An origin of narrow extended structure in the interstellar medium:
an interstellar contrail created by a fast-moving massive object

Turbulent structures are ubiquitously observed in the ISM (e.g., Miville-Deschênes et al., 2016, 2017). To form a linear gaseous structure, it must be formed in a shorter time than eddy turnover time. In this paper we consider a gravitating object moving at supersonic speeds relative to the ISM. For simplicity, we consider the gravitating object moving at a constant velocity through an interstellar gas of uniform density. We describe the hydrodynamical equations in the comoving frame of the gravitating object. In this frame, we expect that the solution is in a steady state. In this section, we neglect the effect of the gas pressure prior to the formation of a shock wave. The effect of the pre-shock pressure will be discussed in §2.3.

The location of the shock $R$ is estimated from the law of conservation of mass along the $z$-axis. We assume a steady state for simplicity, the mass conservation law in the $z$-axis direction is as follows:

where $n_{\rm f}$ and $v_{z}$ are the macroscopic number density inside the shock surface and the velocity along the $z$-axis. This yields

where $s$ is a stagnation point. Although it is difficult to determine accurately the location of the stagnation point, it is known to be $s\approx r_{\rm HL}$(e.g., Matsuda et al., 2015).

As we are interested in the length of the filament, we seek a solution at $z\gg s$. Since the effect of the gravitating object has almost no effect at $z\gg s$, $v_{z}$ is the same as $v_{\infty}$ and then the solution is as follows:

We define the density compression ratio by the shock ${\mathcal{K}}_{\rm sh}$. In the limit of a weak shock, the compressibility is close to 1, and in the limit of a strong shock wave, it is 4, so $1<{\mathcal{K}}_{\rm sh}<4$. Since $n_{\rm f}={\mathcal{K}}_{\rm sh}\kappa_{\rm f}n_{\infty}$ holds, we have

where $z_{\rm f}$ is the $z$ value of the intersection of streamline and shock surface.

As in the case of streamlines, the nondimensionalization is as follows:

where $\tilde{z}_{\rm f}={z_{\rm f}}/{r_{\rm HL}}$.

Considering that the shock is not so strong far from the gravitating object and the post-shock region is actually supported by magnetic pressure in the magnetized ISM (e.g., Inoue & Inutsuka, 2009), ${\mathcal{K}}_{\rm sh}\sim 2.0$ can be used in an approximate calculation.

For the sake of later discussion, we derive the vertical velocity component at the shock wave front $v_{\perp}$ as follows:

By Eqs.(9),(11), and (18), the relation between $\tilde{b}$ and $\tilde{z}_{\rm f}$ is as follows:

The solution is obviously in $0<z_{\rm f}<b^{2}$. As mentioned in §2.2, ${\mathcal{K}}_{\rm sh}=2.0$ is used in this study. From this, we can derive the relationship between dimensionless quantities.

Now, we are interested in the behavior at the point far from the stagnation point. In this case, Eq.(20) is approximated as

From this equation and ${\mathcal{K}}_{\rm sh}=2.0$, the values of $\tilde{z}_{\rm f}$, $\tilde{R}$, $\kappa_{\rm f}$ and $v_{\perp}/v_{\infty}$ in the region far from the stagnation point are as follows:

From this result, $v_{\perp}\approx v_{\theta}(\theta=0)$ in the region far from the stagnation point (see $v_{\theta}(\theta=0)/v_{\infty}=\tilde{z}^{-1/2}$). Since $\kappa_{\rm f}$ does not depend on $\tilde{z}_{\rm f}$, the pressure gradient force can be neglected. The calculation results are shown in Fig.2.

We estimate other basic dimensions of the filament such as the width and the total mass. Since the filament boundary surface depends on the value of $z$, a representative value is taken as the filament width. This study estimates the filament width $w$ as follows:

Next, we approximate the total filament mass $M_{\rm f}$. We denote $b_{\rm max}=b(L_{\rm pt})$, and the accretion rate on the filament $\dot{M}_{\rm f}$ is as follows:

Since the filament formation timescale $\tau=L/v_{\infty}$, $M_{\rm f}$ is obtained as follows:

We derive expressions for various physical quantities of filaments. For simplicity, we ignore $P_{\rm pre}$ and use the approximation $L\approx 2L_{\rm pt}^{\prime}$. In this approximation, the length $L$ follows from Eqs.(23), (25) , (30), and (32) so that

Thus, it is shown that $L$ does not depend on $v_{\infty}$.

Using the approximation $L_{\rm pt}\approx L_{\rm pt}^{\prime}$, the filament width $w$, $b_{\rm max}$ and total mass $M_{\rm f}$ can be estimated similarly as follows:

First, the radius of the gravitating object should be much larger than $b_{\rm max}$ for creating a contrail. Second, if a massive compact object moves at a very low velocity, the object only absorbs surrounding gases and obviously does not create a contrail. In such situations, the stagnation point ($s\approx r_{\rm HL}$) is located far away from the massive object in our framework and the gravitational focusing is negligible. If the relative speed $v_{\infty}$ satisfies $L_{\rm pt}>r_{\rm HL}$, the interstellar contrail is expected to form. We denote such condition by $v_{\infty}>v_{\rm FC}$ where $v_{\rm FC}$ is estimated from the approximation $L_{\rm pt}\approx L_{\rm pt}^{\prime}$ as follows:

The ratio of $L_{\rm pt}$ and $r_{\rm HL}$ is shown in Fig.5.

The Galactic midplane is supposed to be occupied mostly by WNM except in bubbles due to supernova explosions. The estimation in §4.1 corresponds to the case when a massive compact object moves in WNM. This process is very different from the formation of filamentary molecular clouds in a shock-compressed layer (e.g., Abe et al., 2021).

On the other hand, a limited volume of the Galactic disk corresponds to CNM (i.e., cold HI clouds and molecular clouds). If the CNM size is large enough, a massive object of $10^{4}\ {\rm M_{\odot}}$ may create a long filament as large as $100\ {\rm pc}$, which should be compared to those observed in Zucker et al. (2018). If the CNM size is smaller than $2b_{\rm max}$, the filament length is smaller than that estimated in this study. If the CNM is distributed intermittently on the trajectory of the gravitating source, the resulting filament might have spatially disconnected (see Fig.6).

Recent $21\ {\rm cm}$ line observations (e.g., HI4PI Collaboration et al., 2016) have revealed the existence of numerous linear structures along the interstellar magnetic fields in HI clouds (e.g., Clark et al., 2014; Jelić et al., 2018; Turić et al., 2021). It is difficult to determine the distance to those filaments, which is one of the difficulties in understanding their property. Although such filamentary structures seem to be ubiquitous on the celestial plane, the cause of those structures is not fully understood. Thus, it is interesting to consider the possibility of creating filamentary structures by multiple gravitating sources passing through the shell of the Local Bubble. The typical length of observed HI filaments is roughly estimated as $1\text{--}10\ {\rm pc}$ if they are located on the surface of the Local Bubble whose radius is about $50\ {\rm pc}$ (e.g., Zucker et al., 2022). With Eqs. (41) – (44) we can estimate the condition for a gravitating object to form these filaments. For example, we consider a gravitating object passing through the Local Bubble’s shell. If the number density of ambient gas is $20\ {\rm cm^{-3}}$, gravitating objects of $10$, $10^{2}$, and $10^{3}\ {\rm M_{\odot}}$ create filaments with lengths of approximately $0.078$, $0.78$, and $7.8\ {\rm pc}$, respectively. Whereas the number density is $200\ {\rm cm^{-3}}$, gravitating objects of $10$, $10^{2}$, and $10^{3}\ {\rm M_{\odot}}$ create approximately $0.14$, $1.4$, and $14\ {\rm pc}$, respectively. Note that $P_{\rm crit}$ is not a simple linear function of $n_{\infty}$ so the filament length (Eq. (41)) has a nonlinear dependence on $n_{\infty}$. We should note, however, that the thermal condensation is hindered by the interstellar magnetic field (e.g., Inoue & Inutsuka, 2008, 2009), and the length of the interstellar contrail becomes shorter than our model if the strength of the magnetic field is significantly large. It is difficult to treat the effects of magnetic fields in the present analytical framework and we will study them by numerical simulation in the future.

Accretion onto a moving gravitating source is called Bondi–Hoyle–Littleton accretion (Hoyle & Lyttleton, 1939, 1941; Bondi & Hoyle, 1944), and the mass accretion rate to the gravitating source can be estimated by

This value seems to be too small to be observed. A detailed discussion on observability is beyond the scope of this paper (see, e.g., Fukue & Ioroi, 1999; Ogata et al., 2021).