Virial Velocities for Stellar Flybys with Planetary Disks in Star Formation Regions

Stellar encounters in crowded regions such as the birth cloud of the Sun have important effects on the morphology and evolution of the orbital structure of planetary disks (e.g., Craig & Krumholz 2013; Jiménez-Torres et al. 2013; Vincke et al. 2015; Jiménez-Torres 2020). In this context, the determination of the number of stellar encounters with planetary disks is a relevant part of the analysis to reproduce the orbits of objects in the planetary disk. In (Jiménez-Torres 2020, 2021, 2023), we studied the effect of the number of encounters with variables mean free path, virial velocities, and the time during which the cluster is still embedded and stellar encounters can occur. However, the role of the velocity of the passing stars was not discussed, using constant velocities of the order of 2 km s⁻¹.

The velocity dispersion measures in different young stellar clusters show a wide distribution. Foster et al. (2015) show that the velocity dispersion of young members (1–2 Myr) in the cluster NGC 1333 was of the order of 0.92 ± 0.12 km s⁻¹, which are of the order of velocities of members in dense star formation cores. Cottaar et al. (2015) show that members in the cluster IC 348 have a velocity dispersion of the order of 0.6–0.7 km s⁻¹. In the Orion Nebula Cluster, the velocity dispersion of their members is of the order of 3–4 km s⁻¹ (Fűrész et al. 2006, 2008; Tobin et al. 2009). These large differences cause some concern when selecting velocities of the passing stars for the simulations, and in this work, we provide a sensible solution to this situation.

In Jiménez-Torres et al. (2013) and Jiménez-Torres (2022) we provide a mathematical expression to calculate the number of encounters N_e on a delimited cross-sectional area containing a planetary disk for the time T of 2 Myr when close stellar encounters occur in the crowded star formation cluster as a consequence of the cluster development (Pfalzner 2013). This equation is given by

$$\begin{eqnarray}&&{N}_{e}=\pi \cdot {\rho }_{n}\cdot v\cdot T\cdot {r}^{2},\end{eqnarray} \tag{ 1 }$$

where ρ_n corresponds to the stellar number density, v the velocity of passing stars, and r the radius of a cross sectional area σ.

To calculate the velocity v of members star clusters, we assume there are N_s (N_s − 1) ∼$\tfrac{1}{2}$ ${{N}_{s}}^{2}$ pairs of stars in the cluster, where N_s is the number of stars in the cluster (Karttunen et al. 2003). The total potential energy is given by

$$\begin{eqnarray}&&{E}_{P}=-G\displaystyle \frac{{{m}_{s}}^{2}}{{R}_{c}}\displaystyle \frac{{{N}_{s}}^{2}}{2},\end{eqnarray} \tag{ 2 }$$

where m_s is the average mass of stars and R_c is the average distance of each pair. This R_c is described as the effective radius in Portegies Zwart et al. (2010) and Kuhn et al. (2019). The total kinetic energy is given by

$$\begin{eqnarray}&&{E}_{K}=\displaystyle \frac{1}{2}{m}_{s}{v}^{2}{N}_{s}.\end{eqnarray} \tag{ 3 }$$

The virial theorem tells that E_K  = −$\tfrac{1}{2}$ E_P , and solving for the virial velocity v we have

$$\begin{eqnarray}&&{v}^{2}=\displaystyle \frac{G\ {M}_{c}}{2\ {R}_{c}},\end{eqnarray} \tag{ 4 }$$

where M_c is the mass of the cluster. This is applied in calculations in Jiménez-Torres (2020).

In Figure 1 we show results from simulations of encounters with different velocities with a 1 M_⊙ planetary system. We used the same methods as in Jiménez-Torres et al. (2011), Jiménez-Torres et al. (2013).

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Results from simulations with 1.5, 2 km s⁻¹, or 3 km s⁻¹ flyby velocities show quite similar resulting eccentricities and inclinations to each other. In this way, we argue that the assumption of constant velocities for all members in a star cluster is a reasonable first guess because the range of velocities encompasses most of the results found for young open clusters (see above). The most noticeable difference begins to appear for flyby velocities above 5 km s⁻¹, however, these kinds of velocities do not correspond to velocities of members in star formation clusters, but velocities in different galactic regions such as globular clusters (Jiménez-Torres et al. 2013).

According to the results in Figure 1, we do not find significant differences in the resulting orbital elements of the disk's test particles when using flyby velocities up to 3 km s⁻¹. This suggests that virial velocities or constant velocities can be used to simulate conditions of stellar encounters in a typical star formation region with reasonable accuracy. This also provides a solution to the concern of unknown velocities of passing stars, which can be modeled with constant velocities with good reliability in simulations of stellar encounters with planetary disks.

J.J. thanks to the Instituto de Astronomía UNAM, México, and the Kanazawa Institute of Technology, Japan, where this work was conceived. AAC acknowledges financial support from the Severo Ochoa grant CEX2021-001131-S funded by MCIN/AEI/10.13039/501100011033.