Modified Natal Kick Prescription

• Modified natal kick prescription is a physically driven model that quantifies the velocity imparted to compact objects at birth using progenitor and explosion properties.
• It integrates advanced analytical formulations and empirical observations from pulsars, binary systems, and gravitational-wave events to improve upon traditional Maxwellian models.
• The prescription significantly influences binary survival, merger rates, and galactic evolution, guiding simulation techniques and observational studies in astrophysics.

A modified natal kick prescription is a physically motivated, quantitative description of the velocity impulse imparted to a compact object (neutron star, black hole) at birth, reflecting not only the stochastic asymmetries in explosion or mass loss but also its dependence on progenitor properties, fallback, dynamics in binaries, and observational constraints. Traditionally, population synthesis and binary evolution models employed ad hoc statistical distributions (usually single-parameter Maxwellians) to assign natal kicks. Recent advancements have led to physically grounded, parameterized, sometimes stochastic prescriptions, designed to better match gravitational-wave, pulsar, and electromagnetic observations, as well as the outcomes of multi-dimensional core-collapse supernova simulations.

1. Theoretical Foundations and Analytical Formulations

Several formalisms currently exist for the modified natal kick prescription. The core concept is to tie the magnitude—and sometimes direction—of the natal kick $v_{\rm kick}$ to properties such as the ejecta mass $m_{\rm ej}$ and remnant mass $m_{\rm rem}$, and to include an explicit stochastic (“random draw”) component.

Key Generic Prescription Types

Model Type	Kick Dependence	Example Equation / Parameterization
Standard Maxwellian	$p(v)\,dv \propto v^2 e^{-v^2/2\sigma^2}\, dv$	$v_{\rm kick}$ drawn from Maxwellian with $\sigma$ (e.g., 265 km/s)
Conservation-of-momentum (Bray)	$$v_{\rm kick} = \alpha \frac{m_{\rm ej}}{m_{\rm rem}} + \beta$$	$\alpha \sim 115$, $\beta \sim 15$ km/s
Momentum-conserving fallback	$v_{\rm kick} = (1 - f_{\rm fb})\, v_{\rm NS}$	$f_{\rm fb}$ = fallback fraction; $v_{\rm NS}$ from Maxwellian
Physically scaled (MM20)	$$\mu_{\rm kick} = v_{\rm ns} \frac{M_{\rm CO} - M_{\rm NS}}{M_{\rm NS}}$$	$v_{\rm ns}\simeq 520$ km/s; Gaussian scatter with $\sigma_{\rm ns}\simeq0.3$
Probabilistic core-mass mapping	$$v_{\rm kick} \propto (M_{\rm ej}/M_{\rm rem}) + \delta v$$	Distribution parameters tied to CO core mass, outcome stochastic
Bimodal/Composite	$$w\,{\rm Maxwell}(\sigma_1) + (1-w)\,{\rm Maxwell}(\sigma_2)$$	$w\simeq0.2$, $\sigma_1\simeq45$ km/s, $\sigma_2\simeq336$ km/s
Beta/lognormal fit (binary)	Direct fit to binary kick data	Mode $\sim70$ km/s, mean $\sim100$ km/s for binaries

These various prescriptions reflect the growing understanding that natal kicks are influenced by ejecta mass, progenitor structure (e.g., carbon-oxygen core mass), fallback fraction, and binary evolutionary history.

2. Physical Mechanisms and Explosion Physics

Natal kicks arise from two distinct classes of physical mechanisms:

1. Hydrodynamical Asymmetries (e.g., convective “gravitational tug-boat,” anisotropic mass ejection, jet-induced kicks):
   • Large core-collapse supernovae may develop global or stochastic asymmetries; the resulting mass/energy flux imparts a recoil.
   • The “kick by early asymmetrical pairs” (kick-BEAP) mechanism within the jittering jets explosion paradigm attributes the NS kick to momentum imbalance from a small number of dominant jet-launching episodes—sometimes resulting in velocities $\sim$450 km/s and aligning the spin–kick direction (Shishkin et al., 26 Jun 2025).
   • Fallback-suppressed models scale the kick inversely with the fallback mass fraction, such that black holes formed via direct collapse (full fallback) can receive negligible kicks (Banerjee et al., 2019).
2. Neutrino Emission Asymmetries:
   • For direct collapse (“failed supernova”) scenarios, the only significant energy loss is via nearly isotropic neutrino emission. Any natal kick is set by the net dipole anisotropy $\alpha_\nu$, constrained to be $\lesssim 4\%$ for BHs such as VFTS 243 (Vigna-Gómez et al., 2023).
   • The corresponding kick is $$M_{\rm BH}v_{\nu,\rm kick} = \alpha_\nu E_\nu^{\rm tot}/c$$; typical values for complete collapse yield $v_{\rm kick}\sim$ a few km/s, well below neutron star values.

Population synthesis studies must therefore encode both mechanisms, including stochastics and mass/fallback dependencies, and link them with the dynamic outcomes for binaries.

3. Observational Constraints and Empirical Distributions

Empirical kick distributions are calibrated against several distinct populations:

• Isolated Pulsars and Neutron Stars: VLBI and pulsar timing data reveal a broad distribution, often fit by a Maxwellian with $\sigma\sim265$ km/s [Hobbs], but recent kinematic studies favor a log-normal distribution with $\mu=6.38$, $\sigma=1.01$ (peaking near 200 km/s, median $\sim$400 km/s) and no strong evidence for true bimodality in large samples (Disberg et al., 3 Mar 2025). Bimodality is, however, seen in some young-pulsar populations, but such features may reflect sample size limitations.
• Binary Pulsars and Low-Mass Binary Neutron Stars: Proper motions, distances, and radial velocities are used to reconstruct birth kicks. The overall “binary kick” distribution exhibits a lower typical speed (mode near 70 km/s), commonly described by a Beta function fit (O'Doherty et al., 2023). Roughly 19% of NSs in binaries have kicks $\leq$50 km/s, facilitating binary survival.
• Be X-ray Binaries and High-Mass Stellar Binaries: Combined likelihood analyses find that a bimodal prescription ($w=0.2\pm0.1$, $\sigma_1=45^{+25}_{-15}$ km/s, $\sigma_2=336$ km/s) simultaneously explains both the low velocities of Be X-ray binaries and the high velocities of isolated pulsars (Igoshev et al., 2021).
• Black Hole X-ray Binaries and Microlensed BHs: When astrometric, spectroscopic, or microlensing studies provide the three-dimensional motion of a compact object, the derived peculiar velocity serves as a direct proxy for the natal kick. Systems such as MOA-11-BLG-191/OGLE-11-0462 indicate BH natal kicks $\lesssim100$ km/s (Andrews et al., 2022), consistent with supernova origin but higher than pure neutrino kicks. Some BH-XRBs, like H 1705–250, show strong evidence for large natal kicks (median $\sim$295 km/s), implying that the physical mechanisms for high kicks are not exclusive to neutron stars (Brown et al., 2023).

4. Impact on Astrophysical Populations and Galactic Evolution

The choice of natal kick prescription directly constrains:

• Compact Object Merger Rates: Enhanced kicks disrupt binaries, lowering the double neutron star (DNS) and binary black hole (BBH) merger rates. Prescriptions that allow low kicks for ultra-stripped SNe or ECSNe (e.g., $\beta\geq0$ in Bray’s model, or a “weak” Maxwellian component in bimodal models) are needed to reproduce LIGO/Virgo rates and Galactic populations (Richards et al., 2022, Wysocki et al., 2017, Giacobbo et al., 2019).
• Spin–Orbit Misalignments: Natal kicks can tilt the binary orbital plane, producing effective spin parameters ($\chi_{\rm eff}$) close to zero even when natal spins are high, as observed in gravitational-wave events like GW151226 (O'Shaughnessy et al., 2017, Wysocki et al., 2017). Models with $\sigma\sim50$–200 km/s foster the observed diversity of alignments.
• Galactic r-process Enrichment: High kicks can “eject” neutron star mergers out of the star-forming region, reducing the fraction of mergers contributing to r-process enrichment and dramatically increasing the scatter in elements such as [Eu/Mg] at low metallicity (Safarzadeh et al., 2017, Voort et al., 2021). Even with optimistic merger rates, up to 40% of mergers can be chemically irrelevant for their host galaxy if kicks are large.
• Retention in Clusters: Models with fallback-suppressed or collapse-asymmetry-driven kicks retain more BHs in globular clusters, whereas neutrino-driven kicks with no fallback correction expel nearly all remnants (Banerjee et al., 2019).

5. Computational Implementation and Simulation Techniques

State-of-the-art simulation codes (NBODY7, StarTrack, COMPAS, COSMIC, POSYDON) implement modular natal kick schemes:

• Momentum-conserving fallback models: $v_{\rm kick} = (1-f_{\rm fb}) v_{\rm NS}$.
• Parameterized analytical models: $$v_{\rm kick} = \alpha (m_{\rm ej}/m_{\rm rem}) + \beta$$ (Richards et al., 2022).
• Probabilistic and core-mass-dependent models: Distribution draws tied to CO core mass, allowing for “fuzzy” outcomes and remnant mass–kick correlations (Mandel et al., 2020, Kapil et al., 2022).

Observational data calibrate key parameters, e.g. $\alpha=115^{+40}_{-55}$ km/s, $\beta=15^{+10}_{-15}$ km/s, or $v_{\rm ns}\simeq520$ km/s, respectively (notation following (Richards et al., 2022, Kapil et al., 2022)).

Implementation must handle not only kick magnitude but direction (assigned isotropically unless observational data suggest alignment), and must convolve with the full binary orbital parameter space to ensure post-supernova stability. For gravitational-wave predictions, spin evolution and post-kick orbital tilts are computed explicitly.

Example (Bray Model):

$$v_{\rm kick} = \alpha \frac{m_{\rm ej}}{m_{\rm rem}} + \beta$$

with values: $$\alpha = 115^{+40}_{-55}~\mathrm{km\,s}^{-1},\quad\beta = 15^{+10}_{-15}~\mathrm{km\,s}^{-1}$$

6. Observational and Simulation-based Validation

Direct validation approaches include:

• Astrometric Microlensing: Proper motions of microlensed BHs, e.g., MOA-2011-BLG-191, constrain natal kicks to $\lesssim100$ km/s, linking runaway velocity to birth kick within 20% accuracy (Andrews et al., 2022).
• VLBI Proper Motions: Long baseline astrometry (e.g., for AT2019wey) yields precise proper motion and kick magnitude measurements, with MC methods providing robust PKV (potential kick velocity) distributions (Cui et al., 27 Mar 2025).
• Supernova Remnant Morphology & Age: Bipolar “ear” features in SNRs (e.g., S147) signal cumulative jet-driven kicks with velocities $\sim$450 km/s, strongly supporting jet-induced natal kick models (Shishkin et al., 26 Jun 2025).
• Population Synthesis Constraints: Only a small fraction of parameter space in conservation-of-momentum models reproduces all observations (merger rates, eccentricities, velocity distributions, low-ejecta SNe); multi-observable calibration is now standard (Richards et al., 2022).

7. Future Directions, Limitations, and Prospects

The field is proceeding toward maximal physical fidelity:

• Future observations (e.g., Gaia astrometry, more microlensing BHs, improved pulsar surveys) will allow direct, model-independent determination of natal kicks.
• Modeling of three-dimensional, stochastic explosion asymmetries and detailed neutrino radiation-hydrodynamics is increasingly being mapped onto simulation input prescriptions.
• Some systems (notably massive BHs in compact binaries) challenge the naive expectation of negligible kicks in direct-collapse scenarios, necessitating further paper of both small and large-kick cases.
• Consideration of dynamical gravitational wave losses at birth, as well as possible “kick-BEAP” jet contributions, may demand further modifications to the analytical forms used in synthesis codes.
• Wide binaries and retrograde orbits seen in GW-driven inspirals may uniquely trace the history of natal kicks in their progenitors, providing additional, model-discriminating power (Vigna-Gómez, 10 Jul 2025).

The modified natal kick prescription is now recognized as a tightly constrained, multi-parameter function—empirically and physically calibrated—that is essential for modeling the evolution of binaries, the chemical and dynamical evolution of galaxies, and the properties of compact object merger populations across cosmic time.