Instantaneous Rotation Pole (IRP)

• Instantaneous Rotation Pole (IRP) is a kinematic descriptor defining the instantaneous axis of rotation in systems like Earth, comets, and geomagnetic fields.
• Advanced methods including jet morphology analysis, lunar laser ranging, and VLBI fringe frequency are used to determine the IRP in diverse reference frames.
• Estimation challenges arise from measurement uncertainties, model assumptions, and domain-specific limitations that impact the precision of IRP recovery.

Searching arXiv for recent and directly relevant papers on "instantaneous rotation pole" and related Earth/comet pole estimation. The instantaneous rotation pole (IRP) denotes an instantaneous pole associated with a rotating or pole-tracked system, but its exact realization is domain-dependent. In cometary dynamics, the IRP of a comet is “the direction of its spin angular-momentum vector at a given epoch, expressed as a unit vector $p$ in inertial space.” In modern Earth-orientation theory, the position of the Earth’s instantaneous rotation axis relative to the crust is parameterized by the pole offsets $x_p$ and $y_p$. In VLBI-based celestial kinematics, the Instantaneous Rotation Pole is “the point on the celestial sphere toward which the instantaneous rotation axis points—that is, the direction of the vector $\Omega$ in the International Celestial Reference System (ICRS)” (Bolin et al., 2019, Singh et al., 2021, Titov, 7 Jul 2025). A related pole-tracking formulation in geomagnetism defines the magnetic pole at time $t$ as the point $\bigl(\lambda(t),\varphi(t)\bigr)$ for which the horizontal field vanishes, $H\bigl(\lambda(t),\varphi(t),t\bigr)=0$, and then derives an instantaneous pole velocity from spatial and temporal field derivatives (Ivanov et al., 2022).

1. Domain-specific definitions and realizations

The literature uses closely related instantaneous-pole concepts in several technically distinct settings. In comet studies, the IRP is a spin-state quantity inferred from coma morphology. In Earth-orientation work, it is represented either in the terrestrial frame by $x_p,y_p$ or in the celestial frame by angles $X,Y$ linked to the direction of the instantaneous angular-velocity vector. In geomagnetic modeling, the pole is not a mechanical rotation axis but the zero of the horizontal field $H$ (Bolin et al., 2019, Singh et al., 2021, Titov, 7 Jul 2025, Ivanov et al., 2022).

Domain	Definition	Coordinates or variables
Comet 2I/Borisov	Direction of the spin angular-momentum vector at a given epoch	$x_p$0, $x_p$1, $x_p$2
Earth rotation in ITRS	Position of the Earth’s instantaneous rotation axis relative to the crust	$x_p$3, $x_p$4
Earth rotation in ICRS	Direction of $x_p$5 on the celestial sphere	$x_p$6, $x_p$7, $x_p$8
Geomagnetic pole tracking	Point where $x_p$9	$y_p$0, $y_p$1, $y_p$2

This suggests that “IRP” is best understood as a kinematic descriptor whose interpretation depends on the physical system, the adopted frame, and the observable used to recover the pole.

2. Kinematic and coordinate formulations

In the cometary formulation, the pole is a unit vector $y_p$3 in inertial Cartesian coordinates. Once $y_p$4 is determined, right ascension and declination are obtained from

$y_p$5

The same solution can be expressed in ecliptic longitude and latitude by a rotation about the $y_p$6-axis by the obliquity $y_p$7: $y_p$8

$y_p$9

$\Omega$0

Thus

$\Omega$1

These relations provide the standard equatorial–ecliptic conversion for the inferred spin pole (Bolin et al., 2019).

In Earth-orientation theory, if $\Omega$2 is the Earth’s instantaneous angular-velocity vector expressed in the ITRS $\Omega$3 frame, then to first order in the small pole offsets,

$\Omega$4

with $\Omega$5. Equivalently,

$\Omega$6

The terrestrial-to-celestial transformation introduces the polar-motion matrix

$\Omega$7

where

$\Omega$8

Accordingly, $\Omega$9 enter directly into the sequence of rotations between the ITRS and GCRS (Singh et al., 2021).

In the VLBI celestial formulation, the pole direction is derived from the instantaneous angular-velocity vector $t$0. With

$t$1

the IRP coordinates are defined by either the small-angle approximation

$t$2

or the exact form

$t$3

Because $t$4, day-to-day variations in $t$5 map directly into $t$6 (Titov, 7 Jul 2025).

3. Cometary IRP inference from jet morphology

For 2I/Borisov, the IRP is inferred from deep HST/WFC3 imaging acquired on 2019 October 12, 2019 November 16, 2019 December 8, and 2020 January 27, before the March 2020 outburst and fragmentation, thus observing the comet in a relatively undisrupted state. The observations locate $t$7–$t$8 long jet-like structures near the optocenter that appear to change position angles from epoch to epoch. The method rests on four stated assumptions: stationarity of the jet over $t$9–$\bigl(\lambda(t),\varphi(t)\bigr)$0 h, principal-axis rotation, a dominant source centered within $\bigl(\lambda(t),\varphi(t)\bigr)$1 of the spin pole, and continuous activity throughout the full rotation (Bolin et al., 2019).

For each observation $\bigl(\lambda(t),\varphi(t)\bigr)$2, the line-of-sight unit vector $\bigl(\lambda(t),\varphi(t)\bigr)$3 is known from the orbital geometry, and the jet position angle $\bigl(\lambda(t),\varphi(t)\bigr)$4 is measured east of north. In the sky plane,

$\bigl(\lambda(t),\varphi(t)\bigr)$5

After rotation into inertial coordinates,

$\bigl(\lambda(t),\varphi(t)\bigr)$6

Because the true pole must lie in the plane defined by $\bigl(\lambda(t),\varphi(t)\bigr)$7 and $\bigl(\lambda(t),\varphi(t)\bigr)$8, the pole satisfies

$\bigl(\lambda(t),\varphi(t)\bigr)$9

Defining

$H\bigl(\lambda(t),\varphi(t),t\bigr)=0$0

the $H\bigl(\lambda(t),\varphi(t),t\bigr)=0$1-th epoch yields the linear constraint

$H\bigl(\lambda(t),\varphi(t),t\bigr)=0$2

With $H\bigl(\lambda(t),\varphi(t),t\bigr)=0$3 epochs, the pole is estimated by minimizing

$H\bigl(\lambda(t),\varphi(t),t\bigr)=0$4

subject to $H\bigl(\lambda(t),\varphi(t),t\bigr)=0$5, or equivalently by taking the eigenvector of

$H\bigl(\lambda(t),\varphi(t),t\bigr)=0$6

corresponding to its smallest eigenvalue (Bolin et al., 2019).

The error model assumes a $H\bigl(\lambda(t),\varphi(t),t\bigr)=0$7-$H\bigl(\lambda(t),\varphi(t),t\bigr)=0$8 jet position-angle uncertainty $H\bigl(\lambda(t),\varphi(t),t\bigr)=0$9. Plotting all six great circles $x_p,y_p$0 on the celestial sphere shows an intersection zone of roughly $x_p,y_p$1, and the common overlap of the projected uncertainty bands yields the $x_p,y_p$2-$x_p,y_p$3 contour of width $x_p,y_p$4 in both $x_p,y_p$5 and $x_p,y_p$6. Under these assumptions, the fit pole is

$x_p,y_p$7

or in ecliptic coordinates

$x_p,y_p$8

The antipodal solution lies at $x_p,y_p$9 or $X,Y$0, also $X,Y$1. Imaging alone cannot distinguish the antipodes; the choice requires additional information such as sense of rotation or seasonal activity (Bolin et al., 2019).

The same study also finds evidence for possible periodicity in the HST time-series lightcurve on the timescale of $X,Y$2 h with amplitude $X,Y$3 mag, implying a lower limit on $X,Y$4 of $X,Y$5. However, because the light-scattering cross-section is dust-dominated, the small lightcurve variations may not reflect nucleus rotation, and uniquely constraining the pre-Solar System encounter, pre-outburst rotation state may not be possible even with HST resolution and sensitivity (Bolin et al., 2019).

4. Earth-fixed IRP estimation from Lunar Laser Ranging

In the LLR framework, the Earth’s IRP is estimated through the pole offsets $X,Y$6. LLR measures the distance between observatories on Earth and retro-reflectors on the Moon, and a normal point is a statistically weighted average of several hundred to a few thousand single-pulse round-trip time measurements collected over $X,Y$7–$X,Y$8 min. To obtain stable on-night ERP estimates, the analysis groups nights into subsets requiring at least $X,Y$9, $H$0, or $H$1 normal points per night, and also by station composition. The key subsets are All10, All15, OCA10, and OCA15 (Singh et al., 2021).

The LUNAR software performs a Gauss–Markov least-squares adjustment of one-way light-time observations to the dynamical lunar-orbit model, estimating up to $H$2 parameters in a global solution. To avoid strong correlations, $H$3 is estimated separately from $H$4. Within the $H$5 runs, the pole offsets may be estimated together or individually. Observatory velocities are held fixed to ITRF2014. At observation level, each normal point time is corrected for geocentric station and lunar reflector positions, relativistic time transfers, tropospheric delay, and optionally non-tidal station-loading displacement from IMLS (Singh et al., 2021).

The core light-time residual is

$H$6

with partial derivatives entering the normal equations through

$H$7

Closed-form expressions are given in Biskupek (2015) and Hofmann et al. (2018) (Singh et al., 2021).

For post-2000.0 data, the mean $H$8 accuracies of the pole coordinates are reported as follows:

Subset	$H$9 together	Only $x_p$00 / only $x_p$01
All10 ($x_p$02 NP)	$x_p$03 mas, $x_p$04 mas	$x_p$05 mas / $x_p$06 mas
All15 ($x_p$07 NP)	$x_p$08 mas, $x_p$09 mas	$x_p$10 mas / $x_p$11 mas

For All15, the absolute mean differences to IERS 14C04 are $x_p$12 and $x_p$13. The study concludes that modern LLR data permit on-night determinations of the IRP with $x_p$14 precision of $x_p$15 mas in $x_p$16 and $x_p$17 mas in $x_p$18, corresponding to $x_p$19–$x_p$20 cm on the ground, and that correlations between $x_p$21 and $x_p$22 per night fall to $x_p$23–$x_p$24 percent when at least $x_p$25–$x_p$26 normal points per night are available (Singh et al., 2021).

The inclusion of combined IMLS non-tidal loading as station-displacement corrections yields a marginal improvement of about $x_p$27 percent in the $x_p$28 accuracies. For example, in All15 the standard $x_p$29 values $x_p$30 improve to $x_p$31, and the mean $x_p$32 of $x_p$33 improves from $x_p$34 to $x_p$35 (Singh et al., 2021).

5. Celestial IRP estimation from VLBI fringe frequency

A distinct realization of the Earth’s IRP is obtained directly in the ICRS from VLBI fringe frequency. Very Long Baseline Interferometry measures both group delay and fringe frequency, and the latter is presented as a unique tool for direct estimation of the instantaneous Earth angular rotation velocity, which is not accessible with the group delay alone. In the approximation that both stations move only with Earth rotation,

$x_p$36

Using $x_p$37, this becomes

$x_p$38

where $x_p$39 is the baseline vector. Each delay-rate observation therefore provides a linear equation in the three unknowns $x_p$40 (Titov, 7 Jul 2025).

For each $x_p$41 h session, the observation model is written

$x_p$42

or in matrix form

$x_p$43

The solution uses weighted least squares,

$x_p$44

with $x_p$45. The normal equations are $x_p$46 and $x_p$47. In addition to the three components of $x_p$48, the model estimates per-station nuisance parameters, specifically tropospheric delay-rate gradients and hydrogen maser offsets, a total of $x_p$49 per site. Each fringe-frequency observation is assigned $x_p$50 from the correlator’s formal error, after which iterative $x_p$51 editing is applied to remove outliers. Higher-order relativistic terms and mapping-function derivatives for the wet troposphere are included at the modelling stage so that residual systematics in $x_p$52 are $x_p$53 prad/s, and the covariance is

$x_p$54

Typical formal errors are $x_p$55–$x_p$56 prad/s for each component (Titov, 7 Jul 2025).

Over the $x_p$57 years April 1993–April 2024, the method yields daily estimates of $x_p$58. Formal errors in these components are $x_p$59–$x_p$60 prad/s, yielding formal uncertainties in $x_p$61 of order $x_p$62–$x_p$63 mas. The daily magnitude $x_p$64 maps directly into length of day through

$x_p$65

with $x_p$66 and $x_p$67. The VLBI-derived LOD agrees with the IERS C04 LOD at the $x_p$68 ms level, while the IRP $x_p$69 time series track the IAU2000A/2006 precession–nutation model plus EOP offsets at the $x_p$70 mas level. The rms of the $x_p$71 residuals with respect to the IERS solution is $x_p$72 prad/s, corresponding to $x_p$73 mas in $x_p$74 and $x_p$75 (Titov, 7 Jul 2025).

The reported advantages are direct vector measurement of the full three-dimensional $x_p$76, immunity to short-term hydrogen-maser phase flicker or wet-troposphere phase noise in the sense stated in the paper, and the possibility of solving in sub-daily batches down to $x_p$77–$x_p$78 h or shorter, limited by the number of observations per interval. The cited geophysical uses include near-real-time monitoring of Earth rotation irregularities, independent validation of precession–nutation models, and improved separation of axial and polar excitations from atmospheric, oceanic, and core–mantle coupling processes (Titov, 7 Jul 2025).

6. Related instantaneous-pole velocity methods in geomagnetic models

A related, but physically distinct, pole-tracking method is developed for the motion of the magnetic pole using global field models. Let $x_p$79 be the geomagnetic scalar potential expressed in spherical harmonics. On the reference sphere of radius $x_p$80,

$x_p$81

and

$x_p$82

By definition, the magnetic pole at time $x_p$83 is the point for which

$x_p$84

Differentiating the conditions $x_p$85 and $x_p$86 along the pole trajectory produces

$x_p$87

with

$x_p$88

so that

$x_p$89

provided $x_p$90 (Ivanov et al., 2022).

The global models IGRF and COV-OBSx2 provide discrete epochs, and the pole positions and velocities at those nodes are interpolated by a Hermite spline. On each interval $x_p$91, with $x_p$92 and $x_p$93, the longitude is

$x_p$94

and similarly for $x_p$95. The Hermite basis functions guarantee $x_p$96 continuity and preserve the instantaneous-pole velocity information through each node (Ivanov et al., 2022).

The paper reports that synthetic tests on a dipole of known linear motion recover the exact velocity to $x_p$97 for typical $x_p$98, and to $x_p$99 error for $y_p$00 yr steps in the full octupole reduced IGRF test. Real IGRF north-pole velocities rise from $y_p$01 in the mid-1990s to $y_p$02 today, while COV-OBSx2 gives very similar magnitudes but a smoother time series owing to its $y_p$03-yr epoch spacing. Direction uncertainties can reach tens of degrees when interval velocities change rapidly, but the Hermite-spline trajectories closely follow the epoch-to-epoch pole path while preserving a physically plausible, $y_p$04-smooth motion (Ivanov et al., 2022).

This suggests that instantaneous-pole methods extend beyond mechanical rotation-axis estimation to moving-pole problems defined implicitly by field constraints and model derivatives.

7. Ambiguities, error sources, and interpretive limits

The principal limitations of IRP determinations differ by domain. In the 2I/Borisov case, the result depends on the jet-origin assumption: if the observed feature is not produced by a source at or very near the spin pole, the derived pole can be spurious. The nucleus is never seen directly, all light is from dust, and dust dynamical and scattering effects can bias the measured position angle by up to $y_p$05, exactly the level of the reported uncertainty. HST imaging alone also leaves a sense-of-rotation ambiguity, and although the plane convergence suggests principal-axis rotation, a small non-principal-axis component with period $y_p$06 h or low amplitude could still be present undetected. Two other groups using the same HST imagery but different source-location assumptions derived very different pole directions, demonstrating that the IRP is not uniquely constrained without an independent handle on source location and stability (Bolin et al., 2019).

In the LLR case, the achievable nightly precision depends strongly on the number of normal points per night. The results are described as stable when at least $y_p$07–$y_p$08 normal points per night are available, and the addition of non-tidal station-loading corrections produces only a marginal, though consistent, improvement of about $y_p$09 percent. The study also states that LLR pole-motion precision still lags behind VLBI/GNSS at the level of tens of $y_p$10as, even though its long time span and dynamical tie make it a valuable independent check of other ERP series (Singh et al., 2021).

In the VLBI fringe-frequency approach, the formal error budget is at the prad/s level, but the analysis still includes model deficiencies such as troposphere, maser drift, and relativistic terms in the residual term $y_p$11. Systematic effects are handled by explicit modelling, and the reported residual systematics in $y_p$12 are $y_p$13 prad/s. A plausible implication is that the method is limited less by algebraic identifiability than by modelling fidelity and network strength, since the paper states that the formal error can be $y_p$14 prad/s in relative units, or better, if a large international VLBI network is at work (Titov, 7 Jul 2025).

For geomagnetic pole-velocity estimation, the local horizontal-field gradient matrix $y_p$15 must be nonsingular, the spherical-harmonic model truncation neglects external field and rapid variations, and the direction of the inferred velocity can change substantially when epoch spacing is coarse or interval velocities vary rapidly. Reported uncertainties include direction jumps up to $y_p$16 between 1975 and 1980 in IGRF, $y_p$17 in 1945–50 for SMP, and up to $y_p$18 in the worst 1908–1916 COV-OBSx2 segment (Ivanov et al., 2022).

Across these literatures, the common pattern is that the IRP is not observed directly. It is reconstructed from a measurement model—jet morphology, laser ranging geometry, VLBI delay rate, or field zeros—and the scientific meaning of the recovered pole therefore depends on the validity of the physical assumptions built into that model.