# Apparent superluminal velocities and random walk in the velocity space

Abhijit Sen<EMAIL_ADDRESS>Novosibirsk State University, Novosibirsk
630 090, Russia  Z.K. Silagadze<EMAIL_ADDRESS>Budker Institute of
Nuclear Physics and Novosibirsk State University, Novosibirsk 630 090, Russia

###### Abstract

We conjecture that the random walk and the corresponding diffusion in the
relativistic velocity space is an adequate method for describing the
acceleration process in relativistic jets. Considering a simple toy model, the
main features of diffusion in the velocity space are demonstrated in both non-
relativistic and relativistic regimes.

Special relativity; Velocity space; Random walk; Apparent superluminal
velocities

##  I Introduction

Apparent superluminal motion has been found to occur in many astrophysical
phenomena with the relativistic ejecta such as active galactic nuclei
(quasars, radio galaxies, blazars), gamma-ray bursts and micro-quasars 1 ; 2 ;
3 ; 4 . Although these objects are significantly different in their sizes and
properties, the common feature that makes possible the apparent superluminal
motion is that they all support relativistic jets.

In fact, the possibility of apparent superluminal motion was predicted by Rees
5  well before the phenomenon was actually observed thanks to a new very long
baseline interferometry technique, first in the quasar 3C273, and then in many
other sources 2 .

The basic idea behind the apparent superluminal motion is rather elementary:
this effect arises from the Doppler contraction of the arrival times of
photons due to the finite speed of light 6 ; 7 . This Doppler contraction is
most easily explained by the space-time diagram in Fig. 1.

Figure 1: Illustration of the Doppler contraction of the arrival times of
photons.

Suppose a radio-emitting blob moves along world-line $AB$. Two consecutive
radio-emissions at $A$ and $B$ are separated by a time interval $\Delta
t_{e}$. Arrival times of the corresponding radio-waves at $D$ and $C$ are
separated by another time interval $\Delta t_{r}$. Then $BF=c\Delta t_{e}$ and
$CD=BE=c\Delta t_{r}$. If the velocity of the blob is $V$, and its radial
component is $V\cos{\theta}$, then $AF=EF=V\cos{\theta}\,\Delta t_{e}$. It is
clear from Fig. 1 that $c\Delta t_{r}=BF-EF=c\Delta
t_{e}-V\cos{\theta}\,\Delta t_{e}$. Therefore $\Delta t_{r}=\Delta
t_{e}(1-\beta\,\cos{\theta})$. Since the lateral displacement of the blob is
$V\sin{\theta}\,\Delta t_{e}$, the apparent transverse velocity of the blob
(divided by the light velocity $c$) equals to111This formula is valid only for
nearby sources. The apparent transverse velocity of a distant radio source
depends on its redshift $z$ and is reduced by a factor of $1+z$ 2 ; 8 .

$\beta_{app}=\frac{V\sin{\theta}\,\Delta t_{e}}{c\Delta
t_{r}}=\frac{\beta\sin{\theta}}{1-\beta\cos{\theta}}.$ (1)

The apparent lateral motion is superluminal if

$\beta>\frac{1}{\sin{\theta}+\cos{\theta}}=\frac{1}{\sqrt{2}\sin{\left(\frac{\pi}{4}+\theta\right)}}\geq\frac{1}{\sqrt{2}}\approx
0.71.$ (2)

Consequently, apparent superluminal motion is possible only if the source
moves with relativistic speed.

Relativistic jets are produced by accreting compact objects such as neutron
stars or black holes, although the exact mechanism how relativistic outflows
are launched is not yet clear 9 . The most promising candidate is the so
called Blandford–Znajek mechanism 10 ; 11 , which suggests that the jets are
driven by the rotational energy of the black hole, extracted
electromagnetically in the form of a Poynting flux through magnetic fields
penetrating the event horizon of the central black hole.

However, there is still no consensus on the mechanisms that cause acceleration
and collimation of jets. Observations show that jets are subluminal near the
core, then they are gradually accelerated and become relativistic at a
distance relatively remote from the central engine (about a thousand
Schwarzschild radii) 12 .

Possible acceleration mechanisms are mostly related to the propagation of
relativistic shock waves and turbulent structures in the jets and include
diffusive shock acceleration (first-order Fermi acceleration), fast magnetic
reconnection, second-order Fermi acceleration (stochastic acceleration),
Compton rocket 9 ; 13 ; 14 ; 15 . There is, probably, a competition between
different acceleration mechanisms, with stochastic acceleration playing a
significant role 13 . Due to the stochastic nature of the acceleration
process, it was proposed to describe it as diffusion in momentum space 16 ,
assuming that the isotropic and homogeneous phase-space density $f(p,t)$
evolves in accordance with the equation

$\frac{\partial f(p,t)}{\partial t}=\frac{1}{p^{2}}\,\frac{\partial}{\partial
p}\left[p^{2}D(p,t)\,\frac{\partial f(p,t)}{\partial p}\right],$ (3)

where $D(p,t)$ is the momentum-diffussion coefficient.

Our hypothesis is that probably the better description of the stochastic
acceleration process is given by a random walk on the relativistic velocity
space. Even a very simple type of such random walk, an inebriated-astronaut
model, is surprisingly effective in producing relativistic velocities 17  in
the sense that rapidity grows linearly with the number of steps, not as the
square root of the number of steps, as expected from non-relativistic
experience.

##  II Random walk on the non-relativistic velocity space

We suppose that in the comoving rest frame of the particle under stochastic
acceleration it receives isotropic kicks and after each kick its velocity
changes by $\Delta\vec{V}$, such that $\delta
V=|\Delta\vec{V}|=\mathrm{const}$. If at the $(n-1)$-th step the velocity was
$\vec{X}$ and after the $n$-th kick it became $\vec{V}$, then
$\vec{X}+\Delta\vec{V}=\vec{V}$ and

$X=\sqrt{V^{2}+(\delta V)^{2}-2V\delta V\cos{\alpha}}\approx V-\delta
V\cos{\alpha}+\frac{(\delta V)^{2}}{2V}\sin^{2}{\alpha},$ (4)

where $\alpha$ is the angle between $\Delta\vec{V}$ and $\vec{V}$. The
probability that $\alpha$ is in the range $(\alpha,\,\alpha+\Delta\alpha)$ is
$\frac{d\Omega}{4\pi}=\frac{1}{2}\,\sin{\alpha}\,d\alpha$. Therefore, if
$P(V,n)$ is the probability density at the $n$-th step222That is,
$P(V,n)\,d\vec{V}$ is the probability that the velocity at the $n$-th step
will be in the range $(\vec{V},\,\vec{V}+d\vec{V})$., then we can write

$P(V,n)=\int\limits_{0}^{\pi}P\left(V-\delta V\cos{\alpha}+\frac{(\delta
V)^{2}}{2V}\sin^{2}{\alpha},\,n-1\right)\frac{1}{2}\,\sin{\alpha}\,d\alpha.$
(5)

Now we assume that $\delta V\ll V$ and expand up to the quadratic terms in
$\delta V$:

$P\left(V-\delta V\cos{\alpha}+\frac{(\delta
V)^{2}}{2V}\sin^{2}{\alpha},\,n-1\right)\approx P(V,n-1)+\frac{\partial
P}{\partial V}\left(-\delta V\cos{\alpha}+\frac{(\delta
V)^{2}}{2V}\sin^{2}{\alpha}\right)+\frac{1}{2}\,\frac{\partial^{2}P}{\partial
V^{2}}(\delta V)^{2}\cos^{2}{\alpha}.$ (6)

We can substitute (6) into (5), evaluate elementary integrals and obtain

$P(V,n)=P(V,n-1)+\frac{1}{3}\,\frac{(\delta V)^{2}}{V}\,\frac{\partial
P}{\partial V}+\frac{1}{6}\,(\delta V)^{2}\,\frac{\partial^{2}P}{\partial
V^{2}}.$ (7)

If $n\gg 1$, then $P(V,n)-P(V,n-1)\approx\frac{\partial P}{\partial n}$ and
(7) can be rewritten as follows

$\frac{\partial P}{\partial n}=\frac{(\delta
V)^{2}}{6V^{2}}\,\frac{\partial}{\partial V}\left(V^{2}\frac{\partial
P}{\partial V}\right)=\frac{(\delta V)^{2}}{6}\,\nabla^{2}P,$ (8)

where $\nabla^{2}$ is the Laplacian in the velocity space, and the last
equality follows from the fact that $P$ depends only on the magnitude $V$ of
the velocity, not on the angular variables characterizing its direction.

The equation (8) is the same diffusion equation (3), provided that the
momentum is given by its non-relativistic expression $p=mV$ and the diffusion
coefficient $D(p,t)=(\delta V)^{2}/6$ is a constant. We need a solution of
this diffusion equation with the initial condition $P(V,0)=f_{0}\delta(V)$,
which assumes that the initial velocity was zero. Here $f_{0}$ is the
normalization constant to be determined from the normalization condition $\int
P(V,n)\,d\vec{V}=1$, $n\neq 0$333$P(V,0)$ cannot be used to determine the
normalization constant $f_{0}$, because, as can be seen from the explicit form
of $P(V,n)$, $\lim_{n\to 0}\int P(V,n)\,d\vec{V}\neq\int\left(\lim_{n\to
0}P(V,n)\right)d\vec{V}$..

The solution is given by the well-known Euclidean heat kernel 18  and has the
form of a normalized Gaussian

$P(V,n)=(4\pi)^{-3/2}\left(\frac{n\,(\delta
V)^{2}}{6}\right)^{-3/2}\exp{\left(-\frac{3V^{2}}{2n\,(\delta
V)^{2}}\right)}.$ (9)

As we see, the likelihood that the speed increase will far exceed
$\sqrt{n}\,\delta V$ is vanishingly small. As expected, the random walk
efficiency for producing high speeds increases as $\sqrt{n}$.

For the convenience of readers, we present in the appendix the derivation of
the Euclidean heat kernel. Although originally associated with the heat
equation 19 , physical motivation soon receded far into the background, and
heat kernels in their general form became ubiquitous in mathematics 20 ; 21 ;
22 ; 23 .

##  III Random walk on the relativistic velocity space

If $\delta V\ll c$, in the reference frame where initially the particle
velocity is non-relativistic, it will remain non-relativistic for many
consecutive kicks. Therefore, equation (8) remains valid in this frame
locally. It will be globally valid if written in the covariant form 17

$\frac{\partial P}{\partial n}=\frac{(\delta
V)^{2}}{6}\,\nabla^{2}P,\;\;\;\nabla^{2}P=\frac{1}{\sqrt{g}}\frac{\partial}{\partial
V^{\alpha}}\left(\sqrt{g}\,g^{\alpha\beta}\frac{\partial P}{\partial
V^{\beta}}\right),$ (10)

where Einstein’s summation convention is accepted, $g_{\alpha\beta}$ is the
metric tensor in the relativistic velocity space with determinant $g$,
$g^{\alpha\beta}$ is the inverse of the metric tensor, and the expression used
for the Laplacian in curved spaces can be found, for example, in 24 .

Given two relativistic velocities, $\vec{V}$ and $\vec{V}+d\vec{V}$, the
square of the relative velocity can be considered as a line element in three-
dimensional velocity space 25 ; 26 ; 27 . If $\vec{V}_{1}$ and $\vec{V}_{2}$
are two relativistic velocities, and $u_{1}$ and $u_{2}$ are the corresponding
four-velocities, then it easy to see that the square of the relative velocity
is 17 ; 28  (to simplify expressions, $c=1$ is assumed throughout the rest of
the paper)

$[\vec{V}_{1}\ominus\vec{V}_{2}]^{2}=1-\frac{1}{(u_{1}\cdot
u_{2})^{2}}=\frac{(\vec{V}_{1}-\vec{V}_{2})^{2}-[\vec{V}_{1}\times\vec{V}_{2}]^{2}}{(1-\vec{V}_{1}\cdot\vec{V}_{2})^{2}}.$
(11)

Substituting $\vec{V}$ and $\vec{V}+d\vec{V}$, insteed of $\vec{V}_{1}$ and
$\vec{V}_{2}$ in this formula, we get for the line element of the relativistic
velocity frame the following expression

$ds^{2}=\frac{(d\vec{V})^{2}-[\vec{V}\times
d\vec{V}]^{2}}{(1-\vec{V}\cdot\vec{V})^{2}}=\frac{dV^{2}}{(1-V^{2})^{2}}+\frac{V^{2}}{1-V^{2}}\left(d\theta^{2}+\sin^{2}{\theta}\,d\varphi^{2}\right),$
(12)

where the last equality follows from $[\vec{V}\times
d\vec{V}]^{2}=V^{2}d\vec{V}\cdot d\vec{V}-(\vec{V}\cdot d\vec{V})^{2}$ and
$\vec{V}=V(\sin{\theta}\cos{\varphi},\,\sin{\theta}\sin{\varphi},\,\cos{\theta})$.

It turns out that (12) is the Riemannian line element corresponding to the
Beltrami–Klein model $\mathbb{H}^{3}$ of three-dimensional hyperbolic geometry
26 ; 29 .

The natural parameter for Lorentz boosts is rapidity, not velocity 30 . It is
not surprising therefore that (11) simplifies if, instead of the velocity $V$,
we introduce the rapidity $\psi$ through $V=\tanh{\psi}$:

$ds^{2}=d\psi^{2}+\sinh^{2}{\psi}\left(d\theta^{2}+\sin^{2}{\theta}\,d\varphi^{2}\right).$
(13)

Thus, non-zero components of the metric tensor and its determinant are

$g_{\psi\psi}=1,\;\;g_{\theta\theta}=\sinh^{2}{\psi},\;\;g_{\varphi\varphi}=\sinh^{2}{\psi}\,\sin^{2}{\theta},\;\;g=\sinh^{4}{\psi}\,\sin^{2}{\theta}.$
(14)

Because of spherical symmetry, $P(\psi,n)$ does not depend on $\theta$ and
$\varphi$. Therefore, from (10), we get 17

$\frac{\partial P}{\partial n}=\frac{(\delta
V)^{2}}{6}\,\frac{1}{\sinh^{2}{\psi}}\,\frac{\partial}{\partial\psi}\left(\sinh^{2}{\psi}\,\frac{\partial
P}{\partial\psi}\right)=\frac{(\delta
V)^{2}}{6}\left[\frac{\partial^{2}P}{\partial\psi^{2}}+2\,\frac{\cosh{\psi}}{\sinh{\psi}}\,\frac{\partial
P}{\partial\psi}\right].$ (15)

To solve (15) with $P(\psi,0)\sim\delta(\psi)$ initial condition, let us note
that

$\frac{\psi}{\sinh{\psi}}\,\Delta_{0}\,\frac{\sinh{\psi}}{\psi}=\frac{\partial^{2}}{\partial\psi^{2}}+2\,\frac{\cosh{\psi}}{\sinh{\psi}}\,\frac{\partial}{\partial\psi}+1=\Delta+1,$
(16)

and

$e^{-an}\,\frac{\partial}{\partial n}\,e^{an}=\frac{\partial}{\partial n}+a,$
(17)

where $a=\frac{(\delta V)^{2}}{6}$, and $\Delta_{0}$, $\Delta$ are radial
parts of the Euclidean and hyperbolic Laplacians:

$\Delta_{0}=\frac{1}{\psi^{2}}\,\frac{\partial}{\partial\psi}\,\left(\psi^{2}\,\frac{\partial}{\partial\psi}\right),\;\;\Delta=\frac{1}{\sinh^{2}{\psi}}\,\frac{\partial}{\partial\psi}\,\left(\sinh^{2}{\psi}\,\frac{\partial}{\partial\psi}\right).$
(18)

Relations (16) and (17) imply that

$e^{-an}\frac{\psi}{\sinh{\psi}}\left(\frac{\partial}{\partial
n}-a\Delta_{0}\right)\frac{\sinh{\psi}}{\psi}e^{an}=\frac{\partial}{\partial
n}-a\Delta.$ (19)

Therefore, if $P_{0}(\psi,n)$ is a solution of the Euclidean heat equation

$\left(\frac{\partial}{\partial n}-a\Delta_{0}\right)P_{0}(\psi,n)=0$

with the initial condition $P_{0}(\psi,0)\sim\delta(\psi)$, then

$P(\psi,n)=\frac{\psi}{\sinh{\psi}}e^{-an}P_{0}(\psi,n)$

is the solution of the heat equation

$\left(\frac{\partial}{\partial n}-a\Delta\right)P(\psi,n)=0$

in the $\mathbb{H}^{3}$ hyperbolic space with the initial condition
$P(\psi,0)\sim\delta(\psi)$.

Consequently, the required solution of (15) is

$P(\psi,n)=e^{-\frac{(\delta
V)^{2}}{6}\,n}\,\frac{\psi}{\sinh{\psi}}\,(4\pi)^{-3/2}\left(\frac{n\,(\delta
V)^{2}}{6}\right)^{-3/2}\exp{\left(-\frac{3\psi^{2}}{2n\,(\delta
V)^{2}}\right)}.$ (20)

This result agrees, as it should be, with the well-known expression of the
heat kernel in the $\mathbb{H}^{3}$ hyperbolic space 22 ; 31 ; 32 ; 33 .
Besides, in the non-relativistic limit $\psi\approx V\ll 1$, $n\,(\delta
V)^{2}\ll 1$, (20) turns into (9).

The volume element in $\mathbb{H}^{3}$ is $d\vec{V}=\sqrt{g}\,d\psi d\theta
d\varphi$. Therefore, the probability that $\psi$ will be between $\psi$ and
$\psi+d\psi$ is
$\int\limits_{0}^{\pi}d\theta\int\limits_{0}^{2\pi}d\varphi\,\sqrt{g}\,P(\psi,n)\,d\psi=4\pi\sinh^{2}{\psi}\,P(\psi,n)\,d\psi$.
Combined with (20), this implies the following probability density for the
rapidity $\psi$:

$p(\psi,n)=\frac{\psi\,\sinh{\psi}}{\sqrt{4\pi}}\,\left(\frac{n\,(\delta
V)^{2}}{6}\right)^{-3/2}\exp{\left(-\frac{3}{2n\,(\delta
V)^{2}}\left[\psi^{2}+\frac{n^{2}(\delta V)^{4}}{9}\right]\right)}.$ (21)

In the ultra-relativistic limit $\psi\gg 1$,
$\sinh{\psi}\approx\frac{1}{2}e^{\psi}$ and (21) takes the form

$p(\psi,n)=\frac{\psi}{\sqrt{16\pi}}\,\left(\frac{n\,(\delta
V)^{2}}{6}\right)^{-3/2}\,\exp{\left(-\frac{3}{2n\,(\delta
V)^{2}}\left[\psi-\frac{n(\delta V)^{2}}{3}\right]^{2}\right)}.$ (22)

When $n$ increases, the rapidity grows linearly with $n$, not as $\sqrt{n}$,
and becomes more and more concentrated around $\psi=\frac{n(\delta
V)^{2}}{3}$.

##  IV Concluding remarks

The observation of apparent superluminal velocities in astrophysical jets
indicates that particles move relativistically in these jets. To explain
observed features of such jets, in addition to the initial jet acceleration by
means of the Blandford-Znajek and/or some other processes, additional
acceleration mechanisms operating within the jets are required, with the
stochastic acceleration playing probably a significant role 13 . We conjecture
that in this case random walk and the corresponding diffusion in the
relativistic velocity space is an adequate method for describing the
acceleration process in relativistic jets.

The formulation of a consistent theory of relativistic diffusion is a long-
standing fascinating problem in physics 34 ; 35 . The diffusion equation is a
parabolic differential equation and it assumes an infinite propagation speed
for the initial data. Therefore, a relativistic diffusion cannot exist on the
Minkowski space-time. However, as shown by R. M. Dudley 36  it still makes
sense at the level of its tangent bundle (the relativistic velocity space). We
believe that the idea that particles could follow a Brownian motion on the
relativistic velocity space could prove interesting in the context of
astrophysics 37 .

The way we got the heat kernel on $\mathbb{H}^{3}$ is similar (but not
identical) to the use of shift operators (intertwining operators) 38 ; 39 ; 40
relating radial parts of Laplacians on $\mathbb{H}^{2n+1}$ and
$\mathbb{R}^{1}$.

## Acknowledgments

The work is supported by the Ministry of Education and Science of the Russian
Federation.

##  Appendix A Derivation of the Euclidean heat kernel

We want to solve the heat equation

$\frac{\partial P}{\partial t}=a\,\nabla^{2}P,$ (23)

with the initial condition

$P(\vec{r},0)=\delta(\vec{r}).$ (24)

The following method of solution is adapted from 18 . We take
$P(\vec{r},t)=f(x,t)f(y,t)f(z,t)$, where $f(x,t)$ satisfies a one-dimensional
heat equation

$\frac{\partial f}{\partial t}=a\,\frac{\partial^{2}f}{\partial x^{2}},$ (25)

with the initial condition

$f(x,0)=\delta(x).$ (26)

It is convenient to introduce a dimensionless auxiliary function
$\tilde{f}(x,t)$, such that $f(x,t)=\frac{\partial\tilde{f}(x,t)}{\partial
x}$. If we further assume that this function satisfies the same one-
dimensional heat equation (25), but with the different initial condition

$\tilde{f}(x,0)=\Theta(x),$ (27)

where $\Theta(x)$ is the Heaviside step function, then $f(x,t)$ will just
satisfy (25) with the correct initial condition (26).

According to Buckingham’s Pi-theorem 41 , any physical law can be expressed as
a relationship between dimensionless quantities. From $x$, $t$ and $a$, we can
construct only one independent dimensionless quantity
$\tau=\frac{x}{\sqrt{at}}$. Therefore, Pi-theorem implies that

$\tilde{f}(x,t)=g(\tau).$ (28)

Substituting (28) into the one-dimensional heat equation, we get an ordinary
differential equation for the unknown function $g$:

$\frac{d^{2}g}{d\tau^{2}}+\frac{\tau}{2}\,\frac{dg}{d\tau}=0.$ (29)

The general solution of this equation has the form

$g(\tau)=g_{0}+g_{1}\int\limits_{0}^{\tau}e^{-s^{2}/4}ds,$ (30)

where $g_{0}$ and $g_{1}$ are some constants.

The initial condition (27) imply that $g(\infty)=1$ and $g(-\infty)=0$.
Therefore, $g_{0}$ and $g_{1}$ are determined by a linear system

$g_{0}+g_{1}\int\limits_{0}^{\infty}e^{-s^{2}/4}ds=1,\;\;\;g_{0}+g_{1}\int\limits_{0}^{-\infty}e^{-s^{2}/4}ds=0.$
(31)

In particular,

$g_{1}=\left[\int\limits_{0}^{\infty}e^{-s^{2}/4}ds-\int\limits_{0}^{-\infty}e^{-s^{2}/4}ds\right]^{-1}=\frac{1}{2\sqrt{\pi}}.$
(32)

Then

$f(x,t)=\frac{\partial}{\partial
x}\left[g_{0}+g_{1}\int\limits_{0}^{\tau}e^{-s^{2}/4}ds\right]=\frac{1}{2\sqrt{\pi}}\,\frac{\partial\tau}{\partial
x}\,e^{-\tau^{2}/4}=\frac{1}{\sqrt{4\pi
at}}\exp{\left(-\frac{x^{2}}{4at}\right)},$ (33)

and

$P(\vec{r},t)=\left(4\pi
at\right)^{-3/2}\exp{\left(-\frac{r^{2}}{4at}\right)}.$ (34)

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# In-situ detection of Europa’s water plumes is harder than previously
thought.

Rowan Dayton-Oxland<EMAIL_ADDRESS>Hans L. F. Huybrighs
Thomas O. Winterhalder  Arnaud Mahieux  David Goldstein

###### Abstract

Europa’s subsurface ocean is a potential candidate for life in the outer solar
system. It is thought that plumes may exist which eject ocean material out
into space, which may be detected by a spacecraft flyby. Previous work on the
feasibility of these detections has assumed a collisionless model of the plume
particles. New models of the plumes including particle collisions have shown
that a shock can develop in the plume interior as rising particles collide
with particles falling back to the moon’s surface, limiting the plume’s
altitude. These models also assume a Laval nozzle-like vent which results in a
colder plume source temperature than in previous studies, further limiting the
plume’s extent. We investigate to what degree the limited extent of the
shocked plumes reduces the ability of the JUICE spacecraft to detect plume H2O
molecules. Results show that the region over Europa’s surface within which
plumes would be separable from the H2O atmosphere by JUICE (the region of
separability) is reduced by up to a half with the collisional model compared
to the collisionless model. Putative plume sources which are on the border of
the region of separability for the collisionless model cannot be separated
from the atmosphere when the shock is considered for a mass flux case of
100$\,\mathrm{kg\,s^{-1}}$. Increasing the flyby altitude by
100$\,\mathrm{km}$ such that the spacecraft passes above the shock canopy
results in a reduction in region of separability by a third, whilst decreasing
the flyby altitude by 100$\,\mathrm{km}$ increases the region of separability
by the same amount. We recommend flybys pass through or as close to the shock
as possible to sample the most high-density region. If the spacecraft flies
close to the shock, the structure of the plume could be resolvable using the
neutral mass spectrometer on JUICE, allowing us to test models of the plume
physics and understand the underlying physics of Europa’s plumes. As the
altitude of the shock is uncertain and dependent on unpredictable plume
parameters, we recommend flybys be lowered where possible to reduce the risk
of passing above the shock and losing detection coverage, density and
duration.

###### keywords:

Europa; Satellites, atmospheres; Atmospheres, dynamics; Volcanism; Jupiter,
satellites; Satellites, composition

††journal: Icarus

[1]organization=School of Physics and Astronomy, University of Southampton,
addressline= University Road, city=Southampton, postcode=SO17 1BJ, country=UK
[2]organization=ESTEC (ESA), addressline=Keplerlaan 1, city=Noordwijk,
postcode=2201 AZ, country=Netherlands [3]organization=Leiden University,
addressline=Rapenburg 70, city=Leiden, postcode=2311 EZ, country=Netherlands
[4]organization=Space and Planetary Science Centre, Khalifa University,
addressline=Zone 1, city=Abu Dhabi, country=UAE [5]organization=Department of
Mathematics, Khalifa University, addressline=Zone 1, city=Abu Dhabi,
country=UAE [6]organization=Landessternwarte, Universitat Heidelberg,
addressline=Konigstuhl, city=Heidelberg, postcode=D 69117, country=Germany
[7]organization=European Southern Observatory, addressline = Karl-
Schwarzschild-Str 2, city = Garching, postcode = 85748, country =Germany
[8]organization=University of Texas at Austin, addressline=110 Inner Campus
Drive, city=Austin, postcode=78705, state=Texas, country=USA
[9]organization=Royal Belgian Institute of Space Aeronomy, addressline=Av.
Circulaire 3, city=Brussels, postcode=1180, state=Uccle, country=Belgium

An internal shock caused by particle collisions reduces the radial extent of
the model plume, resulting in a decrease in plume density as detected by a
simulated flyby by as much as three orders of magnitude.

This can halve the area of the Europan surface over which plumes can be
differentiated from the H2O atmosphere (region of separability), making plume
detection more difficult.

We recommend Europa flybys pass below the plume canopy altitude, e.g.
300$\,\mathrm{km}$, to maximise the chance of detecting a plume.

Lowering the altitude of the flyby by 100km can increase the chance of
detecting a plume by increasing the region of separability by as much as 32%.

If the flyby passes close to the plume, a neutral mass spectrometer could
resolve the structure of the canopy shock - allowing us to probe the plume
physics.

##  1 Introduction

Europa is thought to have a subsurface water ocean with the potential to
support life, making it one of the most intriguing places in the solar system
to astrobiologists. The physical and chemical characteristics of this ocean
are poorly constrained, but it is thought that tidal heating from Jupiter
sustains a liquid ocean around 100km deep, which could contain the essential
ingredients and energy for life (Nimmo and Pappalardo, 2016; Schubert et al.,
2004). Several observation methods have hinted at the possibility of water
plumes erupting from the surface, potentially ejecting ocean material into
space and providing an opportunity to study its composition (Hand et al.,
2009; Chyba and Phillips, 2002).

Europa’s plumes were first detected by Roth et al. (2014), who reported
evidence of excess UV emission (in the Hydrogen Lyman-$\alpha$ and Oxygen OI
130.4$\,\mathrm{nm}$ spectral emission lines) above the southern hemisphere
from analysis of Hubble images, consistent with a 200km water plume near
Europa’s south pole. Sparks et al. (2016, 2017, 2019) investigated limb-
darkening effects in far ultra-violet Hubble images and also found cases of
possible plume activity, although this has since been contested by Giono et
al. (2020) who showed that at least one of these detections could also be
explained by statistical fluctuations. Recently, studies of Europa’s magnetic
field and plasma wave observations by Jia et al. (2018) and Arnold et al.
(2019) have reported features consistent with interaction between Jupiter’s
co-rotating plasma and plumes, giving independent evidence to support their
existence. Low-albedo geographical regions have also been linked to
cryovolcanic activity by Fagents et al. (2000). Paganini et al. (2020) has
shown possible evidence of a plume through IR spectroscopic measurements with
the Keck telescope. Recent work by Huybrighs et al. (2020) shows that
energetic proton depletion during Galileo’s flyby of Europa is consistent with
charge exchange with a plume, providing evidence from an independent method
and data set from previous examples. However, an artefact in the energetic
proton data prevents a definitive conclusion on the presence of a plume (Jia
et al., 2021; Huybrighs et al., 2021).

The potential to study the Europan ocean for signs of life has encouraged
scientific and public interest in a spacecraft flyby of Europa’s plumes. It is
proposed that spacecraft flybys could investigate the ocean’s composition,
search for organic molecules and perhaps sample microbial life directly
(Lorenz, 2016). In this paper ESA’s JUpiter Icy Moons Explorer (JUICE) mission
is used as an example mission (Grasset et al., 2013); the spacecraft’s
Particle Environment Package (PEP) (Barabash et al., 2013) has been predicted
to be able to detect molecules from the plumes during its two planned flybys
of Europa in the early 2030’s (Huybrighs et al., 2017; Winterhalder and
Huybrighs, 2022). The spacecraft may be capable of ‘tasting’ trace gases and
compounds in the plume using the Neutral Ion Mass spectrometer (NIM)
instrument in the PEP package. Europa Clipper’s MAss SPectrometer for
Planetary EXploration/Europa (MASPEX) instrument is also intended to analyse
the composition of Europa’s atmosphere and possible plumes (Teolis et al.,
2017), when the spacecraft reaches Europa (Brockwell et al., 2016).

Collisionless models have been used to model plumes on Europa (Southworth et
al., 2015; Lorenz, 2016; Teolis et al., 2017; Huybrighs et al., 2017;
Vorburger and Wurz, 2021; Winterhalder and Huybrighs, 2022) and Enceladus
(Smith et al., 2010; Tenishev et al., 2010; Dong et al., 2011). Specifically
in Huybrighs et al. (2017) and Winterhalder and Huybrighs (2022), a simple
collisionless model using Monte Carlo particle tracing was used to simulate
the trajectories for neutrals and ions under Europa’s gravity, Jupiter’s
electric field, and the conventional electric field, for several possible
plume source mass fluxes. These studies show that H2O molecules could be
detected by the NIM instrument from a plume anywhere on the surface. However,
simulations by Berg et al. (2016a) and more recently by Tseng et al. (2022),
which included particle-particle collisions between H2O neutrals resulted in a
canopy shock forming within the plume, as particles falling back to the
surface collide with rising material. This limits the altitude and extent of
the plume. The models presented in Berg et al. (2016a) assume a Laval nozzle-
like vent which results in a colder plume source temperature than in previous
studies, further limiting the plume’s extent. The canopy shock and the lower
source temperature suggest that in reality a spacecraft’s ability to detect
plume molecules will be poorer than previously thought.

While the structure of Europa’s plumes, including the presence of a canopy
shock, has not been determined from the currently available observations, the
presence of canopy shocks has been observed at Io (e.g. in Morabito et al.
(1979); Geissler and McMillan (2008)) and predicted by models (e.g. in Zhang
et al. (2003); McDoniel et al. (2015, 2019)). The plumes on Enceladus do not
exhibit the canopy shape observed at Io and predicted at Europa. This is
because the exhaust velocity of the plumes exceeds the escape velocity of the
moon. The plume particles are therefore not strongly affected by gravity and
thus the collisions between rising and returning particles are not frequent
enough to form the canopy shock (e.g. Yeoh et al. (2017)). Estimates of plume
mass flux range from 5$\,\mathrm{kg\,s^{-1}}$ (Southworth et al., 2015) to
7000$\,\mathrm{kg\,s^{-1}}$ (Roth et al., 2014).

In this study the extent to which using a collisional model with a canopy
shock reduces the potential of detection compared to a collisionless case is
investigated. We determine the effect of particle collisions and the formation
of a canopy shock on the region of separability (i.e. the surface area of
Europa that can be probed by a flyby), and on the peak density detected by the
simulated flyby. The feasibility of detecting particles from previously
observed potential plumes by Roth et al. (2014); Sparks et al. (2016, 2017,
2019); Jia et al. (2018) and Arnold et al. (2019) are investigated. Flybys at
different altitudes are modelled to determine if a lower flyby would
significantly improve the area over Europa’s surface where plumes would be
separable from the atmosphere (the region of separability) and the peak
density of particles encountered.

##  2 Method

The density distribution of the plumes is here simulated using the Direct
Simulation Monte Carlo (Bird, 1994) code PLANET developed at The University of
Texas at Austin. The PLANET code was run with different vent conditions listed
in Table 1, varying the mass flux and turning particle collisions on and off.
Subsequently, the simulated plume densities are used as an input for
simulations of the measurements with the neutral mass spectrometer, NIM, on
JUICE.

###  2.1 Plume Model

PLANET has been previously used for a wide variety of atmospheric
applications, including plume simulations for Europa (Berg et al., 2016a), Io
(McDoniel et al., 2019) and Enceladus (Mahieux et al., 2019; Yeoh et al.,
2017). In this work we simulate the plume H2O molecule number density,
velocity, and kinetic- and rotational-temperature fields using the same
approach as the one described in Mahieux et al. (2019); the reader can find an
extensive description and justification of the computation method in that
paper. A summary is provided here.

A single plume source with a circular vent aperture is considered, pointing
normal to the surface of Europa, outgassing a pure water vapour flow. In DSMC,
the time-dependent movement and collisions of numerical particles, which
represent a large number of real molecules, are computed on a 3D spherical
grid. The grid is defined by collision cells, in which the collisions of the
particles are computed probabilistically. However, only 1/360 of the flow in a
1-degree wedge needs to be simulated because of the cylindrical symmetry of
the problem. The computation is done in stages (see Figure 1), whose sizes
increase as a geometric progression, and such that each stage encompasses the
previous ones. Thus, the cell size is defined such that it resolves the local
flow mean free path. Likewise, the time step needs to resolve the mean
collision time. There are seven stages in the computation. PLANET is run in
each stage until steady state is reached, i.e. that the number of numerical
particles is constant. When steady state is reached, the flow is sampled over
a large number of time steps to reduce the numerical noise, and the numerical
particles crossing the right and top boundaries in Figure 1 are counted. They
are used as initial conditions for the next stage.

We note that, in the present DSMC simulations, the particles can only travel
from a lower stage to an upper stage. This introduces a small error below when
considering Europan plumes, as the particles from the outermost stage that
should fall back to stage 6 are not counted. Note, in our simulations the
outer stage (stage 7) extends from 40-700$\,\mathrm{km}$ from the plume
source, and so fully contains the region where the canopy shock forms at an
altitude of $\sim 400$$\,\mathrm{km}$. As shown in Berg et al. (2016a), this
error can be neglected. The stage parameters for the Default case are given in
the Appendices, Table Appendices.

The collisional model we use has an artefact along the symmetry axis (Figure 1
A) (or $z$-axis in Figure 1 B), where the density is lower at the centre of
the plume than would be expected. This is because there is a statistically
small proportion of simulation particles that have the exactly z-aligned
velocity needed to populate the region along the central axis. None of the
conclusions of this paper are affected by the artefact. The results of Figure
4 and 6 are not affected because we use the maximum density along the entire
trajectory and not the value at closest approach (where the artefact could
occur). In Figure 5, panel B, the density at closest approach is an
underestimate of less than an order of magnitude compared to panel A. This
does not affect the conclusions of the paper.

Figure 1: Panel A: DSMC domain geometry. The vent is the white circle at the
centre. The grid represents the domain that is simulated using the code
PLANET. Panel B: Representation of the staged calculation. The collision cells
are shown in Stage 1 only for clarity.

###  2.2 Simulation cases

For this study, four different DSMC plume simulations were run: a Default
case, a second case with 10 times lower mass flux
(1.04$\,\mathrm{kg\,s^{-1}}$), a third case with 10 times larger mass flux
(104$\,\mathrm{kg\,s^{-1}}$) and a fourth case with no collisions. All cases
have the same size vent and the same water vapour speed and temperature at the
vent exit. An overview of the simulations is shown in Table 1. The first line
refers to the Default case, with collisions turned on. The speed and
temperature parameters were taken from Berg et al. (2016a). The stage sizes
and grids are the same for all runs. In the Low Flux case, the expanding flow
in the neighbourhood of the vent remains collisional only in stage 1 and
reaches nearly the same vertical speed as the collisionless case within a few
vent diameters above the surface. In the High Flux case, the expanding flow
remains collisional up to stage 4. The High Flux plume extends to the same
altitude as the Default case. The Default, Low Flux, and High Flux cases
plumes all rise to approximately the same altitude. As described in Section 1,
estimates from observations show mass fluxes up to $\sim
1000$$\,\mathrm{kg\,s^{-1}}$, so the ’high flux’ case is high relative to the
other models we considered, rather than high relative to the mass fluxes of
the observed of plumes.

A collisionless plume model was also run. The collisionless plume can be
similarly scaled linearly for different mass fluxes. The collisionless case we
expect to be similar to a slightly collisional low-mass flux plume since the
flow is so rarefied that collisions between particles hardly occur at any
altitude. Therefore, descending particles do not collide with rising particles
and a shock wave does not form. Without the canopy shock, the collisionless
plume rises higher and spreads more broadly than a collisional, shock-
constrained plume would. We expect that the geometric (canopy or no-canopy)
nature of Europan plumes is based simply on the mass flow rate and its
influence on whether or not a canopy shock forms. Examples of the plume model
for a collisional and collisionless 100$\,\mathrm{kg\,s^{-1}}$ plume are shown
in Figure 2.

The plume model can be scaled linearly in either the non-shocked (essentially
collisionless) regime (Huybrighs et al., 2017), or in the collisional shocked
case (Berg et al., 2016a). However, during the transition from non-shocked to
shocked (roughly between 1 and 10$\,\mathrm{kg\,s^{-1}}$) the scaling of the
model is not linear as the structure and height changes from fountain-shaped
to the dome-shaped canopy shock. A $\sim 1000$$\,\mathrm{kg\,s^{-1}}$ plume
simulation would yield essentially the same result as a $\sim
100$$\,\mathrm{kg\,s^{-1}}$ plume, just with a density scaling of $\sim\times
10$.

Run Name |  Mass Flux in $\,\mathrm{kg\,s^{-1}}$ |  Vent radius in m |  Water vapour exit speed in $\,\mathrm{m\,s^{-1}}$ |  Water vapour temperature in K |  Angle of the Flow to the normal at the vent in ° |  Particle Collisions
---|---|---|---|---|---|---
Default |  10.4 |  10.2 |  902 |  53 |  0 |  ON
High Flux |  100.4 |  10.2 |  902 |  53 |  0 |  ON
Low Flux |  1.04 |  10.2 |  902 |  53 |  0 |  ON
No Collisions |  100.4 |  10.2 |  902 |  53 |  0 |  OFF
Table 1: Parameters of the vent and of the flow at the vent for the different
cases

(a) Collision-less

(b) Collisional

Figure 2: H2O neutral density in the $xz$ plane of two simulated
100$\,\mathrm{kg\,s^{-1}}$ plumes. The coordinate system is centred on the
centre of Europa with the $z$-axis passing through the plume source. (a) is a
collisionless plume showing a large fountain-like shape, (b) is a collisional
plume showing the canopy shock. The empty disk segment at the bottom of the
figures is the surface of Europa, with the plume source at
$z=\mathrm{R}_{\mathrm{Europa}}$. The canopy shock forms at around
$z=2100$$\,\mathrm{km}$ corresponding to an altitude of $\sim
500$$\,\mathrm{km}$ above the surface.

The lifetime of a plume particle is sufficiently short such that only a small
fraction of the plume particles are ionised by the radiation environment
(including e.g. solar UV-C and the Jovian plasma torus). We therefore assume
the ion component does not affect the outcome of this study and it is
neglected (Huybrighs et al., 2017).

###  2.3 Instrument Simulation

The example JUICE mission flybys are the baseline 3.0
flybys111https://www.cosmos.esa.int/web/spice/spice-for-juice originally
planned to take place in October 2030. At closest approach the spacecraft will
have an altitude of $\sim 400$$\,\mathrm{km}$. Note that since this study is
primarily illustrative and the flybys are still planned to pass over each
hemisphere with a closest approach at 400 km, the conclusions of this paper
are unaffected by the updates to the flybys thereafter. In this study the
southern flyby is taken as a reference trajectory, as it passes over the
southern hemisphere where most of the putative plume locations lie. The JUICE
spacecraft’s PEP package contains the NIM instrument; a high resolution time-
of-flight mass spectrometer, capable of detecting low energy
($<$10$\,\mathrm{eV}$) neutrals and ions. PEP also contains the ion mass
spectrometer, Jovian Dynamics and Composition Analyzer (JDC), which covers the
1$\,\mathrm{eV}$ \- 41$\,\mathrm{keV}$ range.

A 1D model of Europa’s atmosphere is assumed, with a constant water density of
104$\,\mathrm{cm^{-3}}$, which is based on densities above 100$\,\mathrm{km}$
in model D in Shematovich et al. (2005b), which is the same atmospheric model
as used in Huybrighs et al. (2017) and Winterhalder and Huybrighs (2022). The
Shematovich et al. (2005b) model is a 1D collisional Monte-Carlo model that
includes sublimation and sputtering as sources of H2O.

Since we are only looking at altitudes at or above 300$\,\mathrm{km}$ we
ignore the variations in atmospheric density closer to the surface. This
assumption and simplification of the atmosphere is described in Shematovich et
al. (2005b); a summary of the main arguments is as follows. Firstly lateral
density gradients have been predicted and observed between the day and night
side of Europa (Teolis et al., 2017; Plainaki et al., 2010; Roth, 2021) as H2O
is sublimated from the surface during the day and re-adsorbed at night. These
modelled and observed asymmetries are hemispheric and so are at a much larger
scale than the local density increase caused by the plume, meaning that plumes
can be separated from this lateral asymmetry. The atmospheric simplification
also ignores secondary effects playing a role in the distribution of plume
particles caused by interaction with the surface, atmosphere, and radiation
environment such as sputtering/resputtering, bouncing, adsorption/desorption
and photon or electron-impact ionisation, modelled for example in Teolis et
al. (2017). Furthermore, these secondary plume features are ignored as they
are less dense than the primary erupted particles. Since this paper focuses
specifically on the effects of collisions in the plumes, we do not attempt to
simulate all secondary effects.

In this paper a plume detection is defined as the plume signal exceeding NIM’s
noise level. The noise level is determined from instrumental effects and
background radiation represented as a density of 7$\,\mathrm{cm^{-3}}$
following Huybrighs et al. (2017). The density of trace gases, organic
molecules, and H2O molecules must surpass this noise limit to be detected by
NIM. To separate the plume from the background water atmosphere we require the
plume H2O density to exceed the background atmospheric H2O density (at an
altitude of $\geq$100$\,\mathrm{km}$) of $10^{4}$$\,\mathrm{cm^{-3}}$, and
define this point as the atmospheric threshold (Shematovich et al., 2005b).
Trace gases and organic molecules are separable from the atmosphere providing
they exceed both the instrument noise limit and their species’ own atmospheric
background density.

The 2D plume models were converted into a 3D map of H2O neutral density in
Europa’s vicinity by rotating the single slice by 360° via an interpolation
algorithm. The density of particles encountered along the spacecraft
trajectory, as shown in Figure 3, was calculated using the same software tools
as Winterhalder and Huybrighs (2022).

Figure 3: JUICE trajectory through the model 100$\,\mathrm{kg\,s^{-1}}$
collisional plume, with closest approach at an altitude of 400$\,\mathrm{km}$
(at $z=~{}2000\,\mathrm{km}$). The section of the spacecraft trajectory that
we see here covers approximately 15 minutes, and approximate time ticks are
shown in white.

##  3 Results and Discussion

###  3.1 Detected density reduced and concentrated around closest approach

The ability of the spacecraft to detect plumes over the Europan surface is
displayed as detected density maps, showing the peak density detected along
the spacecraft trajectory at the instrument as a function of plume location on
the map. The maps are generated by considering a plume located at any location
on the surface, and determining the peak density by a simulated flyby through
the resulting 3D H2O density map of Europan space for each possible plume
location. To compare the collisional and collisionless plume cases, the two
detected density maps shown in Figure 4 are produced.

The collisional model results in a reduced peak detected density, with
increasing effect the further away from the location of the closest approach.
For example, by inspecting Figure 4, a collisional 100$\,\mathrm{kg\,s^{-1}}$
model plume located at the position of the putative plume detected in Jia et
al. (2018) would lead to a maximum detected density of
$10^{1}$$\,\mathrm{cm^{-3}}$ encountered by the instrument during flyby, while
a collisionless model would suggest a detected density of
$10^{4}$$\,\mathrm{cm^{-3}}$, a difference of three orders of magnitude. For
plumes not located directly below the closest approach, the flyby will detect
orders of magnitude lower H2O molecule density than was previously predicted
from collisionless models. Therefore, plume models that do not include
collisions such as the feasibility studies in Huybrighs et al. (2017) and
Winterhalder and Huybrighs (2022) lead to over-predictions of the plume H2O
density for mass fluxes above 1$\,\mathrm{kg\,s^{-1}}$. Teolis et al. (2017)
also use a collisionless model for a plume with a mass flux of
500$\,\mathrm{kg\,s^{-1}}$. Our models predict a strong canopy shock effect at
this mass flux, and as such the predicted H2O density by Teolis et al. (2017)
from a plume along a spacecraft flyby trajectory is an over-prediction, for
flybys with altitudes of several 100$\,\mathrm{km}$ that will likely pass over
the canopy shock.

The highest density ($>10^{6}$$\,\mathrm{cm^{-3}}$) region is more restricted
to the area of the surface around the closest approach (Figure 4), indicating
that the relationship between density detected and instrument altitude is
stronger in the collisional case. Figure 5 shows how the density changes with
altitude as the spacecraft follows its trajectory in time. The gradient of the
change of density with altitude is steeper in the collisional case as
expected. The density scale factor is defined as the change in altitude at
which the density drops by a factor of $e$. For the 100$\,\mathrm{kg\,s^{-1}}$
mass flux plume, the density scale factor is $\sim$36$\,\mathrm{km}$ using a
collisionless model, and is reduced to just $\sim$28$\,\mathrm{km}$ for a
collisional model. This represents a reduction of 23%, indicating that the
density drops off significantly more quickly when the limiting effect of the
shock is considered.

These two effects of the collisional model, the reduction in peak detected
density and density scale factor, hold for each of the three mass fluxes
studied, but is strongest for the high mass 100$\,\mathrm{kg\,s^{-1}}$ plume,
see Appendices, Figure 9.

Figure 4: Maps illustrating the peak H2O density detectable at the spacecraft
trajectory as a function of plume location over the surface of Europa. For a
plume located at any point on the surface, the colour at that point
corresponds to the peak detected density at the flyby from that plume. The
collisionless (a) and collisional (b) models with a mass flux from the source
of 100$\,\mathrm{kg\,s^{-1}}$ are compared. Approximate locations of putative
plumes from the literature are plotted and labelled in the legend. The white
background on Europa’s surface indicates the regions where the density is less
than $10^{1}\,\mathrm{cm^{-3}}$. This plot illustrates that the density
detected by the spacecraft for a plume not located at the closest approach is
orders of magnitude lower when a collisional model is used, see section 3.1.

###  3.2 Limitation for plume observation time

The plume observation time is defined as the period during the flyby when the
plume density exceeds the atmospheric threshold. For a
100$\,\mathrm{kg\,s^{-1}}$ plume located directly below the closest approach,
the plume observation time for a collisionless model is 10 minutes. This is
reduced to 6 minutes for a collisional plume, a reduction of 40%. This is a
result of the altitude limiting effect of the canopy shock, as there is a
shorter period where the spacecraft is at low enough altitude to detect plume
H2O molecules at a density above the atmospheric threshold.

JUICE’s NIM instrument will only be able to make measurements during the
approach to Europa. When JUICE enters the outbound phase of the flyby,
incoming neutral gas will be blocked from the NIM instrument entrance by the
body of the spacecraft (Galli et al., 2022). Applying the halved NIM time to
our results, the plume time is reduced to 5 minutes for a collisionless model,
and only 3 minutes for the collisional model. Recall that this is for a plume
located below the closest approach and represents a best case scenario; the
plume time will be even shorter for plumes not in this ideal location. This
paper does not fully explore the impact of this restriction on NIM, for
example by calculating the detection from only the first half of the flyby. We
consider an optimal flyby to illustrate the potential of the instruments and
mission. Note that the halved instrument time is a restriction for JUICE
specifically and does not necessarily apply to other spacecraft such as Europa
Clipper, to which this study is also relevant.

The plasma instruments such as the JDC and Energetic Neutral atom imagers in
the PEP package do not have the same limitation and will make measurements
during both phases. JDC may detect H2O+ pickup ions (Huybrighs et al., 2017)
created by electron impact ionisation of plume neutral H2O. We expect that the
collisional model plume will also limit the physical extent of the region from
which pickup ions can be generated as it follows the density distribution of
the neutral component, and therefore would also cause a reduction in plume
detection time for JDC.

(a) No collisions : 100 kg/s

(b) With collisions : 100 kg/s

Figure 5: Variation of H20 density with time during the flyby, with closest
approach at 400km. The plot shows the density of H2O molecules (blue) detected
by the spacecraft in time as it flies through a plume located directly under
the point of closest approach, with the atmospheric limit (the approximate
water density from $\sim 100$$\,\mathrm{km}$ to at least 600$\,\mathrm{km}$ in
the atmosphere from Shematovich et al. (2005b)) and NIM noise limit indicated.
The red line gives the spacecraft’s altitude above the surface in km, the
green line gives the NIM instrument noise limit in particles per cubic
centimetre; the yellow line indicates the atmospheric H2O density below which
plume H2O molecules are not discernible. The density detected by the
spacecraft drops off 23% more quickly with altitude for a collisional plume,
see sections 3.1 and 3.2.

###  3.3 Collisional model gives reduced region of separability

The region of separability can be defined as the area over the surface of
Europa where the H2O density from a model plume (in said area) detected by the
flyby exceeds the atmospheric threshold and is therefore identifiable as
separate from the moon’s atmosphere. Maps showing the region of separability
considering the atmospheric threshold are shown in Figure 6.

Comparing the collisional and collisionless cases there is a reduction in the
area of the region of separability (note that the difference in the extent of
the simulated particles, visible in Figure 2, does not affect the size of the
region of separability). In the high mass flux case of
100$\,\mathrm{kg\>s^{-1}}$, the area of the region of separability is reduced
by 43% for the collisional case compared to the collisionless case. Table 2
shows the area of the region of separability for each case in
$\,\mathrm{km^{2}}$. These data show the region of separability is reduced in
each case, and that higher mass flux plumes suffer greater area reduction. For
the 1$\,\mathrm{kg\,s^{-1}}$ mass flux plume the area is reduced only by 3%.
Hence, at low mass fluxes the detected density is not as greatly affected by
the presence of the shock; this is a result of the shock effect being less
strong and limiting at low mass flux (see Section 2). Therefore, a
collisionless model as used in Huybrighs et al. (2017) and Winterhalder and
Huybrighs (2022) is a useful approximation for modelling plumes with mass
fluxes $\leq 1$$\,\mathrm{kg\,s^{-1}}$. Note 1$\,\mathrm{kg\,s^{-1}}$ plumes
are detectable by JUICE according to Huybrighs et al. (2017); Winterhalder and
Huybrighs (2022). See Appendices, Figure 10 for region of separability plots
for each mass flux.

Due to a lack of measurements, uncertainties exist in the atmospheric profile,
so better understanding of the atmospheric profile is needed to separate
plumes from the atmosphere with high accuracy.

Figure 6: Maps illustrating the H2O molecule density detected in the region of separability, defined as the region in which the detected H2O density exceeds the atmospheric threshold of $10^{4}$ molecules per $\,\mathrm{cm^{3}}$. These maps show the area of the surface of Europa over which JUICE could separate a plume from the moon’s atmosphere (colour), where the greyed out area is everywhere a plume would not be separable by a this flyby. Approximate locations of putative plumes from the literature are plotted on the surface and labelled in the legend. The region of separability is reduced in the collisionless case to about a half of the collisional case, see section 3.3. Mass Flux | Region of separability as % | Ratio
---|---|---
in $\,\mathrm{kg\,s^{-1}}$ | of Europan surface | $\frac{\mathrm{Collisional}}{\mathrm{Collisionless}}$
| Collisional | Collisionless |
100 | 9.34% | 16.50% | 0.57
10 | 7.07% | 11.53% | 0.61
1 | 7.17% | 7.40% | 0.97
Table 2: Comparison of the region of separability for a collisional and
collisionless model at different plume mass fluxes, where the region of
separability is defined as the surface area where the spacecraft is capable of
differentiating plume water molecules from the atmospheric water molecules. In
the 100$\,\mathrm{kg\,s^{-1}}$ case the region of separability is reduced to
57% of the collisionless model, see section 3.3. In the
1$\,\mathrm{kg\,s^{-1}}$ case the collisional and collisionless models agree
with only 3% difference.

###  3.4 Implications for detecting putative plumes

To illustrate the implications of the reduction in area of the region of
separability, the possibility of detecting putative plumes from the literature
is presented in Table 3. Considering the case of a 100$\,\mathrm{kg\,s^{-1}}$
plume, this table shows whether the putative plumes lie within, on the border,
or outwith the region of separability. Putative plumes that lie touching the
border of this region are marked as marginal.

Plume sources | Collisionless | Collisional
---|---|---
Sparks et al. (2016)(A) | m | n
Sparks et al. (2016)(B) | m | n
Sparks et al. (2017)(A) | m | n
Roth et al. (2014) | y | y
Sparks et al. (2017)(B) | n | n
Sparks et al. (2019) | n | n
Jia et al. (2018) | m | n
Arnold et al. (2019) | n | n
Table 3: y Separated from atmosphere ; m marginally separated ; n not
separable.
This table illustrates how plumes which were considered separable using a
collisionless model are not separable when a collisional model is used, for
100$\,\mathrm{kg\,s^{-1}}$ plumes being measured during the southern flyby
over the southern hemisphere, see section 3.4.

There are putative plumes that would be considered to have marginal
possibility of separation using a collisionless model, which are not
detectable when a collisional model is used. This is an illustrative example
of the concrete effect collisional models have on whether these putative
plumes are detectable.

###  3.5 Lowering altitude of flyby increases region of separability

Trajectories at closest approach altitudes of 300$\,\mathrm{km}$ and
500$\,\mathrm{km}$ were tested in addition to the planned JUICE flyby
trajectory at 400$\,\mathrm{km}$. This set of trajectories coincidentally
aligns with the spacecraft passing under, over, and through the top of the
canopy shock. Atmospheric models from Shematovich et al. (2005a) show that the
H2O density in Europa’s atmosphere increases below 300$\,\mathrm{km}$. We do
not consider altitudes below 300$\,\mathrm{km}$ in our analysis, so it is
therefore not necessary to consider this atmospheric density increase.

Table 4 and Figure 7 show the result of different trajectory altitudes on the
region of separability, see Figure 8 in the Appendices. When the spacecraft
flies lower, under the shock canopy altitude, the region of separability is
increased by 32% compared to the 400$\,\mathrm{km}$ trajectory. Similarly,
when the altitude is increased and the spacecraft passes above the shock
canopy the region of separability is reduced by 27%. The density along the
trajectory is affected by an order of magnitude between the 400 and
500$\,\mathrm{km}$ cases, and the width of the density-time peak is reduced
indicating a shorter time period where detections can be made. Wherein during
a 300$\,\mathrm{km}$ flyby the spacecraft may detect plumes within a
14$\,\mathrm{minute}$ window, this is reduced to 9$\,\mathrm{minutes}$ for a
500$\,\mathrm{km}$ flyby.

Altitude at closest | Region of Separability as a | Ratio to
---|---|---
approach in $\,\mathrm{km}$ | fraction of Europa’s surface area | 400km case
300 | 12.4% | 1.32
400 | 9.3% | 1
500 | 6.8% | 0.73
Table 4: Comparison of the region of separability at different altitudes,
based on a collisional 100$\,\mathrm{kg\,s^{-1}}$ case. Increasing or
decreasing the altitude by 100km affects the region of separability by roughly
30% each way, see section 3.5. Figure 7: Region of separability i.e. area over
the surface of Europa over which a plume would be separable from the Europan
atmosphere, plotted against the altitude of the closest approach for three
modelled spacecraft flybys. See section 3.5

The height of the canopy shock of potential plumes is currently poorly
constrained, as previous plume detections have all had high uncertainties
associated with the size of the plume. Simulations for different source mass
fluxes, different vent geometry, or different temperature and velocity
parameters suggest a height somewhere in the 100’s of kilometres range. In the
simulations studied in this paper, the spacecraft trajectory at 400km is
sufficient to pass through the plume shock, but this only represents one
possible set of plume parameters.

###  3.6 Effect of low temperature plume compared to previous studies

Comparing the models presented in this paper to those in Winterhalder and
Huybrighs (2022) and Huybrighs et al. (2017) the collisionless model in this
work (e.g. Figure 6) predicts a smaller region of separability for the same
mass flux. This implies that the model plume in this work has a smaller
physical extent to that used in Winterhalder and Huybrighs (2022); Huybrighs
et al. (2017). Because this discrepancy in plume extent is visible for even
low mass flux and collisionless models where there is no canopy shock, and
because the average velocity in this work is higher
(902$\,\mathrm{m\,s^{-1}}$) than in (Huybrighs et al., 2017; Winterhalder and
Huybrighs, 2022) (460$\,\mathrm{m\,s^{-1}}$) we attribute the difference to
our plume source temperature (53$\,\mathrm{K}$), which is colder than in
previous studies (230$\,\mathrm{K}$ (Huybrighs et al., 2017; Winterhalder and
Huybrighs, 2022) and 150$\,\mathrm{K}$ Teolis et al. (2017)). At cooler source
temperatures, plume molecules ejected have a lower velocity spread as per
their Maxwell-Boltzmann distribution, and thus fewer of them reach distances
further away from the source, reducing the extent of the plume.

The low plume temperature (53$\,\mathrm{K}$) is due to the following physical
argument, as shown in Berg et al. (2016b). According to Berg et al. (2016b),
plume material leaving the vent will be subject to the Laval nozzle effect.
This means that the material will be cooled as it erupts out into space,
dependent on the vent properties. The plume simulation in this study has a
source temperature of 53$\,\mathrm{K}$, based on physical arguments
considering a high speed plume starting at 230$\,\mathrm{K}$ in the vent, then
being cooled by the nozzle effect (Berg et al., 2016a) to 53$\,\mathrm{K}$.
Note that the temperature and exit velocity used in our simulations is the
coldest case put forward by Berg et al. (2016a) i.e. a worst case scenario in
terms of detectability.

Plumes may be colder than expected in previous feasibility studies, making
them more difficult to detect by spacecraft flyby. Plume temperature should be
considered by JUICE and other Europa in-situ missions, as less energetic
plumes will have a smaller physical extent and therefore be less likely to be
detected by spacecraft flyby.

###  3.7 Discerning plume structure

The plume canopy shock is a large structure in comparison with the path of the
spacecraft, and may intersect the trajectory for around 500$\,\mathrm{km}$ (in
$x$, see Figure 3) before and after the closest approach. This translates to
several minutes of plume observation time (see subsection 3.2) which is much
longer than the NIM integration time (5 seconds), giving good temporal (and
therefore spatial) resolution of samples of plume material. Presence of a
plume would dominate the H2O density detected during in-situ detections during
the flyby for a plume located near the closest approach, meaning that any
changes in density throughout the plume would be due to the structure of the
plume itself rather than the background atmosphere. This suggests that the
structure of the plume could be discernible from the density of plume
molecules encountered during the flyby. Measurements of the structure of the
plume would provide an empirical constraint on plume models, by which we can
investigate the underlying physics and typical characteristics of Europan
plumes. This would allow us to investigate yet unknown properties of plumes
such as the temperature, exit velocity, mass flux, and vent characteristics.

##  4 Recommendations for future missions

We recommend that missions such as JUICE and Europa Clipper consider low flyby
altitudes for Europa. For the best chance of detection of any plume, including
those not yet detected, the flyby should take place below the shock altitude
e.g. 300$\,\mathrm{km}$, to give the highest chance of flying through or below
some part of the canopy and making a positive plume detection (i.e. maximising
the region of separability). However, if a specific active plume is targeted,
we recommend to target the shock itself as it is an extended, high density
region which is difficult to miss (as opposed to the dense region directly
above the source, which is a fraction of the size of the shock and therefore
easier to miss).

Putative plume locations from the literature are present primarily around
Europa’s southern hemisphere, close to the track of the southern flyby. The
putative plume detected in Roth et al. (2014) shows the greatest chance of
being detected by JUICE as it is positioned close to the planned closest
approach for the southern flyby. The estimated mass flux of this plume is
$1-7\times 10^{3}$$\,\mathrm{kg\,s^{-1}}$; our models predict that this mass
flux would easily lead to a canopy shock which would constrain the extent of
the plume and therefore the region of separability and plume density (at
altitudes above the canopy).

The JUICE flyby over the southern hemisphere is the better candidate (of the
two planned flybys) for plume detection as it covers more putative plume
candidates, see Winterhalder and Huybrighs (2022). If lowering any of the
flybys is possible the southern flyby should be chosen, as flying low over
putative plumes increases the chance of a positive detection.

Winterhalder and Huybrighs (2022) showed that when using a collisionless
model, lowering the flyby altitude would not result in a significant increase
in the region of separability, so the benefit of a lowered altitude flyby was
only in the increase in density of the detected plume H2O molecules. However,
this study using a collisional model shows that the reduced extent of the
plume due to the canopy shock can lead to a large (43%) reduction in the area
of the region of separability, which can only be offset by lowering the flyby
altitude by 100$\,\mathrm{km}$. Hence, the case study presented in
Winterhalder and Huybrighs (2022) must be re-assessed. In contrast to their
results, we find that lowering the flyby trajectory would increase the
likelihood of separating putative plumes from the atmosphere, when we account
for particle collisions.

Teolis et al. (2017) predicts the H2O atmosphere will be suppressed on the
night side due to surface condensation, but enhanced on the day side due to
sublimation, which was observed in Roth (2021). As the night-side will
therefore have a lower atmospheric water density at the flyby altitude, the
atmospheric threshold will be lower leading to a larger region of
separability. Plumes would therefore be easier to detect by spacecraft flyby
from their H2O signature on the night-side of Europa. The JUICE mission’s
proposed imaging for Europa requires optimal imaging illumination conditions
i.e. a $\beta$-angle (angle between the orbital plane and the sun) of $\geq$
50$\,\mathrm{\degree}$. This means it is likely that most of the JUICE
trajectory will be on the dayside – however the proposed orbits by Europa
clipper should sample the nightside thoroughly (Dougherty et al., 2011; Bayer
et al., 2019).

The canopy altitude of the plume will depend on various parameters, such as
the mass flux, temperature of the flow, exit speed, and spreading angle at the
vent, as well as vent geometry. All of these are poorly constrained on Europa.
Full exploration of the possible canopy altitudes is beyond the scope of this
paper, though a full parameter sweep would be a useful study. Though the plume
height is difficult to constrain, the recommendations presented in this work
would apply to any plume of 100s$\,\mathrm{km}$ scale - the most plausible
case. Models presented by Zhang et al. (2003) and McDoniel et al. (2015)
explore the dependence of canopy height on vent temperature and exit velocity,
but without further constraints on the plume physics it is not currently
possible to give a height range beyond the 100s$\,\mathrm{km}$ scale discussed
here.

##  5 Conclusions

The predicted probability of detecting plume H2O molecules is reduced when
considering a collisional plume model compared to collisionless, for mass
fluxes exceeding 1$\,\mathrm{kg\,s^{-1}}$. For the low mass flux plume
(1$\,\mathrm{kg\,s^{-1}}$ or lower) the collisional model approximates the
collisionless model. In the high mass case for a plume with flux of
100$\,\mathrm{kg\,s^{-1}}$, the useful region of separability is reduced by as
much as half when compared to the collisionless model, and the peak particle
density detected at closest approach can be reduced by as much as an order of
magnitude. Previously suggested putative plume locations that were thought to
be detectable with the southern JUICE flyby may lie outwith the region of
separability when collisions are considered.

Model-based predictions for detecting Europa’s plumes from a spacecraft flyby
must consider particle collisions, as the results obtained using a collisional
model show differences as large as order of magnitude lower density, 23%
steeper altitude density gradient, and 43% reduced region of separability in
the high mass flux (100$\,\mathrm{kg\,s^{-1}}$) case. The nozzle effect
results in much colder plume temperatures than have been used in previous
feasibility studies. Since this limits the extent of the plume, it should also
be considered in future plume modelling - particularly for plume detection
where the physical extent of the plume is vital for a positive
detection/separation.

If a flyby is low enough to pass through or under the plume shock, for a model
100$\,\mathrm{kg\,s^{-1}}$ case, the region of separability is improved by
32%. Conversely, increasing the flyby altitude to 500km reduces the region of
separability by 27%. For a spacecraft aiming to detect plume molecules, a
lower altitude is recommended to improve the likelihood of passing through
plumes originating from the widest possible area of the surface. The
alternative scenario where the spacecraft passes over the shock altitude,
results in a region of separability reduced by almost a third, and peak
density reduced by an order of magnitude.

A flyby passing through the plume shock could give a plume observation window
of 3 minutes (for a model including particle collisions). Compared to JUICE’s
neutral mass spectrometer (NIM) observing cadence of 5 seconds, we see that
the structure of the plume could be resolved, allowing us to probe the
underlying physics of Europa’s plumes. Resolving the structure of the plume
would allow us to compare the observation to plume models and infer plume
parameters such as mass flux, vent characteristics, or plume temperature.

## Declaration of competing interest.

The authors declare that they have no known competing financial interests or
personal relationships that could have appeared to influence the work reported
in this paper.

## Data availability

Data will be made available on request.

## Acknowledgements

We thank the reviewers for their effort in providing their useful and
insightful comments on the manuscript. This work was carried out during the
Leiden/ESA Astrophysics Program for Summer students (LEAPS) 2020 hosted by the
European Space Research and Technology Centre (ESTEC) - European Space Agency
(ESA) and Leiden Observatory. R. Dayton-Oxland is supported by the University
of Southampton and INSPIRE Doctoral Training Program; this work was supported
by the Natural Environmental Research Council [grant number NE/S007210/1]. H.
Huybrighs gratefully acknowledges the support from Khalifa University’s Space
and Planetary Science Center under grant N. KU-SPSC-8474000336, the ESA
research fellowship and the International Space Science Institute (ISSI)
visiting scientist program. T. O. Winterhalder was supported by the University
of Heidelberg and the European Southern Observatory. D. B. Goldstein and A.
Mahieux were supported by the SSW NASA grant with award number 80NSSC21K016.
Plume computations were done at the Texas Advanced Computing Center.

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## Appendices

|  | # vert. | # horiz. | Stg vt. | Stg hz. | Time | Mean coll. | Cell vt. | Cell hz. | Mean free |
---|---|---|---|---|---|---|---|---|---|---|---
Stg | $f_{num}$ | cells | cells | size [m] | size [m] | step [s] | time [s] | size [m] | size [m] | path [m] | $Kn_{grad}$
1 | $9\times 10^{16}$ | 144 | 6 912 | 44.9 | 44.9 | $10^{-4}$ | $5\times 10^{-4}$ | 0.31 | 0.006 | 0.09 | 0.06
2 | $2.3\times 10^{17}$ | 144 | 6 912 | 112.7 | 78.8 | $2\times 10^{-4}$ | 0.0023 | 0.78 | 0.011 | 0.31 | 0.2
3 | $6.7\times 10^{17}$ | 240 | 11 520 | 333.4 | 189.2 | $5\times 10^{-4}$ | 0.0088 | 1.39 | 0.016 | 0.95 | 0.5
4 | $2.3\times 10^{18}$ | 288 | 13 824 | 1 163.7 | 604.3 | 0.002 | 0.095 | 4.04 | 0.043 | 7.63 | 1.7
5 | $9.6\times 10^{18}$ | 384 | 18 432 | 4 790.8 | 2 417.9 | 0.009 | 2.3 | 12.48 | 0.13 | 110.9 | 6.9
6 | $4.7\times 10^{19}$ | 528 | 25 344 | 23 262.4 | 11 653.7 | 0.05 | 82.3 | 22.07 | 0.46 | 2495 | 36.7
7 | $1.8\times 10^{21}$ | 1 008 | 48 384 | 900 000 | 900 000 | 2 | $4\times 10^{6}$ | 892.86 | 18.6 | $6.2\times 10^{7}$ | 3.2
Table 5: Mean values of DSMC parameters in each stage (stg) for the Default
case: $f_{num}$, number of vertical (vt.) and horizontal (hz.) cells,
horizontal and vertical size of each stage, time step, spatially averaged mean
collision time, cell vertical size, spatially averaged mean free path, and
spatially averaged Knudsen number $Kn_{grad}$.
The computational values are mean values over the whole stage. The
computational time step resolves the mean collision in all stages.
Furthermore, the horizontal cell sizes always resolve the mean free path. Note
that the vertical cell size does not resolve the mean free path in stages 1
and 2. However, the flow is close to continuum conditions in those lowest
stages, as the Knudsen number based on the density gradient is very small.
$Kn_{grad}=\lambda_{MFP}\cdot\frac{\|\nabla m_{H_{2}O}\|}{m_{H_{2}O}}$ where
$\lambda_{MFP}$ is the mean free path assuming local thermodynamic equilibrium
and $m_{H_{2}O}$ is the local water vapour mass density. It varies between
$2.42\times 10^{-5}$ and 0.1 in stage 1 and between 0.04 and 0.5 in stage 2.

Figure 8:  Comparison of the region of separability at difference trajectory
altitudes at closest approach. The region of separability for the 500km case
(bottom) above the plume is reduced by a third compared to the default 400km
(middle), where in the 300$\,\mathrm{km}$ case (top) the region of
separability is improved by a third. Figure 9: Plot of the peak H2O density
detected by the NIM instrument during the flyby as a function of the plume
location on the surface, for different plume models consisting of the
collisional and collisionless model of a low mass 1$\,\mathrm{kg\,s^{-1}}$,
10$\,\mathrm{kg\,s^{-1}}$, and 100$\,\mathrm{kg\,s^{-1}}$. For a low mass
plume the effect of the particle-collisions is lesser, so the collisional map
is more similar to the collisionless case than in the high mass plume case,
where the collisional and collisionless plume density maps are drastically
different. This figure illustrates how higher mass flux plumes are more
affected by the addition of particle collisions as the canopy shock effect is
stronger. Figure 10: Region of separability maps for different plume models
including the collisional and collisionless cases for low
1$\,\mathrm{kg\,s^{-1}}$, default 10$\,\mathrm{kg\,s^{-1}}$, and high
100$\,\mathrm{kg\,s^{-1}}$ mass flux plumes. The region of separability is
defined as the area over the surface of Europa where a plume located at that
point would be separable from the atmosphere i.e. it exceeds the atmospheric
threshold. This plot illustrates that the region of separability is reduced
for a collisional plume model, for a low mass plume only by a small amount,
but for a high mass plume the potential region of separability is reduced by a
half.