A Virgo Environmental Survey Tracing Ionised Gas Emission (VESTIGE). XXIII. NGC 4569 past orbit reconstruction with ORFEO

Introduction

Galaxies in dense environments differ systematically from their field counterparts. Dense environments contain a higher fraction of early-type galaxies [36], while cluster spirals have lower atomic gas content [48, 28, 89, 42]; H I-deficient systems can also show reduced molecular-gas reservoirs [40, 8] and star-formation activity [56, 43, 60, 46, 20]. These trends demonstrate the important role of environment in regulating galaxy evolution (see e.g. Peng et al. 2010), but the present-day state of a galaxy does not by itself identify the physical process or interaction history that produced it.

Several mechanisms can drive this transformation (see reviews by Boselli & Gavazzi 2006, 2014; Cortese et al. 2021), and their different physical action gives rise to different characteristic timescales. Gravitational interactions with nearby galaxies and with the global cluster potential can perturb both the stellar and gaseous components [70, 25, 72]; repeated high-speed encounters in particular can produce cumulative transformations over several Gyr [72, 7]. Starvation instead suppresses the replenishment of the cold interstellar medium [59], so star formation declines gradually as the existing gas reservoir is consumed over several Gyr [7, 20]. Ram-pressure stripping (RPS), by contrast, acts directly on the interstellar medium when the pressure exerted by the intracluster medium (ICM) overcomes the gravitational restoring force of the galaxy [47, 15] and can remove gas on sub-Gyr timescales, producing truncated star-forming discs and extended gaseous tails [57, 7, 34, 32, 20, 18, 38, 15, 14].

The timing and effectiveness of environmental processing also depend on the accretion history and orbit of a galaxy. Environmental effects can begin before cluster infall, through pre-processing in groups [65, 35, 37] or suppression of gas replenishment in the cluster outskirts [2, 1], and continue after cluster entry through interactions with other galaxies and with the global cluster potential. RPS is particularly tied to the galaxy trajectory: it becomes more effective where the galaxy encounters denser ICM and high relative velocities, and therefore changes in strength and timing as these conditions vary along the orbit [47, 15].

The present-day properties of a galaxy therefore do not uniquely reveal the interaction history that produced them (see e.g. Cortese et al. 2021). Distinguishing between these environmental pathways requires information that cannot in general be recovered uniquely from the present-day galaxy alone. The central questions are therefore when, and over what timescale, the material now observed in the tail was removed, where the galaxy was along its orbit, and what environmental conditions it encountered.

Extended gaseous tails offer a direct opportunity to address this temporal and orbital problem. Because stripped material can remain observable outside the galaxy disc, its present-day spatial and kinematic structure can retain information on its evolution after removal. In favourable systems, resolved positions and velocities along a tail may therefore preserve a dynamical record of different stages of the recent interaction [39, 61, 90]. The Virgo Environmental Survey Tracing Ionised Gas Emission (VESTIGE), a deep narrow-band H$\alpha$ imaging survey of the Virgo cluster carried out at the Canada–France–Hawaii Telescope (CFHT; [12]), has revealed several galaxies with extended ionised-gas structures associated with ongoing or recent stripping.

NGC 4569 provides a particularly well-suited benchmark. Its ionised-gas tail was originally detected in H$\alpha$ out to $\simeq80$ kpc ([10]), while deeper observations now trace low-surface-brightness emission to at least $\simeq230$ kpc ([90]). New spectroscopy provides LOS velocities for 94 H$\alpha$-emitting regions along the galaxy and tail ([90]), revealing a coherent kinematic sequence and turning this extended structure from a predominantly morphological signature into a resolved spatial and kinematic record of the recent interaction. Previous modelling has also yielded different characteristic epochs for the interaction: Vollmer et al. [93] favoured a peak in ram pressure roughly 300 Myr ago, whereas Boselli et al. [7] inferred a substantially more recent stripping event, of order 100 Myr ago. NGC 4569 therefore provides both the observational leverage needed to attempt a dynamical reconstruction and independent estimates against which its inferred chronology can be assessed.

Previous studies have constrained environmental histories using complementary observables. Spatially resolved stellar populations and star-formation histories provide estimates of quenching timescales ([7, 34]); tail morphology and orientation constrain the projected direction of motion and, with additional assumptions, aspects of the galaxy orbit ([81, 67]); cluster projected phase space provides statistical information on infall, orbital, and stripping stages ([52]); resolved gas spectroscopy traces the dynamics of stripped material ([39, 61, 90]); and the surveying gas distribution constrains the strength and evolutionary stage of stripping ([58]). At the level of individual systems, tailored dynamical and hydrodynamical models combine morphology and kinematics to constrain stripping epochs, wind geometry, ram-pressure histories, and orbital properties ([93, 69, 19, 94]). These tailored models are predominantly forward: candidate interaction histories are specified or explored, evolved, and their predicted observables compared with the data.

Here we introduce the complementary inverse approach implemented in ORFEO (Orbits Reconstruction For Emitting Objects). Rather than starting from a prescribed interaction history and asking whether it reproduces the observed tail, ORFEO uses the resolved positions and LOS velocities of the individual tail tracers to directly constrain the recent galaxy trajectory and tail chronology², without prescribing the interaction history that generated the observed structure. To our knowledge, this is the first attempt to reconstruct the recent dynamical history of a cluster galaxy by directly inverting the resolved spatial and kinematic sequence encoded in its ionised-gas tail.

In this first application of ORFEO, we reconstruct the recent trajectory of NGC 4569 and the age sequence of its observed H$\alpha$ tail over a range of assumed LOS distances. For each reconstruction, the corresponding ram-pressure history is then derived from the inferred trajectory and used to assess whether RPS provides a physically consistent explanation for the observed stripping. As a computationally inexpensive complement to tailored simulations, this inverse approach may ultimately make time-resolved interaction histories accessible for larger samples of stripped galaxies.

The data and ORFEO framework are presented in Sects. 2 and 3, respectively. The reconstruction and its dependence on LOS distance are presented in Sect. 4. The physical assessment of the solutions and comparison with previous models are discussed in Sect. 5. We summarise our conclusions in Sect. 6.

Data

For NGC 4569, we use the complete catalogue of 94 robust H$\alpha$-emitting regions from Sun et al. [90] as direct constraints on the reconstruction. The regions trace the extended ionised-gas tail identified by VESTIGE, a deep narrow-band H$\alpha$ survey of the Virgo cluster carried out with MegaCam at CFHT ([10, 12, 13]). At a resolution of $\simeq3''$, the VESTIGE observations reach a $\sim1\sigma$ limiting surface brightness of $\Sigma(\mathrm{H}\alpha) \simeq1.5 \times10^{-18}$ erg s$^{-1}$ cm$^{-2}$ arcsec$^{-2}$. They trace low-surface-brightness emission to a projected distance of $\simeq230$ kpc, extending well beyond the $\sim80$ kpc bright tail identified in the initial observations ([10, 90]).

NGC 4569

The spectroscopic measurements used in the reconstruction were obtained with the Binospec multi-object spectrograph on the 6.5-m MMT telescope using the 600 lines mm$^{-1}$ grating. The spectra have a typical wavelength coverage of 4900–7200 Å, a dispersion of 0.62 Å pixel$^{-1}$, and a resolving power $R \simeq2800$ for 1″ slits. The measured velocities have a median statistical uncertainty of 5 km s$^{-1}$ and a systematic uncertainty of 6–10 km s$^{-1}$. Each of the 94 regions provides the three observables entering the likelihood: right ascension (RA), declination (DEC), and the observed spectroscopic redshift expressed in velocity units, $cz_{\mathrm{obs}}$. The H$\alpha$-emitting regions span from $cz_{\mathrm{obs}} \simeq-243$ to $+1033$ km s$^{-1}$ and projected distances from approximately 16 to 228 kpc for the fiducial distance $d_0 = 16.5$ Mpc. The nucleus of NGC 4569, which anchors the present-day galaxy position and systemic spectroscopic redshift, is located at (RA, DEC) = (12.61384582 h, 13.1631611°) and has $cz_{\mathrm{obs},0} = -235$ km s$^{-1}$. Figure 1 shows the galaxy nucleus and the observables used in the reconstruction, both in the sky plane and in the $cz_{\mathrm{obs}}$–$D_N$ projection, where $D_N$ is the projected distance from the galaxy nucleus in the parallel-plane approximation. The association of the H$\alpha$-emitting regions with NGC 4569 is supported by their coherent spatial distribution extending from the galaxy and by the continuous trend of recession velocity with projected distance, connecting the tail kinematics to the galaxy’s systemic velocity ([90]).

Vector (PDF)

In addition to the phase-space observables, the dynamical model of the stripped gas described in Sect. 3.2 requires a range of effective gas surface densities, representing the denser material near the disc and the diffuse gas at the outer end of the modelled tail. We use an observationally motivated range for the minimum effective surface density and a fixed dense-disc reference value for the maximum. We set the minimum effective surface density of the diffuse outer-tail gas using the neutral-gas upper-limit scale $N_{\mathrm{H}} \simeq(2$–$5) \times10^{19}\ \mathrm{cm}^{-2}$ inferred by Boselli et al. [10] from the dust non-detection, assuming dust properties and gas-to-dust ratios typical of nearby massive galaxies. This scale is comparable to the VIVA H I sensitivity [30]. For the dense disc gas, we adopt a reference surface density $\Sigma_{\mathrm{disc,ref}} \simeq10\ M_{\odot}\ \mathrm{pc}^{-2}$, representative of the observed total-gas surface density near the present-day disc truncation radius, $R_{\mathrm{strip}} \simeq4.8\ \mathrm{kpc}$, in the radial profile of NGC 4569 derived by Boselli et al. [7] from H I and CO observations [27, 50, 55]. The observed profile is smoothed to 1 kpc resolution and does not provide a formal pointwise uncertainty at this radius. We convert column densities into gas surface densities using $\Sigma_{\mathrm{gas}} = \mu m_{\mathrm{p}}N_{\mathrm{H}}$, where $m_{\mathrm{p}}$ is the proton mass and $\mu= 1.3$ accounts for helium. The quoted tail upper-limit scale motivates the adopted range for the model minimum, $\Sigma_{\mathrm{tail,min}} \in[4.3, 10.9] \times10^{-5}\ \mathrm{g\ cm}^{-2}$. The upper endpoint is fixed to the dense-disc reference value, $\Sigma_{\mathrm{disc,ref}} = 217.4 \times10^{-5}\ \mathrm{g\ cm}^{-2}$. These values set the effective surface-density interval used to model the ram-pressure deceleration of the stripped gas parcels.

Virgo Cluster A

The reconstruction also requires a description of the Virgo Cluster A environment. Its mass distribution sets the gravitational potential in which NGC 4569 and the stripped gas evolve, while the ICM density profile enters the dynamical treatment of the stripped material and the reconstruction of the ram-pressure history. We infer the Virgo Cluster A mass profile from the positions and redshifts of member galaxies selected from the Virgo catalogue of Boselli et al. [20, 14]. The parent catalogue contains 868 cluster members within a projected radius $R \simeq2R_{200}$. Membership is based primarily on spectroscopic redshifts, using in order of priority H I measurements from the ALFALFA survey [45, 49], targeted H I observations compiled in the GoldMine database [41], and optical spectroscopy, mainly from SDSS and NED. We assign galaxies to Virgo substructures following the classification scheme of Boselli et al. [20, 14], which combines sky position and redshift. In regions where a galaxy satisfies the criteria for more than one substructure, we associate it with the smaller structure. Distances to the individual substructures are adopted from Boselli et al. [11]. Early- and late-type galaxies are separated using the morphological classifications of the Virgo Cluster Catalogue (VCC; Binggeli et al. [5]), later updated by Binggeli et al. [4]. For the present analysis, we restrict the dynamical sample to galaxies associated with Cluster A, the M87-centred component that dominates the gravitational potential relevant for the orbit of NGC 4569. We adopt the position of M87, $(\mathrm{RA}_{\mathrm{cl}}, \mathrm{DEC}_{\mathrm{cl}}) = (12.516040\ \mathrm{h}, 12.270700^{\circ})$, as the spatial centre and $d_{\mathrm{cl}} = 16.5\ \mathrm{Mpc}$ as the reference distance of Cluster A [68, 26]. To define the LOS component of this reference frame, we use the observed recession velocity of M87, $cz_{\mathrm{cl}} = 1307\ \mathrm{km\ s}^{-1}$. Relative to the Hubble flow, this corresponds to $v_{\parallel,\mathrm{cl}} = cz_{\mathrm{cl}} - d_{\mathrm{cl}}H_{0} = 102.5\ \mathrm{km\ s}^{-1}$, for $H_{0} = 73\ \mathrm{km\ s}^{-1}\mathrm{Mpc}^{-1}$. Together, these quantities define the M87-centred reference frame used to compute galaxy peculiar velocities and initialise the ORFEO orbit reconstruction. The resulting Cluster A sample contains 285 galaxies with spectroscopic recession velocities in the range $-730 \leq cz \leq3000\ \mathrm{km\ s}^{-1}$, which reflects the adopted membership classification rather than a symmetric velocity cut around $cz_{\mathrm{cl}}$. Of these, 185 are classified as early types and 100 as late types. The dynamical modelling of this sample and the derivation of the NFW potential used by ORFEO are described in Sect. 3.4.1.

To describe the interaction of the stripped gas with the Virgo ICM and to evaluate the ram pressure along the reconstructed trajectory, we adopt a spherically symmetric analytic approximation to the Virgo gas-density profile based on the X-ray analysis of Simionescu et al. [88]. The profile is centred on M87 and is written in terms of the electron density as $n_{\mathrm{e,ICM}}(r) = n_{\mathrm{e},0}\left[1 + (r/r_{\mathrm{c}})^{2}\right]^{-3\beta/2}$, with $n_{\mathrm{e},0} = 3.13 \times10^{-2}\ \mathrm{cm}^{-3}$, $r_{\mathrm{c}} = 13\ \mathrm{kpc}$, and $\beta= 0.47$. This profile is fixed throughout the analysis. When computing ram pressure, we convert the electron density into a gas mass density before evaluating $P_{\mathrm{ram}} = \rho_{\mathrm{ICM}}v^{2}$.

Orbit reconstruction

The ORFEO algorithm reconstructs the orbit of a cluster galaxy by using the phase-space structure of its stripped gas as a dynamical tracer of its recent motion through the cluster environment. For each trial present-day three-dimensional position and velocity in the cluster rest frame, ORFEO integrates the galaxy trajectory in the host-cluster potential. The tail is then constructed by sampling point-like parcels at different lookback times along this trajectory and evolving them to the present under cluster gravity and ram-pressure drag, defining a model that is projected onto sky position and LOS velocity. For a fixed assumed LOS distance, comparison with the observed H$\alpha$-emitting regions constrains the two unknown transverse velocity components of the galaxy.

Orbit equations

We model the galaxy as a test particle moving in the spherically symmetric gravitational potential of the host cluster, $\phi(r) = \phi(r)$. The specific energy and specific angular momentum are conserved,

where $\mathbf{r}$ and $\mathbf{v}$ are the position and velocity in the cluster rest frame, $r = \lvert\mathbf{r}\rvert$, $v = \lvert\mathbf{v}\rvert$, and $m$ is the galaxy mass. Using specific quantities removes the explicit mass dependence from the orbit solution. For each trial state $(\mathbf{r}_0, \mathbf{v}_0)$ at time $t_0$, ORFEO computes $e$ and $\ell$, rotates the coordinate system so that the $z$-axis is aligned with $\ell$, and solves the motion in the $(x,y)$ orbital plane. In this frame, the position is described by the polar coordinates $r(t) = \sqrt{x(t)^2 + y(t)^2}$ and $\theta(t) = \operatorname{atan2}[y(t),x(t)]$. Defining the effective radial potential $u_{\mathrm{eff}}(r) = \phi(r) + \frac{\ell^2}{2r^2}$, $\ell= \lvert\ell\rvert$, energy conservation gives the radial evolution as

For radial orbits, $\ell= 0$ and $\theta(t) = \theta_0$. For non-radial orbits, ORFEO determines the radial turning points from $e-u_{\mathrm{eff}}(r) = 0$, distinguishing between unbound trajectories, $e \ge0$, and bound trajectories, $e < 0$. At each turning point, the sign of the radial velocity is reversed, while the angular coordinate evolves as

The trajectory is then transformed back to the original cluster-centred reference frame.

To reconstruct the past orbit, ORFEO integrates the time-reversed trajectory forward from the present state, using the reversed peculiar velocity. The resulting solution is then mapped back to physical time and velocity, such that $t = 0$ at the present epoch, $t < 0$ in the past, and $\mathbf{v}(t)$ has the physical velocity sign.

Gaseous tail

ORFEO adopts a one-dimensional orbital-spine approximation in which the large-scale tail spine follows the reconstructed galaxy path, while ram-pressure drag causes stripped material to lag along it. This approximation is motivated by hydrodynamical simulations of galaxies passing through cluster pericentre. Across the orbits and disc inclinations examined by Roediger & Brüggen [84]; Roediger & Brüggen [82], the stripped gas forms wakes extending broadly along the galaxy’s orbit, despite flaring and oscillations within the wake. These results motivate using the galaxy trajectory to describe the collective tail spine, rather than the detailed trajectories of individual gas elements. We therefore use the reconstructed trajectory from Sect. 3.1 to model the large-scale tail geometry and its LOS-velocity trend, without resolving the detailed remanentary or multiphase structure of the stripped gas or the gas-removal mechanism itself. The scope of this approximation is discussed in Appendix A.3.

We parametrise the reconstructed trajectory by its arc-length coordinate $\gamma(t)$ so that $\mathbf{r}(t) = \mathbf{r}[\gamma(t)]$. We discretise the stripped gas into parcels, each indexed by an assigned stripping time $t_{\mathrm{strip}} \le0$. A parcel initialised at $t_{\mathrm{strip}}$ is placed at the galaxy position along the reconstructed trajectory and assigned the corresponding galaxy bulk orbital velocity along the trajectory; no local disc-rotation component is added,

where dots denote time derivatives. The subsequent motion of the parcel along the tail spine is governed by

The first term in (6) is the cluster gravitational acceleration projected along the reconstructed trajectory,

where $\mathrm{d}\mathbf{r}/\mathrm{d}\gamma$ is the unit tangent because $\gamma$ is an arc-length coordinate. The second term describes ram-pressure drag. For a parcel moving through the ICM, we write the characteristic drag timescale as

with $\mathbf{r}_{\mathrm{gas}}(t) = \mathbf{r}[\gamma_{\mathrm{gas}}(t)]$ and $v_{\mathrm{gas}}(t) = \lvert\dot{\gamma}_{\mathrm{gas}}(t)\rvert$. Here $\rho_{\mathrm{ICM}}$ is the local ICM mass density, $C_{\mathrm{d}} = 1$ is adopted as an effective order-unity drag coefficient, following commonly used astrophysical gas-drag prescriptions (Roediger & Brüggen [84]; Bekki [3]), while $\Sigma_{\mathrm{gas}}(t;t_{\mathrm{strip}})$ is the effective gas surface density of the parcel. Since the drag acceleration depends on the ratio $C_{\mathrm{d}}/\Sigma_{\mathrm{gas}}$, the normalisation of $C_{\mathrm{d}}$ is degenerate with the adopted surface-density scale. The parcels are point-like Lagrangian tracers and no internal cloud geometry is assigned to them. Neglecting bulk motions of the ICM, $v_{\mathrm{gas}}$ is the relevant parcel–ICM speed, so that the drag term in (6) corresponds to the standard ram-pressure acceleration $C_{\mathrm{d}}\rho_{\mathrm{ICM}}v_{\mathrm{gas}}^2/(2\Sigma_{\mathrm{gas}})$, projected along the tail spine.

The effective surface density entering (8) parametrises differences in the net parcel–ICM coupling associated with the gas region of origin and unresolved post-stripping evolution. It is assigned at $t_{\mathrm{strip}}$ and held fixed during the subsequent parcel motion, $\Sigma_{\mathrm{gas}}(t;t_{\mathrm{strip}}) = \Sigma_{\mathrm{gas}}(t_{\mathrm{strip}})$. It should therefore be interpreted as an effective drag parameter rather than as a resolved surface-density history of an individual parcel. This effective coupling may also partly represent unresolved release physics and the galaxy’s gravitational attraction, which is not explicitly included in (6), as discussed in Appendix A.3. We anchor the assigned sequence to the observationally motivated surface-density interval introduced in Sect. 2.1, bounded by the dense-disc reference value, $\Sigma_{\mathrm{disc,ref}}$, and the adopted minimum effective surface density for the diffuse outer-tail gas, $\Sigma_{\mathrm{tail,min}}$. The assigned surface density approaches $\Sigma_{\mathrm{disc,ref}}$ for the most recently stripped parcels, while older parcels are assigned progressively lower values down to $\Sigma_{\mathrm{tail,min}}$. For $t_{\mathrm{strip}} < -t_{\mathrm{scale}}$, the assigned value is kept fixed at $\Sigma_{\mathrm{tail,min}}$, avoiding an extrapolation below the adopted surface-density floor. Defining $\Delta\Sigma\equiv\Sigma_{\mathrm{disc,ref}}-\Sigma_{\mathrm{tail,min}}$, we use

Here $t_{\mathrm{max}}$ is the maximum lookback time over which model parcels are sampled, chosen to be longer than the expected tail age. Equation (9) is the simplest interpolation between the two observationally motivated surface-density anchors, introducing no additional shape parameter; it is not a resolved density-evolution equation for an individual parcel. The parameter $t_{\mathrm{scale}}$ in Eq. (9) sets the timescale over which the assigned surface-density sequence spans the interval between the dense-disc and diluted-tail reference values. Since it is not known a priori, we determine $t_{\mathrm{scale}}$ through the scan described in Appendix B.2, selecting the preferred fit among solutions for which $t_{\mathrm{scale}}$ and $T_{\mathrm{tail}}$ remain consistent within their uncertainties. The optimisation procedure, the validation of the effective surface-density prescription against hydrodynamical simulations, and the propagation of the associated uncertainty are described in Appendices A and B.2.

For each trial orbit, the integration of Eq. (6) gives the present-day position and velocity of all stripped parcels along the reconstructed trajectory. These quantities are mapped back to three-dimensional space through $r(\gamma)$ and projected onto sky position and LOS velocity. The resulting sequence defines the ORFEO model tail, which is then compared with the H$\alpha$-emitting regions described in Sect. 2.1. The scope of the gaseous-tail approximation and the interpretation of the parcel stripping times are summarised in Appendix A.3.

Data preparation and orbit fitting

Equations (1)–(4) determine the galaxy trajectory once its present-day position and velocity in the cluster rest frame are specified. For a galaxy observed in projection, however, this state is only partially known: the measured quantities $(\mathrm{RA}_0,\mathrm{DEC}_0,z_{\mathrm{obs},0})$ fix the sky position and observed recession velocity, but not the two Cartesian velocity components transverse to the LOS, $\nu_{\perp,1}$ and $\nu_{\perp,2}$. In addition, unless an independent estimate of the galaxy distance $d_0$ is adopted, $z_{\mathrm{obs},0}$ cannot be decomposed uniquely into Hubble-flow and LOS peculiar-velocity contributions. ORFEO turns this incomplete information into a constrained inverse problem by exploring the missing components of the present-day state and selecting those for which the reconstructed orbit, combined with the gas-tail model, reproduces the observed spatial and kinematic spine of the stripped gas.

The observed position and recession velocity of NGC 4569 define the present-day boundary condition of the galaxy orbit, while the likelihood is evaluated from the tail measurements $\{\mathrm{RA}_i,\mathrm{DEC}_i,z_{\mathrm{obs},i}\}_{i=1,\ldots,N}$, described in Sect. 2.1. The cluster centre, distance, LOS velocity, and gravitational potential described in Sects. 2.2 and 3.4 define the rest frame in which the orbit is integrated. For an assumed distance, the observed recession velocity is converted into the LOS peculiar velocity needed to initialise the orbit through

where $\nu_{\parallel}$ is measured along the LOS and $c$ is the speed of light. In this work, $d_0$ is treated as an external assumption rather than as a continuous fit parameter: we adopt $d_0=d_{\mathrm{cl}}$ in the fiducial reconstruction and repeat the analysis over a discrete set of assumed distances in Sect. 4.2. For each fixed $d_0$, the fitted parameter vector is therefore $\boldsymbol{\Omega}=(\nu_{\perp,1},\nu_{\perp,2})$. The LOS peculiar velocity is fixed by Eq. (10) and included in the orbit integration. In the fixed Virgo-centred frame, $\nu_{\perp,1}>0$ points west and $\nu_{\perp,2}>0$ north. The $\nu_{\mathrm{RA}}$ and $\nu_{\mathrm{DEC}}$ quoted later describe the same reconstructed three-dimensional velocity projected onto the local east–north tangent plane at NGC 4569; the small numerical differences arise because this local tangent basis is not identical to the fixed Virgo-centred transverse basis. The corresponding present-day position and velocity are then shifted to the cluster rest frame before orbit integration. Given a trial $\boldsymbol{\Omega}$, ORFEO reconstructs the galaxy orbit, evolves the stripped gas parcels as described in Sect. 3.2, and projects the resulting model tail back onto the observable space $(\mathrm{RA},\mathrm{DEC},z_{\mathrm{obs}})$. The projected model tail is then interpolated as a function of the gas-parcel time coordinate $t$, giving the continuous model predictions $\mathrm{RA}^{\mathrm{m}}(t;\boldsymbol{\Omega})$, $\mathrm{DEC}^{\mathrm{m}}(t;\boldsymbol{\Omega})$, and $z_{\mathrm{obs}}^{\mathrm{m}}(t;\boldsymbol{\Omega})$. The model recession velocity is computed from the model distance and LOS peculiar velocity through Eq. (10). For each observed emitting region $i$, we identify the corresponding point on the model tail by finding the value $t_i^*$ that minimises the dimensionless distance

with $t_i^*=\arg\min_t D_i^2(t;\boldsymbol{\Omega})$. The agreement between the model tail and the data is then quantified through the $\chi^2$-like statistic

The quantities $\sigma_{\mathrm{RA}}$, $\sigma_{\mathrm{DEC}}$, and $\sigma_{z_{\mathrm{obs}}}$ are effective dispersions, not measurement errors on individual regions. This choice reflects the goal of the fit: the H$\alpha$-emitting regions trace an extended tail with finite width and internal velocity scatter, while the model describes its central spine. We therefore estimate these dispersions from the observed scatter of the tail around its mean spatial and kinematic trends, and use them both to associate observed regions with the model tail and to weight the residuals in Eq. (12). With this definition, $\chi^2_{\mathrm{tail}}$ measures how well a trial orbit reproduces the large-scale position and velocity structure of the stripped gas.

The best-fitting parameters, $\Omega_{\mathrm{BF}}$, are obtained by minimising Eq. (12) with the Nelder–Mead algorithm (Nelder & Mead [75]). This parameter set fixes the present-day cluster-frame state of the galaxy and therefore defines its reconstructed past orbit. We explore the solution space through $\Delta\chi^2_{\mathrm{tail}}(\Omega)=\chi^2_{\mathrm{tail}}(\Omega)-\chi^2_{\mathrm{tail,min}}$ profiles and surfaces, which show how the reconstruction quality changes away from the best-fitting solution. The construction of the uncertainty budget from these profiles, and the propagation of the systematic contribution associated with the uncertainty on $t_{\mathrm{scale}}$, are described in Appendix § 8.§ 2. A validation of the $\chi^2$-like reconstruction procedure on a controlled toy model is presented in Appendix § 8.

Gravitational potential

The orbit reconstruction requires specifying the gravitational field of the host cluster. In ORFEO, this field enters as an external spherically symmetric potential, $\phi(r)$, which determines the acceleration acting on the galaxy and on the stripped gas parcels. This separates the orbit-fitting problem from the calibration of the cluster mass profile: the reconstruction can be run once a spherical potential has been specified from independent dynamical, X-ray, or lensing constraints. In the present application, we adopt a Navarro–Frenk–White (NFW) profile (Navarro et al. [74]), fully specified by the virial mass $M_{200}$ and the scale radius $R_{\mathrm{s}}$, calibrated for Virgo Cluster A through the dynamical modelling described in Sect. 3.4.1.

For an NFW profile, the cumulative mass distribution is

where $c_{200}=r_{200}/R_{\mathrm{s}}$ is the concentration. The corresponding gravitational potential, defined to vanish at infinity, can be written as

where $G$ is the gravitational constant. Equivalently, this expression corresponds to an NFW scale density $\rho_{\mathrm{s}}$ defined by

The radius $r_{200}$ is defined as the radius within which the mean enclosed density is 200 times the critical density of the Universe,

Once $M_{200}$ and $R_{\mathrm{s}}$ are specified, Eqs. (13)–(16) fully determine the cluster potential entering the orbit integration.

Mass profile reconstruction with MG-MAMPOSSt

To calibrate the NFW potential defined above for Virgo Cluster A, we infer $M_{200}$ and $R_{\mathrm{s}}$ from the projected phase-space distribution of cluster member galaxies. We use the MG-MAMPOSSt algorithm (Pizzuti et al. [79]), an extension of the MAMPOSSt method (Mamon et al. [62]), which constrains the mass profile, tracer-density profile, and velocity-anisotropy profile of a galaxy cluster under the assumptions of spherical symmetry and dynamical equilibrium.

Published estimates of the Virgo Cluster A mass vary significantly, reflecting the different tracers and assumptions used to model the cluster, from X-ray hydrostatic analyses to galaxy kinematics (e.g. [66, 88, 54] (Conselice et al. 2001)). Because ORFEO integrates galaxy orbits, we calibrate its gravitational potential directly from the projected phase-space distribution of cluster galaxies, which constrains the mass profile governing their motions, rather than adopting a single literature value. To assess the dependence on the tracer population, we perform the analysis for both the full member-galaxy sample and the early-type subsample described in Sect. 2.2. The full analysis, reported in Appendix C, explores different combinations of mass model, projected tracer-density profile, and velocity-anisotropy prescription. Here we retain the NFW solutions, matching the mass profile adopted for the ORFEO potential. We adopt as a fiducial the solution obtained from the early-type Virgo Cluster A sample using an NFW mass profile, a generalised Osipkov–Merritt anisotropy profile, and a projected Hernquist tracer-density profile. The early-type population is expected to trace the virialised Virgo Cluster A component more closely. The corresponding $M_{200}$ and $R_{\mathrm{s}}$, listed in Table 1, define the fiducial potential, while the full-sample NFW solution is shown for comparison to illustrate the dependence on tracer selection.³

$M(r)$ model	$\beta(r)$ model	$N(r)$ model	sample	$r_{200}$ [Mpc]	$M_{200}$ [$10^{14}\,M_{\odot}$]	$R_s$ [Mpc]	$r_{\nu}$ [Mpc]	$\mathcal{A}_{\infty}$	$\mathcal{A}_0$	$\Delta$BIC
NFW	gOM	pHer	ET	$1.63^{+0.32}_{-0.21}$	$5.4^{+2.9}_{-2.3}$	$0.36^{+0.32}_{-0.07}$	$0.50^{+0.12}_{-0.06}$	$1.70^{+1.86}_{-1.23}$	$1.34^{+1.96}_{-0.98}$	0.25
NFW	gOM	pHer	ALL	$1.86^{+0.30}_{-0.26}$	$7.5^{+3.8}_{-3.0}$	$0.54^{+0.61}_{-0.45}$	$0.57^{+0.07}_{-0.09}$	$1.53^{+1.80}_{-1.02}$	$1.22^{+1.74}_{-0.87}$	0.51

Results

We use ORFEO to test whether the observed spatial and LOS-velocity structure of the H$\alpha$ tail of NGC 4569 can be described by a coherent large-scale stripping sequence in the Virgo Cluster A potential. We first consider a fixed-distance reference case, adopting the canonical Virgo Cluster A distance $d_0=d_{\mathrm{cl}}=16.5$ Mpc in Sect. 4.1, consistent with the observational analysis of NGC 4569 and its tail (Sun et al. [90]). We then relax this assumption in Sect. 4.2 and repeat the reconstruction over the LOS-distance range spanned by published estimates. Varying $d_0$ changes both the three-dimensional position of NGC 4569 in the Virgo potential and the peculiar velocity inferred from its observed recession velocity, allowing us to quantify how the uncertain LOS position propagates into the reconstructed orbital phase and ram-pressure evolution.

Reconstruction at the Virgo Cluster A distance

We first test the sensitivity of the fixed-distance reconstruction to the adopted minimum tail surface density, $\Sigma_{\mathrm{tail,min}}$, entering the tail parametrisation in Eq. (9). At $d_0=d_{\mathrm{cl}}=16.5$ Mpc, we vary

$\Sigma_{\mathrm{tail,min}}$ over 13 values spanning $[4.3, 17.3] \times10^{-5}$ g cm$^{-2}$, covering the observationally motivated range discussed in Sect. 2.1 and extending above it for robustness. For each value, we monitor the reduced goodness of fit, $\chi^2/\nu$, with $\nu= 2N - 2 = 186$; the tail age $T_{\mathrm{tail}}$⁴; the signed time of maximum ram pressure, $t_{\mathrm{Pram,peak}}$; and the present-day velocity amplitude $|v|$ in the cluster reference frame. Figure 2 shows the dependence of these quantities on $\Sigma_{\mathrm{tail,min}}$. Across the inspected grid, the reduced goodness of fit varies only at the few-$10^{-3}$ level around $\chi^2/\nu\simeq0.597$. The inferred tail age, peak ram-pressure time, and present-day velocity amplitude remain consistent within their uncertainties and show no systematic trend, indicating that the fixed-distance reconstruction is only weakly sensitive to $\Sigma_{\mathrm{tail,min}}$ over the explored range. We therefore adopt the central value of the observationally motivated interval, $\Sigma_{\mathrm{tail,min}} = 7.6 \times10^{-5}$ g cm$^{-2}$; this choice is observationally motivated rather than selected by variations in the fit quality.

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Fiducial orbit

With the fiducial tail parametrisation fixed, we now reconstruct the dynamics of NGC 4569 at the canonical Virgo Cluster A distance. Figure 3 shows the fiducial reconstruction at $d_0 = 16.5$ Mpc.

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The reconstructed spine follows the projected orientation of the observed H$\alpha$ tail in the sky plane and reproduces the monotonic increase of $c z_{\mathrm{obs}}$ with projected distance from the galaxy in the $c z_{\mathrm{obs}}$–$D_{\mathrm{N}}$ projection. This LOS phase-space agreement is the main constraint on the reconstruction: the fitted sequence must reproduce the kinematic ordering of the stripped gas, not only its projected spatial distribution. The scatter of the observed tracers around the reconstructed spine is expected because the model describes the large-scale tail as a one-dimensional sequence rather than resolving its width and internal structure.

The best-fitting parameters, listed in Table 2, place NGC 4569 on a high-velocity but energetically bound orbit in the adopted Virgo Cluster A potential, with $e = v^2/2 + \phi< 0$ and a velocity dominated by the transverse component $v_{\perp,1}$. Along the reconstructed orbit, ORFEO assigns a model lookback time to each observed H$\alpha$ region, shown by the colour coding in Figure 3. These times show a clear spatial ordering: tracers close to the disc form a compact group assigned to the most recent 5 Myr, $t_{\mathrm{strip}} > -5$ Myr, whereas the extended tail follows the reconstructed spine and is assigned model lookback times older than $\simeq75$ Myr. This apparent separation may partly reflect the spatial sampling of the observed H$\alpha$ regions, but within the fitted sequence the extended tail is not associated with the most recent stripping times. The maximum lookback time assigned to the fitted H$\alpha$ tracers gives $T_{\mathrm{tail}} = 250^{+37}_{-19}$ Myr. This age refers to the fitted ionised tail, not to the full duration of gas removal from NGC 4569. Older stripped material may exist beyond the currently detectable H$\alpha$ phase and therefore does not enter the present reconstruction. The fiducial solution therefore interprets the extended H$\alpha$ tail as a dynamically ordered structure sampling a substantial fraction of the recent reconstructed orbit rather than only its most recent phase.

Quantity	Value
Orbit and tail quantities
$v_{\perp,1}$	$-1876^{+275}_{-214}\ {\rm km\ s^{-1}}$
$v_{\perp,2}$	$61^{+80}_{-83}\ {\rm km\ s^{-1}}$
$T_{\rm tail}$	$250^{+37}_{-27}\ {\rm Myr}$
Ram-pressure quantities
$t_{\rm Pram,peak}$	$-163^{+11}_{-15}\ {\rm Myr}$
$P_{\rm ram,peak}$	$(3.20^{+0.68}_{-0.73})\times10^{-11}\ {\rm dyn\ cm^{-2}}$
$P_{\rm ram}(-T_{\rm tail})$	$(2.20^{+0.57}_{-0.41})\times10^{-11}\ {\rm dyn\ cm^{-2}}$
$P_{\rm ram}(0)$	$(1.27^{+0.52}_{-0.19})\times10^{-11}\ {\rm dyn\ cm^{-2}}$
$t_{\rm end}$	$\simeq-75\ {\rm Myr}$

Ram-pressure history

We define $P_{\rm ram}(t)=\rho_{\rm ICM}[D_{\rm cluster}(t)]v^2(t)$, where $D_{\rm cluster}(t)$ is the reconstructed three-dimensional distance from the Virgo Cluster A centre, $v(t)$ is the velocity amplitude in the Virgo Cluster A rest frame, and $\rho_{\rm ICM}$ is evaluated from the Virgo gas-density model described in Sect. 2.2. For $t\leq0$, we use the reconstructed past trajectory; for $t>0$, we use the forward integration from the best-fitting present-day state. Figure 4 shows that the fiducial reconstruction places NGC 4569 after the maximum of its ram-pressure event. The peak occurs at $t_{\rm Pram,peak}=-163^{+11}_{-15}\ {\rm Myr}$, near the closest approach to the Virgo Cluster A centre, $D_{\rm cluster}=305^{+19}_{-25}\ {\rm kpc}$, when the galaxy reaches $|v|=2839^{+383}_{-191}\ {\rm km\ s^{-1}}$. Although this velocity exceeds the characteristic virial velocity of Virgo Cluster A, the virial velocity is not an upper limit for an infalling galaxy, and the reconstructed orbit remains bound in the fiducial MG-MAMPOSSt potential. The high velocity is therefore a notable requirement of the fitted tail phase-space structure.

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Taking the ram pressure at the model lookback time of the oldest fitted tail tracer, $P_{\rm ram}(-T_{\rm tail})$, as a reference, we find $P_{\rm ram,peak}>P_{\rm ram}(-T_{\rm tail})>P_{\rm ram}(0)$. Thus, the present-day ram pressure is lower than that along the reconstructed orbit at the age of the oldest fitted H$\alpha$ tracer. Since stripping is expected to

proceed broadly from the outer towards the inner disc, where the gas surface density and restoring force are higher, continued removal of the remaining gas would generally require a ram pressure at least comparable to, and likely higher than, the level associated with earlier stripping phases. The ordering $P_{\rm ram}(-T_{\rm tail})>P_{\rm ram}(0)$, together with the expected stronger binding of the retained disc gas, therefore suggests that the main disc-stripping episode has ended by the present epoch. In this interpretation, stripping continues after the ram-pressure maximum until the declining pressure becomes insufficient to remove the remaining disc gas, while more diffuse material may still be removed. The last post-peak crossing of this reference level occurs at $t_{\rm end}\simeq-75\ {\rm Myr}$, approximately coinciding with the youngest model ages assigned to the extended tail in Sect. 4.1.1. We adopt this crossing as an indicative end of the main disc-stripping episode. A quantitative test of this interpretation using the restoring force at the observed disc truncation radius is presented in Sect. 5.1.

Exploring distance assumptions

Recent surface-brightness-fluctuation (SBF) measurements by Cantiello et al. [26] place NGC 4569 at larger distances than the canonical Virgo Cluster A value, with the catalogue reference estimate $d_{\rm ref}=17.47\pm1.06\ {\rm Mpc}$, derived using their preferred SBF-colour calibration, and $d_{\rm mean}=18.09\pm1.23\ {\rm Mpc}$, obtained by averaging the estimates from all available calibrations. To assess how different assumed distances change the inferred orbit and ram-pressure history of NGC 4569, we therefore repeat the full reconstruction over an interval encompassing both $d_{\rm cl}=16.5\ {\rm Mpc}$ and the SBF-based estimates.

Figure 5 shows the reconstruction for 36 fixed distances spanning $d_0\in[15.0,18.5]\ {\rm Mpc}$ in steps of $0.1\ {\rm Mpc}$, refitting $v_{\perp,1}$ and $v_{\perp,2}$ at each value. The reduced robust goodness of fit remains close to $\chi^2_{\rm robust}/\nu\sim0.54$–$0.64$ over most of the scan, showing that the projected spatial and LOS-velocity structure of the H$\alpha$ tail can be reproduced over a broad range of assumed distances. The fit quality alone therefore does not select a unique distance for NGC 4569. The first clear change appears in the timing of the ram-pressure maximum. At low assumed distances, $t_{\mathrm{Pram,peak}} < 0$, so NGC 4569 is at present after the maximum of its ram-pressure event, as in the fiducial solution. As $d_0$ increases, $t_{\mathrm{Pram,peak}}$ rises steadily, crosses the present epoch around 17.1 Mpc, and becomes positive at larger distances. This crossing corresponds to a change in the present-day cluster-centric radial velocity from outward to inward motion, placing the galaxy on the post-pericentre and pre-pericentre branches, respectively (Appendix § 10). In the high-distance solutions, the galaxy is therefore reconstructed before peak ram pressure, with the peak shifted up to $\sim0.8$–$1.0$ Gyr into the future near the upper end of the explored range. The same distance interval also contains a narrow window of unbound or marginally bound solutions. This identifies $d_0 \simeq16.8$–$17.1$ Mpc as a transition interval in the reconstructed orbital phase, rather than as a sharply measured critical distance. The tail age and velocity amplitude support the same transition but are less regular. Along the low-distance branch, $T_{\mathrm{tail}}$ generally decreases from $\simeq0.46$ Gyr at 15.5 Mpc to $\simeq0.25$ Gyr at the fiducial distance. Across $d_0 \simeq16.8$–$17.1$ Mpc, it reaches $\simeq0.17$–$0.28$ Gyr before increasing to $\simeq0.6$ Gyr on the high-distance branch, where inferred ages are typically 0.4–0.6 Gyr with larger uncertainties. The reconstructed $|v|$ likewise increases along the low-distance branch, becomes highly structured across the transition, and settles onto a lower-amplitude high-distance branch. The sign change of $t_{\mathrm{Pram,peak}}$ therefore remains the clearest indicator of the transition, while $T_{\mathrm{tail}}$ and $|v|$ near and beyond it should be interpreted more cautiously. The reconstructed eccentricities provide further context for the plausibility of these orbital families through comparison with recently infalling satellites in cosmological simulations ([44]); the comparison is presented in Appendix § 10.

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The ram-pressure diagnostics in Figure 6 show how the conditions for disc stripping vary across the reconstructed orbital families. As in Sect. 4.1.2, the ordering $P_{\mathrm{ram}}(-T_{\mathrm{tail}}) > P_{\mathrm{ram}}(0)$, combined with the expected stronger binding of the retained gas, qualitatively favours a configuration in which the main disc stripping episode has ended. This condition is satisfied on the low-distance branch up to $d_0 \simeq16.6$ Mpc, where the reconstructed peak pressure is of order $10^{-10}$ dyn cm$^{-2}$ and exceeds the present-day value. Across $d_0 \simeq16.6$–$17.1$ Mpc, $P_{\mathrm{ram,peak}}$, $P_{\mathrm{ram}}(-T_{\mathrm{tail}})$, and $P_{\mathrm{ram}}(0)$ become comparable, coinciding with the orbital-phase transition identified above. At larger distances, $P_{\mathrm{ram}}(-T_{\mathrm{tail}})$ drops by about an order of magnitude below $P_{\mathrm{ram}}(0)$, while the ram-pressure maximum shifts to the future. The pressure ordering thus complements the peak-time diagnostic by describing the evolution of the stripping conditions. Near the transition, the present pressure can remain above $P_{\mathrm{ram}}(-T_{\mathrm{tail}})$ even after the ram-pressure maximum has passed.

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The distance scan therefore yields two evolutionary families rather than a unique distance for NGC 4569, with the central SBF estimates of Cantiello et al. [26] lying on the high-distance, pre-peak side of the transition. The existence of this change in evolutionary interpretation is robust, whereas its exact boundary and the detailed properties of the high-distance branch should not be over-interpreted because the reconstructed $T_{\mathrm{tail}}$, $|v|$, and pressure diagnostics become increasingly branch-dependent across the transition. In Sect. 5.1, we test the physical plausibility of these solutions against an independent stripping threshold for NGC 4569.

Discussion

To interpret the reconstructed ram-pressure histories in terms of gas removal from NGC 4569, we compare them with the pressure required to strip gas at the observed disc truncation radius, estimated independently from the galaxy structural and gas properties.

Physical plausibility of the distance-dependent solutions

The distance scan in Sect. 4.2 yields two families of formally acceptable reconstructions: a low-distance post-peak branch and a high-distance branch in which the ram-pressure maximum lies in the future. The effective pressure required for gas removal depends on both the wind–disc geometry and the gas distribution [e.g. 83]. A fully orbit-dependent threshold would therefore require combining the reconstructed wind–disc angle (Appendix § 11) with an evolving gas-disc model and a geometry-dependent stripping efficiency. For a uniform comparison among the reconstructed orbits, we instead adopt a common physical reference: the pressure required to strip gas at the observed disc truncation radius of NGC 4569.

To estimate this pressure scale, we adopt a Gunn–Gott-type restoring-force criterion [47]. Stripping at a radius $R_{\mathrm{strip}}$ is expected when the external ram pressure exceeds the maximum vertical restoring force per unit area acting on the gas,

Here, $g_z(R_{\rm strip},z)$ is the vertical gravitational acceleration generated by the adopted disc mass model. We compute $g_z$ with galpy Bovy (2015). We model the stellar, atomic, and molecular discs with exponential surface-density profiles,

where $i = \star, {\rm H\,I}, {\rm H}_2$, and $r_{h,i}$ is the corresponding scale radius. For an exponential disc normalised to a total mass $M_i$ and truncated at $R_{\max}$, the central surface density is

In practice, $R_{\max}$ is chosen sufficiently large that the normalisation is effectively equivalent to integrating the exponential profile to infinity.

For NGC 4569, we evaluate Eq. (17) at the observed truncation radius, $R_{\rm strip} = 4.8\ {\rm kpc}$. We adopt a stellar scale radius $r_h = 0.59R_e = 3.7\ {\rm kpc}$, using $R_e = 6.26\ {\rm kpc}$ from Muñoz-Mateos et al. [73], and a stellar mass $M_\star= 10^{10.74}\ M_\odot$ [14]. The pre-stripping atomic gas mass is estimated from the H I deficiency: we adopt the observed mass $M_{\rm H\,I} = 10^{8.90}\ M_\odot$ and deficiency H I-def = 1.08 from GOLDMine [41], updated to $d_0 = 16.5\ {\rm Mpc}$. Together, these give a pre-stripping mass $M_{\rm H\,I} \simeq10^{9.98}\ M_\odot$. Motivated by the systematically larger radial extent of H I discs relative to stellar discs in nearby spiral galaxies (e.g. Broeils & Rhee [22], we adopt an H I disc scale radius 1.8 times larger than the stellar one. We estimate the pre-stripping molecular gas mass as $M_{{\rm H}_2} \simeq10^{10.0}\ M_\odot$, corresponding to $M_{{\rm H}_2}/M_{\rm H\,I} \simeq1$, using the scaling relations of Boselli et al. [9], and assume that this component follows the stellar scale radius. The truncated CO distribution suggests that the interaction may also have removed molecular gas [7, 8], so the present-day mass need not represent the pre-stripping reservoir. The lower VERTICO estimate, $M_{{\rm H}_2} \simeq10^{9.58}\ M_\odot$ [23], is consistent with this interpretation.

We propagate the uncertainties with a Monte Carlo procedure, assuming 0.15 dex uncertainty on $M_\star+ M_{{\rm H}_2}$, 0.3 dex on $M_{\rm H\,I}$, motivated by the scatter of the H I-deficiency calibration, and 20% uncertainty on the adopted scale radii. The resulting threshold distribution has a median and 16th–84th percentile interval of

This value provides an external physical scale against which the ORFEO ram-pressure histories can be tested. The threshold uncertainty reflects uncertainties in the stellar mass, pre-stripping gas content, and disc scale radii, which determine the local restoring pressure, and is conditional on the adopted disc profiles and stripping criterion. The reconstructed pressure uncertainties propagate those in the fitted velocity components and $t_{\rm scale}$; additional systematic uncertainties associated with the cluster potential, ICM density profile, and effective tail prescription are not included.

Figure 7 compares the stripping threshold with three characteristic pressure levels from the distance-dependent reconstructions: $P_{\rm ram,peak}$, $P_{\rm ram}(0)$, and $P_{\rm ram}(-T_{\rm tail})$. These probe complementary aspects of the inferred history. $P_{\rm ram,peak}$ tests whether the reconstructed orbit reaches conditions sufficient to produce the observed disc truncation, $P_{\rm ram}(0)$ whether such conditions are present today, and $P_{\rm ram}(-T_{\rm tail})$ whether they were already reached at the model lookback time of the oldest fitted H$\alpha$ tracer. We quantify the comparison through $\log_{10}[P_i(d_0)/P_{\rm ram,th}]$, where $P_i = \{P_{\rm ram,peak}, P_{\rm ram}(0), P_{\rm ram}(-T_{\rm tail})\}$; positive values indicate pressures above the adopted stripping threshold. The central pressure comparison favours somewhat lower distances than the fiducial value. The propagated uncertainties and the dependence on the adopted disc and cluster models nevertheless limit the precision of a distance constraint based on this diagnostic.

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Table 3 summarises the compatibility limits. At the $1\sigma$ level, only reconstructed past pressures on the low-distance branch are compatible with the stripping threshold: $P_{\rm ram,peak}$ remains compatible up to $d_0 = 16.8\ {\rm Mpc}$ and $P_{\rm ram}(-T_{\rm tail})$ up to $d_0 = 16.5 Mpc, whereas $P_{\mathrm{ram}}(0)$ remains below the threshold throughout the scan. Thus, within the nominal uncertainty budget, the present-day interaction is not sufficient to strip gas from the remaining disc at its observed truncation radius, whereas the reconstructed past event is compatible with such stripping. This comparison is conservative for the onset of stripping, since the gaseous disc was likely more extended at earlier times and its outer gas would therefore have required a lower ram pressure to be removed. The low-distance solutions therefore support a post-stripping interpretation and imply that the fitted H$\alpha$ tail may trace a later observable phase of an episode that began earlier.

Pressure diagnostic	$1\sigma$	$3\sigma$
$P_{\mathrm{ram,peak}}$	16.8	18.1
$P_{\mathrm{ram}}(-T_{\mathrm{tail}})$	16.5	17.8
$P_{\mathrm{ram}}(0)$	–	17.2

At the $3\sigma$ level the comparison is less restrictive, but the physical preference for the low-distance branch remains. $P_{\mathrm{ram,peak}}$ is compatible with the stripping threshold up to $d_0 \simeq18.1$ Mpc; however, for $d_0 \gtrsim17.1$ Mpc this peak occurs in the future and therefore cannot be associated with the event responsible for the observed disc truncation. Independent constraints from the truncated young stellar populations place the associated quenching/stripping episode at lookback times of order $10^8$ yr (e.g. [7, 34, 10]). In the same high-distance regime, neither $P_{\mathrm{ram}}(-T_{\mathrm{tail}})$ nor $P_{\mathrm{ram}}(0)$ is robustly compatible with the threshold, apart from small branch-sensitive oscillations. The high-distance histories therefore do not naturally provide the pressure required at an epoch consistent with the independently inferred truncation time.

The threshold comparison also clarifies the physical implications of the three distance anchors considered here. At the fiducial Virgo Cluster A distance, $d_0 = 16.5$ Mpc, the reconstructed past ram-pressure event is compatible with stripping at the observed disc radius, whereas $P_{\mathrm{ram}}(0)$ remains below the threshold. The SBF-based distances from Cantiello et al. [26] are less favoured by this physical consistency test: $d_{\mathrm{ref}} = 17.47$ Mpc is compatible only within the broader $3\sigma$ envelope, while $d_{\mathrm{mean}} = 18.09$ Mpc lies at the upper edge of the $3\sigma$ peak-pressure compatibility range. At both distances, however, the reconstructed ram-pressure maximum occurs in the future and therefore cannot explain the past disc truncation. High-distance solutions can therefore reproduce the tail phase space but do not naturally explain the disc truncation and H$\alpha$ tail at an epoch consistent with the independent stellar-population constraints. Independent constraints do not fully resolve this distance ambiguity. Boselli et al. [10], referring to the TRGB-based Virgo infall analysis of Karakas et al. (2014), noted that NGC 4569 may lie at $D \gtrsim17$ Mpc, close to or beyond the transition between the reconstructed branches. Conversely, Köppen et al. [58] classified NGC 4569 as a past stripper from the surviving H I disc and the inferred past-to-present ram-pressure contrast, independently supporting the post-stripping interpretation of the low-distance branch. Thus, the available distance estimates alone do not uniquely select between the reconstructed families.

Comparison with previous models of NGC 4569

NGC 4569 has long been interpreted as one of the clearest Virgo examples of advanced ram-pressure stripping. A key timing constraint was obtained by Boselli et al. [7], who modelled the multi-wavelength radial profiles of NGC 4569, including the gas, H$\alpha$, UV, optical, and near-infrared emission. They showed that starvation alone cannot reproduce the observed truncation of the gas and star-forming discs, whereas ram-pressure stripping can account for the radial quenching pattern. Their best-fitting model dates the peak of the stripping event to $\simeq100$ Myr ago. The comparatively regular stellar disc, despite the strongly disturbed gaseous component, favours a predominantly hydrodynamical origin for the tail, although an additional gravitational contribution from an interaction with another galaxy or with the cluster potential cannot be excluded.

The low-distance branch of ORFEO is consistent with this independent constraint. For $d_0 \leq16.5$ Mpc, the reconstruction places NGC 4569 after the maximum of its ram-pressure history, with $t_{\mathrm{pr,peak}} \leq-163$ Myr. Combined with the threshold analysis, which indicates that the present-day ram pressure is insufficient to strip gas from the remaining disc at its observed truncation radius, this supports interpreting the low-distance solutions as both post-peak and post-stripping. The Boselli et al. and ORFEO timescales are not identical measurements: the former is inferred from radial truncation and stellar-population gradients in the disc, whereas the latter describes the orbital ram-pressure history and the chronology of the observed ionised tail. A complementary constraint is provided by Sun et al. [90], who estimated that gas along the extended H$\alpha$ tail may require $\sim500$ Myr to decelerate towards the Virgo systemic velocity. This timescale is more directly comparable with $T_{\mathrm{tail}}$ than with $t_{\mathrm{pr,peak}}$, although the estimates rely on different dynamical assumptions.

This interpretation is also broadly consistent with the H I modelling of Vollmer et al. [93], based on 21-cm observations of the atomic gas distribution and kinematics. Their N-body/sticky-particle simulations favoured a post-peak scenario over ongoing stripping, with the imposed ram-pressure pulse peaking a few $10^2$ Myr before the best-matching snapshot. We therefore use the Vollmer et al. model as independent support for the post-stripping state, while the Boselli et al. radial-profile and stellar-population analysis provides the more direct comparison for the timing of disc truncation.

The simulation approach of Vollmer et al. [93] and the reconstruction performed here provide complementary information. Tailored simulations follow the hydrodynamical response of the gas disc and stripped material to an assumed ram-pressure history. ORFEO does not model this hydrodynamical response; instead, it uses the observed phase-space structure of the tail to reconstruct the galaxy orbit, tail chronology, and ram-pressure history. In this sense, tailored simulations predict the gas response to a prescribed interaction history, whereas ORFEO infers the recent dynamical history directly from the observed tail and can provide constrained input for such simulations.

The H$\alpha$ morphology of Boselli et al. [10] also suggested a geometrical interpretation of the interaction. They argued for a projected west-to-east orbit and proposed that NGC 4569 entered the cluster from behind, encountering the dense Virgo environment north of M87 at a cluster-centric distance of $\gtrsim230$ kpc. For $d_0 = 16.5$ Mpc, the ORFEO reconstruction gives $v_{\mathrm{RA}} = +1915\ \mathrm{km\ s^{-1}}$ and $v_{\mathrm{DEC}} = +73\ \mathrm{km\ s^{-1}}$ after projection of the reconstructed three-dimensional velocity onto the local east–north tangent plane at NGC 4569. The projected motion is therefore dominated by the RA component, corresponding to a mostly west-to-east trajectory with only a small south-to-north component; this remains true across the low-distance branch. The reconstructed closest approach likewise remains within a few $10^{2}$ kpc, with $D_{\mathrm{cluster}} = 305^{+19}_{-25}$ kpc in the fiducial solution. ORFEO thus provides a quantitative dynamical counterpart to the morphology-based picture, extending it to the motion of NGC 4569 through the Virgo ICM. Combining the reconstructed orbit with the adopted orientation of the NGC 4569 disc also yields the wind–disc angle as a derived, rather than independently fitted, quantity; its evolution for the fiducial solution is presented in Appendix E.

Conclusions

We have presented the first application of ORFEO to NGC 4569, using the resolved spatial and LOS-velocity structure of its H$\alpha$ tail to reconstruct the recent galaxy trajectory and tail chronology. Combined with the adopted Virgo Cluster A potential, ICM density profile, and gas surface-density constraints, the reconstruction yields the corresponding ram-pressure history for an assumed galaxy LOS distance. Our main results are as follows.

1. Placing NGC 4569 at the canonical Virgo Cluster A distance, $d_0 = 16.5$ Mpc, ORFEO reconstructs a coherent phase-space sequence for the ionised tail, reproducing both its projected orientation and the monotonic LOS-velocity trend with distance from the galaxy. The fitted tail age is $T_{\mathrm{tail}} = 250^{+37}_{-19}$ Myr, while the ram-pressure maximum occurred at $t_{\mathrm{Pram,peak}} = -163^{+11}_{-15}$ Myr, close to the reconstructed nearest approach to the Virgo Cluster A centre at $305^{+19}_{-25}$ kpc. The negative value of $t_{\mathrm{Pram,peak}}$ places the fiducial solution after peak ram pressure, while the ordering $P_{\mathrm{ram,peak}} > P_{\mathrm{ram}}(-T_{\mathrm{tail}}) > P_{\mathrm{ram}}(0)$ shows that the present-day ram pressure is lower than at the model epoch of the oldest fitted tail tracer.

2. The LOS-distance scan does not determine a unique distance for NGC 4569, but separates comparably fitting reconstructions into two evolutionary families. Low-distance solutions, $d_0 \lesssim16.6$ Mpc, place the galaxy after peak ram pressure. At larger distances, the reconstruction moves onto a pre-peak branch, in which the ram-pressure maximum lies in the future and the present-day interaction is still strengthening. The transition occurs around $d_0 \simeq16.8$–17.1 Mpc and is identified most robustly by the sign change of $t_{\mathrm{Pram,peak}}$, although the detailed chronology of the high-distance branch is less stable.

3. An external Gunn–Gott stripping-threshold comparison provides additional physical discrimination between these comparably fitting orbital families and favours the low-distance branch. At the $1\sigma$ level, $P_{\mathrm{ram,peak}}$ remains compatible with stripping at the observed disc truncation radius up to $d_0 = 16.8$ Mpc, while $P_{\mathrm{ram}}(-T_{\mathrm{tail}})$ remains compatible up to $d_0 = 16.5$ Mpc; $P_{\mathrm{ram}}(0)$ remains below the threshold throughout the scan. At higher distances, the compatibility weakens and, beyond the transition, the reconstructed ram-pressure maximum lies in the future, making these solutions less natural as an explanation for a disc truncation independently dated to the past. The low-distance branch therefore provides the preferred post-stripping interpretation of NGC 4569.

4. The favoured low-distance reconstruction places previous constraints on NGC 4569 within a common dynamical picture. Its peak-pressure epoch complements the $\sim100$ Myr disc-truncation and stellar-population constraint of Boselli et al. [7], while the $\sim500$ Myr deceleration timescale estimated by Sun et al. [90] for extended H$\alpha$ tail provides a more direct comparison with $T_{\mathrm{tail}}$. The same reconstruction quantitatively recovers the mostly west-to-east projected motion inferred from the H$\alpha$ morphology of Boselli et al. [10], while the post-stripping model of Vollmer et al. [93] provides independent supporting context.

The method is not intrinsically restricted to H$\alpha$: resolved H$\mathrm{I}$ tails provide the same basic observables—projected position and LOS velocity—while directly constraining the gas column density entering the drag model. Other gas phases could also be used if their large-scale kinematics trace the stripped wake and an appropriate effective surface-density prescription can be defined. The minimum useful tail extent is not universal, because the constraining power depends on the spatial and velocity baseline, tracer sampling, and observational uncertainties; controlled truncation-and-recovery tests will be needed to quantify this limit for broader applications of ORFEO.

This first application shows that extended ionised tails can be used as dynamical tracers of the recent interaction histories of cluster galaxies. Despite its simplified treatment of the stripped gas, ORFEO reconstructs the orbit, tail chronology, and ram-pressure evolution of NGC 4569 in only $\sim10$–11 min of wall-clock time. These reconstructed histories can provide physically motivated constraints for future tailored simulations and, when applied to larger samples, a statistical view of where, when, and under which ICM conditions gas removal occurs in clusters.

Acknowledgements. We are grateful to the whole CFHT team who assisted us in the preparation and in the execution of the observations and in the calibration and data reduction. We acknowledge financial support from “Programme National de Cosmologie et Galaxies” (PNCG) funded by CNRS/INSU-IN2P3-INP, CEA and CNES, France. This research has made use of the NASA/IPAC Extragalactic Database (NED) which is operated by the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration and of the GOLDMine database (http://goldmine.inaf.it/) ([41]). M. S. acknowledges the support from the NSF grant 2407821.

Gaseous tail modelling

Tail construction

This section details the numerical implementation of the one-dimensional orbital-spine model introduced in Sect. 3.2. Let $\mathbf{x}_{\rm gal}(t)$ denote the galaxy orbit. We define the signed arc-length coordinate along this trajectory as

Writing the same trajectory as $\mathbf{x}(\gamma)$, the position of a gas parcel is $\mathbf{x}_{\rm gas}(t)=\mathbf{x}[\gamma_{\rm gas}(t)]$, where $\gamma_{\rm gas}(t)$ evolves under drag and gravity. The total acceleration is decomposed as

where $\mathbf{a}_{\rm gal}$ traces the gravitational field of the cluster, the second term accounts for ram-pressure drag, and $\mathbf{a}_{\rm track}$ is the acceleration required to keep the gas confined to the orbit. Since we impose geometric confinement, we do not compute $\mathbf{a}_{\rm track}$ explicitly; instead, we retain only the tangential component of the acceleration.

Defining the unit tangent vector along the orbit as $\hat{\mathbf{t}}(t)=\frac{d\mathbf{x}_{\rm gal}/dt}{\lVert d\mathbf{x}_{\rm gal}/dt\rVert}$, the projected gravitational acceleration induced by the cluster gravitational potential becomes $a_{\parallel}(t)=\mathbf{a}_{\rm gal}(t)\cdot\hat{\mathbf{t}}(t)$, which we interpolate as a function of arc-length. The motion of the gas is then governed by the one-dimensional differential equation

Here $\tau(t;t_{\rm strip})$ is the ram-pressure drag timescale defined in Eq. (8). For each parcel, the assigned surface density remains fixed, while the local ICM density and parcel speed vary along its trajectory, making $\tau$ time-dependent. We integrate Eq. (A.3) using a leapfrog scheme. At each time step, the gas velocity is reconstructed as

ensuring that the motion remains tangent to the orbit. This reduces the dynamics of each stripped parcel to a single degree of freedom.

An ensemble of parcels is evolved from its assigned $t_{\rm strip}$ to $t=0$ using the initial conditions of Eq. (5). Present-day positions and velocities are obtained by interpolating the orbit at $\gamma_{\rm gas}(0)$, defining the model tail used for comparison with simulations or observations.

Benchmark of the effective drag prescription

We benchmark the effective post-release drag prescription against the $T_{\min}=300$ K hydrodynamical simulation of Tonnesen & Bryan [92], hereafter T10. The simulation adopts a constant ICM wind with $v_{\rm ICM}=1413$ km s$^{-1}$ and $P_{\rm ram}=6.4\times10^{-12}$ dyn cm$^{-2}$. We reproduce this configuration with a straight-line trajectory at $v=-v_{\rm ICM}$ and compare the resulting longitudinal tail structure with the simulation. As in Sect. 3.2, we assign parcel surface densities between a dense-gas anchor, $\Sigma(R_{\rm strip})$, and a minimum tail value, $\Sigma_{\rm tail,min}$. Under the constant-wind conditions of T10, the ICM density is fixed, while the drag timescale in Eq. (8) varies through the assigned surface density and the instantaneous parcel velocity. We fix $\Sigma(R_{\rm strip})=6.5\times10^{-4}$ g cm$^{-2}$, corresponding to $N_{\rm H}\simeq3\times10^{20}$ cm$^{-2}$ and consistent with T10, and compare three values of $\Sigma_{\rm tail,min}$. The “optimal” case, $5.2\times10^{-5}$ g cm$^{-2}$, is calibrated to match both the simulated tail extent and its 250 Myr age, while $4.3\times10^{-5}$ and $1.1\times10^{-4}$ g cm$^{-2}$ bracket the observationally motivated range adopted for NGC 4569 and test the response to stronger and weaker effective drag, respectively. Figure A.1 compares the (T10) simulation at $t=250$ Myr with the reconstructed tails. All three prescriptions reproduce the velocity gradient of the dense inner tail, with increasing differences towards larger distances while remaining within the characteristic velocity scatter of the simulated wake. The calibrated case matches the 250 Myr tail age by construction; its independent test is therefore the agreement of the simplified drag model with the simulated velocity–distance structure. The lower- and upper-density bounds yield $T_{\rm tail}=244$ and 305 Myr, respectively, corresponding to a spread of $\simeq60$ Myr. Appendix B tests separately how uncertainty in the adopted tail-density scale propagates into the inverse recovery.

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Scope of the gaseous-tail model

ORFEO is an inverse model of the large-scale phase-space spine of the observed tail, rather than a forward model of gas removal from the disc. Parcels are sampled at assigned stripping times $t_{\rm strip}\leq0$, independently of a ram-pressure threshold or stripping-efficiency criterion. Thus, $t_{\rm strip}$ labels a model parcel along the present-day tail rather than predicting the physical instant at which that gas became unbound. Accordingly, $T_{\rm tail}$ is the age of the oldest fitted H$\alpha$ parcel and need not coincide with the onset of gas removal.

At $t_{\rm strip}$, each parcel is initialised at the galaxy position with the galaxy bulk orbital velocity. The model does not include finite launch radius or azimuth, an evolving stripping radius, motion through the galaxy potential during detachment, or disc rotation. Parcel release is also independent of wind–disc inclination and does not distinguish between direct ram-pressure displacement, edge stripping, or Kelvin–Helmholtz removal [85]. The wind–disc angle derived in Appendix § 11 therefore characterises the reconstructed geometry but does not enter the release prescription.

After release, parcels remain confined to the one-dimensional orbital spine and evolve under projected cluster gravity and effective ram-pressure drag. The galaxy’s own gravitational attraction is neglected in this post-release evolution. Where its component along the spine opposes drag, it reduces the net gas deceleration. This effect may be partly represented by a larger effective surface density and hence a longer drag timescale, although the prescribed coupling cannot generally reproduce the full time dependence of the omitted force. Tail width, lateral accelerations, turbulence, filamentation, expansion or fragmentation, multiphase evolution, and ICM bulk motions are not resolved. Unresolved changes in scalar post-release gas–ICM coupling are represented through the effective $C_{\mathrm{d}}/\Sigma_{\mathrm{gas}}$ prescription, whereas coherent launch-side offsets or systematic transverse displacements are not. Such systematic displacements of the collective tail spine would affect the recovered orbit and chronology, which are therefore conditional on the adopted orbital-spine approximation. Quantifying these effects requires calibrating departures from the galaxy trajectory against hydrodynamical simulations of curved galaxy orbits.

Observed H$\alpha$ regions are associated independently with the nearest location on the model tail in (RA, DEC, $cz_{\mathrm{obs}}$) through Eq. (11). No requirement is imposed that every model segment be detectable in H$\alpha$, and no hard association distance is applied; mismatches are instead penalised continuously by their spatial and kinematic residuals. Gas older than the oldest fitted H$\alpha$ tracer may therefore be absent from the observed sequence. The model also does not separate hydrodynamical from possible tidal contributions, and its orbital interpretation applies to the large-scale component associated with the galaxy–ICM interaction.

Likelihood validation

The validation suite presented in this appendix was developed using the earlier 83-region version of the observational catalogue. Accordingly, the controlled mock is sampled with 83 tracers, and the $t_{\mathrm{scale}}$ diagnostic in Appendix § 8.§ 2 illustrates the corresponding 83-region reconstruction. The results reported in the main text have instead been recomputed using the final 94-region catalogue. The reconstruction algorithm and validation procedure are unchanged. We test the inverse-reconstruction framework on a controlled mock with known orbital parameters and tail chronology. The purpose is not to reproduce the full complexity of the observations, but to determine whether the fitting

Quantity	$\Delta$	$\sigma$
$v_{\perp,1}$ (km s$^{-1}$)	$-23$	58
$v_{\perp,2}$ (km s$^{-1}$)	28	35
$T_{\mathrm{tail}}$ (Myr)	15	17
$T_{\mathrm{ram,max}}$ (Myr)	1	7

Vector (PDF)

procedure recovers the injected quantities from a sparsely sampled tail with realistic spatial and kinematic scatter.

We describe the Virgo mass profile with a Navarro–Frenk–White (NFW) profile with $R_{200}=1623.3$ kpc, $M_{200}=5.3\times10^{14}\,M_\odot$, and $R_{\mathrm{s}}=112$ kpc. For the ICM, we adopt the single-$\beta$ electron-density profile of Simionescu et al. [88],

with $n_{\mathrm{e},0}=3.13\times10^{-2}\,\mathrm{cm}^{-3}$, $r_{\mathrm{c}}=13$ kpc, and $\beta=0.47$. The gas surface-density prescription is set by the surface density at the stripping radius and by an observationally motivated minimum tail surface density. Specifically, we adopt $N_{\mathrm{H,gas}}(R_{\mathrm{strip}})=1\times10^{21}\,\mathrm{cm}^{-2}$ as the dense-gas anchor at the present-day disc truncation radius and $N_{\mathrm{H,tail,min}}=3.2\times10^{19}\,\mathrm{cm}^{-2}$ as the adopted minimum effective column density in the mock tail. These values are consistent with the dense-disc reference and the adopted tail reference range discussed in Sect. 2.1 and are converted to surface density through $\Sigma_{\mathrm{gas}}=\mu m_{\mathrm{p}}N_{\mathrm{H}}$, with $\mu=1.3$. The resulting tail is truncated at a maximum lookback time of 300 Myr to emulate a system that began stripping at a fixed epoch in the past. It is then converted into observable quantities—RA, DEC, projected distance from the galaxy nucleus ($D_{\mathrm{nucleus}}$), and LOS velocity ($cz_{\mathrm{obs}}$)—and perturbed to reproduce the effective spatial and kinematic dispersion observed around the mean tail trends. The different dispersions adopted in RA and DEC reflect the anisotropic width of the observed tail in these projections. Gaussian noise is added with standard deviations of $0.0001$ h in RA, $0.1^\circ$ in Dec, and 120 km s$^{-1}$ in velocity. A random subset of 83 tail points is then selected, matching the sampling density of the earlier observational catalogue on which this validation was developed.

To test our model, we apply our likelihood framework to this perturbed dataset, scanning the ($v_{\perp,1},v_{\perp,2}$) parameter space while fixing the galaxy position. The goal is threefold: assess whether the tail morphology and kinematics are reproduced; test how well the time-dependent ram pressure is recovered; and determine whether the input initial conditions and derived stripping timescales are retrieved within realistic uncertainties. Figure B.1 shows the $\Delta\chi^2$ surface in the $(v_{\perp,1}, v_{\perp,2})$ plane. The black dot marks the best-fitting value, while the nominal 1–3$\sigma$ two-parameter contours are shown as black solid lines; the 4–5$\sigma$ contours are shown as white dashed lines. For this realisation, the recovered velocities are $v_{\perp,1} = -477 \pm58$ km s$^{-1}$ and $v_{\perp,2} = 272 \pm35$ km s$^{-1}$, consistent with the true values (−500, 300) km s$^{-1}$ used to generate the tail, marked by a blue star.

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Completing the picture, Fig. B.2 shows the best-fit model for the tail compared with the mock dataset. The data points (coloured dots) are shown in the $RA - DEC$ plane (top) and as LOS velocity versus projected distance from the nucleus (bottom). Colours encode the model lookback time assigned to each mock tracer by the reconstruction. The best-fit reconstructed curve (solid line, colour-coded by lookback time) closely follows the unperturbed generator model (light-blue dashed line), reproducing both the on-sky morphology and the $v_{\mathrm{los}}$–$D_{\mathrm{nucleus}}$ relation. The maximum reconstructed lookback time recovers the injected tail age of 300 Myr to within 15 Myr. Going beyond the pure orbital information, Fig. B.3 shows the retrieved ram-pressure history, computed as $P_{\mathrm{ram}}(t) = \rho_{\mathrm{ICM}}[r(t)]\,v^2(t)$, for the injected (dashed) and reconstructed (solid) orbits. To test the orbital extrapolation beyond the interval directly sampled by the mock tail, we extend the reconstructed orbit beyond 300 Myr. The reconstructed curve follows the injected one closely over the full interval, with only a mild underestimation near the peak. This agreement supports the reliability of the reconstructed ram-pressure history under the controlled assumptions of the test.

Vector (PDF)

In addition to the initial condition, we focus on two derived timescales: the tail age, $T_{\mathrm{tail}}$, defined as the maximum lookback time assigned to any tail point, and the time of peak ram pressure, $T_{\mathrm{Pram,max}}$. These are not fit parameters, but are computed a posteriori from the best-fit trajectory. Table B.1 summarises the deviations from the true values and their uncertainties. For this realisation, the recovered $T_{\mathrm{tail}}$ and $T_{\mathrm{Pram,max}}$ differ from their injected values by 15 and 1 Myr, respectively, both within the nominal uncertainties reported in Table B.1. Together with the close match in Fig. B.3, this shows that the reconstruction recovers both the injected timescales and the ordering of the tail tracers despite their independent nearest-point association.

Sensitivity to input gas density

Whereas Appendix A.2 benchmarks the effective drag prescription against an independent hydrodynamical simulation, here we test how an incorrect $\Sigma_{\mathrm{tail,min}}$ propagates into the inverse recovery of the orbit and chronology. We consider $N_{\mathrm{H,tail,min}}^{\mathrm{low}} = 2 \times10^{19}\ \mathrm{cm}^{-2}$ and $N_{\mathrm{H,tail,min}}^{\mathrm{high}} = 5 \times10^{19}\ \mathrm{cm}^{-2}$, bracketing the value used to generate the mock. As in the validation against the T10 simulation (Appendix A), and for the same column densities, we find that underestimating the density (blue dashed line) does not degrade the fit, which still reproduces both the morphology in the $RA - -Dec$ plane and the velocity gradient. In contrast, overestimating the density (red dotted line) leads to a visibly poorer match, particularly at large distances from the nucleus—consistent with the trend shown in Fig. A.1.

We quantify the relative agreement of the ORFEO predictions with the input dataset using the same reduced $\chi^2$-like statistic adopted in the main analysis. Using $\nu= 2N - 2 = 164$ as a common normalisation for the 83-point mock, we obtain $\chi^2_{\mathrm{true}}/\nu= 0.35$ (reference model), $\chi^2_{\mathrm{low}}/\nu= 0.35$ (low density), $\chi^2_{\mathrm{high}}/\nu= 0.57$ (high density). These values are used as comparative indicators of fit quality rather than as an exact $\chi^2$ probability calibration. The reference and low-density cases are indistinguishable according to this diagnostic, whereas the high-density assumption produces a substantially poorer fit. In this controlled mock, the diagnostic distinguishes the reference and low-density prescriptions from the incompatible high-density case.

To further assess whether an incorrect assumption on $\Sigma_{\mathrm{tail,min}}$ biases the relevant quantities estimated by ORFEO, we perform the complete likelihood analysis for the three density cases, focusing on $(v_{\perp,1}, v_{\perp,2})$, $T_{\mathrm{tail}}$, and $T_{\mathrm{Pram,max}}$. The results are summarised in Table B.2, expressed as deviations from the expected values $\Delta$. For comparison, the corresponding results obtained with the true density are reported in Table B.1.

Vector (PDF)

Quantity	$\Delta_{\mathrm{low}}$	$\sigma_{\mathrm{low}}$	$\Delta_{\mathrm{high}}$	$\sigma_{\mathrm{high}}$
$v_{\perp,1}\,(\mathrm{km\,s}^{-1})$	-8	63	105	40
$v_{\perp,2}\,(\mathrm{km\,s}^{-1})$	-5	37	-81	24
$T_{\mathrm{tail}}\,(\mathrm{Myr})$	-13	17	108	17
$T_{\mathrm{Pram,max}}\,(\mathrm{Myr})$	0	7	-12	15

The lower-density assumption leaves the recovered orbit and chronology consistent with the injected values within the nominal $1\sigma$ uncertainties. The higher-density case instead produces significant deviations, most notably $|\Delta T_{\mathrm{tail}}| = 108$ Myr. Together with its poorer fit statistic, this shows that the high-density prescription does not recover the known orbit and chronology of the controlled mock. This does not imply that a comparable separation between the age of the oldest observable tail material and peak ram pressure would be intrinsically unphysical in a real system, since older stripped gas may no longer be detectable. Here, however, the imposed onset and orbital history are known, so the chronology provides an additional diagnostic of model mismatch. In the analysis of NGC 4569, we therefore explore

different column-density assumptions and assess both their relative fit quality and the physical plausibility of the reconstructed chronology.

Sensitivity to the stripping timescale

The effective gas-surface-density prescription adopted by ORFEO requires a trial effective surface-density evolution timescale $t_{\mathrm{scale}}$, which sets the temporal normalisation of (9). Since $t_{\mathrm{scale}}$ is not known a priori, it is selected through an iterative scan coupled to the orbit reconstruction. For each trial $t_{\mathrm{scale}}$, ORFEO returns the best-fitting orbital parameters, fit statistic, and reconstructed $T_{\mathrm{tail}}$. Because $t_{\mathrm{scale}}$ sets the interval over which the assigned surface-density sequence reaches the tail-density scale, we retain solutions for which $t_{\mathrm{scale}}$ and $T_{\mathrm{tail}}$ are consistent within their uncertainties. The requirement $t_{\mathrm{scale}} \simeq T_{\mathrm{tail}}$ is therefore an internal consistency criterion of the adopted surface-density parametrisation, rather than an independent physical measurement of the stripping onset. The optimisation is performed in two stages. We first carry out a coarse scan over a broad range of plausible $t_{\mathrm{scale}}$ values. At each value of $t_{\mathrm{scale}}$, we minimise $\chi^2$ over the two transverse velocity components and assess the numerical quality of the resulting minimum on a local $8 \times8$ grid spanning $\pm3$ Hessian standard deviations. To prevent poorly resolved or boundary-truncated minima from driving the selection, we define

with

Here $I_{\mathrm{H}}$, $I_{\mathrm{G}}$, and $I_{\mathrm{edge}}$ flag an invalid Hessian, failure of the local-grid calculation, and intersection of the $\Delta\chi^2 \le2.30$ region with the grid boundary, respectively; $B$ is the minimum boundary value of $\Delta\chi^2$, and $N$ is the number of grid points within $\Delta\chi^2 \le2.30$. The penalty acts only as a numerical-quality safeguard: for accepted, well-resolved solutions $P = 0$, so that $\chi^2_{\mathrm{robust}} = \chi^2$ and their ranking is unchanged. The penalty is not included in the reported goodness-of-fit or confidence intervals. If several local minima are present, we retain the basin with the lowest $\chi^2_{\mathrm{robust}}$ among the numerically accepted solutions for which $t_{\mathrm{scale}}$ remains compatible with $T_{\mathrm{tail}}$ within its local uncertainty estimate. A second, refined scan is then performed within this basin to determine the preferred stripping timescale more precisely. The uncertainty associated with the input choice of $t_{\mathrm{scale}}$ is propagated into the final reconstructed quantities as an additional systematic contribution, $\sigma_{\mathrm{sys}}$, estimated from the spread of the best-fitting parameters over the $1\sigma$ interval of the allowed $t_{\mathrm{scale}}$ range. The reported within-model uncertainty is then obtained by combining these terms in quadrature,

Here $\sigma_{\mathrm{tot}}$ denotes the combined uncertainty within the adopted cluster, ICM, and tail model; uncertainties associated with changing these structural assumptions are not included.

Figure B.5 illustrates the $t_{\mathrm{scale}}$ optimisation using the earlier 83-region reconstruction of NGC 4569 at $d_0 = 16.5$ Mpc and $\Sigma_{\mathrm{tail,min}} = 7.6 \times10^{-5}\ \mathrm{g\ cm^{-2}}$. The figure is included to document the selection procedure; the results reported in Sect. 4.1 are obtained from the final 94-region catalogue. The coarse scan identifies the preferred numerically accepted basin, and the subsequent fine scan refines $t_{\mathrm{scale}}$ within that region. The residual $t_{\mathrm{scale}} - T_{\mathrm{tail}}$ confirms that the adopted value satisfies the internal consistency criterion. The uncertainty associated with $t_{\mathrm{scale}}$ is then propagated into the reported ORFEO constraints.

Full MG-MAMPOSSt analysis

The reconstruction of the orbit of NGC 4569 requires an estimate of the mass profile of Cluster A. We derive it using the MG-MAMPOSSt code ([78, 79]), an extension of the MAMPOSSt algorithm ([62]) that simultaneously constrains the total gravitational potential and orbital-anisotropy profile of galaxy clusters from the kinematics of their member galaxies.

MAMPOSSt (Modelling Anisotropy and Mass Profile of Spherical Observed Systems) assumes spherical symmetry and treats galaxies as collisionless tracers of the gravitational potential. Under these assumptions, the dynamics are governed by the spherical Jeans equation,

Here, $r$ is the three-dimensional radial distance from the cluster centre, $\nu(r)$ is the tracer number-density profile, $\sigma_r$ is the radial velocity dispersion, and $\phi$ is the total gravitational potential. The velocity anisotropy is defined as $\beta\equiv1 - (\sigma_\theta^2 + \sigma_\varphi^2)/(2\sigma_r^2)$, where $\sigma_\theta^2$ and $\sigma_\varphi^2$ are the tangential and azimuthal velocity-dispersion components, respectively. In spherical symmetry, $\sigma_\theta^2 = \sigma_\varphi^2$, and the anisotropy profile simplifies to $\beta= 1 - \sigma_\theta^2/\sigma_r^2$.

MAMPOSSt and MG-MAMPOSSt take as input parametric forms for the gravitational potential, tracer number-density profile, and velocity-anisotropy profile. A maximum-likelihood analysis then determines the parameter set that best describes the distribution of cluster members in projected phase space, $(R,v_{\parallel})$. Here, $R$ is the cluster-centric distance projected on the sky and $v_{\parallel}$ is the galaxy LOS velocity in the cluster rest frame.

We denote by $\boldsymbol{\theta}_{\nu}$, $\boldsymbol{\theta}_{\beta}$, and $\boldsymbol{\theta}_{m}$ the parameter vectors describing the tracer number-density profile, velocity anisotropy, and gravitational potential, respectively. For each parameter set $\boldsymbol{\Theta} = \{\boldsymbol{\theta}_{\nu},\boldsymbol{\theta}_{\beta},\boldsymbol{\theta}_{m}\}$, the code computes the sum of the log-probabilities $\ln q(R_i,v_{\parallel,i}\mid\boldsymbol{\Theta})$ of finding a galaxy at the projected-phase-space position $(R_i,v_{\parallel,i})$. A detailed description of the algorithm is given by Mamon et al. [62]; here we briefly summarise the main steps used to obtain $\ln q$:

• Given $\boldsymbol{\Theta}$, the Jeans equation (Eq. (C.1)) is solved to obtain $\sigma_r^2(r)$.

• Assuming a functional form for the three-dimensional velocity distribution $f(\boldsymbol{v})$, MAMPOSSt integrates over the two velocity components perpendicular to the LOS to obtain the LOS velocity distribution $h(v_{\parallel},R,r)$.

• The probability $q$ is then obtained by integrating $h(v_{\parallel},R,r)$ along $r$, weighted by the projected tracer surface density $\Sigma(R)$.

Our version of MG-MAMPOSSt assumes a Gaussian three-dimensional velocity distribution. In this case, $h(v_{\parallel},R,r)$ is also Gaussian, with velocity dispersion

MG-MAMPOSSt extends the models available in MAMPOSSt to additional parametrisations of the gravitational potential, including modified-gravity and dark-energy scenarios that are viable at cosmological scales (see [78]), and to general velocity-anisotropy profiles that encompass a wide range of member-galaxy orbital distributions. A recent version of the code also includes a module that incorporates velocity-dispersion measurements of the brightest cluster galaxy (BCG) into the fit, providing tighter constraints on the slope of cluster dark-matter profiles (see [6]).

The Virgo cluster as a whole is not dynamically relaxed (see e.g. Schindler et al. 1999; Conselice et al. 2001; Boselli et al. 2014c). Since the Jeans analysis assumes that galaxies are collisionless tracers of the potential in dynamical equilibrium, departures from equilibrium may introduce systematic effects in the mass determination. This bias is particularly important when reconstructing the mass profile in modified gravity [80]. However, the tests performed by Mamon et al. [62] showed that MAMPOSSt can recover the virial mass $M_{200}$ reasonably well in standard gravity, even for unrelaxed systems and velocity distributions that deviate from the Gaussian assumption. In addition, we restrict our analysis to Virgo Cluster A, the subsystem that dominates the diffuse X-ray emission of Virgo [87]. Previous studies indicate that Cluster A is close to dynamical equilibrium when traced by its early-type galaxy population [54] (Conselice et al. 2001). The right panel of Fig. C.1 compares the LOS velocity distributions of the full member catalogue and the early-type subsample.

$M(r)$ model	$\beta(r)$ model	$N(r)$ model	sample	$r_{200}$ [Mpc]	$M_{200}$ [$10^{14}\,M_{\odot}$]	$r_s$ [Mpc]	$r_{\nu}$ [Mpc]	$\mathcal{A}_{\infty}$	$\mathcal{A}_0$	$\Delta\mathrm{BIC}$
Her	gOM	pHer	ET	$1.66^{+0.28}_{-0.29}$	$5.2^{+2.9}_{-2.4}$	$0.86^{+0.48}_{-0.75}$	$0.50^{+0.06}_{-0.12}$	$1.68^{+1.92}_{-1.24}$	$1.68^{+1.91}_{-0.92}$	0
Her	gT	pHer	ET	$1.57^{+0.37}_{-0.21}$	$5.1^{+3.0}_{-2.4}$	$0.57^{+0.67}_{-0.49}$	$0.51^{+0.05}_{-0.13}$	$1.68^{+1.93}_{-1.24}$	$1.26^{+1.98}_{-0.90}$	0.67
Her	gOM	pNFW	ET	$1.51^{+0.27}_{-0.15}$	$4.5^{+2.0}_{-1.7}$	$0.68^{+0.39}_{-0.70}$	$0.18^{+0.04}_{-0.07}$	$1.69^{+1.92}_{-1.24}$	$1.29^{+1.94}_{-0.93}$	10.52
NFW	gOM	pHer	ET	$1.63^{+0.32}_{-0.21}$	$5.4^{+2.9}_{-2.3}$	$0.36^{+0.32}_{-0.07}$	$0.50^{+0.12}_{-0.06}$	$1.70^{+1.86}_{-1.23}$	$1.34^{+1.96}_{-0.98}$	0.25
NFW	gT	pHer	ET	$1.57^{+0.40}_{-0.16}$	$5.4^{+3.1}_{-2.4}$	$0.23^{+0.43}_{-0.20}$	$0.50^{+0.12}_{-0.05}$	$1.72^{+1.90}_{-1.27}$	$1.34^{+1.87}_{-0.86}$	1.14
NFW	gOM	pNFW	ET	$1.45^{+0.33}_{-0.24}$	$4.5^{+1.9}_{-1.6}$	$0.27^{+0.22}_{-0.24}$	$0.19^{+0.04}_{-0.08}$	$1.70^{+1.87}_{-1.24}$	$1.30^{+1.95}_{-0.94}$	10.90
Bur	gOM	pHer	ET	$1.71^{+0.23}_{-0.30}$	$5.3^{+2.8}_{-2.2}$	$0.36^{+0.07}_{-0.24}$	$0.50^{+0.12}_{-0.06}$	$1.62^{+1.91}_{-1.19}$	$1.36^{+1.91}_{-1.00}$	1.00
Bur	gT	pHer	ET	$1.58^{+0.34}_{-0.19}$	$5.2^{+2.8}_{-2.2}$	$0.17^{+0.15}_{-0.14}$	$0.50^{+0.12}_{-0.06}$	$1.66^{+1.91}_{-1.22}$	$1.30^{+1.95}_{-0.94}$	0.94
Bur	gOM	pNFW	ET	$1.57^{+0.20}_{-0.19}$	$4.5^{+1.8}_{-1.5}$	$0.17^{+0.06}_{-0.06}$	$0.14^{+0.13}_{-0.10}$	$1.61^{+1.87}_{-1.16}$	$1.41^{+1.98}_{-1.06}$	11.70
Her	gOM	pHer	ALL	$1.85^{+0.26}_{-0.24}$	$7.4^{+3.2}_{-2.7}$	$1.04^{+0.75}_{-0.64}$	$0.57^{+0.07}_{-0.09}$	$1.51^{+1.79}_{-1.02}$	$1.24^{+1.72}_{-0.87}$	0.43
NFW	gOM	pHer	ALL	$1.86^{+0.30}_{-0.26}$	$7.5^{+3.8}_{-3.0}$	$0.54^{+0.61}_{-0.45}$	$0.57^{+0.07}_{-0.09}$	$1.53^{+1.80}_{-1.02}$	$1.22^{+1.74}_{-0.87}$	0.51
Bur	gOM	pHer	ALL	$1.85^{+0.28}_{-0.26}$	$7.3^{+3.3}_{-2.8}$	$0.27^{+0.23}_{-0.19}$	$0.57^{+0.07}_{-0.09}$	$1.31^{+1.49}_{-0.85}$	$1.45^{+1.83}_{-1.04}$	0

In this work, we apply MG-MAMPOSSt to infer the dynamical mass profile of Cluster A using two datasets: the full member catalogue and a subsample restricted to early-type galaxies. The full sample contains 285 cluster members, selected as described in Sect. 2.2, within a projected radius of 1.8 Mpc from the cluster centre, identified with the position of M87. Of these, 185 are classified as early-type galaxies. Figure C.1 shows the projected phase space (left) and LOS velocity distributions (right) of the full sample over $R \in[0, 1.9]$ Mpc, in blue, and of the early-type subsample, in red.

In principle, MG-MAMPOSSt can jointly fit the three model components $\nu(r)$, $\beta(r)$, and $\phi(r)$ in a parameter space of up to 10 dimensions. However, the galaxy number-density profile $\nu(r)$ can be constrained independently from the projected phase-space data using the unbinned method of Sarazin [86]. We consider two models for the projected tracer profile: a projected NFW profile (pNFW) and a projected Hernquist profile (pHer; see below). Each depends on a scale radius $r_{\nu}$ and a normalisation. Since the normalisation factors out of the expression for $\sigma_r^2(r)$, only $r_{\nu}$ enters the MG-MAMPOSSt analysis.

For the cluster gravitational potential, we consider three mass-profile models: the NFW profile, adopted as our reference model; the Hernquist profile [51],

and the Burkert profile [24],

All three models are described by two free parameters: the mass $M_{200}$ (or, equivalently, the radius $r_{200}$) and the scale radius $r_s$, both fitted by MG-MAMPOSSt. Note that $r_s$ does not have the same meaning for the different mass-profile parametrisations. For the NFW model, $r_s \equiv R_s \equiv r_{-2}$, where $R_s$ is the notation used in Sect. 3.4 and $r_{-2}$ is the radius at which the logarithmic slope of the density profile equals $-2$. For the Hernquist and Burkert models, $r_s=2r_{-2}$ and $r_s=(2/3)r_{-2}$, respectively.

Finally, we adopt two parametrisations for $\beta(r)$ (see, e.g., Mamon et al. [64]): a generalised Tiret profile [91], hereafter gT,

and a generalised Osipkov–Merritt model [76, 71], hereafter gOM,

Here, $\beta_0$ and $\beta_{\infty}$ represent the inner and outer values of the anisotropy, respectively, and $r_{\beta}$ is a characteristic scale radius. In both cases, we set $r_{\beta}=r_{-2}$ of the mass profile, as suggested by numerical simulations [63]. In MG-MAMPOSSt, we work with the rescaled parameters $\mathcal{A}_0=(1-\beta_0)^{-1/2}$ and $\mathcal{A}_{\infty}=(1-\beta_{\infty})^{-1/2}$, which are equal to unity for isotropic orbits.

Results

To characterise the total mass profile of Cluster A, we examine the combinations of $M(r)$, $\nu(r)$, and $\beta(r)$ models described above. We also test the dependence of the inferred mass profile on tracer selection by analysing both the full galaxy sample and the early-type subsample. We sample the parameter space $\boldsymbol{\Theta} = \{r_{200}, r_{\mathrm{s}}, r_{\nu}, A_{\infty}, A_{0}\}$ using an MCMC chain of 110 000 steps. We discard the first 10 000 steps as burn-in and use the remaining 100 000 samples for posterior inference, adopting flat, uninformative priors for all parameters. The number-density scale radius $r_{\nu}$ can be constrained independently of MG-MAMPOSSt, so we first fit the projected galaxy distribution $R_i$ assuming the pHer and pNFW models. We then allow $r_{\nu}$ to vary in the MG-MAMPOSSt analysis, adopting the $2\sigma$ interval from this preliminary fit as its prior range.

Our results are summarised in Table C.1, where we report the constraints on the free parameters for each combination of mass, number-density, and anisotropy models. The last column reports the Bayesian Information Criterion difference (BIC; see e.g. Mamon et al. [64]) relative to the best-fitting model for each sample. A difference $\Delta\mathrm{BIC} > 6$ is generally taken as strong evidence against the model with the larger BIC.

The pNFW tracer number-density model is strongly disfavoured relative to the pHer profile. For the early-type sample, the Hernquist mass model combined with the gOM anisotropy profile gives the lowest BIC. However, the differences relative to the NFW and Burkert mass profiles are not statistically significant ($\Delta\mathrm{BIC} < 1$). We therefore retain the NFW profile as our reference mass model.

Figure C.2 shows the marginalised distributions of $M_{200} = [100H^{2}(z)/G]r_{200}^{3}$ for the three mass profiles fitted to the early-type sample, together with the corresponding distribution for the reference NFW model fitted to the full galaxy sample (dotted curve). In all cases, a pHer tracer number-density profile and a gOM anisotropy profile are assumed.

The constraints on $M_{200}$ obtained with the three mass models agree closely, indicating limited sensitivity of the inferred cluster mass to the tested mass-profile parametrisations. For the NFW model, we obtain

at $2\sigma$, in agreement with Kashibadze et al. [54]. When the full sample (ET+LT) is used, the NFW mass inferred by MG-MAMPOSSt is larger, $M_{200} = 7.5^{+3.8}_{-3.0} \times10^{14}\,\mathrm{M}_{\odot}$. This suggests that the late-type population may be farther from dynamical equilibrium than the lenticular and elliptical populations, as discussed above. The MG-MAMPOSSt best-fitting values for the full-sample analysis are listed in the last three rows of Table C.1.

Additional orbital diagnostics

We complement the distance scan in Sect. 4.2 with two diagnostics: the orbital eccentricity and the present-day cluster-centric radial velocity. Figure D.1 shows their dependence on the assumed LOS distance.

Following Gill et al. [44], we define the eccentricity as $e \equiv1-r_{\mathrm{peri}}/r_{\mathrm{apo}}$, where $r_{\mathrm{peri}}$ and $r_{\mathrm{apo}}$ are the turning radii of the reconstructed galaxy orbit in the adopted cluster potential. This definition applies to bound solutions; unbound orbits have no apocentre. Solutions at $d_{0} \simeq16.5$–$16.7$ Mpc and $17.2$–$17.6$ Mpc, including the fiducial case, have $e \simeq0.7$–$0.9$. For the first-passage interpretation considered here, the relevant comparison is the recently infalling satellite population identified by Gill et al. [44], for which they report a mean eccentricity of approximately 0.8. Their simulations also show that eccentricity tends to decrease with successive orbits. The reconstructed eccentricities in these distance intervals are therefore comparable to those of recently infalling satellites. Other bound solutions approach unity, implying more extreme apocentre-to-pericentre ratios. This comparison complements the boundedness diagnostic but does not select a unique distance.

The present-day cluster-centric radial velocity is

where $\boldsymbol{R}_{0}$ and $\boldsymbol{V}_{0}$ are the present-day galaxy position and velocity relative to the Virgo Cluster A centre. The low-distance solutions have $v_{r,\mathrm{cl}}(0) > 0$, corresponding to outward motion after pericentre, whereas the high-distance solutions have $v_{r,\mathrm{cl}}(0) < 0$, corresponding to inward motion before pericentre. The zero crossing near $d_{0} \simeq17.1$ Mpc coincides with the reconstructed ram-pressure maximum crossing the present epoch. The broader transition interval $d_{0} \simeq16.8$–$17.1$ Mpc encompasses the accompanying changes in the fitted orbital parameters and boundedness; only near its upper edge is the galaxy reconstructed close to pericentre at the present epoch. Each distance corresponds to a separately fitted orbit.

Wind–disc geometry

The reconstructed orbit also constrains the orientation of the ICM wind relative to the disc of NGC 4569. We construct the disc-normal vector, $\boldsymbol{n}_{\mathrm{disc}}$, in the ICRS frame by adopting a disc inclination to the LOS $i = 65^{\circ}$, corresponding to the average of the available photometric and kinematic estimates [48, 5, 29], and a position angle $\mathrm{PA} = 24^\circ$ [93]. The disc orientation is assumed to remain fixed over the reconstructed interval. Neglecting bulk motions of the ICM, the wind velocity in the galaxy rest frame is $\mathbf{W}(t) = -\mathbf{v}_{\mathrm{gal}}(t)$. We define the wind–disc angle as

Vector (PDF)

where the absolute value accounts for the twofold orientation ambiguity of the disc normal. With this convention, $\theta_{\mathrm{wd}} = 0^\circ$

Vector (PDF)

Vector (PDF)

corresponds to a face-on interaction and $\theta_{\mathrm{wd}} = 90^\circ$ to an edge-on interaction.

Figure E.1 shows the geometry of the fiducial reconstruction at $d_0 = 16.5$ Mpc. During the interval sampled by the fitted H$\alpha$ tail, $\theta_{\mathrm{wd}}$ increases from $\theta_{\mathrm{wd}}(-T_{\mathrm{tail}}) = 59.3^\circ$ to $\theta_{\mathrm{wd}}(0) = 67.5^\circ$. The interaction therefore remains oblique, but closer to edge-on than face-on, throughout the reconstructed tail history. This is broadly consistent with the inclined stripping configuration adopted by Vollmer et al. [93].

The dependence on the assumed LOS distance is shown in Fig. E.2, where $\theta_{\mathrm{wd}}$ is evaluated both at the lookback time assigned to the oldest fitted H$\alpha$ material and at the present epoch. Although the detailed angle varies with $d_0$, particularly across the transition between the two orbital branches, it remains above approximately $50^\circ$ throughout the explored range. The interaction is therefore consistently reconstructed as oblique and closer to edge-on. The distance dependence of $\theta_{\mathrm{wd}}$ is not used as an independent distance constraint, but provides the wind geometry associated with each reconstructed orbit for future tailored hydrodynamical simulations.