Explosive Ejections Generated by Gravitational Interactions

We first consider the motion of one of these cloudlets in the frame of reference of the colliding object (the system xy, see Figure 1). In this frame of reference, the cloudlet approaches the massive object with a velocity v₀ (at infinity), along an initial straight trajectory with an impact parameter ξ. Considering that initial gravitational interaction is small enough to be neglected, the mechanical energy of the incident particle is positive (equal to ${v}_{0}^{2}/2$ per unit mass) and the trajectory is a hyperbole.

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We define the gravitational radius ξ₀ as

$$\begin{eqnarray}&&{\xi }_{0}=\displaystyle \frac{{GM}}{{v}_{0}^{2}},\end{eqnarray} \tag{ 1 }$$

and the characteristic time t₀,

$$\begin{eqnarray}&&{t}_{0}=\displaystyle \frac{{\xi }_{0}}{{v}_{0}}.\end{eqnarray} \tag{ 2 }$$

Using these quantities, we adimensionalize distances with ξ₀, velocities with v₀, and times with t₀.

Then, the hyperbolic trajectory of a cloudlet, in polar coordinates, is given by (Calkin 1989; see also Cantó et al. 2011),

$$\begin{eqnarray}&&r=\displaystyle \frac{{\xi }^{{\prime} 2}}{1-\cos \theta +\xi ^{\prime} \sin \theta },\end{eqnarray} \tag{ 3 }$$

where r and $\xi ^{\prime}$ are the dimensionless distance to the particle and impact parameter, respectively.

In Equation (3), the angle θ is measured with respect to the $+\hat{x}$ direction (see Figure 1). From this equation, we can observe that for r → ∞, the cloudlet moves along incoming and outcoming asymptotic straight lines, which are given by the angles θ equal to 0 and θ_∞. This last sentence can be written as

$$\begin{eqnarray}&&\cos {\theta }_{\infty }=\displaystyle \frac{1-{\xi }^{{\prime} 2}}{1+{\xi }^{{\prime} 2}}.\end{eqnarray} \tag{ 4 }$$

We set the scattering angle, i.e., the angle between the incoming and outcoming directions of the movement, as α_m = θ_∞ − π (see Figure 1). From Equation (4), we obtain:

$$\begin{eqnarray}&&\cos {\alpha }_{m}=\displaystyle \frac{{\xi }^{{\prime} 2}-1}{{\xi }^{{\prime} 2}+1}.\end{eqnarray} \tag{ 5 }$$

Our next step is to obtain the x− and y− components of the cloudlet's velocity as a function of θ and $\xi ^{\prime}$. The dimensionless velocity can be written in polar coordinates as

$$\begin{eqnarray}&&{\bf{u}}={u}_{r}\hat{r}+{u}_{\theta }\hat{\theta }=\displaystyle \frac{{dr}}{d\tau }\hat{r}+r\displaystyle \frac{d\theta }{d\tau }\hat{\theta }=\displaystyle \frac{d\theta }{d\tau }\left(\displaystyle \frac{{dr}}{d\theta }\hat{r}+r\hat{\theta }\right),\end{eqnarray} \tag{ 6 }$$

where τ is the dimensionless time. In this problem, the angular momentum is conserved, which is given by

$$\begin{eqnarray}&&\xi ^{\prime} ={{ru}}_{\theta }={r}^{2}\displaystyle \frac{d\theta }{d\tau }.\end{eqnarray} \tag{ 7 }$$

Then, we finally obtain u_x and u_y , putting Equations (3) and (7) in Equation (6), and writing the directions $\hat{r}$ and $\hat{\theta }$ as a functions of $\hat{x}$, $\hat{y}$, and θ:

$$\begin{eqnarray}&&{\bf{u}}={u}_{x}\hat{x}+{u}_{y}\hat{y}=-\displaystyle \frac{1}{\xi ^{\prime} }(\xi ^{\prime} +\sin \theta )\hat{x}+\displaystyle \frac{1}{\xi ^{\prime} }(1-\cos \theta )\hat{y}.\end{eqnarray} \tag{ 8 }$$

Equation (8) can be used to obtain the velocity module $u=\sqrt{{u}_{x}^{2}+{u}_{y}^{2}}$ and, after some algebra, we obtain the mechanical energy conservation equation,

$$\begin{eqnarray}&&\displaystyle \frac{1}{2}{u}^{2}-\displaystyle \frac{1}{r}=\displaystyle \frac{1}{2}.\end{eqnarray} \tag{ 9 }$$

Besides, we find a relationship between d τ and d θ, combining Equations (3) and (7), which is given by

$$\begin{eqnarray}&&d\tau =\displaystyle \frac{{\xi }^{{\prime} 3}}{{\left(1-\cos \theta +\xi ^{\prime} \sin \theta \right)}^{2}}d\theta ,\end{eqnarray} \tag{ 10 }$$

which can be integrated to give

$$\begin{eqnarray}&&\tau +c=\mathrm{ln}\left[1+\xi ^{\prime} \cot \displaystyle \frac{\theta }{2}\right]+\displaystyle \frac{1+{\xi }^{{\prime} 2}}{2\left[1+\xi ^{\prime} \cot \tfrac{\theta }{2}\right]}-\displaystyle \frac{\xi ^{\prime} }{2}\cot \displaystyle \frac{\theta }{2},\end{eqnarray} \tag{ 11 }$$

with c as an integration constant. By choosing τ = 0 for an angle θ = π in the limit $\xi ^{\prime} =0$ we find that $c=\tfrac{1}{2}$.

Defining $\mu =\left[1+\xi ^{\prime} \cot \tfrac{\theta }{2}\right]$, the relation between the position angle, e.g., the angle θ, and the time τ is given by

$$\begin{eqnarray}&&\tau =\mathrm{ln}[\mu ]+\displaystyle \frac{1+{\xi }^{{\prime} 2}}{2\mu }-\displaystyle \frac{\mu }{2}.\end{eqnarray} \tag{ 12 }$$

For an angle θ = π, according to Equations (3) and (12), the particle crosses the x-axis (θ = π) at a time and position that depend on the impact parameter, such that

$$\begin{eqnarray}&&{\tau }_{* }=\displaystyle \frac{{\xi }^{{\prime} 2}}{2}\end{eqnarray} \tag{ 13 }$$

$$\begin{eqnarray}&&{x}_{* }=-\displaystyle \frac{{\xi }^{{\prime} 2}}{2}.\end{eqnarray} \tag{ 14 }$$

Note that, correctly, for ξ = 0, τ_* = x_* = 0.

In this subsection, we adopt the frame of reference where both the observer and the cloudlets are at rest with respect to each other, and the massive particle moves with velocity v₀ (see Figure 1). In this frame of reference, the cloudlet has coordinates

$$\begin{eqnarray}&&x^{\prime} =x+\tau ,y^{\prime} =y,\end{eqnarray} \tag{ 15 }$$

and velocities

$$\begin{eqnarray}&&{u}_{x}^{{\prime} }={u}_{x}+1,{u}_{y}^{{\prime} }={u}_{y}.\end{eqnarray} \tag{ 16 }$$

From this last equation and considering Equation (8) we find that the velocity is given by

$$\begin{eqnarray}&&u^{\prime} =\displaystyle \frac{{\left[2(1-\cos \theta )\right]}^{1/2}}{{\xi }^{{\prime} }}.\end{eqnarray} \tag{ 17 }$$

Then, the direction in which the cloudlet moves after interacting with the massive particle and far away from it, i.e., for θ = θ_∞ (see Equation (4)), forms an angle $\pi -{\alpha }_{m}^{{\prime} }$ with respect to $\hat{x}$ (see Figure 1(b)). This angle is obtained by doing

$$\begin{eqnarray}&&\cos (\pi -{\alpha }_{m}^{{\prime} })=-\cos {\alpha }_{m}^{{\prime} }=\displaystyle \frac{{u}_{x}^{{\prime} }}{u^{\prime} }=\displaystyle \frac{-\sin {\theta }_{\infty }}{\sqrt{2(1-\cos {\theta }_{\infty })}}.\end{eqnarray} \tag{ 18 }$$

Considering Equation (4) into Equation (18), we get

$$\begin{eqnarray}&&{\alpha }_{m}^{{\prime} }=\arccos \left(-\displaystyle \frac{1}{{\left(1+{\xi }^{{\prime} 2}\right)}^{1/2}}\right).\end{eqnarray} \tag{ 19 }$$

Finally, the position ${x}_{* }^{{\prime} }$ when the particle crosses the x-axis is

$$\begin{eqnarray}&&{x}_{* }^{{\prime} }={x}_{* }+{\tau }_{* }=0,\end{eqnarray} \tag{ 20 }$$

which is independent of ξ (see Equations (14) and (13)), and therefore every particle crosses the x-axis at the same point although at a different time.

Combining Equations (17) and (19) we obtain:

$$\begin{eqnarray}&&u=\displaystyle \frac{2}{{\left(1+{\xi }^{{\prime} 2}\right)}^{1/2}}.\end{eqnarray} \tag{ 21 }$$

Using the dimensional expressions we have:

$$\begin{eqnarray}&&v=\displaystyle \frac{2{v}_{0}}{{\left(1+{\xi }^{{\prime} 2}\right)}^{1/2}}.\end{eqnarray} \tag{ 22 }$$

Consider a cluster of radius ξ_c containing N_T clumps. The density distribution is

$$\begin{eqnarray}&&n(r)={{Ar}}^{\alpha }.\end{eqnarray} \tag{ 23 }$$

Then,

$$\begin{eqnarray}&&{N}_{T}={\int }_{0}^{{\xi }_{c}}4\pi {r}^{2}n(r){dr}=\displaystyle \frac{4\pi A}{3+\alpha }{\xi }_{c}^{3+\alpha }\end{eqnarray} \tag{ 24 }$$

for α ≠ −3 and thus,

$$\begin{eqnarray}&&A=\displaystyle \frac{(3+\alpha ){N}_{T}}{4\pi {\xi }_{c}^{3+\alpha }}.\end{eqnarray} \tag{ 25 }$$

Let ${dN}(\xi )$ be the number of clumps with an impact parameter between ξ and ξ + d ξ. We define the distribution g(ξ) of impact parameters such that

$$\begin{eqnarray}&&{dN}(\xi )={N}_{T}g(\xi )d\xi .\end{eqnarray} \tag{ 26 }$$

Clearly,

$$\begin{eqnarray}&&{\int }_{0}^{{\xi }_{c}}g(\xi )d\xi =1\end{eqnarray} \tag{ 27 }$$

where ξ_c is the maximum impact parameter.

From Figure 2 we find

$$\begin{eqnarray}&&{dN}(\xi )=2\pi \xi d\xi \left(2{\int }_{0}^{{x}_{m}}n(x){dx}\right)\end{eqnarray} \tag{ 28 }$$

where ${x}_{m}={\left({\xi }_{c}^{2}-{\xi }^{2}\right)}^{1/2}$ and $n(x)={{Ar}}^{\alpha }=A{\left({\xi }^{2}+{x}^{2}\right)}^{\alpha /2}.$

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Let

$$\begin{eqnarray}&&{\int }_{0}^{{x}_{m}}n\,{dx}=A{\int }_{0}^{{x}_{m}}{({\xi }^{2}+{x}^{2})}^{\alpha /2}\,{dx}={{AI}}_{\alpha }\end{eqnarray} \tag{ 29 }$$

where

$$\begin{eqnarray}&&{I}_{\alpha }={\int }_{0}^{{x}_{m}}{({\xi }^{2}+{x}^{2})}^{\alpha /2}\,{dx}.\end{eqnarray} \tag{ 30 }$$

Then, from Equations (28) and (30), we can express g(ξ) as

$$\begin{eqnarray}&&g(\xi )=4\pi A\xi {I}_{\alpha }/{N}_{T}=\displaystyle \frac{(3+\alpha )\xi }{{\xi }_{c}^{3+\alpha }}{I}_{\alpha },\end{eqnarray} \tag{ 31 }$$

where we have used Equation (26).

Let f(v) be the velocity distribution function, N_T f(v)dv = N_T g(ξ) d ξ, where the left side is the number of clumps with velocities between v and v + dv. Then,

$$\begin{eqnarray}&&f(v)=g(\xi )\left|\displaystyle \frac{d\xi }{{dv}}\right|.\end{eqnarray} \tag{ 32 }$$

Clearly,

$$\begin{eqnarray}&&{\int }_{{v}_{\min }}^{{v}_{\max }}f(v){dv}=1\end{eqnarray} \tag{ 33 }$$

From Equation (22) we obtain

$$\begin{eqnarray}&&\left|\displaystyle \frac{{dv}}{d\xi }\right|=\displaystyle \frac{2\xi {\xi }_{0}{v}_{0}}{{\left({\xi }^{2}+{\xi }_{0}^{2}\right)}^{1/2}}=\displaystyle \frac{4{v}_{0}^{2}{\xi }_{0}^{2}}{\xi {v}^{3}}\end{eqnarray} \tag{ 34 }$$

to finally express f(v) as

$$\begin{eqnarray}&&f(v)=\displaystyle \frac{4(3+\alpha ){v}_{0}^{2}{\xi }_{0}^{2}}{{\xi }_{c}^{3+\alpha }{v}^{3}}{I}_{\alpha }\end{eqnarray} \tag{ 35 }$$

where we have used Equation (33).

According to Equation (22), the distribution function for the velocity has a lower limit:

$$\begin{eqnarray}&&{v}_{\min }=\displaystyle \frac{2{v}_{0}}{{\left(1+{\xi }_{c}^{{\prime} 2}\right)}^{1/2}},\end{eqnarray} \tag{ 36 }$$

and an upper limit:

$$\begin{eqnarray}&&{v}_{\max }=2{v}_{0}.\end{eqnarray} \tag{ 37 }$$

Additionally, the velocity distribution given by Equation (35) has a maximum and/or a minimum given by the condition

$$\begin{eqnarray}&&{\left[\displaystyle \frac{{{dI}}_{\alpha }}{{dv}}\right]}_{{v}_{* }}=\displaystyle \frac{3{I}_{a}({v}_{* })}{{v}_{* }}\end{eqnarray} \tag{ 38 }$$

for any of maximum or minimum at v_*.

We consider the following particular cases:

$$\begin{eqnarray}&&\alpha =0,I={x}_{m}\end{eqnarray} \tag{ 39 }$$

$$\begin{eqnarray}&&\alpha =-1,\quad I=\mathrm{arcsinh}({x}_{m}/\xi )\end{eqnarray} \tag{ 40 }$$

$$\begin{eqnarray}&&\alpha =-2,\quad I=\displaystyle \frac{1}{\xi }\arctan ({x}_{m}/\xi ).\end{eqnarray} \tag{ 41 }$$

Then, a cluster with uniform density, i.e., α = 0, implies three different velocities that characterize the velocity distributions v_min, v_max, and ${v}_{* ,\max }$ where

$$\begin{eqnarray}&&{v}_{* ,\max }=\displaystyle \frac{4{v}_{0}}{\sqrt{3(1+{\xi }_{c}^{{\prime} 2})}}.\end{eqnarray} \tag{ 42 }$$

So,

$$\begin{eqnarray}&&\displaystyle \frac{{v}_{* ,\max }}{{v}_{\min }}=\displaystyle \frac{2}{\sqrt{3}}\approx 1.1547,\end{eqnarray} \tag{ 43 }$$

$$\begin{eqnarray}&&{f}_{* }=\displaystyle \frac{9\sqrt{3}(1+{\xi }_{c}^{{\prime} 2})}{32{\xi }_{c}^{{\prime} 3}{v}_{0}},\end{eqnarray} \tag{ 44 }$$

and

$$\begin{eqnarray*}&&f({v}_{\max })=\displaystyle \frac{3}{2{v}_{0}{\xi }_{c}^{{\prime} 2}}.\end{eqnarray*}$$

This velocity distribution is almost flattened in magnitude. Then, to explore the spatial distribution, numerical simulations are performed and discussed in the next section.

In order to test the analytical study, we have performed several N-body simulations of a particle of M_* = 10 M_⊙ "colliding" with a cluster of N_T small mass particles to reproduce the velocity distributions derived in the last section. The free parameters of the simulations are α and v₀, and the impact parameter of the massive particle respect to the cluster center is y*.

We selected ξ_c = 0.67 au as the cluster radius, which is a small region considering the size of a protostellar envelope. However, a larger size would represent a weaker interaction where the internal forces should be considered. The number of total particles is N_T = 200, which, approximately, have a typical separation of around 0.11 au. The cluster is centered at the origin in the simulations and we consider a spatial distribution (the number of clumps per unit volume) of the form

$$\begin{eqnarray}&&n(r)=\displaystyle \frac{(3+\alpha ){N}_{T}}{4\pi {\xi }_{c}^{3+\alpha }}{r}^{\alpha },\end{eqnarray} \tag{ 45 }$$

where we analyze the case α = 0, and for an homogeneous distribution α = −1 and α = −2.

This cluster must be in dynamical equilibrium, then, assuming that every particle has a circular orbit around the distribution center of mass, the orbital velocity of a particle at a radius r is

$$\begin{eqnarray}&&v(r)=\sqrt{\displaystyle \frac{{N}_{T}{M}_{i}G}{{\xi }_{c}}}{\left(\displaystyle \frac{r}{{\xi }_{c}}\right)}^{\displaystyle \frac{\alpha +2}{2}},\end{eqnarray} \tag{ 46 }$$

where M_i is their individual mass and G is the gravitational constant. The analytical model considers the cluster particles at rest, so we choose M_i = 10⁻¹⁰ M_⊙ to have a very small orbital velocity ∼10⁻² km s⁻¹ that, effectively, allows us to use zero velocity for every cluster particle. Also, the force between two cluster particles is negligible. We use a random number generator that chooses a number η uniformly distributed in the interval [0,1]. The value is related to the radial distance as

$$\begin{eqnarray}&&r={\xi }_{c}{\eta }^{1/(3+\alpha )},\end{eqnarray} \tag{ 47 }$$

from which we can sample ξ as a function of the random number η (for more detail see Rodríguez-González et al. 2007). We assigned random directions to the position vector of each particle.

We also included a single massive particle with a mass of 10 M_⊙, moving toward the particle distributions with v₀ = 100, 200, 300, and 400 km s⁻¹. At t = 0, the massive particle starts moving from the Cartesian point (−10 au, y*, 0), in a direction parallel to the x-axis toward the clump distribution. Table 1 shows the initial velocity v₀ and the initial position y* over the y-axis of the massive particle and the exponent α of the distribution of particles at rest for the simulations presented in this paper.

Table 1. Numerical Simulation Models of a 10 M_⊙ Particle Moving at v₀ toward a Cluster with a Number Density Parameter α from the Point (−10 au, y*, 0)

Models	v0	y*	α
	(km s−1)
v200R0	200	0	0
v200R1	200	0	−1
v200R2	200	0	−2
v100R1	100	0	−1
v300R1	300	0	−1
v400R1	400	0	−1
v200R1s05	200	$\tfrac{1}{2}{\xi }_{c}$	−1
v200R1s1	200	ξc	−1
v200R0m	200	0	0

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In order to obtain a statistically significant result to compare with the velocity distribution, we have developed sets of 10 random distributions in each of our models. The stability of our numerical solver is presented in the Appendix.

First of all, the consideration of negligible-mass particles allowed us to ignore the potential energy of the cluster, since the kinetic energy of the massive particle is greater by several orders of magnitude and in this approximation the massive particle is not affected by the interaction. For the low-mass particles, there is a very fast interaction that ejects them in almost every direction; Figure 3 shows the velocity v_∞, constant after the interaction, related to the distance, in astronomical units, at an evolutionary time of 500 yr for the v200R1 model (open circles) and the homologous expansion model (solid line) with final distance of 4 × 10⁴ au at 500 yr. The average difference between the dynamical age of each point in the simulation and the model is ∼0.65 yr, corresponding to the time when the low-mass particles reach v_∞. As we can see, the numerical model follows a Hubble-type law, however, the small difference is due to: (a) the time it takes (∼0.25 yr) for the massive particle to get to the center of the particle cluster and leave it, considering v₀ = 200 km s⁻¹, and (b) the time for the low-mass particles and the massive particle to get their terminal velocities (∼0.4 yr), after their dynamic interaction.

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Figure 4's upper panels show the velocity as a function of the impact parameter ξ; the solid line is the solution presented in Equation (22) and the open circles are each of the low-mass particles from our numerical models. The lower panels show the histograms obtained by integrating the distribution function (Equation (35)) for bins with a width Δv = 10 km s⁻¹, as the solid line, and the numerical solution is showed in bins. The models v200R0, v200R1, and v200R2, corresponding to models with a initial massive particle velocity of 200 km s⁻¹ and particle distribution with α = 0, −1, and −2 (left, center, and right columns, respectively), are presented. Figure 4 shows a very good agreement between the numerical and analytical results; the minimum velocity is given by Equation (36). For the models with v₀ = 200 km s⁻¹ we use ξ_c = 3ξ₀ (three times the gravitational radius), therefore ${v}_{\min }=126.4$ km s⁻¹, shown in these three models (v200R0, v200R1 and v200R2) independently of α. The maximum velocity is given by Equation (37), in this case it is 400 km s⁻¹ for particles with ξ = 0. The velocity of our numerical models tends to this value and it is more evident in the model with α = −2, where the particles are more concentrated at the center of the cluster; in this case, the velocity distribution is vertically asymptotic in v_max. In the constant-density case, α = 0, ${v}_{* ,\max }=145.47\,\mathrm{km}\,{{\rm{s}}}^{-1}$ and for this α the distribution does not have a minimum (local or global). For the cases where α = −1 there are no analytical expressions for the position of the maximum or minimum in the velocity distribution, but this can be obtained semianalytically, for the case of α = −1 ${v}_{* ,\max }=151.2\,\mathrm{km}\,{{\rm{s}}}^{-1}$ and ${v}_{* ,\min }=375.1\,\mathrm{km}\,{{\rm{s}}}^{-1}$, and for α = −2 ${v}_{* ,\max }=163.5\,\mathrm{km}\,{{\rm{s}}}^{-1}$ and ${v}_{* ,\min }=308\,\mathrm{km}\,{{\rm{s}}}^{-1}$. These analytical values are in good agreement with the numerical ones shown in Figure 4.

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Moreover, we ran models with different velocities, fixing M_* and α = −1, therefore the gravitational radius is different in each of our models: ξ₀ = 0.85, 0.22, 0.095, and 0.053 au for v100R1, v200R1, v300R1, and v400R1, respectively. These models, with the exception of v200R1, are shown in Figure 5. The description of the rows (and the plots in them) is the same as Figure 4 except that the columns correspond to the models (V100R1, V300R1, and V400R1, are in the left, center, and right columns, respectively), whereas v200R1 is in the second column of Figure 4. Like in Figure 4, the analytical solution is in accordance with the numerical solution. For all these models we ran the numerical simulations using ξ_c = 0.66 au, which is three times the gravitational radius of models with v₀ = 200 km s⁻¹. As one expects, the particles of the V100R1 model are distributed in a cluster with a radius smaller than its gravitational one. Therefore the velocity distribution has values around the maximum speed, which is ${v}_{\max }=200\,\mathrm{km}\,{{\rm{s}}}^{-1}$ in this model. In our models with higher initial velocities, which means a lower gravitational radius, the cluster is distributed in a larger volume, because we have fixed the radius of the cluster and the initial impact parameters of the particles are larger, so the velocity distribution has a similar form but one distributed in a larger range of velocities (corresponding at a maximum velocity on each model).

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It is also important to analyze the distribution of the tangential velocities when the massive particle does not pass through the center of the particles' distribution. In the models v200R1s05 and v200R1s1, the massive particle goes through the cluster distribution at half the radius of the cluster on the y-axis and on the edge of the cluster, on the y-axis as well. Figure 6 shows the v200R1s05 model in the upper row and v200R1s1 in the lower row, where the right column shows the velocity as a function of the impact parameter and the left column shows the histogram of the tangential velocities. As one can see, the velocities of the particles of v200R1s1 are minors, because the impact parameters of the particles are larger than for v200R1s05's particles, or for the case where the massive particle crosses the center of the distribution where the particle density is greater, because it increases toward the center of the cluster. The plots show a maximum plane-of-the-sky velocity of about 100 and 90 km s⁻¹ for v200R1s05 and v200R1s1, respectively (see Table 1).

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Using the results of the v200R1 model we obtain the projection, on the plane of the sky, of the position and velocity for each particle in our models. To achieve this, we must take into account the rotation projections on the plane of the sky to get the projected positions and velocities for each one of the particles. We use the rotation matrices R_x and R_z , which are rotations with an angle θ and ϕ in the x- and z-axes.

Using the projected velocity, one can calculate the tangential velocities (the plane-of-the-sky velocities):

$$\begin{eqnarray}&&{v}_{t}=\sqrt{{v}_{{\rm{tot}}}^{2}-{v}_{r}^{2}},\end{eqnarray} \tag{ 48 }$$

with ${v}_{{\rm{tot}}}=\sqrt{{v}_{x}^{2}+{v}_{y}^{2}+{v}_{z}^{2}}$; the radial velocities of each of the particles is given by the projected z-velocity,

$$\begin{eqnarray}&&{v}_{r}={v}_{z}^{{\prime} }.\end{eqnarray} \tag{ 49 }$$

For this analysis, we present the result using the rotation angles θ = 30° and ϕ = 30°. These angles are selected in order to obtain a wide tangential velocity distribution with a maximum value of around 200 km s⁻¹, like the case of Figure 7 in RO19b. It is important to note that it is not the aim of this paper to explore in detail the combination of angles and/or the precise initial clump distribution that reproduce the initial velocities proposed for this object. In that case, one must consider other effects, such as the gas dynamics, but we are interested in proposing this kind of explosion as a possible mechanism for the formation of this type of object.

Figure 7 shows the velocity as a function of distance (upper-left panel), the total velocity, radial velocity, and tangential velocity distributions (upper-right, lower-left, and lower-right panels, respectively).

Using Equation (49), we have calculated the radial velocity distribution in a range of velocities between about −110 and 270 km s⁻¹, with maximum values of about −100 and 150 km s⁻¹. Using these radial velocities in Equation (48), we obtained the velocity on the sky plane for each of the particles in the v200R1 numerical simulation set. The tangential velocity distribution of this model has a maximum value of about 150 km s⁻¹, and the shape of this tangential velocity distribution is similar to the initial velocity distribution presented in Figure 7 of RO19b, even when the distribution is spread over a large velocity range.

However, the mass of clumps in the region of Orion BN/KL can be estimated by using the total mass of the moving gas in the region and, for simplicity, dividing it by the total number of current observed clumps. Additionally, RO19b predicted the initial mass of the clumps (see Figure 6 of that paper). The initial mass of each individual clump is around 10⁻² M_⊙, and the total mass of these particles is comparable with the mass of the more massive particle (the star mass, i.e., 10 M_⊙). The effects of the interaction between the low-mass particles and their contribution in the global motion of this event are not considered in the first models presented in this work. Nevertheless, the high velocity of the massive particle, and its momentum, play a more important role than the mass of the particles. In order to prove this, we ran a final model, v200R0m, where each of the low-mass particles have a mass of 0.01 M_⊙. We have assigned a random direction for the orbital velocity, assuming a circular motion according to Equation (46). The particles are in a quasi-equilibrium with each other, while the massive particle collides with this cluster and interacts with it. We have calculated the random positions and orbit directions of each of the particles following this procedure:

• 1.  
  For each particle, assign a random radius using

  $$\begin{eqnarray}&&r={\xi }_{c}{\eta }^{1/(\alpha +3)}\end{eqnarray} \tag{ 50 }$$

  where η is a uniform random number between 0 → 1.
• 2.  
  We can calculate the Cartesian coordinates, x, y, and z, using

  $$\begin{eqnarray}\begin{array}{rcl}x & = & r[\sin (\theta )\cos (\phi )],\\ y & = & r[\sin (\theta )\sin (\phi )],\\ z & = & r[\cos (\theta )]\end{array}\end{eqnarray} \tag{ 51 }$$

  where $\theta =\arccos (2{\eta }_{t}-1)$ and ϕ = 2π η_p , and η_t and η_p are uniform random numbers between 0 → 1.
• 3.  
  We assigned an orbital velocity, v_r , using Equation (46).
• 4.  
  We calculated v_x , v_y , and v_z using

  $$\begin{eqnarray}\begin{array}{rcl}{v}_{x} & = & {v}_{r}[\cos (\phi )\cos (\theta )\sin (\chi )+\sin (\phi )\cos (\chi )]\\ {v}_{y} & = & {v}_{r}[\cos (\theta )\sin (\phi )\sin (\chi )-\cos (\phi )\cos (\chi )]\\ {v}_{z} & = & -{v}_{r}\sin (\theta )\sin (\chi )\end{array}\end{eqnarray} \tag{ 52 }$$

  where χ = 2π η_x and η_x is a uniform random number between 0 → 1.

Similar to previous models, we have run a set of 10 simulations using different random distributions. In this model we used a uniform distribution where the massive particle has an initial velocity of 200 km s⁻¹. Figure 8 shows the histogram of the total velocity for the v200R0m model as the solid line, the v200R0 model (with clumps of negligible mass) is shown by the dashed line, and the dashed–dotted line is the analytical solution (Section 2). The shape of these histograms is similar, but the v200R0m model presents a maximum of the distribution in a small velocity, about 130 km s⁻¹, 20 km s⁻¹ lower than the v200R0 model. The minimum velocity of any particle is about 80 km s⁻¹, this is also 30 km s⁻¹ lower than the v200R0 model; also, the particles with higher velocities are larger than in the model where there are negligible low-mass particles. The maximum velocity obtained in our numerical simulation is very similar to the maximum velocity predicted by RO19b (see Figure 7 in that paper), but the shape of the velocity distribution does not fully agree with that predicted in RO19b. However, as shown in the v200R1c0.5 and v200R1c1 models (Figure 6), the projected velocity distribution is also a function of the position through which the massive particle cross the initial particle distribution (Section 3.3) and the projection angle where the observed velocities are calculated (Section 4.1). However, a study of these parameters, for this particular object, is outside the scope of this paper and will be addressed in subsequent works.

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Finally, Figure 9 shows the position and velocities of a single simulation of the v200R0m model, with projection angles of θ = 0 and ϕ = 30, 45, and 60, for the upper, middle, and bottom panels, respectively. In this figure we plotted the tangential velocity of all the particles using red or blue arrows for positive and negative radial velocities, respectively. We also present, in the upper panel, the position of each particle when they crossed the y-axis (where they were blown away by dynamic interaction, according to Equation (14)). As one can see, the particles are ejected from a very small volume (about 0.3 au), which is insignificant with the size of the event after 500 years (about 4 × 10⁴ au) and is in accordance with the observations that suggest a single ejection point, at least with the current resolution. Thus, these types of explosions seem to be ejected in a singular place in space, as well as the Orion BN/KL event.

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However, the number of particles approaching or moving away from us is dependent on the projected angle, but the explosion is not at all isotropic in the x-axis, which is the axis of movement of the massive particle. But the morphology of the explosion is similar to the one observed in Orion BN/KL. It is important to note that the massive particle in the Orion BN/KL explosion should be a runaway massive star with a velocity of at least 150 km s⁻¹, but the dynamical interaction of the massive star with clumps with a total mass comparable to the star's mass could substantially decrease the velocity of the star at the end of the interaction. However, it is not the goal of this work to study the dynamic interaction of low-mass particles.