1.Introduction

^{De Young & Axford (1967)} derived the so-called “plasmon” solution, which consists of the balance between the gas pressure within a decelerating (or accelerating) isothermal clump of gas and the ram-pressure of the environment into which it is travelling. This simple solution still continues to be used to model the dynamics of different astrophysical flows involving cometary clumps (see, e.g., ^{Rivera-Oríz et al. 2019a}, b; ^{Veilleux et al. 1999}; ^{De Young 1997}).

Modified versions of the plasmon solution of ^{De Young & Axford (1967)} have been obtained including the effects of:

The formation of “tails” by wind/clump interactions was explored analytically by ^{Dyson, Hartquist & Biro (1993)}.

Numerical simulations show that plasmon-style “interstellar bullet” flows are highly unstable, with rather intense fragmentation of the plasmon configuration (see, e.g., ^{Klein et al. 2003}; ^{Raga et al. 2007}). It can be argued that if a high speed flow rapidly disrupts a liquid droplet (see, e.g., ^{Nicholls & Ranger 1969}), a gas cloud would be disrupted with even greater ease.

On the other hand, it is clear that some astrophysical flows (e.g., the Orion “fingers” around the BN-KL object, see ^{Rivera-Ortíz et al. 2019}a) do show the characteristics predicted by the “braking plasmon” analytic model. In a recent series of papers, ^{Pittard et al. (2009}, ²⁰¹⁰⁾ and ^{Goldsmith & Pittard (2017}, 2019) show that clumps with high clump to environment density ratios can be substantially braked (or accelerated, depending on the reference system) before fragmenting and mixing with the environment. This “dense clump regime” was also explored by ^{Rivera et al. (2019b)}, who carried out a comparison between an analytic plasmon model and numerical simulations. At least for such dense gas clumps, it appears that the plasmon model of ^{De Young & Axford (1967)} is still relevant.

In the present paper we consider the effect of the environmental gas pressure on the structure of a plasmon. This pressure will of course have an important effect for a plasmon moving at a relatively low Mach number (with respect to the environmental sound speed). Also, even in the case of a high Mach number plasmon, the environmental pressure will have an important effect in the plasmon “wings”, where the bow shock becomes highly oblique.

The paper is organized as follows. In § 2, we present the new plasmon model, and derive a full analytic solution for the shape of the plasmon. In § 3 we derive the equation of motion for the modified plasmon, and integrate it numerically to determine the velocity and position of the plasmon as a function of time. Finally, we present a discussion of the results in § 4.

2.The Plasmon Model

We consider the situation shown in the schematic diagram of Figure 1. In a cylindrical reference frame moving with the plasmon, the surrounding environment (of density ρa, pressure P _a and isothermal sound speed ca=Pa/ρa) impinges on the plasmon with a velocity v _a (this is the relative velocity between the environment and the plasmon).

In our model, we assume that the environment is isothermal, with a position-independent sound speed c _a . We also assume that the plasmon is isothermal, but allow it to have a different sound speed c₀. This choice is appropriate for a dense clump in an outflow from a young star, travelling within a higher temperature, neutral or partially ionized environment.

We assume that at any time in its evolution, the internal pressure stratification of the decelerating (or accelerating) plasmon instantaneously relaxes to the hydrostatic equilibrium, so that:

for an isothermal gas, where

with c₀ being the isothermal sound speed and α the acceleration/deceleration of the plasmon. An exploration of the validity of equation 1 is presented in Appendix B.

As discussed in Appendix A, the pressure of the shocked environment in contact with the plasmon is approximately given by:

where ρa, v _a and P _a are the ambient density, velocity and pressure (respectively) and α=arctan(dz/dr) is the angle between the impinging flow and the normal to the surface of the plasmon (see Figure 1). Therefore, the pressure P₀ at the head of the plasmon (see equation 1) is:

with

where M is the Mach number calculated with the velocity v _a of the plasmon and the isothermal sound speed c _a of the environment.

Now, setting P(z)=Ps (equations 1 and 3) and considering that dz/dr = tan α (see Figure 1), we obtain the differential equation

with β given by equation (5).

From the second equality of equation (6) we obtain:

with

It is clear that the plasmon solution ends at a finite z_m, for which α=π/2 (so that also ω=tanα→∞). From equation (7) we obtain:

In order to obtain the z(r) solution, we first consider the first equality of equation (6):

where ω is defined in equation (8). Also, from equation (7) we have:

Combining equations (10-11) we obtain the differential equation:

which can be integrated to obtain:

with ω given by equation (8). Equations (8) and (13) are then the solution for the shape of a plasmon interacting with an environment with a non-zero gas pressure.

This solution has the following limiting cases:

z≪1

z→zm (see equation 9):

β→0

Equation (16) is the plasmon solution of De Young & Axford (1967).

In Figure 2, we plot the r(z) solutions (equations 8 and 13) for different values of β. It is clear that the length-to-width ratio of the plasmon grows as a function of decreasing β. This effect can be quantified by calculating the length-to-width ratio L/W=zm/(2rm) from equations (9) and (15):

where for the second equality, we have used the M2=1/β relation. This Mach number dependence of the length-to-width ratio of the plasmon is shown in Figure 3.

It is of interest to have an analytic expression for calculating the Mach number M of the plasmon flow as a function of the observed L/W length-to-width ratio. As equation (17) does not have an analytic inversion, we propose the fit:

In Figure 3, we also show M _a vs. M/L solution, as well as its relative deviation (Ma-M)/M (with M given by equation 17) with respect to the exact solution. In the bottom frame of Figure 3, we see that for the M=1→100 Mach number range, the relative error of the interpolation of equation (18) is smaller than 2%.

Finally, we calculate the mass of the plasmon. To do this, we combine equations (1), (4) and (8) to obtain

with ρ0=P0/c02 being the density at the head of the plasmon. Now, using equations (19), (11) and (13) we can calculate the mass M_p of the plasmon:

with

We have not been able to carry out this integral analytically. However, in the limits of low and high β one obtains:

This latter, β >> 1, limit corresponds to a highly subsonic flow, for which our model is probably not appropriate.

A good analytic approximation for β in the full 0→∞ range is:

with

Figure 4 (top) shows the dimensionless mass m obtained from a numerical integration of equation (21) as a function of β, as well as the analytic approximation m _a . The bottom plot shows the relative deviation between these two solutions, and we can see that the approximate analytic solution has deviations of less that 3% relative to the exact (i.e., numerical) solution.

3.The Equation of Motion for the Plasmon

The plasmon’s equation of motion can be straightforwardly derived noting that the deceleration of the plasmon is a=c02/H (see equation 2), and then writing H in terms of the plasmon mass using equation (20). We then obtain:

where we have also used equation (4). We note that in this equation, β=ca2/va2.

Equation (26) was derived assuming that the plasmon has a time-independent mass. This is of course not necessarily true, since

A parametrization of the “detrainment” of material from a plasmon has been obtained by ^{Rivera-Ortíz et al. (2019}b) by comparing a mass-losing plasmon analytic model with numerical simulations. Through their combined analytic and numerical aproach, these authors estimate a characteristic timescale

for substantial mass loss from the plasmon. A mass conserving plasmon model is therefore appropriate only for evolutionary times <tm.

We first define dimensionless variables:

where x is the position of the plasmon, and

In terms of these dimensionless variables, equation (26) takes the form:

This is the equation of motion for the plasmon, and a numerical integration is presented in § 3.

For v'≫1, we can set 1+v'2≈v'2, and

(see equation 22 and 25) in the second equality of equation (30). With an initial condition (the initial Mach number of the plasmon), we integrate this equation to obtain:

It is clear that v'→0 as t'→t'0, with

We can integrate again to obtain the dimensionless position of the plasmon as a function of time:

where we have assumed that x'(t'=0)=0. This solution has been previously derived, e.g., by ^{Cabrit & Raga (2000)}.

In Figure 5 we present a comparison of the “large Mach number” analytical solution (equations 32 and 34) with a full, numerical integration of equation (30) for a plasmon with a v'(t'=0)=M0=10 initial Mach number. It is clear that initially the two solutions are most similar, and that they start diverging when the plasmon slows down to v'≈1. While the analytic, high Mach number solution (which is really not applicable in this low v´ regime) shows a plasmon which stops at a time t₀ (see equation 33), the full solution gives a plasmon that gradually slows down but does not stop at a finite distance.

Figure 5 also shows the axial extent z_m (equation 9) and the length-to-width ratio (L/W, given by equation 17) of the plasmon as a function of time. The length z_m initially grows with time, reaches a peak (of ≈1.42l₀) and then slowly decreases for times t>4l0/ca. The length-to-width ratio decreases monotonically from a L/W≈0.8 down to an asymptotic value of ≈0.32.

4.Discussion

We have derived an analytic solution for the problem of an isothermal plasmon travelling supersonicaly within an environment with a non-zero pressure.

For a hypersonic flow with M >>1, our model coincides with the ^{De Young & Axford (1967)} plasmon solution.

Interestingly, we find that even for large values of M the non-zero environmental pressure produces a cut-off for the plasmon, which is terminated at the distance z_m and cylindrical radius r_m from the plasmon head given by equations (9) and (15). Thus cutoff results in rather stubby plasmons (see Figure 2), unless one goes to very high Mach number flows.

We find that the length-to-width ratio of the plasmon solution (equations 17 and 18) has values ranging from ≈0.4 to ≈1.5 for Mach numbers M=1→100 (see Figure 3). Therefore, the extended wings of the ^{De Young & Axford (1967)} solution will basically never be formed in a real astrophysical flow, unless it has an extremely high Mach number. We find an analytical expression (equation 18) that can be used to derive an estimate for the flow Mach number from the observed length-to-width ratio of a cometary clump.

We also integrated the equation of motion for the new plasmon solution, and for high Mach numbers we find (not surprisingly) a time-dependent position and velocity which are similar to the ones found from the ^{De Young & Axford (1967)} plasmon solution. When the flow reaches a Mach number of ≈3, the new solution starts to separate from the De Young & Axford model, with the plasmon slowing down more slowly, and never stopping completely (while the De Young & Axford plasmon stops at a finite distance along its direction of motion).

Finally, we would like to point out an important qualitative result obtained from our new model. The ratio between the extent along the symmetry axis of the plasmon z_m and the scale-height H (equation 9) only has a logarithmic dependence on β=1/M2 (see equation 5). Therefore, for a wide range of possible values of M _a , we will have H ≈z_m, so that the gas within the plasmon will not be strongly stratified (as for this, one would need many pressure scale heights fitting within the length of the plasmon). This is a feature that is found when one tries to fit clumps obtained in numerical simulations with an analytic plasmon solution (see, e.g., ^{Raga et al. 1998}).