-noise ratio

$S=|\Delta\mu|/\sigma_{\Delta\mu}$

used to classify detections (Kervella et al., 2019).

In the UrHip catalog, the anomaly is defined more specifically as the difference between a short-baseline Hipparcos+URAT1 motion and the Hipparcos proper motion alone,

$$\Delta\mu\equiv \mu_{\rm UrHip}-\mu_{\rm Hipparcos},$$

evaluated per coordinate and tested against the combined uncertainty $\sigma_{\Delta\mu}$ (Frouard et al., 2015). There the anomaly is interpreted primarily as evidence for non-linear photocenter motion caused by an unseen companion.

In the Large Magellanic Cloud context, the quantity of interest is not a stellar reflex wobble but a bias in the recovered center-of-mass proper motion. After subtracting modeled rotation and perspective terms from the observed field motions, Bekki defines

$$\Delta\mu\equiv \langle\mu_{\rm obs}\rangle-\mu_{\rm true} = \frac{1}{N}\sum_{i=1}^{N}\delta\mu_i,$$

with variance

${\rm Var}(\Delta\mu)=\sigma^2/N,$

so that the anomaly is a field-sampling bias driven by residual local random motions (Bekki, 2011).

In Galactic and extragalactic reference-frame work, “proper motion anomaly” can refer to a coherent systematic term. For the newly derived line-of-sight contribution to secular aberration drift, the extra stellar proper motion is

$$\Delta\vec\mu=-\,\frac{1}{c}(\vec r\cdot\vec V_0)\,\vec\mu,$$

where $\vec V_0$ is the Solar System barycenter velocity and $$\mu_{\rm HG}=(\vartheta_{\rm G2}-\vartheta_{\rm Hip})/(t_{\rm G2}-t_{\rm Hip}),$$0 is the stellar direction vector (Liu et al., 2024). In this usage, the anomaly is not evidence for binarity but for an unmodeled frame effect.

These definitions are mathematically similar—each is a residual—but physically heterogeneous. A common misconception is to treat all proper-motion anomalies as direct signs of companions. The cited literature shows that anomalies may instead reflect internal kinematics of a galaxy, observer acceleration, stellar acceleration in the Galactic potential, or variability-induced photocenter shifts (Bekki, 2011, Liu et al., 2024, Khamitov et al., 2023).

2. Field-averaged biases in the Large Magellanic Cloud

Recent Hubble Space Telescope studies of the Large Magellanic Cloud measured two-component proper motions in typically small sets of HRC/ACS fields and corrected each field for disk rotation and perspective. Bekki emphasized that these corrections do not remove local random stellar motions in a thick, velocity-dispersed disk, so the mean of $$\mu_{\rm HG}=(\vartheta_{\rm G2}-\vartheta_{\rm Hip})/(t_{\rm G2}-t_{\rm Hip}),$$1 fields can differ significantly from the true center-of-mass proper motion (Bekki, 2011). In his formulation,

$$\mu_{\rm HG}=(\vartheta_{\rm G2}-\vartheta_{\rm Hip})/(t_{\rm G2}-t_{\rm Hip}),$$2

and after subtracting the modeled correction $$\mu_{\rm HG}=(\vartheta_{\rm G2}-\vartheta_{\rm Hip})/(t_{\rm G2}-t_{\rm Hip}),$$3,

$$\mu_{\rm HG}=(\vartheta_{\rm G2}-\vartheta_{\rm Hip})/(t_{\rm G2}-t_{\rm Hip}),$$4

The observational center-of-mass proper motion is then the mean over fields.

The key result is that the residual field term need not average to zero when $$\mu_{\rm HG}=(\vartheta_{\rm G2}-\vartheta_{\rm Hip})/(t_{\rm G2}-t_{\rm Hip}),$$5 is small. In Bekki’s $$\mu_{\rm HG}=(\vartheta_{\rm G2}-\vartheta_{\rm Hip})/(t_{\rm G2}-t_{\rm Hip}),$$6-body models, a fiducial LMC with a thick stellar disk, scale height $$\mu_{\rm HG}=(\vartheta_{\rm G2}-\vartheta_{\rm Hip})/(t_{\rm G2}-t_{\rm Hip}),$$7, Toomre $$\mu_{\rm HG}=(\vartheta_{\rm G2}-\vartheta_{\rm Hip})/(t_{\rm G2}-t_{\rm Hip}),$$8, and maximum circular velocity $$\mu_{\rm HG}=(\vartheta_{\rm G2}-\vartheta_{\rm Hip})/(t_{\rm G2}-t_{\rm Hip}),$$9 has central one-dimensional dispersions $\Delta\mu=\mu_{\rm G2}-\mu_{\rm HG},$0 and $\Delta\mu=\mu_{\rm G2}-\mu_{\rm HG},$1, corresponding at $\Delta\mu=\mu_{\rm G2}-\mu_{\rm HG},$2 to $\Delta\mu=\mu_{\rm G2}-\mu_{\rm HG},$3 (Bekki, 2011). The deviation between the observed and true center-of-mass proper motions can be as large as $\Delta\mu=\mu_{\rm G2}-\mu_{\rm HG},$4 ($\Delta\mu=\mu_{\rm G2}-\mu_{\rm HG},$5).

Because $\Delta\mu=\mu_{\rm G2}-\mu_{\rm HG},$6, the root-mean-square bias scales as $\Delta\mu=\mu_{\rm G2}-\mu_{\rm HG},$7. Bekki gives explicit values for the fiducial case: for $\Delta\mu=\mu_{\rm G2}-\mu_{\rm HG},$8, $\Delta\mu=\mu_{\rm G2}-\mu_{\rm HG},$9 ($S=|\Delta\mu|/\sigma_{\Delta\mu}$0); for $S=|\Delta\mu|/\sigma_{\Delta\mu}$1, $S=|\Delta\mu|/\sigma_{\Delta\mu}$2 ($S=|\Delta\mu|/\sigma_{\Delta\mu}$3); and for $S=|\Delta\mu|/\sigma_{\Delta\mu}$4, $S=|\Delta\mu|/\sigma_{\Delta\mu}$5 ($S=|\Delta\mu|/\sigma_{\Delta\mu}$6) (Bekki, 2011). The $S=|\Delta\mu|/\sigma_{\Delta\mu}$7-body experiments confirm the same $S=|\Delta\mu|/\sigma_{\Delta\mu}$8 scaling and indicate that several hundred fields are required to suppress the anomaly below $S=|\Delta\mu|/\sigma_{\Delta\mu}$9.

This usage of proper motion anomaly is methodologically important because it separates deterministic bulk corrections from stochastic residuals. A common misconception is that accurate rotation and perspective modeling is sufficient. Bekki’s analysis shows that “perfect” bulk correction still leaves a residual $$\Delta\mu\equiv \mu_{\rm UrHip}-\mu_{\rm Hipparcos},$$0 set by the local dispersion, so reliable recovery of the LMC center-of-mass motion requires large $$\Delta\mu\equiv \mu_{\rm UrHip}-\mu_{\rm Hipparcos},$$1 as well as accurate kinematic modeling (Bekki, 2011).

3. Astrometric binarity, reflex motion, and multiplicity censuses

In stellar multiplicity studies, proper-motion anomaly is primarily a tracer of orbital reflex motion of the photocenter. The UrHip catalog combined Hipparcos positions at epoch $$\Delta\mu\equiv \mu_{\rm UrHip}-\mu_{\rm Hipparcos},$$2 with URAT1 positions at mean epoch $$\Delta\mu\equiv \mu_{\rm UrHip}-\mu_{\rm Hipparcos},$$3, over an effective baseline of $$\Delta\mu\equiv \mu_{\rm UrHip}-\mu_{\rm Hipparcos},$$4, to compute improved proper motions for $$\Delta\mu\equiv \mu_{\rm UrHip}-\mu_{\rm Hipparcos},$$5 stars (Frouard et al., 2015). The anomaly criterion was

$$\Delta\mu\equiv \mu_{\rm UrHip}-\mu_{\rm Hipparcos},$$6

in at least one coordinate. After excluding $$\Delta\mu\equiv \mu_{\rm UrHip}-\mu_{\rm Hipparcos},$$7 Hipparcos binaries and $$\Delta\mu\equiv \mu_{\rm UrHip}-\mu_{\rm Hipparcos},$$8 Tycho-2–Hipparcos $$\Delta\mu\equiv \mu_{\rm UrHip}-\mu_{\rm Hipparcos},$$9-binaries, the single-star subset contained $\sigma_{\Delta\mu}$0 objects, of which $\sigma_{\Delta\mu}$1 new candidates were flagged as likely astrometric binaries (Frouard et al., 2015).

The Hipparcos–Gaia framework generalized this approach. For nearby stars within $\sigma_{\Delta\mu}$2, Kervella et al. analyzed $\sigma_{\Delta\mu}$3 stars and also presented a catalog for $\sigma_{\Delta\mu}$4 of the Hipparcos catalog ($\sigma_{\Delta\mu}$5 stars) (Kervella et al., 2018). They adopted a $\sigma_{\Delta\mu}$6 detection criterion and reported, within $\sigma_{\Delta\mu}$7, $\sigma_{\Delta\mu}$8 for $\sigma_{\Delta\mu}$9 of stars, $$\Delta\mu\equiv \langle\mu_{\rm obs}\rangle-\mu_{\rm true} = \frac{1}{N}\sum_{i=1}^{N}\delta\mu_i,$$0 for $$\Delta\mu\equiv \langle\mu_{\rm obs}\rangle-\mu_{\rm true} = \frac{1}{N}\sum_{i=1}^{N}\delta\mu_i,$$1, and $$\Delta\mu\equiv \langle\mu_{\rm obs}\rangle-\mu_{\rm true} = \frac{1}{N}\sum_{i=1}^{N}\delta\mu_i,$$2 for $$\Delta\mu\equiv \langle\mu_{\rm obs}\rangle-\mu_{\rm true} = \frac{1}{N}\sum_{i=1}^{N}\delta\mu_i,$$3. The median proper-motion-anomaly uncertainty was $$\Delta\mu\equiv \langle\mu_{\rm obs}\rangle-\mu_{\rm true} = \frac{1}{N}\sum_{i=1}^{N}\delta\mu_i,$$4, corresponding to $$\Delta\mu\equiv \langle\mu_{\rm obs}\rangle-\mu_{\rm true} = \frac{1}{N}\sum_{i=1}^{N}\delta\mu_i,$$5, where

$$\Delta\mu\equiv \langle\mu_{\rm obs}\rangle-\mu_{\rm true} = \frac{1}{N}\sum_{i=1}^{N}\delta\mu_i,$$6

(Kervella et al., 2018). This made it possible both to detect companions and to place upper limits on their masses and separations.

Gaia EDR3 improved the same methodology. In the EDR3 PMa catalog, $$\Delta\mu\equiv \langle\mu_{\rm obs}\rangle-\mu_{\rm true} = \frac{1}{N}\sum_{i=1}^{N}\delta\mu_i,$$7 used a $$\Delta\mu\equiv \langle\mu_{\rm obs}\rangle-\mu_{\rm true} = \frac{1}{N}\sum_{i=1}^{N}\delta\mu_i,$$8 Hipparcos-to-Gaia baseline, Monte Carlo propagation of Hipparcos and Gaia covariance matrices, and a significance threshold of $$\Delta\mu\equiv \langle\mu_{\rm obs}\rangle-\mu_{\rm true} = \frac{1}{N}\sum_{i=1}^{N}\delta\mu_i,$$9 (Kervella et al., 2021). The median PMa uncertainty was ${\rm Var}(\Delta\mu)=\sigma^2/N,$0, corresponding to ${\rm Var}(\Delta\mu)=\sigma^2/N,$1, a factor ${\rm Var}(\Delta\mu)=\sigma^2/N,$2 improvement in PMa ${\rm Var}(\Delta\mu)=\sigma^2/N,$3 relative to DR2 (Kervella et al., 2021). The catalog contained ${\rm Var}(\Delta\mu)=\sigma^2/N,$4 Hipparcos stars with ${\rm Var}(\Delta\mu)=\sigma^2/N,$5 measurements; ${\rm Var}(\Delta\mu)=\sigma^2/N,$6 stars had significant PMa detections (${\rm Var}(\Delta\mu)=\sigma^2/N,$7), ${\rm Var}(\Delta\mu)=\sigma^2/N,$8 had common proper-motion bound candidate companions, ${\rm Var}(\Delta\mu)=\sigma^2/N,$9 had $$\Delta\vec\mu=-\,\frac{1}{c}(\vec r\cdot\vec V_0)\,\vec\mu,$$0, and $$\Delta\vec\mu=-\,\frac{1}{c}(\vec r\cdot\vec V_0)\,\vec\mu,$$1 stars ($$\Delta\vec\mu=-\,\frac{1}{c}(\vec r\cdot\vec V_0)\,\vec\mu,$$2 of the Hipparcos sample) exhibited at least one binarity signal when PMa, CPM, and RUWE were combined (Kervella et al., 2021).

Variable-star applications demonstrate the astrophysical reach of PMa beyond nearby dwarfs. In Gaia DR2, $$\Delta\vec\mu=-\,\frac{1}{c}(\vec r\cdot\vec V_0)\,\vec\mu,$$3 classical Cepheids and $$\Delta\vec\mu=-\,\frac{1}{c}(\vec r\cdot\vec V_0)\,\vec\mu,$$4 RR Lyrae stars present in both Hipparcos and Gaia were tested (Kervella et al., 2019). Using $$\Delta\vec\mu=-\,\frac{1}{c}(\vec r\cdot\vec V_0)\,\vec\mu,$$5 as strong detections, $$\Delta\vec\mu=-\,\frac{1}{c}(\vec r\cdot\vec V_0)\,\vec\mu,$$6 as clear detections, and $$\Delta\vec\mu=-\,\frac{1}{c}(\vec r\cdot\vec V_0)\,\vec\mu,$$7 as suspicious candidates, the study found $$\Delta\vec\mu=-\,\frac{1}{c}(\vec r\cdot\vec V_0)\,\vec\mu,$$8 significant PMa detections and $$\Delta\vec\mu=-\,\frac{1}{c}(\vec r\cdot\vec V_0)\,\vec\mu,$$9 additional candidates among Cepheids, and $\vec V_0$0 significant detections and $\vec V_0$1 candidates among RR Lyrae stars (Kervella et al., 2019). For $\vec V_0$2 binary Cepheids with known spectroscopic orbits, the PMa vectors were combined with radial velocities to infer the orientation of the orbital plane and derive companion masses. The inferred true binary fraction was $\vec V_0$3 for classical Cepheids, while RR Lyrae stars showed a raw detection fraction of $\vec V_0$4 (Kervella et al., 2019).

This family of methods shares a single premise: a single star should move approximately linearly, whereas a companion induces non-linear photocenter motion. The main limitations are also consistent across studies: sensitivity declines for small parallaxes, for orbital periods shorter than the Hipparcos or Gaia observing windows, and when luminous companions invalidate the assumption that the photocenter coincides with the primary (Kervella et al., 2019, Kervella et al., 2021).

4. Secular aberration, Galactic acceleration, and reference-frame systematics

Proper-motion anomalies are not confined to orbital perturbations. At microarcsecond-per-year precision, coherent observer-induced or Galactic-potential-induced terms become measurable. The VLBA Extragalactic Proper Motion Catalog showed that quasars and other distant radio sources are not fixed on the sky at this level. Using $\vec V_0$5 extragalactic proper motions measured over $\vec V_0$6 with VLBI, the secular aberration drift was detected at $\vec V_0$7 significance (Truebenbach et al., 2017). Modeled as a spheroidal dipole, the fit yielded a square-root dipole power of $\vec V_0$8, a Cartesian amplitude of $\vec V_0$9, and an apex at $$\mu_{\rm HG}=(\vartheta_{\rm G2}-\vartheta_{\rm Hip})/(t_{\rm G2}-t_{\rm Hip}),$$00, consistent within $$\mu_{\rm HG}=(\vartheta_{\rm G2}-\vartheta_{\rm Hip})/(t_{\rm G2}-t_{\rm Hip}),$$01 with the Galactic center direction (Truebenbach et al., 2017). The paper also emphasized that a tight no-net-rotation constraint can partially absorb the dipole.

For stars in the Milky Way, Liu, Xie, and Zhu derived aberrational proper motions generated by both the Solar System barycenter acceleration and stellar acceleration in the Galactic potential (Liu et al., 2013). With the Galactic aberration constant $$\mu_{\rm HG}=(\vartheta_{\rm G2}-\vartheta_{\rm Hip})/(t_{\rm G2}-t_{\rm Hip}),$$02, they found that within $$\mu_{\rm HG}=(\vartheta_{\rm G2}-\vartheta_{\rm Hip})/(t_{\rm G2}-t_{\rm Hip}),$$03 of the Galactic center the systematic proper motion can exceed $$\mu_{\rm HG}=(\vartheta_{\rm G2}-\vartheta_{\rm Hip})/(t_{\rm G2}-t_{\rm Hip}),$$04 under a flat rotation curve, but is limited to about $$\mu_{\rm HG}=(\vartheta_{\rm G2}-\vartheta_{\rm Hip})/(t_{\rm G2}-t_{\rm Hip}),$$05 for a more realistic linearly rising core rotation curve (Liu et al., 2013). They also showed that the Kovalevsky formulation is not appropriate when the orbital period is only a fraction of the light time from the star to the Solar System barycenter, and that for such short-period stars the aberration shifts orbital phase on the sky rather than the inferred orbit shape.

A further refinement was derived for stellar proper motions in Gaia DR3. In addition to the classical galactocentric-acceleration term,

$$\mu_{\rm HG}=(\vartheta_{\rm G2}-\vartheta_{\rm Hip})/(t_{\rm G2}-t_{\rm Hip}),$$06

an extra secular aberration drift arises from the change in the line-of-sight direction itself,

$$\mu_{\rm HG}=(\vartheta_{\rm G2}-\vartheta_{\rm Hip})/(t_{\rm G2}-t_{\rm Hip}),$$07

(Liu et al., 2024). This term tends to decrease observed proper motions for stars with Galactic longitudes between $$\mu_{\rm HG}=(\vartheta_{\rm G2}-\vartheta_{\rm Hip})/(t_{\rm G2}-t_{\rm Hip}),$$08 and $$\mu_{\rm HG}=(\vartheta_{\rm G2}-\vartheta_{\rm Hip})/(t_{\rm G2}-t_{\rm Hip}),$$09, and increase them in the remaining region. If ignored, it induces an additional proper motion of $$\mu_{\rm HG}=(\vartheta_{\rm G2}-\vartheta_{\rm Hip})/(t_{\rm G2}-t_{\rm Hip}),$$10 for $$\mu_{\rm HG}=(\vartheta_{\rm G2}-\vartheta_{\rm Hip})/(t_{\rm G2}-t_{\rm Hip}),$$11 stars and $$\mu_{\rm HG}=(\vartheta_{\rm G2}-\vartheta_{\rm Hip})/(t_{\rm G2}-t_{\rm Hip}),$$12 for $$\mu_{\rm HG}=(\vartheta_{\rm G2}-\vartheta_{\rm Hip})/(t_{\rm G2}-t_{\rm Hip}),$$13 stars in Gaia DR3, comparable to or larger than typical formal uncertainties at $$\mu_{\rm HG}=(\vartheta_{\rm G2}-\vartheta_{\rm Hip})/(t_{\rm G2}-t_{\rm Hip}),$$14 (Liu et al., 2024).

In this domain, a proper-motion anomaly is a correlated frame effect rather than a source-specific perturbation. The practical consequence is that both secular aberration contributions must be modeled if the stellar reference frame is to remain consistent with the extragalactic realization of the ICRS (Liu et al., 2024).

5. Variability-induced motion and extragalactic “proper motion imitations”

A distinct class of apparent proper-motion anomalies arises when the source photocenter moves because the brightness distribution changes within the instrumental resolution element. For unresolved or marginally resolved AGNs and quasars in Gaia, this is the variability-induced motion effect (Khamitov et al., 2023). Gaia fits a single centroid to each transit; when a transient occurs within the $$\mu_{\rm HG}=(\vartheta_{\rm G2}-\vartheta_{\rm Hip})/(t_{\rm G2}-t_{\rm Hip}),$$15 Gaia resolution element, the instantaneous photocenter is displaced toward the transient, and over the $$\mu_{\rm HG}=(\vartheta_{\rm G2}-\vartheta_{\rm Hip})/(t_{\rm G2}-t_{\rm Hip}),$$16-year EDR3 baseline the centroid motion can mimic a linear proper motion.

The formal model in the AGN transient analysis uses a fast-rise, exponential-decay flare profile. With transient flux $$\mu_{\rm HG}=(\vartheta_{\rm G2}-\vartheta_{\rm Hip})/(t_{\rm G2}-t_{\rm Hip}),$$17, quiescent AGN flux $$\mu_{\rm HG}=(\vartheta_{\rm G2}-\vartheta_{\rm Hip})/(t_{\rm G2}-t_{\rm Hip}),$$18, and angular offset $$\mu_{\rm HG}=(\vartheta_{\rm G2}-\vartheta_{\rm Hip})/(t_{\rm G2}-t_{\rm Hip}),$$19, the centroid shift is

$$\mu_{\rm HG}=(\vartheta_{\rm G2}-\vartheta_{\rm Hip})/(t_{\rm G2}-t_{\rm Hip}),$$20

where $$\mu_{\rm HG}=(\vartheta_{\rm G2}-\vartheta_{\rm Hip})/(t_{\rm G2}-t_{\rm Hip}),$$21, and the apparent proper motion is

$$\mu_{\rm HG}=(\vartheta_{\rm G2}-\vartheta_{\rm Hip})/(t_{\rm G2}-t_{\rm Hip}),$$22

(Khamitov et al., 2023). A transient at $$\mu_{\rm HG}=(\vartheta_{\rm G2}-\vartheta_{\rm Hip})/(t_{\rm G2}-t_{\rm Hip}),$$23 with peak brightness equal to the AGN and decay time $$\mu_{\rm HG}=(\vartheta_{\rm G2}-\vartheta_{\rm Hip})/(t_{\rm G2}-t_{\rm Hip}),$$24 yields a peak photocenter shift $$\mu_{\rm HG}=(\vartheta_{\rm G2}-\vartheta_{\rm Hip})/(t_{\rm G2}-t_{\rm Hip}),$$25, and a drop of $$\mu_{\rm HG}=(\vartheta_{\rm G2}-\vartheta_{\rm Hip})/(t_{\rm G2}-t_{\rm Hip}),$$26 over the first year of decay, corresponding to $$\mu_{\rm HG}=(\vartheta_{\rm G2}-\vartheta_{\rm Hip})/(t_{\rm G2}-t_{\rm Hip}),$$27 (Khamitov et al., 2023).

The SRG/eROSITA–Gaia EDR3 cross-match produced a catalog of $$\mu_{\rm HG}=(\vartheta_{\rm G2}-\vartheta_{\rm Hip})/(t_{\rm G2}-t_{\rm Hip}),$$28 spectroscopically confirmed extragalactic sources in the eastern Galactic hemisphere, including $$\mu_{\rm HG}=(\vartheta_{\rm G2}-\vartheta_{\rm Hip})/(t_{\rm G2}-t_{\rm Hip}),$$29 sources with redshifts measured at the RTT-150 telescope (Khamitov et al., 2023). The proper motions of these objects are formally significant in Gaia, but their extragalactic nature is confirmed. The paper argues that such anomalies can be explained by transient events near AGN nuclei or quasars, including supernovae outbursts, tidal destruction events in AGNs with double nuclei, variability of large-mass supergiants, and OB associations in the field of view of a variable-brightness AGN (Khamitov et al., 2023).

This usage directly counters a common misinterpretation: significant Gaia proper motions of extragalactic sources need not indicate real bulk transverse motion. They may instead be astrometric artifacts of time-variable photocenters.

6. Extended and context-dependent usages

The phrase also appears in specialized contexts where the anomaly is defined relative to a population expectation rather than a catalog baseline. In high proper motion X-ray binaries, Maccarone et al. used “proper motion anomaly” to denote a peculiar transverse speed significantly larger than the typical peculiar speeds of comparable binaries and, in some cases, larger than the local Galactic rotation speed (Maccarone et al., 2014). For IGR J17544–2619, the measured proper motion components $$\mu_{\rm HG}=(\vartheta_{\rm G2}-\vartheta_{\rm Hip})/(t_{\rm G2}-t_{\rm Hip}),$$30 and $$\mu_{\rm HG}=(\vartheta_{\rm G2}-\vartheta_{\rm Hip})/(t_{\rm G2}-t_{\rm Hip}),$$31 imply $$\mu_{\rm HG}=(\vartheta_{\rm G2}-\vartheta_{\rm Hip})/(t_{\rm G2}-t_{\rm Hip}),$$32 and $$\mu_{\rm HG}=(\vartheta_{\rm G2}-\vartheta_{\rm Hip})/(t_{\rm G2}-t_{\rm Hip}),$$33 at $$\mu_{\rm HG}=(\vartheta_{\rm G2}-\vartheta_{\rm Hip})/(t_{\rm G2}-t_{\rm Hip}),$$34, with the paper quoting a peculiar velocity of about $$\mu_{\rm HG}=(\vartheta_{\rm G2}-\vartheta_{\rm Hip})/(t_{\rm G2}-t_{\rm Hip}),$$35 (Maccarone et al., 2014). For 2A 1822–371, $$\mu_{\rm HG}=(\vartheta_{\rm G2}-\vartheta_{\rm Hip})/(t_{\rm G2}-t_{\rm Hip}),$$36 gives $$\mu_{\rm HG}=(\vartheta_{\rm G2}-\vartheta_{\rm Hip})/(t_{\rm G2}-t_{\rm Hip}),$$37 at $$\mu_{\rm HG}=(\vartheta_{\rm G2}-\vartheta_{\rm Hip})/(t_{\rm G2}-t_{\rm Hip}),$$38 and $$\mu_{\rm HG}=(\vartheta_{\rm G2}-\vartheta_{\rm Hip})/(t_{\rm G2}-t_{\rm Hip}),$$39 at $$\mu_{\rm HG}=(\vartheta_{\rm G2}-\vartheta_{\rm Hip})/(t_{\rm G2}-t_{\rm Hip}),$$40 (Maccarone et al., 2014). Here the anomaly traces supernova-driven mass loss, natal kicks, and residual eccentricity.

In microlensing, the term can denote an unexpectedly large lens-source relative proper motion,

$$\mu_{\rm HG}=(\vartheta_{\rm G2}-\vartheta_{\rm Hip})/(t_{\rm G2}-t_{\rm Hip}),$$41

rather than a residual between catalogs (Shin et al., 2021). In OGLE-2019-BLG-1058, the finite-source model yielded $$\mu_{\rm HG}=(\vartheta_{\rm G2}-\vartheta_{\rm Hip})/(t_{\rm G2}-t_{\rm Hip}),$$42, $$\mu_{\rm HG}=(\vartheta_{\rm G2}-\vartheta_{\rm Hip})/(t_{\rm G2}-t_{\rm Hip}),$$43, $$\mu_{\rm HG}=(\vartheta_{\rm G2}-\vartheta_{\rm Hip})/(t_{\rm G2}-t_{\rm Hip}),$$44, and $$\mu_{\rm HG}=(\vartheta_{\rm G2}-\vartheta_{\rm Hip})/(t_{\rm G2}-t_{\rm Hip}),$$45, unusually large compared with typical bulge–bulge events of $$\mu_{\rm HG}=(\vartheta_{\rm G2}-\vartheta_{\rm Hip})/(t_{\rm G2}-t_{\rm Hip}),$$46–$$\mu_{\rm HG}=(\vartheta_{\rm G2}-\vartheta_{\rm Hip})/(t_{\rm G2}-t_{\rm Hip}),$$47 (Shin et al., 2021). A marginal terrestrial parallax signal initially suggested a free-floating-planet candidate in the disk, but direct measurement of the source proper motion,

$$\mu_{\rm HG}=(\vartheta_{\rm G2}-\vartheta_{\rm Hip})/(t_{\rm G2}-t_{\rm Hip}),$$48

showed that the large $$\mu_{\rm HG}=(\vartheta_{\rm G2}-\vartheta_{\rm Hip})/(t_{\rm G2}-t_{\rm Hip}),$$49 was caused by an extreme source proper motion; the posterior then shifted to a low-mass $$\mu_{\rm HG}=(\vartheta_{\rm G2}-\vartheta_{\rm Hip})/(t_{\rm G2}-t_{\rm Hip}),$$50–$$\mu_{\rm HG}=(\vartheta_{\rm G2}-\vartheta_{\rm Hip})/(t_{\rm G2}-t_{\rm Hip}),$$51 stellar lens in the bulge (Shin et al., 2021).

These examples show that proper motion anomaly is not a uniquely standardized observable across astronomy. The phrase consistently identifies a mismatch between observed angular motion and an adopted reference expectation, but the reference may be a catalog baseline, a bulk center-of-mass model, an inertial frame, a population kinematic prior, or a fixed-photocenter assumption. The interpretation is therefore inseparable from the astrometric model against which the motion is judged (Kervella et al., 2018, Bekki, 2011, Khamitov et al., 2023).